PhD Thesis

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MSc in Electrical Engineering and Information Technology, ETH Zurich, Switzerland ..... 11.3.2 Solution of the MPC Problem by Distributed Optimization Techniques111 ... a modern control methodology, which is particularly suited for constrained ... wünschenswert, dass die lokalen Entscheidungen kooperativ, d.h. im Sinne ...
Diss. ETH No. 21834

Stability and Computations in Cooperative Distributed Model Predictive Control A dissertation submitted to ETH ZURICH

for the degree of Doctor of Sciences

presented by Christian Conte MSc in Electrical Engineering and Information Technology, ETH Zurich, Switzerland born 26.09.1983 citizen of Zurich

accepted on the recommendation of Prof. Dr. Manfred Morari (examiner) Prof. Dr. Francesco Borrelli (co-examiner)

2014

C 2014 Christian Conte

All Rights Reserved ISBN 978-3-906031-66-8

For Karl, Lotty, Martina and Priska

Acknowledgments During my time at the Automatic Control Laboratory (IfA) I was lucky to have the support of a number of outstanding people. The following is to express my gratitude to them, as it is mostly for them that doing a PhD turned out to be a great experience for me. First and foremost, my deep gratitude goes to Manfred Morari for being a great supervisor and for giving me the opportunity to do my PhD at his lab. He always impressed me with his drive, his dedication and his charisma. I strongly appreciate him as an independently thinking person who always tries to live up to his own high standards. Thanks for your guidance and your support in my work. I also owe my gratitude to Francesco Borrelli for being co-referee in my PhD committee and for providing a number of helpful comments on this thesis. Thanks also for giving me the opportunity to do my Master’s thesis at your lab at UC Berkeley, without which I would have probably never ended up doing a PhD at IfA. I was very lucky to meet Colin Jones at IfA, who awakened my interest in optimization already in my Master’s years and continuously supported me in my studies. I was always impressed by his unprecedented speed at generating ideas and expressing neat mathematical formalism on whiteboards, notebooks and wherever else you can write on. Thanks for your help, which was invaluable especially in the first two years of my PhD, when I struggled to shape scope and direction of my research. I also want to express my gratitude to all people I collaborated with during my PhD. A huge thanks goes to Melanie Zeilinger, whom I collaborated with in almost all of my work. Thank you for always finding time to discuss research with me, both on a strategic and on a very technical level. Thanks also for all the detailed and precise feedback on my paper drafts and thesis chapters, your inputs were absolutely invaluable. My very special thanks goes to Vedrana Spudi´c, whom I had a very enriching and productive collaboration during her half-year visit at ETH. Thank you for teaching me the basics of wind power and thanks for bringing the most positive spirit to work every day. Thanks also to Sean Summers, Tyler Summers and John Lygeros for an interesting collaboration on novel teaching concepts. Thanks especially to Sean for demonstrating how far you can get with a bit of Californian optimism. I’m also grateful to Sergio Grammatico, who shared some interesting ideas with me in the last year of my PhD.

I also want to say thanks to a number of people whom I did not work with directly, but who made spending time at IfA interesting and fun. Thanks to Alex Domahidi for being an outstanding officemate, notably in ETH’s best-situated office, for several years. Thanks to Stephan Huck and Christoph Zechner for countless interesting discussions at work and elsewhere. Thanks to Aldo Zgraggen for dragging me around lake Zurich by bike and thanks to him and Tony Wood for an outstanding time in Florence. Thanks to Paul Goulart and Stefan Richter for valuable advices and thanks to Davide Raimondo for proof-reading parts of this thesis. Thanks also to Andreas Hempel, Joe Warrington, Tobi Baltensperger, Nik Kariotoglou, Juan Jerez, Georg Schildbach, Claudia Fischer, Urban M¨ader, David Sturzenegger, Marko Tanaskovi´c and generally to all IfA members for creating the great atmosphere which makes IfA an amazing place to work at. Last but not least, I owe my gratitude to my family, Mom, Dad, Martina and Priska, as well as to my brother Sergej, who supported me all the way through, even though I usually failed at trying to explain my work to them. Thanks for always listening and believing in me all the same.

Christian Conte Zurich, July 2014

Contents Abstract

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1 Introduction 1.1 Outline and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preliminaries

2 Notation and Definitions 2.1 Notation . . . . . . . . . . . . 2.2 Definitions . . . . . . . . . . . 2.2.1 Set Related Definitions 2.2.2 Function Definitions . .

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3 System Theory and Control 3.1 Constrained LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Networks of Constrained LTI Systems . . . . . . . . . . . . . . . . . . . . 3.3 Set Invariance and Lyapunov Stability . . . . . . . . . . . . . . . . . . . .

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4 Convex Optimization 4.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Common Convex Optimization Problems . . . . . . . . . . . . 4.2.1 Linear Program . . . . . . . . . . . . . . . . . . . . . 4.2.2 Quadratic Program . . . . . . . . . . . . . . . . . . . 4.2.3 Semidefinite Program . . . . . . . . . . . . . . . . . . 4.3 Duality in Convex Optimization . . . . . . . . . . . . . . . . . 4.4 Distributed Convex Optimization . . . . . . . . . . . . . . . . 4.4.1 Structured Convex Optimization Problems . . . . . . . 4.4.2 Distributed Fast Gradient Method . . . . . . . . . . . . 4.4.3 Distributed Alternating Direction Method of Multipliers

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vi 5 Model Predictive Control 5.1 Nominal Model Predictive Control for Regulation . . . 5.2 Robust Model Predictive Control for Regulation . . . 5.2.1 Robust MPC according to [Mayne et al., 2005] 5.2.2 Robust MPC according to [Chisci et al., 2001] 5.3 Model Predictive Control for Reference Tracking . . .

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6 Distributed Model Predictive Control - A Brief Survey 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Non-cooperative Distributed MPC Methods . . . . . . . 6.3 Cooperative Distributed MPC Methods . . . . . . . . . 6.4 Tabular Overview of Distributed MPC Methods . . . . .

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II Cooperative Distributed MPC: Stability based on Distributed Invariance 45 7 Nominal Cooperative Distributed MPC for Regulation 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . 7.3 Distributed Invariance and Stability . . . . . . . . . . . . . . 7.3.1 Structured Terminal Cost Function . . . . . . . . . . 7.3.2 Time-Varying Distributed Terminal Set for Regulation 7.4 Distributed Synthesis . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Distributed Synthesis of a Structured Terminal Cost . 7.4.2 Distributed Synthesis of a Distributed Terminal Set . . 7.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Robust Cooperative Distributed MPC for Regulation 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Distributed Synthesis for Robust Cooperative Distributed MPC . . 8.3.1 Structured RPI Sets . . . . . . . . . . . . . . . . . . . . . 8.3.2 Distributed Constraint Tightening . . . . . . . . . . . . . . 8.4 Robust Cooperative Distributed MPC: Closed-loop Operation . . . 8.4.1 Robust Distributed MPC according to [Mayne et al., 2005] 8.4.2 Robust Distributed MPC according to [Chisci et al., 2001] .

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Contents 8.5 8.6

vii Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Cooperative Distributed MPC for Reference Tracking 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem Statement . . . . . . . . . . . . . . . . . . 9.3 Time-Varying Distributed Invariant Set for Tracking . 9.4 Distributed Synthesis of a Distributed Invariant Set for 9.5 Numerical Example . . . . . . . . . . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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III Computational Aspects of Distributed Optimization in MPC 87 10 Computational Performance of Distributed Optimization in 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Setup of the Computational Study . . . . . . . . . . . . . 10.2.1 System Setup . . . . . . . . . . . . . . . . . . . . 10.2.2 MPC Setup . . . . . . . . . . . . . . . . . . . . . 10.2.3 Performance Evaluation and Interpretation . . . . . 10.3 Computational Results and Discussion . . . . . . . . . . . . 10.3.1 Coupling Strength and Stability . . . . . . . . . . . 10.3.2 Coupling Topology . . . . . . . . . . . . . . . . . . 10.3.3 Initial State . . . . . . . . . . . . . . . . . . . . . 10.3.4 Number of Connected Systems . . . . . . . . . . . 10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Cooperative Distributed MPC for Wind Farms 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Control of Wind Farms . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Operating Modes for Wind Farms . . . . . . . . . . . . . . . . . . 11.2.2 Operation of Wind Turbines as a Part of Wind Farms . . . . . . . 11.2.3 Dynamics Relevant to Wind Farms and Wind Farm Control System Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Control Oriented Wind Farm Modeling . . . . . . . . . . . . . . . 11.2.5 Control Objective and Constraints . . . . . . . . . . . . . . . . . . 11.3 Cooperative Distributed MPC for the Control of Wind Farms . . . . . . . 11.3.1 Formulation of the distributed MPC Control Law . . . . . . . . . .

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11.3.2 Solution of the MPC Problem by Distributed Optimization Techniques111 Setup of the Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . 113 11.4.1 Simulation Scenarios and Performance Evaluation . . . . . . . . . 113 11.4.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 11.5.1 Closed-Loop Performance of Distributed MPC Controllers . . . . . 115 11.5.2 Convergence Properties of the Distributed Optimization Methods . 119 11.5.3 Performance of Distributed MPC under Suboptimal Solutions . . . 120 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Conclusion and Outlook

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Bibliography

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Curriculum Vitae

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Abstract The main theme of this thesis is the development of cooperative distributed model predictive control (MPC) methods for large-scale networks of constrained dynamic systems. MPC is a modern control methodology, which is particularly suited for constrained systems and has proven successful in practice. For large-scale networks of systems, which are often subject to communication constraints, MPC controllers have to be operated in a distributed way, i.e. each system in the network has to take local control decisions based on local measurements and communication with neighboring systems. Moreover, for networks of systems with a common objective function, it is desirable for the systems to take their control decisions cooperatively, which implies the need for cooperative distributed MPC. Distributed optimization is a well-established methodology which allows the combination of the cooperative and the distributed aspect within the MPC framework. Specifically, one finite-horizon optimal control problem, in the following referred to as MPC problem, can be formulated for the whole network of systems, and it can be solved by a distributed optimization method at each time step. This thesis is concerned with issues arising from the use of distributed optimization in MPC. In the first part of the thesis, distributed optimization based cooperative distributed MPC controllers for networks of linear systems are presented. All controllers guarantee stability and feasibility in closed-loop. The first controller presented is a nominal MPC controller. Closed-loop stability and feasibility are guaranteed by adapting well-established methodologies from the centralized MPC literature. Specifically, the global MPC problem is equipped with a suitably designed terminal cost, which is a Lyapunov function for the unconstrained system, and a terminal set, which is positively invariant (PI). In order to make the MPC problem amenable to distributed optimization algorithms, the terminal cost is designed as a separable function and the terminal set is a Cartesian product of local sets, which are time-varying. Specific synthesis methods for terminal cost and set are presented, where these methods can be executed in a distributed fashion themselves. In the following, two cooperative distributed MPC controller are presented, which extend the nominal one described above. The first is a robust MPC controller for networks of linear systems subject to bounded additive noise, the second is an MPC controller for reference tracking. In both cases, well established methodologies from the centralized MPC literature

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Abstract

are adapted for the use in a cooperative distributed setup, where the MPC problem is solved by distributed optimization. For distributed robust MPC, the main additional ingredients are structured robust positive invariant (RPI) sets, as well as constraint tightening methods, which can be executed in a distributed way. For distributed reference tracking MPC, the main additional ingredient is an invariant set for tracking, again designed as a Cartesian product, and again equipped with a synthesis method that can be carried out in a distributed fashion. In the second part of the thesis, computational aspects of specific distributed optimization methods in MPC are investigated. In particular, the performance of these methods on the MPC problem, i.e. the number of iterations to convergence, is computationally analyzed under various system setups and operational modes. A first study contains computational results for general networks of linear systems, which are controlled by standard nominal cooperative distributed MPC controllers. In the computational scenarios considered, various system properties, such as the strength of the dynamic coupling between the systems or the network topology, are varied. The results show that the performance of distributed optimization is sensitive to changes in these properties. In particular, as a general qualitative observation, the performance decreases in cases where coordination among the systems in the network is crucial, which usually manifests in Lagrange multipliers of large magnitude. These observations could be confirmed in a wind farm application study. In particular, it is shown that the performance of the distributed optimization methods decreases in operational conditions where the power production has to be dynamically reallocated across the wind farm. This is the case for example when there is not enough wind to fulfill the farm-wide power production requirements.

Zusammenfassung Diese Dissertation befasst haupts¨achlich mit der Entwicklung von kooperativen, verteilten, modellbasierten pr¨adiktiven Reglern (engl.: Model Predictive Controller, MPC) f¨ur grosse Netzwerke von dynamischen Systemen. MPC ist eine moderne Regelungsmethodik f¨ur Systeme, welche Bedingungen bez¨uglich ihrer Eing¨ange und Zust¨ande unterliegen. Diese Methodik wird erfolgreich in der Praxis eingesetzt. Da Netzwerke von Systemen oft Kommunikationsbeschr¨ankungen unterliegen, m¨ussen MPC Regler verteilt funktionieren, d.h. jedes System macht seine lokalen Regelentscheidungen basierend auf lokalen Messungen und Kommunikation mit benachbarten Systemen. Ausserdem ist es f¨ur viele Netzwerke w¨unschenswert, dass die lokalen Entscheidungen kooperativ, d.h. im Sinne einer globalen Zielfunktion, getroffen werden. Verteilte Optimierung ist eine etablierte Methodik, mit der der kooperative und der verteilte Aspekt innerhalb eines MPC Reglers kombiniert werden k¨onnen. Dabei wird das Optimierungsproblem, welches den MPC Regler f¨ur das globale Netzwerk von Systemen definiert und im Folgenden MPC Problem genannt wird, in jedem Zeitschritt durch eine verteilte Optimierungsmethode gel¨ost. Im ersten Teil der Arbeit werden kooperative, verteilte MPC Regler pr¨asentiert, deren MPC Probleme so formuliert sind, dass sie von verteilten Optimierungsmethoden gel¨ost werden k¨onnen. Des Weiteren weisen alle diese Regler Garantien bez¨uglich der Stabilit¨atsund Beschr¨ankungskonformit¨at des geschlossenen Regelkreises auf. Der erste pr¨asentierte Regler ist ein nominaler MPC Regler f¨ur Netzwerke linear Systeme. Stabilit¨at und Beschr¨ankungskonformit¨at werden durch die Adaptierung etablierter Methoden aus der Literatur u¨ber zentralisiertes MPC erm¨oglicht. Insbesondere weist das globale MPC Problem einen Endkostenterm auf, der durch eine Lyapunov Funktion f¨ur das nicht beschr¨ankte System beschrieben wird. Des Weiteren weist das Problem eine Endzustandsbedingung auf, welche durch eine positiv invariante (PI) Menge charakterisiert ist. Um das MPC Problem mit verteilten Optimierungsmethoden l¨osen zu k¨onnen wird die Endkostenfunktion so gew¨ahlt, dass sie separierbar ist. Ausserdem wird die Endzustandsmenge als karthesisches Produkt lokaler Mengen gew¨ahlt, wobei diese lokalen Mengen zeitvariant sind. Spezifische Synthesemethoden, sowohl f¨ur die Endkostenfunktion als auch f¨ur die Endzustandsmenge werden pr¨asentiert, wobei alle diese Methoden selbst auf eine verteilte Weise ausgef¨uhrt werden k¨onnen.

xii Im weiteren Verlauf der Arbeit werden zwei kooperativ verteilte MPC Regler vorgestellt, welche den oben vorgestellten Regler erweitern. Der erste ist ein robuster MPC Regler f¨ur Netzwerke von linearen Systemen, welche beschr¨ankten, additiven St¨orungen unterliegen, der zweite ist ein MPC Regler zur Referenzfolge. In beiden F¨allen werden etablierte Methoden aus der Literatur u¨ber zentralisiertes MPC adaptiert, um MPC Probleme zu formulieren, welche Stabilit¨atsgarantien aufweisen und durch verteilte Optimieruntsmethoden gel¨ost werden k¨onnen. Im Falle des robusten MPC Reglers bestehen die, im Vergleich zum nominalen Fall, neu ben¨otigten Komponenten aus strukturierten robust positiv invarianten (RPI) Mengen, sowie Beschr¨ankungsverkleinerungsmethoden, welche verteilt ausgef¨uhrt werden k¨onnen. Im Falle des Referenzfolge MPC Reglers besteht die, im Vergleich zum nominalen Fall, neu ben¨otigte Komponente aus einer strukturierten invarianten Referenzfolgemenge. Diese wird, wie die positiv invariante Menge im nominalen Fall, als karthesisches Produkt gew¨ahlt und kann ebenfalls mit einer verteilten Methode synthetisiert werden. Im zweiten Teil der Arbeit werden berechnungsspezifische Aspekte der verteilten Optimierung in der MPC Regelung untersucht. Insbesondere wird die Leistung von verteilten Optimierungsmethoden bei der L¨osung von MPC Problemen, d.h. die Anzahl der Iterationen bis zur Konvergenz, f¨ur verschiedene Systemeigenschaften und Operationsmodi untersucht. Eine erste Studie befasst sich mit allgemeinen Netzwerken linearer Systeme, welche durch nominale, kooperativ verteilte MPC Regler geregelt werden. In den ber¨ucksichtigten Szenarien werden verschiedene Eigenschaften dieser Netzwerke variiert, wie beispielsweise die Kopplungsst¨arke zwischen den Systemen oder die Netzwerktopologie. Die Resultate zeigen, ¨ dass die Leistung der verteilten Optimierung sensitiv auf diese Anderungen reagiert. Insbesondere konnte qualitativ beobachtet werden, dass die Leistung der verteilten Optimierungsmethoden abnimmt, je wichtiger die Kooperation zwischen den Systemen im jeweiligen Fall ist, was sich normalerweise durch Lagrange Multiplikatoren mit grossen Absolutwerten ¨aussert. Diese Beobachtungen konnten in einer Windfarm Anwendungsstudie best¨atigt werden. Insbesondere konnte beobachtet werden, dass die Leistung der verteilten Optimierungsmethoden in spezifischen F¨allen abnimmt, in denen die Energieproduktion in der Windfarm dynamisch neu verteilt werden muss. Dies ist beispielsweise der Fall wenn nicht genug Wind vorhanden ist, um ein vorgegebenes Produktionsziel zu erf¨ullen.

1 Introduction Large-scale networks of dynamically coupled systems are increasingly present in the modern world. Examples are power grids, transportation systems or supply chains. A property common to all of these networks is the fact that they consist of a large number of systems, which are subject to communication constraints. Traditionally, such networks were controlled in a decentralized way, where each system acts autonomously, taking decisions solely based on ˇ locally available measurements [Sandell et al., 1978], [Siljak, 1991]. However, with powerful and inexpensive communication and computation hardware increasingly available, it is more natural to resort to distributed control schemes, where the systems in the network communicate with each other. A particular distributed control scheme, which has gained popularity in recent years, is distributed model predictive control (MPC). This scheme is the distributed extension to centralized MPC, which is a popular control methodology mainly due to its ability to deal with constraints in a systematic manner, and which has proven particularly successful in the process control industry [Rawlings and Mayne, 2009]. The theory of centralized MPC has been extensively studied and numerous controller formulations and synthesis methods, e.g. for robust regulation and reference tracking, have been proposed. In comparison, the theory of distributed MPC is less mature. Available distributed MPC methods can mainly be divided into cooperative, e.g. [Stewart et al., 2010], and non-cooperative approaches, e.g. [Jia and Krogh, 2002]. Cooperative distributed MPC approaches usually rely on distributed optimization, which can be used to solve a network-wide MPC problem in a distributed way. Distributed optimization has its origin in the dual ascent method, originally proposed by [Arrow et al., 1958], which solves the dual of a convex optimization problem by a firstorder method. A large variety of distributed optimization techniques, based both on primal and dual decomposition schemes, have been proposed over the years, see [Bertsekas and Tsitsiklis, 1989] for a summary of the basic concepts. Cooperative distributed MPC based on distributed optimization offers all the benefits of centralized MPC, including a large body of rigorous theory. However, if the global MPC problem should be solved by distributed optimization, it is required to be appropriately structured. This is non-trivial if closed-loop stability guarantees are required, since such guarantees usually rely on specific components in the MPC problem, i.e. a terminal cost

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1 Introduction

function and a terminal constraint [Mayne et al., 2000]. These components are not generally structured, even if the given system dynamics and constraints are. A possible remedy is the use of trivial conditions in the MPC problem, e.g. constraining the terminal state to match the origin, as suggested in [Venkat et al., 2005]. Such trivial conditions might however induce considerable conservatism. Therefore, the theoretical part of this thesis focuses on formulations of MPC problems with reduced conservatism, which still define stabilizing control laws, and which are structured such that distributed optimization methods are applicable. As a first theoretical contribution presented in this thesis, existing MPC controllers, available from the literature on centralized MPC, are adapted such that the MPC problem can be solved by distributed optimization, while all stability guarantees still hold. This requires a structured design of those components in the MPC problem, which are responsible for the closed-loop stability guarantees. For nominal MPC, these components are the terminal cost function and the terminal constraint. Using a result from [Jokic and Lazar, 2009], it is shown that a structured terminal set can be built as the Cartesian product of local sets, where these local sets correspond to time-varying level sets of local terminal cost functions. In robust MPC, as proposed in [Mayne et al., 2005] and [Chisci et al., 2001], the main component required for stability is a robust positively invariant set. In this thesis, it is shown how such sets can be formulated in a structured way, such that distributed optimization is applicable to the resulting MPC problem. In reference tracking MPC, stability can be achieved by use of a terminal invariant set for tracking. In this thesis, it is shown that a structured invariant set for tracking can be formulated using similar techniques as in the nominal case. As a second theoretical contribution of this thesis, distributed synthesis methods are proposed for all the controller formulations outlined above. All the methods presented apply to networks of linear systems with quadratic cost, which are subject to polytopic constraints. Distributed synthesis is important due to the absence of a global coordinator in large-scale networks of systems. Therefore, all the proposed methods are designed such that they can be executed under the same communication constraints, under which the resulting controller is operated in closed-loop. Distributed synthesis is a meaningful methodology, considering that networks of systems might be subject to unit failures, or have time-varying topologies for other reasons. Such setups have recently been studied in the context of plug-and-play MPC, see [Riverso et al., 2012], [Zeilinger et al., 2013]. A known issue for cooperative distributed MPC is the fact that distributed optimization methods may converge slowly for badly conditioned problems. While upper bounds on the required number of iterations exist for some first-order methods, see [Richter, 2012], these bounds are sometimes impractically conservative. In this thesis, the issue is approached

1.1 Outline and Contribution

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from a computational angle. In particular, a computational study on distributed optimization in MPC is presented, which illustrates the performance sensitivity of two common distributed optimization methods to fundamental properties of networked dynamic systems, such as coupling strength, coupling topology, stability of the systems and network size. As a computational contribution, the study illustrates how system properties can affect the performance of distributed optimization methods. Along the same lines, the potential of distributed optimization in cooperative distributed MPC for output power tracking in wind farms is assessed. In this wind farm operating mode, a reserve in the farm-wide available power, which arises due to a farm-wide power output command imposed by the transmission system operator, is cooperatively used to reduce the farm-wide structural fatigue. Satisfactory closed-loop performance under cooperative distributed MPC is verified in simulations. Furthermore, it is shown that the convergence speed of the used distributed optimization methods mainly depends on the farm-wide amount of available power. In particular, the convergence speed decreases, as the lack of available power makes the coordination of the individual turbines in the farm more difficult. Nevertheless, for all wind scenarios simulated, satisfactory closed-loop performance was observed under only a few tens of distributed optimization iterations per time step.

1.1 Outline and Contribution Part I focuses on the background material required for the thesis. In Chapter 2, notation specifics and general definitions are introduced. In Chapter 3, existing concepts related to system theory and control are revisited. In Chapter 4, principles of convex optimization are introduced, where problem decomposition techniques and distributed solution methods are emphasized. Chapter 5 introduces MPC related concepts, including nominal, robust and reference tracking formulations. Finally, Chapter 6 contains a survey on distributed MPC approaches available in the literature. In Part II, formulations of cooperative distributed MPC controllers, which guarantee stability in closed-loop and which can be operated by distributed optimization are proposed. In Chapter 7, this is done for nominal distributed MPC, where the main contribution is the formulation of a novel distributed positively invariant set, which is used as a timevarying terminal set in the MPC problem. In Chapter 8, which is on robust distributed MPC, the main contribution is the formulation of structured robust positively invariant sets, as well as the design of distributed constraint tightening methods. In Chapter 9, which is on cooperative distributed reference tracking MPC, the main contribution lies in the formulation and distributed synthesis of a structured terminal invariant set for tracking. For all controllers proposed, as additional contributions, distributed synthesis methods are

4

1 Introduction

presented for networks of linear systems subject to polytopic input and state constraints. Part III is on the computational performance of distributed optimization methods in cooperative distributed MPC. In the computational study presented in Chapter 10, which is, to the best of the authors’ knowledge, the first study of this kind for distributed MPC, the contribution lies in a number of computational results. These results suggest that the convergence speed of distributed optimization methods in MPC decreases if the coordination of the networked systems is“hard”, a property which usually manifests in Lagrange multipliers of large magnitude. This observation is confirmed in a wind farm application study in Section 11.2. The first contribution of this study is, to the best of the authors’ knowledge, the first presentation of a cooperative distributed MPC controller for wind farms. The second contribution are computational results, which document the dependence of the distributed optimization performance on the farm-wide available power.

1.2 Publications The work presented in this thesis is based on the following publications, which were all realized in collaboration with colleagues. Chapter 7 is largely based on the publication Distributed Synthesis and Control of Constrained Linear Systems. C. Conte, N. R. Voellmy, M. N. Zeilinger, M. Morari, C. N. Jones, Proceedings of the American Control Conference, Montreal, Canada, pp. 6017 – 6022, June 2012. [Conte et al., 2012b]. Chapter 8 is largely based on the publication Robust Distributed Model Predictive Control of Linear Systems. C. Conte, M. N. Zeilinger, M. Morari, C. N. Jones, Proceedings of the European Control Conference, Zurich, Switzerland, pp. 2764 – 2769, July 2013. [Conte et al., 2013c]. Chapter 9 is largely based on the publication Cooperative Distributed Tracking MPC for Constrained Linear Systems: Theory and Synthesis. C. Conte, M. N. Zeilinger, M. Morari, C. N. Jones, Proceedings of the 52nd Conference on Decision and Control, Florence, Italy, pp. 3812 – 3817, December 2013. [Conte et al., 2013a]. Chapter 7, Chapter 8 and Chapter 9 contain elements of the publication A Framework for Cooperative Distributed Model Predictive Control. C. Conte, M. N. Zeilinger, M. Morari, C. N. Jones, submitted to Automatica, December 2013. [Conte et al., 2013b].

1.2 Publications

5

Chapter 10 is largely based on the publication Computational Aspects of Distributed Optimization in Model Predictive Control. C. Conte, T. H. Summers, M. N. Zeilinger, M. Morari, C. N. Jones, Proceedings of the 51st Conference on Decision and Control, Maui, USA, pp. 6819 – 6824, December 2012. [Conte et al., 2012a]. Finally, Chapter 11 is largely based on the following publication, which was realized in an equal contribution collaboration with Vedrana Spudi´c: Cooperative Distributed Model Predictive Control for Wind Farms. V. Spuci´c, C. Conte, M. Baoti´c, M. Morari, submitted to Optimal Control Applications and Methods, July 2013. [Spudi´c et al., 2013]. Publications which contain topics related to this thesis, which are not discussed, are: Benchmarking Large Scale Distributed Convex Quadratic Programming Algorithms. A. Kozma, C. Conte, M. Diehl, submitted to Optimization Methods and Software, July 2013. [Kozma et al., 2013]. Formation Flight of Micro Aerial Vehicles without Infrastructure. T. N¨ageli, C. Conte, A. Domahidi, M. Morari, O. Hilliges, submitted to the International Conference on Intelligent Robots and Systems, February 2014. [N¨ageli et al., 2014].

Part I Preliminaries

2 Notation and Definitions 2.1 Notation Sets ∅ N N0 Z R R≥0 Rn Rm×n

The empty set Set of natural numbers Set of natural numbers including 0 Set of integer numbers Set of real numbers Set of non-negative real numbers Set of n-dimensional vectors with real entries Set of m × n-dimensional matrices with real entries

Set operators Let S1 , . . . , SM be sets |S1 | S1 \ S2 S1 ∪ S2 S1 ∩ S2 S1 ⊕ S2 LM i=1 Si S1 S2 S1 × S2 QM i=1 Si

Cardinality of S1 Set difference, S1 \ S2 = {s|s ∈ S1 and s ∈ / S2 } Union of sets, S1 ∪ S2 = {s|s ∈ S1 or s ∈ S2 } Intersection of sets, S1 ∩ S2 = {s|s ∈ S1 and s ∈ S2 } Minkowski sum, S1 ⊕ S2 = {s1 + s2 |s1 ∈ S1 , s2 ∈ S2 } L Minkowski sum over M sets, M i=1 Si = S1 ⊕ . . . ⊕ SM Minkowski (Pontryagin) difference, S1 S2 = {s|s + s2 ∈ S1 , ∀s2 ∈ S2 } Cartesian product of sets, S1 × S2 = {(s1 , s2 )|s1 ∈ S1 , s2 ∈ S2 } Q Cartesian product of M sets, M i=1 Si = S1 × . . . × SM

Algebraic operators Let T ∈ Rn×n be a quadratic matrix, S ∈ Rn×n a symmetric matrix, v ∈ Rn a vector and s ∈ R a scalar.

10 bsc dse kv k kv k2 kv k∞ λmin (S) λmax (S) S≥0 S>0 kv kS kT k2

2 Notation and Definitions Floor function, bsc = max{m ∈ Z|m ≤ s} ceiling function, dse = min{m ∈ Z|m ≥ s} Any vector norm p l2 vector norm (Euclidean norm), kv k2 = v12 + . . . + vn2 l∞ vector norm, kv k∞ = maxi∈{1,...,n} {|v1 |, . . . , |vn |} Minimum eigenvalue of a symmetric matrix S Maximum eigenvalue of a symmetric matrix S Positive semi-definite matrix, S ≥ 0 ⇔ v T Sv ≥ 0 ∀v 6= 0 ⇔ λmin (S) ≥ 0 Positive definite matrix, S > 0 ⇔ v T Sv > 0 ∀v 6= 0 ⇔ λmin (S) > 0 √ Scaled Euclidean norm, kv kS = v T Sv , S ≥ 0 p l2 matrix norm (Euclidean norm), kT k2 = λmin (AT A)

Other Let Si ∈ Rni ×mi , i ∈ {1, . . . , M} be matrices and si ∈ Rni , i ∈ {1, . . . , M} be vectors. diagi∈{1,...,M} (Si ) coli∈{1,...,M} (si ) coli∈{1,...,M} (Si ) In

Block-diagonal matrix with blocks Si Vector consisting of stacked subvectors si Stacked matrices Si , all having an equal number of columns Identity matrix in Rn×n

2.2 Definitions 2.2.1 Set Related Definitions Definition 2.1 (Convex set). A set S ⊆ Rn is convex if for any pair (s1 , s2 ) ∈ S × S and any λ ∈ [0, 1] it holds that λs1 + (1 − λ)s2 ∈ S. Definition 2.2 (Affine set). A set S ⊆ Rn is affine if for any pair (s1 , s2 ) ∈ S × S and any λ ∈ R it holds that λs1 + (1 − λ)s2 ∈ S. Definition 2.3 (Affine hull). The affine hull aff S of a set S ∈ Rn is defined as the intersection of all affine sets in Rn that contain S. Definition 2.4 (open -Ball). An open -Ball with radius  > 0 around a point xc ∈ Rn is defined as B (xc ) = {x ∈ Rn |kx − xc k < } . (2.1) Definition 2.5 (Interior). The interior of a set S ⊆ Rn is defined as int S = {s ∈ S|∃ > 0, B (s) ⊆ S} .

(2.2)

2.2 Definitions

11

Definition 2.6 (Relative interior). The relative interior of a set S ⊆ Rn is defined as relint S = {s ∈ S|∃ > 0, B (s) ∩ affS ⊆ S} .

(2.3)

Definition 2.7 (Open/closed set). A set S ∈ Rn is open if S = relint S. It is called closed if its complement S c = {s ∈ Rn |s ∈ / S} is open. Definition 2.8 (Bounded set). A set S ∈ Rn is bounded if there exists a scalar radius r < ∞ and a point x ∈ Rn such that S ⊆ Br (x). Definition 2.9 (Compact set). A set S ∈ Rn is compact if it is both closed and bounded. Definition 2.10 (Halfspace). A halfspace H ⊆ Rn is defined by a vector a ∈ Rn and a scalar b ∈ R, as H = {x ∈ Rn |aT x ≤ b} . (2.4) Definition 2.11 (Polytope). A polytope in P ⊆ Rn is defined as the intersection of m halfspaces in Rn , hence P = {x ∈ Rn |Hx ≤ k} , (2.5) where H ∈ Rm×n and k ∈ Rm . Definition 2.12 (Ellipsoid). An ellipsoid E ⊆ Rn is defined by a shape matrix P ≥ 0, P ∈ Rn×n , a centerpoint x0 ∈ Rn and a radius r ∈ R≥0 as E = {x ∈ Rn |(x − x0 )T P (x − x0 ) ≤ r } .

(2.6)

2.2.2 Function Definitions Definition 2.13 (Convex function). A function f (·) : D → R is convex if its domain D ⊆ Rn is a convex set and if for any pair (x1 , x2 ) in D × D and any λ ∈ [0, 1], it holds that f (λx1 + (1 − λ)x2 ) ≤ λf (x1 ) + (1 − λ)f (x2 ) . (2.7) Furthermore, f (·) is strictly convex if the inequality in (2.7) is strictly fulfilled for any pair (x1 , x2 ) in D × D and any λ ∈ (0, 1), and it is strongly convex with convexity parameter µ > 0 if the function f (x) − µ2 kxk22 is convex on D. Definition 2.14 (Concave function). A function f (·) : D → R is concave if the function −f (x) is convex. Moreover, it is strictly or strongly concave if −f (x) is strictly or strongly convex respectively.

12

2 Notation and Definitions

Definition 2.15 (Lipschitz continuous function). A function f (·) : D → Rn with domain D ⊆ Rm is Lipschitz continuous if for any pair (x1 , x2 ) ∈ D × D it holds that kf (x1 ) − f (x2 )k ≤ Lkx1 − x2 k .

(2.8)

Definition 2.16 (Support function). The support function σS (·) : Rn → R of a set S ⊆ Rn is defined as σS (a) = supx∈S aT x. Definition 2.17 (K-class function). A function β(·) : R≥0 → R≥0 belongs to class K if it is continuous, strictly increasing and if β(0) = 0. Definition 2.18 (K∞ -class function). A function β(·) : R≥0 → R≥0 belongs to class K∞ if it belongs to class K and if limx→∞ β(x) = ∞.

3 System Theory and Control In this section, some fundamental definitions and properties of discrete-time linear timeinvariant (LTI) systems are introduced. The discussion is restricted to concepts which are relevant for this thesis. For more details, the reader is referred to standard textbooks, such as [Callier and Desoer, 1991].

3.1 Constrained LTI Systems Throughout this thesis, discrete-time constrained LTI systems of the form x(k + 1) = Ax(k) + Bu(k) + w (k) ∀k ≥ 0 y (k) = Cx(k) ∀k ≥ 0

(3.1a) (3.1b)

are considered, where x(k) ∈ Rn is the state of the system at time k, u(k) ∈ Rm is its input, y (k) ∈ Rp its output and w (k) ∈ Rn an additive disturbance. The dynamics of the system are captured by the system matrix A ∈ Rn×n , as well as the input map B ∈ Rn×m . Both the input and the state of system (3.1) are subject to constraints of the form x(k) ∈ X , u(k) ∈ U

∀k ≥ 0,

(3.2)

where both X ⊆ Rn and U ⊆ Rm are convex sets which contain the origin in their interior. The disturbance is assumed to be bounded as w (k) ∈ W

∀k ≥ 0,

(3.3)

where the set W ⊆ Rn is convex and contains the origin. If no disturbance is acting on the system, i.e. W = {0}, the system can be written as x(k + 1) = Ax(k) + Bu(k) ∀k ≥ 0

(3.4)

and is denoted as nominal. Moreover, in case the input of a nominal system is given as a function κ(·) : Rn → Rm of the current state, the resulting closed-loop system x(k + 1) = Ax(k) + Bκ(x(k)) ∀k ≥ 0

(3.5)

14

3 System Theory and Control

is denoted as autonomous. If the time index k is not needed for the context, the short-hand notation x + = Ax + Bu + w y = Cx

(3.6a) (3.6b)

is used for system (3.1), and respectively for systems (3.4) and (3.5). For all nominal LTI systems considered in this thesis, the following assumption is made. Assumption 3.1. The pair (A, B) is stabilizable. Assumption 3.1 implies the fact that a linear control law κ(x) = Kx, K ∈ Rm×n exists, such that all eigenvalues of the matrix A + BK lie strictly within the unit circle of the complex plane.

3.2 Networks of Constrained LTI Systems By a network of constrained linear systems, we denote a set of coupled dynamic systems, which are subject to local constraints. Specifically, this means that for a global dynamic system (3.6), i.e. the whole network, a partition into M non-overlapping subsystems exists, each of which has a local state xi ∈ Rni , a local input ui ∈ Rmi and a local output yi ∈ Rpi and is subject to a local disturbance wi ∈ Rni . The dimensions of these local vectors are such that M M M X X X ni = n, mi = m, pi = p , (3.7) i=1

i=1

i=1

and the global state vector can be written as x = coli∈{1,...,M} (xi ), the global input as u = coli∈{1,...,M} (ui ), the global output as y = coli∈{1,...,M} (yi ) and the global disturbance as w = coli∈{1,...,M} (wi ). Throughout this thesis, we mainly consider networks of systems with local dynamics of the form xi+

=

M X

Aij xj + Bi ui + wi ,

(3.8a)

Cij xj ,

(3.8b)

j=1

yi =

M X j=1

and local constraint sets and disturbance bounds of the form xi ∈ Xi , ui ∈ Ui , wi ∈ Wi

∀i ∈ {1, . . . , M} ,

(3.9)

3.2 Networks of Constrained LTI Systems

15

where all Xi and Ui are convex sets which contain the origin in their interior, and all Wi are convex sets which contain the origin. Note that in the dynamics formulated in (3.8), only coupling in the state is present. Nevertheless, input coupled networks of system can be brought into form (3.8) by a simple reformulation, i.e. by considering the original inputs as additional states and the changes in the original inputs as the inputs. The dynamics (3.8) are used to define neighboring systems. Definition 3.1 (Neighboring Systems). System j is a neighbor of system i if Aij 6= 0 or Cij 6= 0. The set of all neighbors of i , including i itself, is denoted as Ni . The states and outputs of all systems j ∈ Ni are denoted as xNi = colj∈Ni (xj ) ∈ RnNi and yNi = colj∈Ni (yj ) ∈ RpNi respectively. The local systems (3.8) can thus be equivalently written, in compact form, as xi+ = ANi xNi + Bi ui , yi = CNi xNi

∀i ∈ {1, . . . , M} ,

(3.10)

where ANi ∈ Rni ×nNi and CNi ∈ Rpi ×nNi . Throughout the thesis, we make the following assumption. Assumption 3.2 (Communication). Two systems i and j can communicate in a bidirectional way, if i ∈ Nj or j ∈ Ni . Naturally, a network of dynamic systems (3.8) has a global system matrix A and a global output map C consisting of blocks Aij ∈ Rni ×mj and Cij ∈ Rpi ×nj for each pair (i, j) ∈ {1, . . . , M}2 . Furthermore, it has a global input map B = diagi∈{1,...,M} (Bi ) ∈ Rn×m , which is block-diagonal. Moreover, using (3.9), the global state and input constraints can be written as x ∈ X = X1 × . . . × XM ⊆ Rn , u ∈ U = U1 × . . . × UM ⊆ Rm ,

(3.11)

and similarly, the global disturbance bound can be written as w ∈ W = W1 × . . . × WM ⊆ Rn .

(3.12)

For all networks of nominal LTI systems considered in this thesis, the following assumption is made. Assumption 3.3. The pair (A, B) is stabilizable by a structured linear state feedback control law of the form κ(x) = Kx = coli∈{1,...,M} (KNi xNi ) , where K ∈ Rm×n and where KNi ∈ Rmi ×nNi

∀i ∈ {1, . . . , M}.

(3.13)

16

3 System Theory and Control

The structured stabilizability in Assumption 3.3 is a generalization of the more common notion of decentralized stabilizability, where the matrix K is block-diagonal. Decentralized stabilizability has been extensively studied over the past decades. Corresponding conditions for linear systems can be found e.g. in [Wang and Davison, 1973], based on fixed modes ˇ in the open left half of the complex plane, or in [Siljak, 1991], based on combinatorial conditions on the off-diagonal blocks of the system matrix A. These blocks essentially describe the dynamic interconnections among the systems in the network.

3.3 Set Invariance and Lyapunov Stability The main theoretical results presented in this thesis are related to stability of constrained networked LTI systems. In order to obtain such results, set invariance and Lyapunov stability theory are of paramount importance. Therefore, the corresponding methodology is introduced in the following. All definitions stated in this section can be found in similar form for continuous-time systems in standard textbooks such as [Vidyasagar, 1993]. For details on Lyapunov stability of discrete-time systems, the reader is referred to the appendix of [Rawlings and Mayne, 2009]. First, definitions for positively invariant and robust positively invariant sets are introduced, which can be found in similar form in e.g. [Blanchini, 1999]. Definition 3.2 (Positively invariant (PI) set). A set XPI ⊆ Rn is denoted positively invariant for an autonomous closed-loop system x + = Ax + Bκ(x), if x ∈ XPI ⇒ x + ∈ XPI .

(3.14)

Definition 3.3 (Robust positively invariant (RPI) set). A set XRPI ⊆ Rn is denoted robust positively invariant for a closed-loop system x + = Ax + Bκ(x) + w , if x ∈ XRPI ⇒ x + ∈ XRPI

∀w ∈ W .

(3.15)

In the following, the concept of asymptotic stability is introduced. This concept is based on a PI set which contains a stable and globally attractive equilibrium point. To simplify the following explanations, we define this equilibrium point to be the origin. Definition 3.4 (Stable origin). Let XPI be a PI set containing the origin. For an autonomous system (3.5), the origin is stable if, for any  > 0 there exists a δ > 0, such that for all x(0) ∈ XPI it holds that kx(0)k ≤ δ ⇒ kx(k)k ≤  ∀k ≥ 0 .

(3.16)

3.3 Set Invariance and Lyapunov Stability

17

Definition 3.5 (Attractive origin). Let XPI be a PI set for the autonomous system (3.5), containing the origin. The origin is attractive within XPI if kx(k)k → 0 as k → ∞. Definition 3.6 (Asymptotic stability). Let XPI be a PI set containing the origin. The origin is an asymptotically stable equilibrium for the autonomous system (3.5), if it is both stable and attractive within XPI . In the following, it is shown how asymptotic stability can be shown using Lyapunov functions. Lyapunov functions are characterized by K∞ -class functions, which were introduced in Definition 2.18. Definition 3.7 (Lyapunov function). Let XPI be a PI set, which contains a neighborhood N (0) of the origin in its interior, for the autonomous system (3.5). Let furthermore β1 (·), β2 (·) and β3 (·) be K∞ -class functions. A function V (·) : XPI → R≥0 with V (0) = 0 is a Lyapunov function in XPI if β1 (kxk) ≤ V (x) ∀x ∈ XPI

(3.17a)

V (x) ≤ β2 (kxk) ∀x ∈ N (0)

(3.17b)

V (Ax + Bκ(x)) − V (x) ≤ −β3 (kxk) ∀x ∈ XPI

(3.17c)

The construction of a Lyapunov function is a direct way to show asymptotic stability for an autonomous system of type (3.5). Theorem 3.1 (Asymptotic stability by Lyapunov function). Given a Lyapunov function for system (3.5) in XPI , the origin is an asymptotically stable equilibrium with region of attraction XPI .

4 Convex Optimization In this chapter, some basic concepts of convex optimization are introduced. The emphasis is on theory and methods relevant for this thesis. For more details, the reader is referred to standard textbooks such as [Boyd and Vandenberghe, 2004], [Ben-Tal and Nemirovski, 1987].

4.1 Terminology A general mathematical optimization problem consists of the minimization of an objective function f (·) : Rn → R, the argument of which is constrained to lie in a set Z ⊆ Rn , hence

min f (z)

(4.1a)

s.t. z ∈ Z .

(4.1b)

z

In principle, any objective function and constraint set can be combined into an optimization problem. The resulting problem might however be intractable due to its complexity or due to the lack of an efficient solution method. Therefore, this thesis mainly relies on techniques related to convex optimization problems, which exhibit the simplifying property that any local optimum is also a global one. In Section 4.2, a number of common convex optimization problems are introduced for which efficient solution methods exist. All of these problems will be used in the technical chapters of this thesis to formulate control related problems. Throughout the thesis, the following terminology will be consistently used: If the set Z is non-empty, problem (4.1) is called feasible, otherwise it is called infeasible. Moreover, if Z = Rn , the problem is called unconstrained. A point z¯ is called feasible if z¯ ∈ Z, and it is called infeasible otherwise. If there is a feasible sequence of points zk ∈ Z, where f (zk ) → −∞ as k → ∞, the problem is called unbounded. If the problem is feasible and not unbounded, Z ∗ denotes the set of optimizers (minimizers), which is the set of points z ∗ at which the optimum (minimum) f ∗ = f (z ∗ ) of problem (4.1) is attained. An optimization problem is called convex, if f (z) is a convex function according to Definition 2.13 and if Z is a convex set according to Definition 2.1.

20

4 Convex Optimization

4.2 Common Convex Optimization Problems In the following, the definitions of a number of common convex optimization problems are introduced. All of these problems can be solved efficiently by available software and are relevant in later chapters of this thesis.

4.2.1 Linear Program A linear program (LP) is characterized by a linear objective function to be minimized, where the minimization is subject to affine inequality constraints. A possible LP representation is given as min bT z

(4.2a)

s.t. Hz ≤ k ,

(4.2b)

z

where the constraint set Z = {z ∈ Rn |Hz ≤ k} in (4.2b) is a polytope characterized by the pair (H ∈ Rm×n , k ∈ Rm ), see Definition 2.11. Provided the polytope defined by the pair (H, k) is non-empty and the LP (4.2) is not unbounded, at least one point in the set of minimizers Z ∗ lies on a vertex of the polytope Z.

4.2.2 Quadratic Program A quadratic program (QP) is characterized by a quadratic objective function to be minimized, where the minimization is subject to affine inequality constraints. A possible QP representation is given as min z T Az + bT z

(4.3a)

s.t. Hz ≤ k ,

(4.3b)

z

where the matrix A ∈ Rn×n is required to be positive semidefinite in order for problem (4.3) to be convex. Note that if A = 0, the problem reduces to an LP. A quadratic optimization problem which is additionally subject to quadratic inequality constraints is referred to as a quadratically constrained quadratic program (QCQP) and is given as min z T Az + bT z

(4.4a)

s.t. z T Pi z + qiT z + ri ≤ 0 ∀i ∈ {1, . . . , m} ,

(4.4b)

z

4.2 Common Convex Optimization Problems

21

where Pi ∈ Rn×n , qi ∈ Rn and ri ∈ R for all i ∈ {1, . . . , m}. The matrices A and Pi , ∀i ∈ {1, . . . , M}, are required to be positive semidefinite in order for problem (4.4) to be convex. Note that if all Pi are zero, the problem reduces to a QP, and if additionally A is zero, the problem reduces to an LP.

4.2.3 Semidefinite Program A semidefinite program (SDP) can be characterized by a linear objective function to be minimized, where the minimization is subject to a linear matrix inequality (LMI) constraint.

min bT z

(4.5a)

z

s.t. P0 +

n X

zi Pi ≥ 0 ,

(4.5b)

i=1

where zi ∈ R ∀i ∈ {1, . . . , n}, z = [z1 , . . . , zn ]T ∈ Rn , Pi ∈ Rn×n ∀i ∈ {0, . . . , n} and where the constraint (4.5b) is an LMI, hence the left hand side of (4.5b) is constrained to be positive semidefinite. An SDP is convex, since the set of positive semi-definite matrices is a convex cone. Note that any affine inequality constraint, as well as any convex quadratic inequality constraint, can equivalently be represented by LMIs [Ben-Tal and Nemirovski, 1987]. This implies that any LP, QP or QCQP can equivalently be formulated as an SDP. A particular class of SDP is given by the determinant maximization problem. This class is relevant for this thesis since the determinant of a positive definite matrix Q ∈ Rn×n is proportional to the volume of the ellipsoid E = {x ∈ Rn |x T Qx ≤ 1}. Therefore, determinant maximization has many applications in the computation of ellipsoidal invariant sets for dynamic systems. In a determinant maximization problem, the determinant of a linear matrix-valued function G(·) : Rn → Sm×m is maximized, where Sm×m is the set + + of positive semidefinite matrices, and where the maximization is subject to constraints on z. The determinant of G(z) is a non-convex operator, its logarithm however, is concave [Boyd and Vandenberghe, 2004]. Therefore, a determinant maximization problem can be equivalently reformulated as the convex problem min − log det G(z) z

s.t. P0 +

n X

zi Pi ≥ 0 .

(4.6a) (4.6b)

i=1

It is worthwile to note that the log det-operator is separable in case its argument is a block-diagonal matrix.

22

4 Convex Optimization

Lemma 4.1. For a block-diagonal matrix-valued function G(·) : Rn → Sm×m , G(z) = + diagi∈{1,...,M} (Gi (z)), it holds that log det(G(z)) =

M X

log det(Gi (z)) .

(4.7)

i=1

Proof. The proof follows from basic properties of the determinant and the logarithm operator. Since G(z) is block-diagonal, its determinant can be written as det(G(z)) =

M Y

det(Gi (z)) .

(4.8)

i=1

Due to the inherent properties of the logarithm operator, we directly get ! M M Y X log det(G(z)) = log det(Gi (z)) = log det(Gi (z)) . i=1

(4.9)

i=1

4.3 Duality in Convex Optimization Consider a convex optimization problem of the form (4.1), where the constraint set is given as Z = {z ∈ Rn |fineq (z) ≤ 0, feq (z) = 0}, where fineq (·) : Rn → Rm and feq (·) : Rn → Rp . This problem has an associated dual problem, which reads as max d(ν, λ) ,

(4.10)

d(ν, λ) = min L(z, ν, λ) ,

(4.11)

ν≥0,λ

where d(ν, λ) is given as z

and L(z , ν, λ) is the Lagrangian L(z, ν, λ) = f (z) + ν T fineq (z) + λT feq (z) .

(4.12)

Note that the dual function is the pointwise minimum over a family of affine functions in ν and λ. Therefore, the dual function is concave, even if the primal problem is not convex. For primal feasible points z¯ ∈ Z, the Lagrangian, with positive multipliers ν ≥ 0, yields a lower bound to the primal function value, i.e. L(¯ z , ν, λ) ≤ f (¯ z ) ∀(¯ z ∈ Z, ν ≥ 0, λ ∈ Rp ) ,

(4.13)

4.4 Distributed Convex Optimization

23

since ν T fineq (¯ z ) + λT feq (¯ z ) ≤ 0. Consequently, also the dual function d(ν, λ) can be used to underestimate the primal function value as d(ν, λ) = min L(z, ν, λ) ≤ f (¯ z ) ∀(¯ z ∈ Z, ν ≥ 0, λ ∈ Rp ) . z

(4.14)

Since this property evidently also holds for the primal optimizer z ∗ ∈ Z, we conclude that the maximum of the dual problem (4.10) is always an underestimator of the minimum of the associated primal problem (4.1), hence d ∗ = max d(ν, λ) ≤ min f (z) = f ∗ . ν≥0

z∈Z

(4.15)

The underestimation property (4.15) of the dual problem is commonly referred to as weak duality. In some cases even strong duality holds, i.e. d ∗ = max d(ν, λ) = min f (z) = f ∗ . ν≥0

z∈Z

(4.16)

One condition under which strong duality holds is Slater’s condition, which requires that there exists a point z in the relative interior of Z. Furthermore, strong duality always holds for LP s and QP s, provided they are feasible and not unbounded.

4.4 Distributed Convex Optimization For the constrained convex optimization problems introduced in Section 4.2, a variety of solution methods exist in the literature (see e.g. [Bertsekas, 1999], [Boyd and Vandenberghe, 2004]). Prominent approaches are e.g. active set methods, interior-point methods and firstorder methods. In the context of this thesis, which is concerned with large-scale systems subject to communication constraints, first-order methods are particularly appealing due to two reasons. First, addressing the large-scale aspect, the iterations of first-order methods are often inexpensive regarding memory and computations. Second, addressing the distributed aspect, many first-order methods can be executed in a distributed way without central coordination. For these reasons, the remainder of this chapter focuses on distributed firstorder methods for solving convex optimization problems. In Section 4.4.1, a particularly structured formulation for convex optimization problems is introduced, which facilitates the discussion of distributed optimization methods. In the following, two first-order optimization methods and their distributed variants are introduced. Specifically, in Section 4.4.2, a dual decomposition based fast gradient method (DDFG) is discussed, and in Section 4.4.3, the alternating direction method of multipliers (ADMM) is introduced. These two methods do by no means exhaustively cover the distributed optimization techniques available in the literature. DDFG was chosen due to its favorable

24

4 Convex Optimization

theoretical convergence guarantee, and ADMM due to the fact that it is known to often converge fast in practice (see e.g. [Summers and Lygeros, 2012], [Kraning et al., 2013]). However, other methods, such as the subgradient method [Bertsekas, 1999], the Jacobi method used in [Stewart et al., 2010], or the primal block-coordinate descent method proposed in [Necoara and Clipici, 2013], are applicable as well if the problem satisfies the method-specific assumptions. For a comprehensive overview on distributed optimization concepts and methods, the reader is referred to [Bertsekas and Tsitsiklis, 1989].

4.4.1 Structured Convex Optimization Problems The distributed solution of convex optimization problems is meaningful only if there exists a decomposition of either the primal or the dual problem. A decomposition commonly refers to the segmentation of the problem’s objective function and constraints into separate problem units, which are coupled in a (preferrably sparse) neighbor-to-neighbor fashion. In this section, we will introduce a particular decomposition scheme for the primal problem, under which a variety of distributed optimization techniques are conveniently applicable. First, we define the notion of a coupling graph. Definition 4.1 (Coupling Graph). of vertices V = {1, . . . , |V|} and a neighbors of a vertex i , including i set of outgoing neighbors of vertex

A directed coupling graph G(V, E) is defined by a set set of directed edges E ⊆ V × V. The set of incoming itself, is denoted as Ni = {j ∈ V|(j, i ) ∈ E} ∪ i . The ¯i = {j ∈ V|(i, j) ∈ E}. i , without i , is denoted as N

Remark 4.1. Note that in context of networked dynamic systems, a coupling graph can be defined through the coupled system dynamics. In particular, each system in the network corresponds to a vertex, and dynamically coupled neighboring systems, as defined in Definition 3.1, correspond to directed edges which connect the vertices. In this thesis, convex optimization problems of the form (4.1) are considered, for which a decomposition along a coupling graph G(V, E) exists, which fulfills the following three properties: (i) A non-overlapping decomposition of the global variable vector exists, i.e. z = coli∈V (zi ), where zi ∈ Rni is denoted as the global variables associated with vertex i ∈ V. (ii) The constraint set Z can be decomposed into |V| sets Zi , each of which depends on the vector yi = colj∈Ni (zj ) ∈ RnNi only. This vector is denoted as the local variables of vertex i and contains the variables of vertex i and its incoming neighbors. P|V| (iii) The objective function is separable into terms fi (yi ), i.e. f (z) = i=1 fi (yi ).

4.4 Distributed Convex Optimization

25

If a general convex optimization problem (4.1) exhibits the three properties listed above, it is denoted decomposable and can equivalently be written as min

|V| X

y1 ,...,y|V| ,z

fi (yi )

(4.17a)

i=1

s.t. yi ∈ Zi yi = Ei z

∀i ∈ V , ∀i ∈ V .

(4.17b) (4.17c)

Note that in problem (4.17), the problem is formulated in the global variables z = coli∈V (zi ) as well as in the local variables y1 , . . . , y|V| . The matrices Ei , similar to permutation matrices, have entries in {0, 1} with exactly one entry per row equal to 1. For each i ∈ {1, . . . , |V|}, the matrix Ei generates the local variables of vertex i , i.e. yi , from the global variables z, and in particular from those parts zj in z, which are associated with neighboring vertices j ∈ Ni . Hence, each Ei is composed of block-rows Eij , i.e. Ei = colj∈Ni (Eij ), where the non-zero entries of each Eij map to zj in (4.17c). The advantage of the formulation (4.17) is the fact that it simplifies the application of distributed optimization techniques. Most distributed optimization schemes rely on first-order methods and a decomposition of either the primal problem (4.1) or the dual problem (4.10). The advantage of primal decomposition schemes is the fact that they often yield feasible primal solutions in every iteration. However, first-order methods usually require a projection on the set of feasible variables in every iteration. The projection operation, if the set of feasible variables is not particularly simple to be projected onto, is generally as hard as the original optimization problem itself and thus often renders primal decomposition schemes unsuitable. In this case, dual decomposition schemes are still viable since the set of feasible dual variables can be projected onto efficiently, even if the primal constraints do not allow so. The drawback of dual schemes is the fact that they yield primal feasible solutions only asymptotically. Nevertheless, since most problems of interest for this thesis are characterized by constraint sets, which can not be projected onto efficiently, we focus on dual decomposition based techniques in the following. For the purpose of distributed optimization, it suffices to build the partial Lagrangian of problem (4.17), where “partial” is meant w.r.t. the equality constraint (4.17c), i.e. Lp (y , z, λ) =

|V| X

fi (yi ) + λTi (yi − Ei z) ,

(4.18)

i=1

where λ = coli∈V (λi ). The partial dual function, in order to yield the same lower bound properties as the dual function (4.10), is defined using domain constraints on yi , ∀i ∈

26

4 Convex Optimization

{1, . . . , M}, i.e. dp (λ) =

min

yi ∈Zi ∀i∈{1,...,M}, z

Lp (y , z, λ) .

(4.19)

The partial Lagrangian (4.18) is separable into terms in (yi , zi ), which are coupled through λ only, i.e. Lp (y , z, λ) =

|V| X

Lp,i (yi , zi , λ) =

i=1

|V| X

X

fi (yi ) + λTi yi −

λTji Eji zi ,

(4.20)

¯i ∪i j∈N

i=1

where for each i it holds that λi = colj∈Ni (λij ), and λij denotes the multipliers associated with the copying of zj , i.e. the global variables of vertex j, into yi , i.e. the local variables of vertex i . Similarly, the partial augmented Lagrangian, which is used instead of the partial Lagrangian in some first-order methods, is separable into terms coupled through λ and z only, i.e. La (y , z, λ) =

|V| X

La,i (yi , z, λ) =

i=1

|V| X i=1

ρ fi (yi ) + λTi (yi − Ei z) + kyi − Ei zk22 . 2

(4.21)

As a consequence of the separable partial Lagrangian (4.20), the partial dual function (4.19) can be written as dp (λ) =

|V| X i=1

min Lp,i (yi , zi , λ) .

yi ∈Zi ,zi

(4.22)

Hence, evaluating dp (·) at a particular λ corresponds to the solution of |V| uncoupled constrained optimization problems. As shown in e.g. [Bertsekas, 1999], a subgradient of dp (λ) is given as g(λ) = coli∈V (gi (λi )) = coli∈V (yi∗ (λ) − colj∈Ni (Eij zj∗ (λ))) ∈ ∂dp (λ) ,

(4.23)

where ∂dp (λ) is the subdifferential of dp (λ), i.e. the set of the subgradients of dp (λ), and yi∗ (λ) and zi∗ (λ), for each i ∈ {1, . . . , kVk}, are defined as (yi∗ (λ), zi∗ (λ)) = arg min Lp,i (yi , zi , λ) . yi ∈Zi ,zi

(4.24)

Equation (4.23) illustrates a fundamental property, which is utilized by most distributed optimization methods based on dual decomposition: First-order information on the dual function can be obtained by local computations and neighbor-to-neighbor communication only, no global coordination is required. In other words, the partial dual function (4.22) can be evaluated by parallel local computations, and the byproduct (yi∗ (λ), zi∗ (λ)) of this evaluation can be used to build a subgradient (4.23) by neighbor-to-neighbor communication. This property enables dual decomposition based distributed optimization methods such as DDFG and ADMM, which are introduced in the next two sections.

4.4 Distributed Convex Optimization

27

4.4.2 Distributed Fast Gradient Method The fast gradient method, originally proposed by Nesterov in 1983, is a first-order optimization method with a theoretical upper bound on the convergence rate. This bound is of order O( k12 ), where k is the iteration count, and it is proven to be the lowest upper bound for convex optimization problems with L-smooth objective functions, see [Nesterov, 2004] for details. Since most problems considered in this work exhibit a smooth dual function, or can be mildly modified in order to do so, the fast gradient method is of interest for this thesis. Given the structured convex optimization problem (4.17), its partial dual problem (4.19) can be solved the constant step scheme II of the fast gradient method [Nesterov, 2004], which is stated in a distributed form in Algorithm 4.1. Note that in this algorithm, L denotes the Lipschitz constant of the gradient ∇dp (λ) of the dual function (see [Richter et al., 2011] for details) and yii denotes vertex i ’s copy of its own global variable zi . Algorithm 4.1 Dual Decomposition based Fast Gradient Method (DDFG) √

Require: Lipschitz constant L of ∇dp (λ), α = 1: ∀i ∈ V in parallel: 2: repeat 3: (yi+ , zi+ ) = arg minyi ∈Zi ,zi =yii Lp,i (yi , zi , η) 4: communicate zi+ to all j ∈ Ni 1 + + 5: λ+ i = ηi + L (yi − Ei z ) √ 6: α+ = α2 ( α2 + 4 − α) 7: β = α(1−α) α2 +α+ + 8: ηi+ = λ+ i + β(λi − λi ) 9: until convergence

5−1 2 ,

λ=η=0

In [Richter et al., 2011], is was shown that the bound ( r  ) L ∗ kλ k2 − 2 , 0 q ≤ max 2 

(4.25)

holds on the number of iterations q to reach an accuracy dp (λ∗ ) − dp (λ) ≤  in the dual objective function. Since the bound (4.25) depends on the Lagrange multiplier λ∗ maximizing the partial dual function (4.19), it can only be verified a-posteriori. Nevertheless, it indicates that more iterations are required, if the magnitude of kλ∗ k is large. Remark 4.2. If the local objective functions fi (yi ) are strongly convex for each i ∈ V, Algorithm 4.1 converges to a dual optimizer λ∗ at a rate of order O( k12 ). The primal optimizer coli∈V (yi∗ , zi∗ ) is found as a byproduct.

28

4 Convex Optimization

4.4.3 Distributed Alternating Direction Method of Multipliers The alternating direction method of multipliers [Boyd et al., 2010], is a parallelizable version of the method of multipliers [Hestenes, 1969], which was originally designed as a robust version of the dual ascent method by [Arrow et al., 1958]. While DDFG is only applicable to problems with smooth dual functions, ADMM also handles problems with general concave dual functions. A distributed version of ADMM, which can be used to solve problem (4.17), is stated explicitly in Algorithm 4.2. Algorithm 4.2 Alternating Direction Method of Multipliers (ADMM) Require: Initial λi = 0 for each i ∈ V, z = 0 1: ∀i ∈ V in parallel: 2: repeat 3: yi+ = arg minyi ∈Zi La,i (yi , z, λ) 4: communicate yi+ to all j∈ Ni  P 5: zi+ = |N¯i1|+1 j∈N¯i ∪i EjiT yji+ + 1ρ λji ¯i 6: communicate zi+ to all j ∈ N + + λ+ i = λi + ρ(yi − Ei z ) 8: until convergence

7:

Remark 4.3. According to [Boyd et al., 2010], particular convergence properties can be made for Algorithm 4.2 under the following assumptions. (i) The functions fi (yi ) are closed, proper and convex ∀i ∈ V. (ii) The unaugmented partial Lagrangian (4.18) has a saddle point, i.e. there exists a point (x ∗ , y ∗ , λ∗ ) such that Lp (x ∗ , y ∗ , λ) ≤ Lp (x ∗ , y ∗ , λ∗ ) ≤ Lp (x, y , λ∗ ), ∀(x, y , λ) .

(4.26)

Assumption (i) implies that local solutions in the update steps 3 and 5 exist. Moreover, assumption (ii) implies that strong duality holds in problem (4.17). Under assumptions (i) and (ii), as Algorithm 4.2 progresses, the residuals yi − Ei z converge to zero asymptotically P for all i ∈ V, and the objective value i∈V fi (yi ) asymptotically converges to the primal optimum.

5 Model Predictive Control Model predictive control (MPC) is a well-established control methododology for constrained systems. Numerous contributions on practical and theoretical aspects of MPC have been published and the methodology has reached considerable maturity. For an in-depth overview of the field, the reader is referred to the textbooks [Maciejowski, 2000], [Camacho and Bordons, 2004], [Rawlings and Mayne, 2009] or the survey papers [Garcia et al., 1989], [Morari and Lee, 1999], [Rawlings, 2000]. In MPC, the control law is defined through a finite-horizon optimal control problem, which is in this thesis simply referred to as an MPC problem. In closed-loop operation, this problem is solved in a receding horizon fashion. Hence, at every point in time, the solution of the finite-horizon problem for the current initial state is found, but only the first element of the optimal input trajectory is applied to the plant. Over the years, several variants of MPC have been developed, covering aspects such as nominal and robust regulation, as well as reference tracking. In the following sections, particular MPC methodologies relevant to this thesis are introduced. This includes specifically formulations and stability properties for nominal MPC in Section 5.1, robust MPC in Section 5.2 and reference tracking MPC in Section 5.3.

5.1 Nominal Model Predictive Control for Regulation The nominal MPC control law for regulation is defined through the finite horizon optimal control problem ∗

V (x) = min Vf (x(N)) + u

N−1 X

l(x(k), u(k))

(5.1a)

k=0

s.t. x(0) = x ,

(5.1b)

x(k + 1) = Ax(k) + Bu(k) ∀k ∈ {0, . . . , N − 1} ,

(5.1c)

(x(k), u(k)) ∈ X × U

(5.1d)

x(N) ∈ Xf .

∀k ∈ {0, . . . , N − 1} ,

(5.1e)

30

5 Model Predictive Control

Both the stage cost l(x, u) and the terminal cost Vf (x) are positive definite convex functions and the terminal set Xf ⊆ Rn is convex, compact and contains the origin in its interior. Furthermore, u = {u(0), . . . , u(N − 1)} denotes an input trajectory over N steps, which together with x(0) = x uniquely defines a state trajectory x = {x(0), . . . , x(N)}. The input trajectory u∗ (x) = {u ∗ (0), . . . , u ∗ (N − 1)} denotes the minimizer of problem (5.1) and its first element is used to define the state feedback control law κMPC (x) = u ∗ (0), which leads to the nonlinear autonomous closed-loop system x + = Ax + BκMPC (x) .

(5.2)

XN ⊆ Rn denotes the domain of κMPC (x), i.e. the set of initial states x, for which a feasible solution to problem (5.1) exists. Sufficient conditions for asymptotic stability of (5.2) on the domain XN are given in the following theorem. Theorem 5.1 ( [Mayne et al., 2000]). Let κf (·) : Xf → Rm be a terminal control law and β1 (·), β2 (·) and β3 (·) K∞ class functions. If for all x ∈ Xf the conditions x ∈ X , κf (x) ∈ U, Ax + Bκf (x) ∈ Xf ,

(5.3a)

β1 (||x||) ≤ Vf (x) ≤ β2 (||x||) ,

(5.3b)

Vf (Ax + Bκf (x)) − Vf (x) ≤ −l(x, κf (x)) ≤ −β3 (||x||)

(5.3c)

hold, then the autonomous system x + = Ax + BκMPC (x) is asymptotically stable with domain of attraction XN . Condition (5.3a) implies that Xf is a feasible PI set and conditions (5.3b) and (5.3c) imply that Vf (x) is a Lyapunov function in Xf for the autonomous system x + = Ax + Bκf (x) .

(5.4)

5.2 Robust Model Predictive Control for Regulation In this section, two well-established robust MPC methods with stability guarantees under bounded disturbances are introduced. In the first approach, which was proposed in [Mayne et al., 2005], the nominal constraints are tightened by an RPI set. This RPI set is furthermore explicitly used in the MPC problem to plan feasible and converging tubes of predicted trajectories. In the second approach, which was proposed in [Chisci et al., 2001], the nominal state and input constraints are tightened over the prediction horizon of an MPC problem in a step-by-step fashion, taking all possible disturbance scenarios for the given horizon into account. Under an RPI terminal set, convergence of the state to the minimum RPI set for the given dynamics and the given disturbance bound is guaranteed in closed-loop. In the next two sections, the two approaches are briefly outlined.

5.2 Robust Model Predictive Control for Regulation

31

5.2.1 Robust MPC according to [Mayne et al., 2005] In the approach proposed in [Mayne et al., 2005], a nominal MPC problem is used to plan the center trajectory of a state prediction tube with horizon N. The problem reads as Vr1∗ (x)

= min x(0),u

N−1 X

l(x(k), u(k)) + Vf (x(N))

(5.5a)

k=0

s.t. x(0) ∈ x ⊕ Z ,

(5.5b)

x(k + 1) = Ax(k) + Bu(k) ∀k ∈ {0, .., N − 1}, (x(k), u(k)) ∈ X¯ × U¯ ∀k ∈ {0, .. , N − 1} ,

(5.5d)

x(N) ∈ X¯f ,

(5.5e)

(5.5c)

where Z is an RPI set for the system x + = (A + BKt )x + w , w ∈ W, where the linear state feedback gain Kt stabilizes the pair (A, B). In order to guarantee recursive feasibility of problem (5.5), the nominal state and input constraints are tightened as X¯ = X Z, U¯ = U Kt Z, X¯f ⊆ X¯ ,

(5.6)

and X¯f is chosen as a PI set. Problem (5.5) is used to plan an optimal input trajectory u∗ (x) = {u ∗ (0), . . . , u ∗ (N −1)} and an optimal initial state x ∗ (0), which together uniquely define a state trajectory x∗ (x) = {x ∗ (0), . . . , x ∗ (N)}, the initial point of which does not generally coincide with the current system state x. The robust control law is defined as κr1 (x) := u ∗ (0) + Kt (x − x ∗ (0)) ,

(5.7)

and XNr 1 denotes the domain of κr1 (x), hence the set of states x for which a feasible solution to problem (5.5) exists. Note that u∗ (x) and x∗ (x) define the center trajectory of a prediction tube with cross-section Z, containing all state trajectories of length N under control law (5.7) and disturbances w ∈ W. Hence, ∀k ∈ {0, . . . , N − 1}, it holds that x ∈ x ∗ (k) ⊕ Z ⇒ Ax + B(u ∗ (k) + Kt (x − x ∗ (k)) + w ∈ x ∗ (k + 1) ⊕ Z .

(5.8)

Theorem 5.2 (Theorem 1 in [Mayne et al., 2005]). Let κf (·) : X¯f → Rm be a terminal control law under which X¯f is a PI set for the system x + = Ax + Bκf (x). Let furthermore β1 (·), β2 (·) and β3 (·) be K∞ -class functions. If ∀x ∈ X¯f it holds that ¯ Ax + Bκf (x) ∈ X¯f , x ∈ X¯, κf (x) ∈ U,

(5.9)

β1 (kxk) ≤ Vf (x) ≤ β2 (kxk) ,

(5.10)

Vf (Ax + Bκf (x)) − Vf (x) ≤ −l(x, κf (x)) ≤ −β2 (kxk) ,

(5.11)

then the state of the system x + = Ax + Bκr1 (x) + w , w ∈ W, asymptotically converges to Z with region of attraction XNr 1 .

32

5 Model Predictive Control

5.2.2 Robust MPC according to [Chisci et al., 2001] In the approach proposed in [Chisci et al., 2001], a nominal state trajectory is planned by solving the MPC problem N−1 X

Vr2 (x) = min v

l(x(k), v (k)) + Vf (x(N))

(5.12a)

k=0

s.t. x(0) = x ,

(5.12b)

x(k + 1) = (A + BKt )x(k) + Bv (k) ∀k ∈ {0, . . . , N − 1} , (x(k), v (k) + Kt x(k)) ∈ X¯k × U¯k ∀k ∈ {0, . . . , N − 1} , x(N) ∈ X¯f ,

(5.12c) (5.12d) (5.12e)

where the state and input constraints are tightened separately for each prediction step as X¯k = X

M k−1



j

(A + BKt ) W

∀k ∈ {0, . . . , N} ,

(5.13a)

j=0

U¯k = U Kt

M k−1

j

(A + BKt ) W

 ∀k ∈ {0, . . . , N − 1} .

(5.13b)

j=0

Note that X¯f is chosen as an RPI set, for which it holds that X¯f ⊆ X¯N . Problem (5.12) is used to plan an optimal input trajectory v∗ (x) = {v ∗ (0), . . . , v ∗ (N − 1)}, which uniquely defines an optimal state trajectory x∗ (x) = {x ∗ (0), . . . , x ∗ (N)}. The robust control law is defined as κr2 (x) := vr2∗ (0) + BKt x ,

(5.14)

and XNr 2 denotes the domain of κr2 (x), hence set of x for which a feasible solution to problem (5.12) exists. The particular constraint tightening approach (5.13) and the choice of X¯f as an RPI set ensures that problem (5.12) is recursively feasible, even under additive disturbances w ∈ W (for details and proofs, see [Chisci et al., 2001]). Furthermore, convergence to a neighborhood of the origin in closed-loop is guaranteed, as stated in the following theorem.

Theorem 5.3 (Theorem 8 in [Chisci et al., 2001]). The state of the system x + = Ax + Bκr2 (x) + w converges to the minimum RPI set of the system x + = (A + BKt )x + w , w ∈ W, with region of attraction XNr2 .

5.3 Model Predictive Control for Reference Tracking

33

5.3 Model Predictive Control for Reference Tracking The objective in reference tracking is to steer the system to an equilibrium point (xs , us ), at which the system output matches a reference yt . At this equilibrium point, the steady-state condition " #" # " # A − In B xs 0 = (5.15) C 0 us yt is satisfied, where the first row of (5.15) represents the equilibrium condition for the state and input pair (xs , us ). The second row in (5.15) ensures consistency of the equilibrium state xs with the output reference yt . Note that an arbitrary reference output yt can only be attained if the left hand side matrix in (5.15) has full row rank. Furthermore, a tracking problem is said to be well-posed if for a given yt , a pair (xs , us ) is uniquely defined, e.g. by the minimum norm point (xs , us ) fulfilling (5.15). In the literature on tracking MPC, two main problems have been considered in recent years. The first problem is offset-free MPC [Muske and Badgwell, 2002], [M¨ader and Morari, 2010], the second is tracking MPC in presence of potentially infeasible references. In the following, the tracking MPC approach proposed in [Limon et al., 2008] is introduced, which handles infeasible references and can furthermore be combined with techniques which ensure offset-free tracking, as shown in [Limon et al., 2010]. The main idea in that approach is the introduction of an artificial admissible reference output ys , which uniquely defines an equilibrium pair (xs , us ) by the MPC problem V ∗ (x) = min +Vf (x(N) − xs ) + kys − yt k2T + u,xs ,us

N−1 X

l(x(k) − xs , u(k) − us )

(5.16a)

k=0

s.t. x(0) = x ,

(5.16b)

x(k + 1) = Ax(k) + Bu(k) ∀k ∈ {0, . . . , N − 1} ,

(5.16c)

(x(k), u(k)) ∈ X × U

(5.16d)

∀k ∈ {0, . . . , N − 1} ,

xs = Axs + Bus ,

(5.16e)

ys = Cxs ,

(5.16f)

(x(N) − xs , xs , us ) ∈ Xtr .

(5.16g)

Problem (5.16) is used to plan an optimal input trajectory u∗ (x) = {u ∗ (0), . . . , u ∗ (N −1)}, which uniquely defines an optimal state trajectory x∗ (x) = {x ∗ (0), . . . , x ∗ (N)}. Furthermore, the optimal solution to problem (5.16) contains a pair of steady states and inputs (xs∗ (x), us∗ (x)). The tracking MPC control law is defined as κtr (x) = u ∗ (0) ,

(5.17)

34

5 Model Predictive Control

and the set XN denotes the domain of κtr (x), hence the set of states x for which problem (5.16) is feasible. Note that for any x ∈ XN , the pair (xs∗ (x), us∗ (x)) yields an admissible steady-state output ys∗ (x), even if the actual reference yt is inadmissible. Consider now the augmented state variable vector z = [∆x T , xsT , usT ]T , with ∆x = x −xs . The corresponding augmented dynamic system can be written as     A 0 0 B     + z =  0 In 0  z +  0  (u − us ) = Al z + Bl (u − us ) . (5.18) 0 0 Im 0 This augmented system is used to define an invariant set for tracking, which is essential to guarantee that the closed-loop system x + = Ax +Bκtr (x) converges to a stable equilibrium. Definition 5.1 (Invariant Set for Tracking). Consider a control law κf (·) : Rn → Rm and the closed-loop system z + = Al z + Bl κf (∆x). The set Xtr ⊆ R2n+m is an admissible invariant set for tracking for this system if for each z ∈ Xtr , it holds that xs + ∆x ∈ X , us + κf (∆x) ∈ U,

(5.19a)

Al z + Bl κf (∆x) ∈ Xtr .

(5.19b)

Theorem 5.4 (Theorem 1 in [Limon et al., 2010]). Let κf (·) : Rn → Rm be a terminal control law and let β1 (·), β2 (·) and β3 (·) be K∞ -class functions, such that β1 (kxk) ≤ Vf (x) ≤ β2 (kxk)

(5.20a)

Vf (Ax + Bκf (x)) − Vf (x) ≤ −β3 (kxk) .

(5.20b)

Suppose furthermore, that for the augmented dynamics z + = Al z + Bl κf (∆x), Xtr is an admissible invariant set for tracking. Then, for any x ∈ XN , the system output converges to the admissible output ys minimizing kys − yt k2T . Remark 5.1. The tracking control law (5.17) guarantees feasibility and convergence in closed-loop, even if the left hand side matrix in (5.15) does not have full rank. At convergence, as specified in Theorem 5.4, the output might however not attain the originally devised reference yt , but the artificial reference ys . Remark 5.2. In [Limon et al., 2008] and the subsequent work of the authors thereof, the subspace of equilibrium points (xs , us ) is parameterized as [xsT , usT ]T = Mθ θ, where θ ∈ Rm . This parametrization is not used in this thesis, since the matrix Mθ is generally dense and the distributed MPC approach advocated in this work depends on sparsely structured optimization problems, which allow for solution by distributed optimization.

6 Distributed Model Predictive Control - A Brief Survey 6.1 Introduction Due to the success of MPC in the process industry in the 1980s, large-scale complex process plants were increasingly controlled by several MPC controllers in a decentralized way. However, while unconstrained decentralized control had been a field of active research for several years and numerous analysis and design method were readily available, see e.g. [Sandell ˇ et al., 1978] and [Siljak, 1991], systematic methods for constrained controllers such as MPC were not available. This situation motivated early-stage studies on decentralized MPC. In [Acar, 1995] for instance, a decentralized MPC controller was analyzed for a network of unconstrained state-coupled linear systems, which is stabilizable by a decentralized linear state-feedback controller. Specifically, the paper states that for such systems a horizon length exists, for which the coupled system is asymptotically stable in closed-loop under a decentralized MPC controller. Furthermore, several application examples related to control of irrigation canal systems were documented, e.g. [Sawadogo et al., 1998]. Years later, a more systematic approach on decentralized MPC was taken in [Magni and Scattolini, 2006], which proposed methodology for a network of nonlinear systems subject to additive disturbance. In this methodology, the local MPC controllers ignore the dynamic coupling in the local prediction and instead lump the coupling effect into the disturbance signal. It is shown that if the disturbance sequence is decaying and if the combination of coupling strength and initial state magnitude is sufficiently small, the closed-loop system is asymptotically stable. A similar approach was taken in [Raimondo et al., 2007], where the disturbance sequence is however not assumed to be decaying. As a consequence, asymptotic stability can not be guaranteed. However, the closed-loop system is shown to be input-tostate stable, again under the condition that coupling strength and initial state magnitude are sufficiently small. Despite the practical relevance of decentralized MPC, the number of dedicated contributions remained relatively small to date. With decentralized MPC being readily used in large-scale chemical processes, some authors started to point out that, while a completely centralized MPC approach was not

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6 Distributed Model Predictive Control - A Brief Survey

realistic due to scalability and fault-tolerance reasons, some level of coordination between the installed decentralized MPC controllers could be beneficial. In particular, it was hoped that by such a distributed MPC approach, efficiency and operating range of the plant could be increased, while its operational cost could be decreased [Lu, 2000], [Zhu and Henson, 2002], [Lu, 2003], [Rawlings and Stewart, 2008]. Moreover, various authors started to point out that a distributed MPC approach could be beneficial for a variety of applications other than chemical processes. All of these applications are large-scale networks of systems, which are subject to communication constraints in some way, e.g. power systems [Venkat et al., 2008], transportation networks [Negenborn et al., 2008], [Camponogara et al., 2002] and supply chains [Dunbar and Sesa, 2007]. As a consequence, distributed MPC theory gained considerable attention in the control systems community. Conceptually, distributed MPC is an extension to decentralized MPC. Same as in a decentralized approach, a globally defined system is controlled by several local controllers. However, some controllers in the network exchange information, while, as opposed to a centralized approach, no global coordination authority exists (see Fig. 6.1 for a schematic illustration of these fundamental controller setups). Within this general definition of distributed MPC, existing methods available in the literature vary considerably in the ways local controllers are operated and information is shared and utilized. The most fundamental distinction of paradigms is as to whether a method is cooperative or non-cooperative. In a cooperative approach, the systems in the global network choose their control inputs w.r.t. a global performance objective. In contrast, in a non-cooperative approach, the systems choose their inputs with a local objective in mind.

(a) Decentralized

(b) Distributed

(c) Centralized

Figure 6.1: Fundamental controller setups.

The distinction between a cooperative and a non-cooperative distributed MPC method is not always clear to make. The operational paradigm of a fully cooperative approach aims at exploiting the full operational range of the global system, in order to improve the global system performance. It is in this sense close to centralized MPC, except for the fact that the information flow is distributed. Due to the global relevance of local control inputs,

6.2 Non-cooperative Distributed MPC Methods

37

cooperative methods are usually communication intensive, since information needs to be propagated across the network in every timestep. In contrast, the operational paradigm of a fully non-cooperative approach is close to the one of decentralized MPC. Hence, the controllers act mainly based on local information, global optimality is out of the picture and communication is usually only used to ensure feasibility and stability. Consequently, non-cooperative methods are usually less communication intensive than cooperative ones. Naturally, there is some middle ground between the two paradigms of cooperative and noncooperative distributed MPC. Nevertheless, this survey relies mainly on this distinction, as similarly done in the recent survey [Christofides et al., 2013]. Note that besides cooperative and non-cooperative, various other criteria can be used to classify distributed MPC controllers. In the following, the criteria used later in this survey are introduced. Some methods, at a given time-step, find their control inputs in iterative communication rounds, while others find them in one non-iterative round. Iterative methods are usually cooperative and often rely on distributed optimization, while non-iterative ones are usually non-cooperative. Moreover, while in most methods the systems in the network update their inputs in parallel, some methods rely on sequential updates. Finally, some approaches are exclusively designed for networks of systems with decoupled dynamics (a setup which, in the control literature, is sometimes referred to as control of multi-agent systems), while other approaches are able to cope with coupled dynamics. In the latter case one can furthermore distinguish between state-coupled systems, which are coupled in the state and possibly also in the input, and input-coupled systems, coupled in their input only. In the following sections, significant distributed MPC approaches available in the literature are surveyed and categorized. The technical details of particular approaches are briefly outlined and discussed. The aim of the chapter is not an exhaustive survey of all published contributions to distributed MPC. Instead, it aims at presenting the main paradigms found in the distributed MPC literature, each accompanied by the discussion of a number of corresponding significant contributions. For a more exhaustive overview and additional details, the reader is referred to the two existing survey papers on distributed MPC, i.e. [Scattolini, 2009] and [Christofides et al., 2013]. Furthermore, a number of recent results were collectively published in the textbook [Maestre and Negenborn, 2014].

6.2 Non-cooperative Distributed MPC Methods Non-cooperative Methods without Constructive Stability Guarantee The methods summarized in this section are based on networked local MPC controllers, which act according to a local performance objective. Neighboring controllers exchange

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6 Distributed Model Predictive Control - A Brief Survey

some information at each time step in a single communication round. The stability properties of the controllers discussed can only be verified a-posteriori, since the controller formulations do not contain elements, which can be synthesized in a stabilizing way constructively. In [Jia and Krogh, 2001], and likewise in [Camponogara et al., 2002], a method for a network of state-coupled LTI systems was proposed. In this method, each system is controlled by an MPC controller, which relies on the neighboring state trajectories predicted at the previous time step. For asymptotic stability, an explicit constraint is imposed on the first control input, which is supposed to contract the norm of the local state. However, since the existence of a contracting control input, satisfying this constraint, is guaranteed only if a local controllability condition w.r.t to the coupling dynamics is satisfied, stability of the closed-loop system can only be verified, not constructed. Furthermore, the authors consider an unconstrained setup and it is not clear, how state and input constraints could be included in the formulation in a straight-forward way. In [Keviczky et al., 2006], a method for a network of uncoupled LTI systems with shared objective functions and coupled constraints was presented. In particular, each local system is controlled by an MPC controller, which considers an objective function and constraints which depend on the states and inputs of the local system and all its neighbors. Furthermore, at each time step, the local controllers predict the input and state trajectories of themselves and their neighbors.This setup naturally results in prediction inconsistencies among neighboring systems. The authors present conditions on the cost and the initial states of neighboring systems, under which the network of systems is asymptotically stable in closed-loop. These conditions encode the requirement that at any point in time, the increase of the global MPC cost inflicted by the prediction inconsistencies must be smaller than the global stage cost. However, the conditions only translate into a tractable controller synthesis problem if the systems are unconstrained.

Non-Cooperative Methods with Constructive Stability Guarantee The approaches discussed in this section are based on networked local MPC controllers, which act according to a local performance objective and exchange neighboring information at each time step in a single communication round. For each of these approaches, constructive methods exist by which controllers with closed-loop stability guarantee can be synthesized. In most cases, this means that the local controllers are designed to be robust under uncertain neighboring behavior. In [Jia and Krogh, 2002], a method for a network of state-coupled LTI systems was presented, in which each system is subject to local state and input constraints. At every point in time, each system decides on an outer bound on its future state and input trajectories,

6.2 Non-cooperative Distributed MPC Methods

39

and communicates these bounds to all neighbors. Each system then relies on a locally robust MPC controller, which solves a min-max problem at every time step to find the locally optimal control input under worst-case neighboring behavior within the received bounds. Naturally, for the approach to work, each system needs to impose the bounds it communicated to its neighbors explicitly as state and input constraints in its own MPC problem. Recursive feasibility and convergence to a robust positively invariant set can be shown under the assumption that all local prediction bound sequences are contractive as time progresses. Two conceptually similar approaches were proposed in [Dunbar, 2007] (see also [Dunbar and Murray, 2006]) and [Farina and Scattolini, 2012]. In [Dunbar, 2007], a method for a network of state-coupled continuous-time nonlinear systems is presented. The approach relies on an a-priori known globally feasible finite horizon state trajectory and a known decentralized terminal control law, by which this trajectory can be asymptotically extended into the origin without violating constraints. The local MPC controllers, at each point in time, impose explit local constraints on their deviation from this feasible trajectory. If these constraints are tight enough under the given dynamic coupling, asymptotic stability of the closed-loop system can be guaranteed. In [Farina and Scattolini, 2012], state coupled LTI systems are considered, for each of which a local tube MPC controller (based on the theory presented in [Mayne et al., 2005]) is designed, which is robust against dynamic coupling to neighboring systems. Similarly to [Dunbar, 2007], the coupling signals are considered to be deviations from an a-priori known globally feasible state trajectory. Thus, in order to guarantee asymptotic stability, these deviations have to be explicitly constrained in every local MPC controller. Both approaches share the issue of requiring a feasible global state trajectory prior to operation. Methods based on Iterative Non-Cooperative Games The methods summarized in this section are based on a global MPC problem, a solution to which is found at each time step by an iterative non-cooperative game among the systems in the network. Specifically, in every iteration, each system optimizes its own variables w.r.t. the global objective, while keeping the neighboring variables constant at the values from the previous iteration. Thus, after every local optimization, a system has to communicate its solution to its neighbors. The iterative procedure corresponds to a non-cooperative game and converges to a Nash equilibrium, if a stable Nash equilibrium exists. MPC schemes based on non-cooperative games were suggested for instance in [Du et al., 2001], [Li et al., 2005], [Zhang and Li, 2007] and [Mercang¨oz and Doyle III, 2007]. One theoretical issue with non-cooperative games in distributed MPC is the fact that the game does not converge, if for the problem under consideration a stable Nash equilibrium

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6 Distributed Model Predictive Control - A Brief Survey

does not exist. In [Du et al., 2001], [Li et al., 2005] and [Zhang and Li, 2007], eigenvalue based conditions on the convergence of the non-cooperative game, as well as similar conditions on the stability of the resulting closed-loop system, were presented. These conditions however only apply to unconstrained systems. Corresponding techniques for constrained systems are not available to date. Another theoretical issue is the fact that standard MPC stability results, relying on the optimal MPC cost to be a Lyapunov function, do not apply since a Nash equilibrium is usually not an optimum of the global MPC problem. This fact was pointed out in [Venkat et al., 2005] and it was argued that if a distributed MPC controller is designed based on an iterative procedure, this procedure should better be cooperative. Such cooperative controllers are discussed within the next section.

6.3 Cooperative Distributed MPC Methods Primal Decomposition based Distributed MPC with Sequential Updates The approaches surveyed in this section rely on a globally defined MPC problem. In closedloop operation, the systems optimize their inputs in a sequential fashion, keeping the variables of the neighboring systems constant. At each time step, either one system optimizes its inputs, or several systems do so in a sequential fashion. In [Richards and How, 2007] and [Trodden and Richards, 2010], two similar methods for a network of uncoupled LTI systems were proposed. In particular, the systems are subject to local state and input constraints, additive disturbance, as well as coupled constraints on their outputs. The methods are based on a global MPC problem, for which a globally feasible trajectory is assumed to be known prior to operation. Furthermore, a decentralized terminal control law is assumed to be available, by which this trajectory can be shifted in a feasible and stable way. At each time step in closed-loop operation, exactly one system optimizes its state and input variables, considering the shifted neighboring trajectories. Robust feasibility and stability of the closed-loop system is shown using well-established robust MPC techniques, namely the constraint tightening approach [Chisci et al., 2001] in [Richards and How, 2007], and the tube MPC approach [Mayne et al., 2005] in [Trodden and Richards, 2010]. Another sequential method was proposed in [Liu et al., 2009] for a fully coupled nonlinear continuous-time system, which is controlled by a number of coordinated controllers. It is assumed that at each time step in closed-loop, all controllers update their inputs in a sequential fashion. The controllers themselves are designed such that the first controller already stabilizes the complete system. The remaining controllers are then merely improving the performance, while ensuring at the same time that the stability guarantee is not lost. The

6.3 Cooperative Distributed MPC Methods

41

provided stability guarantee builds on centralized Lyapunov-based MPC techniques [Mhaskar et al., 2006]. In this approach, a nominally stabilizing control law and a corresponding Lyapunov function for the closed-loop system is assumed to be known. In the MPC problem, it is then explicitly enforced that the Lyapunov decrease under the MPC control law is at least as large as the one obtained by the nominally stabilizing control law. A partially sequential method for a network of input coupled constrained linear systems was presented in [Maestre et al., 2011b]. The method is based on a global MPC problem and it is denoted as “partially sequential”, since neighboring systems can not update their control inputs synchronously. The approach relies on a stabilizing decentralized control law, which can be used to obtain a feasible and improving input trajectory by shifting the trajectory from the previous time step. On top of that, systems are allowed to propose new input trajectories, which are accepted if they result in an improvement of the sum of the MPC cost contributions of all neighboring systems. Cost contributions of non-neighboring systems are not affected by proposals, since the dynamics are coupled in the inputs only. The approach is stabilizing, since the shifted trajectory is stabilizing, and so are, consequently, feasible improvements to it. A drawback of all the sequential methods summarized in this section is the fact that no global optimization is carried out and the controllers are run on suboptimal solutions. Therefore, in the next two sections, distributed MPC methods based on distributed optimization algorithms are summarized. Distributed optimization yields globally optimal solutions which can be obtained by distributed computations and local communication.

Primal Decomposition based Distributed MPC with Parallel Updates The methods summarized in this section are based on a global MPC problem, which is decomposed into a number of coupled local problems. A globally optimal solution to these local problems is then found at every time step by distributed optimization techniques, which directly work on this primal decomposition of the problem. The first approach in this direction was presented in [Camponogara et al., 2002], where it is pointed out that one of the main advantages of solving a global MPC problem by distributed optimization is the fact that theoretical guarantees for centralized MPC are directly inherited by the resulting distributed MPC controller. The distributed optimization method proposed to solve the global MPC problem however relies on an iterative coordinate descent scheme, where in each iteration a sequential round of computations needs to be performed. This characteristic can be prohibitive if the number of systems in the network is large. Another early distributed MPC method based on a primal decomposition was proposed in [Venkat et al., 2005] for a network of input-coupled systems. The work was later extended to

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6 Distributed Model Predictive Control - A Brief Survey

state coupled systems [Venkat et al., 2008], systems with coupled input constraints [Stewart et al., 2010], nonlinear systems [Stewart et al., 2011], and reference tracking [Ferramosca et al., 2013]. The proposed method is based on the primal decomposition of a global MPC problem, which is solved at every time step by a distributed optimization scheme described in [Bertsekas and Tsitsiklis, 1989]. In this scheme, all subproblems iteratively optimize their own variables of the global MPC problem in parallel, keeping the neighboring variables at the most recent update. An update of the global solution is constructed in every iteration as a convex combination of all local solutions, whereas in order to build this convex combination, only neighboring systems need to communicate with each other. The approach has the beneficial property that, given an initial feasible solution, all subsequent iterations are feasible and improving. This property is useful to show closed-loop stability under suboptimal solutions of the MPC problem [Scokaert et al., 1999]. However, since the method can not cope with equality constraints, the MPC problem usually needs to be condensed. In case of state-coupled dynamics, condensing undoes the decomposable structure of the global MPC problem. Primal decomposition based distributed MPC approaches using other distributed optimization schemes were suggested in e.g. [Johansson et al., 2008] and [Necoara and Clipici, 2013]. In [Johansson et al., 2008], an incremental subgradient method is used for distributed optimization, whereas in [Necoara and Clipici, 2013] a parallel coordinate descent method is used. Both of these distributed optimization methods suffer from the fact that at each iteration, a projection operation on the set of feasible variables is required. Such an operation is computationally expensive, unless the constraint sets are simple to be projected onto. While this is the case for some problems, it is not for networks of systems which are both state-coupled and state-constrained. A methodology which can cope with both of these properties is given by dual decomposition based distributed optimization. Available cooperative distributed MPC methods based on this methodology are discussed in the next section. Dual Decomposition based Distributed MPC The methods discussed in this section rely on a global MPC problem, the dual of which is decomposed and solved by a distributed optimization method in every time step. The main advantage of dual decomposition, compared to primal decomposition, is the fact that dual variables are always subject to constraint sets, which are simple to be projected onto. Thus, the projection operations inherent to first-order optimization methods do not pose an issue when solving the dual. On the downside, the primal solution, obtained while iteratively solving the dual problem, converges to feasibility only asymptotically, as the dual solution approaches the optimum. Nevertheless, distributed MPC based on dual decomposition has

6.3 Cooperative Distributed MPC Methods

43

proven practical in various applications, such as e.g. irrigation canal systems [Negenborn et al., 2008], building control [Ma et al., 2012] and wind farms [Spudi´c et al., 2013]. An early work on dual decomposition based distributed MPC was proposed in [Wakasa et al., 2008], where the use of the subgradient method is proposed to solve the decomposed dual of the MPC problem for a network of systems connected in a chain topology. Since convergence speed of the subgradient method can be a problem in dual decomposition based MPC, a number of contributions were concerned with the use of accelerated firstorder methods. In [Necoara et al., 2008], it was pointed out that the fast gradient method, as proposed in [Nesterov, 2004], exhibits a dual convergence rate guarantee which is an order of magnitude better than the one of the subgradient method, i.e. O( k12 ) instead of O( k1 ). Since the dual function is required to be smooth in order for this rate guarantee to hold, the authors suggest the application of the smoothing technique proposed in [Nesterov, 2005] to the primal MPC problem. This technique renders a convex primal optimization problem strongly convex, which implies a smooth corresponding dual function. Another accelerated gradient scheme, based on the work presented in [Beck and Teboulle, 2009], was proposed in [Giselsson et al., 2013] for strongly convex MPC problems. This scheme also guarantees a dual convergence rate of O( k12 ). Additionally, the same convergence rate can be shown for the primal solution, if the primal constraints fulfill the Mangasarian-Fromovitz constraint qualification condition. This condition requires that there exists a strictly feasible point for the primal inequality constraints, while the linear equality constraints are defined by linearly independent vectors. As mentioned previously, a disadvantage of dual decomposition based first-order methods is the fact that primal feasibility is only attained asymptotically. Thus, when stopping the method after a finite number of iterations, no primal feasible solution for the global MPC problem can be guaranteed. This property is in conflict with standard stability results for MPC, see [Mayne et al., 2000] for details, which strongly rely on feasibility. Therefore, the feasibility question has been addressed by several authors in context of dual decomposition based distributed MPC. In [Doan et al., 2011], a result from [Nedi´c and Ozdaglar, 2009] was revisited, which states that for inequality constrained convex optimization problems, where the inequality constraints fulfill the Slater condition, the dual level sets can be explicitly bounded. When solving such a problem by the subgradient method, this result can be used to construct bounds on the primal constraint violation, as well as the primal suboptimality, in every iteration. These bounds in turn can be used to tighten the primal MPC problem in such a way, that a feasible and improving primal solution to the original problem is guaranteed to be attained after an a-priori known finite number of iterations. The main drawback of the approach is the fact that the proposed constraint tightening is time-varying and requires global coordination. In [Giselsson and Rantzer, 2013], another constraint tightening scheme

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was proposed, which guarantees a feasible and improving primal solution after an a-priori unknown, but finite, number of iterations. Using a stopping criterion, which is presented alongside the tightening scheme, these two properties can be checked online. The drawback of the approach is, similarly to [Doan et al., 2011], the global coordination which is required to evaluate the stopping criterion.

6.4 Tabular Overview of Distributed MPC Methods In Tab. 6.1, the theoretical distributed MPC methods discussed in this survey chapter are summarized and categorized. For the categorization, the criteria introduced in the introduction of this chapter are used. Additionally, a statement regarding the capability of the methods to handle state and input constraints, possibly coupled, is amended. Distributed MPC Method [Acar, 1995] [Jia and Krogh, 2001] [Du et al., 2001] [Camponogara et al., 2002] [Jia and Krogh, 2002] [Venkat et al., 2005] [Li et al., 2005] [Keviczky et al., 2006] [Dunbar and Murray, 2006] [Magni and Scattolini, 2006] [Zhang and Li, 2007] [Dunbar, 2007] [Richards and How, 2007] [Raimondo et al., 2007] [Mercang¨ oz and Doyle III, 2007] [Johansson et al., 2008] [Venkat et al., 2008] [Wakasa et al., 2008] [Necoara et al., 2008] [Liu et al., 2009] [Stewart et al., 2010] [Trodden and Richards, 2010] [Doan et al., 2011] [Maestre et al., 2011a] [Farina and Scattolini, 2012] [Necoara and Clipici, 2013] [Giselsson and Rantzer, 2013] [Ferramosca et al., 2013]

Cooperation Scheme decentralized non-coop. non-coop. coop. non-coop. coop. non-coop. non-coop. non-coop. decentralized non-coop. non-coop. coop. decentralized non-coop. coop. coop. coop. coop. coop. coop. coop. coop. coop. non-coop. coop. coop. coop. tracking

Computation and Communication parallel, non-iter. parallel, non-iter. parallel, iter. sequential, iter. parallel, non-iter. parallel, iter. parallel, iter. parallel, non-iter. parallel, non-iter. parallel, non-iter parallel, iter. parallel, non-iter. sequential, non-iter. parallel parallel, iter. parallel, iter. parallel, iter. parallel, iter. parallel, iter. sequential, non-iter. parallel, iter. sequential, non-iter. parallel, iter. sequential, iter. parallel, non-iter. parallel, iter. parallel, iter. parallel, iter.

Coupling in the Dynamics state coupled state coupled state coupled state coupled state coupled input coupled input coupled uncoupled uncoupled state coupled input coupled state coupled uncoupled state coupled state coupled uncoupled state coupled uncoupled state coupled state coupled input coupled uncoupled state coupled input coupled state coupled input coupled state coupled input coupled

Coupling in the Constraints unconstrained unconstrained unconstrained state and input, coup. state and input, uncoup. input, uncoup. unconstrained state and input, coup. input, decoup. input, decoup. unconstrained state and input, decoup. state and input, coup. state and input, decoup. input, uncoup. input, uncoup. input, decoup. input, decoup. state and input, uncoup. input, uncoup. input, coup. state and input, coup. state and input, coup. state and input, uncoup. state and input, uncoup. input, uncoup. state and input, uncoup. state and input, coup.

Table 6.1: Classification of existing distributed MPC approaches.

Part II Cooperative Distributed MPC: Stability based on Distributed Invariance

7 Nominal Cooperative Distributed MPC for Regulation 7.1 Introduction Non-centralized control of unconstrained nominal systems has been a field of active research for decades. In particular, decentralized control approaches have received considerable attention. In [Sandell et al., 1978] for instance, an early survey on decentralized control ˇ methods can be found and in [Siljak, 1991], vector Lyapunov function based analysis and synthesis techniques for decentralized methods are described. Moreover, in [Desoer and G¨unde¸s, 1986] a parametrization of all stabilizing decentralized controllers for a given plant was presented. As for distributed control techniques, it was shown in [Ho and Chu, 1972], that an optimal distributed controller is linear if and only if the communication structure is partially nested. Hence, every system needs to have full information of all states by which it is asymptotically affected, otherwise the optimal controller is nonlinear. Furthermore, it was shown in [Rotkowitz and Lall, 2006], that the synthesis problem for optimal distributed linear controllers is nonconvex in the general case. It was, however, shown that the synthesis problem is convex in the special case when the communication structure of a system is quadratically invariant to its coupling structure. Non-centralized control of constrained nominal systems, mostly by distributed MPC, has also received considerable attention in the last years. In Chapter 6, a survey on available distributed MPC techniques is given. However, in order to guarantee constraint satisfaction and stability in a distributed environment, distributed MPC approaches are often considerably more conservative than centralized ones. In non-cooperative distributed MPC, it is common to impose restrictive constraints on input and state trajectories of the local systems. Under such constraints, the influence of neighboring systems can be accounted for as a bounded disturbance, as done e.g. in [Jia and Krogh, 2002] or [Farina and Scattolini, 2012]. Even in cooperative distributed MPC, where each local control law is defined by the same global MPC problem, which is solved by distributed optimization at each time step, it is common to use conservative terminal sets to guarantee closed-loop stability. In particular, the use of a single point, i.e. the origin, as a terminal set was proposed in [Venkat

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et al., 2005], and the use of decentralized RPI sets, which consider dynamic coupling as disturbance, was proposed in [Maestre et al., 2011b]. In this chapter, a less conservative way to guarantee stability under cooperative distributed MPC is presented. Main Contributions The stability guarantee presented in this chapter relies on a novel notion of distributed invariance, which is used in the global MPC problem to impose a terminal state constraint for each system in the network. In particular, the terminal state of each system is constrained to lie in a local terminal set, the size of which is allowed to vary in time in order to reduce conservatism. To this end, a recent contribution on structured Lyapunov functions proposed in [Jokic and Lazar, 2009] is utilized. The introduced notion of distributed invariant sets is conceptually similar to the one recently presented in [Rakovi´c et al., 2010]. However, while for these sets no practical synthesis method is available, for the ones presented in this chapter an intuitive and practical synthesis method, which is directly linked to the system dynamics, is proposed. This synthesis method can be carried out in a fully distributed way, without central coordination. The resulting distributed MPC controller offers a significantly larger region of attraction compared to a controller using a single-point terminal set. Outline The structure of the chapter is as follows: In Section 7.2, the problem considered in this chapter is stated in detail. In Section 7.3, terminal cost and set based stability conditions for cooperative distributed MPC are derived by adapting an idea on structured control Lyapunov functions proposed in [Jokic and Lazar, 2009]. Subsequently, in Section 7.4, a corresponding distributed synthesis method for a controller fulfilling these stability conditions is proposed, for which the dynamics are required to be linear, the cost functions quadratic and the constraints polytopic. Finally, in Section 7.5, the performance of the resulting controller is demonstrated in a numerical example.

7.2 Problem Statement In nominal MPC for linear systems, where the state feedback control law is defined through an MPC problem of form (5.1), closed-loop stability can be enforced by specifically designing a number of components in this problem. A common way to guarantee stability is to design the terminal cost function Vf (x) to be a Lyapunov function for the unconstrained system within a terminal PI set Xf , see [Mayne et al., 2000] for details. Given these components, it can be shown that the MPC problem is recursively feasible in closed-loop. Moreover, using

7.3 Distributed Invariance and Stability

49

the optimal MPC cost (5.1a) as a Lyapunov function, it can be shown that the closed-loop system is asymptotically stable. This thesis is concerned with a cooperative distributed MPC framework, in which the global MPC problem is solved online by distributed optimization at every time step. However, in order for distributed optimization methods to be applicable under the present communication constraints (see Assumption 3.2), the MPC problem needs to be structured appropriately. Since in particular the terminal cost and the terminal set in (5.1) are not naturally structured, the main challenge is to design these elements in a structured way, without sacrificing their stabilizing properties and without introducing excessive conservatism. An intuitive approach is to aim for a decomposition of the terminal cost and the terminal set into exactly one unit per system in the network, hence Vf (x) =

M X

Vf,i (xi ) ,

(7.1a)

i=1

Xf (α1 , . . . , αM ) = Xf,1 (α1 ) × . . . × Xf,M (αM ) ,

(7.1b)

where for each i ∈ {1, . . . , M}, Vf,i (·) : Rni → R≥0 is the local terminal cost contribution and Xf,i (αi ) ⊆ Rni is a parametrized local terminal set. For each of these sets, the parameter αi is used to adjust the set size. In Section 7.3, a way to define a terminal cost and terminal sets according to (7.1) is presented, such that recursive feasibility and asymptotic stability of the global closed-loop system is guaranteed. Moreover, in Section 7.4, corresponding distributed synthesis methods are presented.

7.3 Distributed Invariance and Stability 7.3.1 Structured Terminal Cost Function A na¨ıve way to choose a terminal cost function, which is structured as defined in (7.1a), would be to require the function Vf,i (xi ) to decrease for each system i in every time step. Such an approach, however, would be very conservative. Consider for instance a system i , whose state xi rests in the origin, where Vf,i (xi ) = 0. If the state xj of a neighboring system j ∈ Ni is nonzero, xi will necessarily be driven away from the origin, causing Vf,i (xi ) to increase. Therefore, as proposed in [Jokic and Lazar, 2009], it is desirable to allow a local terminal cost contribution to increase, as long as at the same time the global terminal cost decreases. Theorem 7.1 (Theorem III.4 in [Jokic and Lazar, 2009]). Let Xf ⊆ Rn be a PI set for system (5.4) under the control law κf (x) = coli∈{1,...,M} (κNi (xNi )). If there exist functions

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7 Nominal Cooperative Distributed MPC for Regulation

Vf,i (xi ), γi (xNi ), κNi (xNi ) and li (xNi , κNi (xNi )), as well as K∞ -class functions β1,i (·), β2,i (·) and β3,i (·) such that ∀x = coli∈{1,...,M} (xi ) ∈ Xf it holds that β1,i (||xi ||) ≤ Vf,i (xi ) ≤ β2,i (||xi ||) ∀i ∈ {1, . . . , M} , β3,i (||xNi ||) ≤ li (xNi , κNi (xNi )) ∀i ∈ {1, . . . , M} , Vf,i (xi+ )

− Vf,i (xi ) ≤ −li (xNi , κNi (xNi )) + γi (xNi ) ∀i ∈ {1, . . . , M} ,

M X

γi (xNi ) ≤ 0 ,

(7.2a) (7.2b) (7.2c) (7.2d)

i=1

PM then the function Vf (x) = i=1 Vf,i (xi ) is a Lyapunov function for system (5.4) and system (5.4) is asymptotically stable on Xf . If the conditions (7.2) are fulfilled, then the functions Vf,i (xi ), ∀i ∈ {1, . . . , M}, can be used as local terminal cost contributions in cooperative distributed MPC.

7.3.2 Time-Varying Distributed Terminal Set for Regulation PM Given a structured Lyapunov function Vf (x) = i=1 Vf,i (xi ), fulfilling conditions (7.2), any feasible level set Xfglob = {x ∈ Rn |Vf (x) ≤ α} ⊆ X thereof is invariant and could be considered as a terminal set for a stabilizing MPC controller. However, since we aim at solving the MPC problem by distributed optimization, it is difficult to use the set Xfglob directly, since it is defined via the global coupling constraint Vf (x) =

M X

Vf,i (xi ) ≤ α .

(7.3)

i=1

Such constraints can be handled by distributed optimization, but usually only by considerably increasing the number of variables in the local optimization problems, see e.g. [Schizas et al., 2008]. Therefore, in this chapter, we aim at designing the global terminal set as a Cartesian product of locally defined sets, as introduced in (7.1b). A design, which is based on a novel notion of distributed invariance, is proposed in the following. Consider for every system a level set of the local terminal cost function Vf,i (xi ), which satisfies the conditions (7.2), as Xf,i (αi ) = {xi ∈ Rni |Vf,i (xi ) ≤ αi } ,

(7.4)

P where M i=1 αi ≤ α. Using these local level sets, a global terminal set Xf (α1 , . . . , αM ) = Xf,1 (α1 ) × . . . × Xf,M (αM ) can be defined. However, for static local level set values αi , invariance of the set Xf (α1 , . . . , αM ) does not hold. In particular, (x1 , . . . , xM ) ∈ Xf (α1 , . . . , αM ) ; x + ∈ Xf (α1 , . . . , αM ) ,

(7.5)

7.3 Distributed Invariance and Stability

51

since, as time evolves, the state of any system i might leave Xf,i (αi ) due to an increase in the function Vf,i (xi ). A remedy for this issue is to update the sizes of the local terminal sets according to the dynamics (7.6) α+ i = αi + γi (xNi ) , where γi (xNi ) is the element in condition (7.2c) which relaxes the local decrease condition on the terminal cost contribution Vf,i (xi ). The recursion (7.6) leads to dynamic local terminal sets Xf,i (αi ) for every system i ∈ {1, . . . , M}. The following discussion will show invariance and recursive feasibility of the Cartesian product of the local terminal sets (7.4), if their sizes are updated using the recursion (7.6). Theorem 7.2. For any i ∈ {1, . . . , M}, under the set dynamics (7.6), the local terminal set Xf,i (αi ) = {xi ∈ Rni |Vf,i (xi ) ≤ αi } is invariant and non-empty at all times, hence (i) xi ∈ Xf,i (αi ) ⇒ xi+ ∈ Xf,i (α+ i ) (ii) 0 ≤ αi ⇒ 0 ≤ α+ i Proof. Consider xi ∈ Xf,i (αi ) and positivity of Vf,i (xi ), which implies that 0 ≤ Vf,i (xi ) ≤ αi . It holds that 0 ≤ Vf,i (xi+ ) ≤ Vf,i (xi ) − li (xNi , κNi (xNi )) + γi (xNi ) ≤ Vf,i (xi ) + γi (xNi ) ≤ αi + γi (xNi ) = α+ i ,

(7.7)

where the first inequality is given by positivity of Vf,i (·) and the second as well as the third by (7.2) in Theorem 7.1. Inequality (7.7) implies both (i) and (ii). Corollary 7.1. Let Xfglob = {x ∈ Rn |Vf (x) ≤ α} be chosen such that x ∈ Xfglob ⇒ x ∈ X and Kf x ∈ U. Let furthermore Xf,i (αi ) = {xi ∈ Rni |Vf,i (xi ) ≤ αi } for each i ∈ {1, . . . , M}. Then, the implication glob + Xf,1 (α1 ) × . . . × Xf,M (αM ) ⊆ Xfglob ⇒ Xf,1 (α+ 1 ) × . . . × Xf,M (αM ) ⊆ Xf

(7.8)

holds and consequently, the Cartesian product Xf,1 (α1 ) × . . . × Xf,M (αM ) is recursively feasible w.r.t. state and input constraints. PM Proof. Xf,1 (α1 ) × . . . × Xf,M (αM ) ⊆ Xfglob implies i=1 αi ≤ α. Using the update rule (7.6) and Theorem 7.1, we see that M X i=1

α+ i

=

M X i=1

αi +

M X i=1

γi (xNi ) ≤

M X

αi ≤ α ,

i=1

which implies property (7.8). Feasibility of the Cartesian product is given since Xfglob is feasible.

52

7 Nominal Cooperative Distributed MPC for Regulation

The combination of Theorem 7.2 and Corollary 7.1 shows that the set Xf,1 (α1 ) × . . . × Xf,M (αM ) is invariant and recursively feasible. Hence, if this set is used as a terminal set PM in combination with Vf (x) = i=1 Vf,i (xi ) in the global MPC problem (5.1), asymptotic stability of the resulting closed-loop system (5.2) is guaranteed. A summary of online distributed MPC is described in Algorithm 7.1. Algorithm 7.1 Online distributed MPC, executed at every system in parallel 1: Measure local state xi . 2: Solve MPC problem (5.1) by distributed optimization. 3: Apply the input obtained in step 2. ∗T ∗ 4: Update αi : α+ i = αi + xNi (N)Γi xNi (N). 5: Go to step 1.

Remark 7.1. Another approach towards dynamic invariant sets for distributed systems was recently proposed in [Rakovi´c et al., 2010]. However, while the set dynamics proposed in [Rakovi´c et al., 2010] are state-independent, the dynamics presented in this chapter depend on the current state of the system and can therefore adapt to specific scenarios. Furthermore, for the sets proposed in [Rakovi´c et al., 2010], no practical synthesis method is available. In contrast, sets as formulated in this section can be efficiently synthesized in a distributed fashion for linear systems subject to polytopic constraints, as will be shown in the next section.

7.4 Distributed Synthesis In this section, a distributed synthesis procedure for a cooperative distributed MPC controller for a network of linear systems subject to polytopic constraints is presented. This procedure consists of two consecutive steps. In the first step, a separable quadratic terminal cost P T function Vf (x) = M i=1 xi Pf,i xi , fulfilling the conditions (7.2), is computed. Thereafter, in the second step, local terminal sets Xf,i (αi ) are found, first by computing the largest global level set Xfglob = {x ∈ Rn |Vf (x) ≤ α} ⊆ X , and second by allocating the global PM level set value α to the local sets, as initial set sizes αi , such that i=1 αi ≤ α, e.g. by setting αi = α/M for each i ∈ {1, . . . , M}. Both of the steps can be performed by distributed optimization, hence, no central coordination is required at any point in the synthesis procedure. In the following, consider local state and input constraint sets of the form Xi = {xi ∈ R |Gi xi ≤ gi } and Ui = {ui ∈ Rmi |Hi ui ≤ hi } for all i ∈ {1, . . . , M}, where Gi ∈ Rlx,i ×ni , ni

7.4 Distributed Synthesis

53

gi ∈ Rlx,i , Hi ∈ Rlu,i ×mi and hi ∈ Rlu,i . Furthermore, all cost function terms are considered to be quadratic, hence li (xNi , ui ) = xNT i QNi xNi + uiT Ri ui ,

(7.9a)

Vf,i (xi ) = xiT Pf,i xi , γi (xNi ) = xNT i ΓNi xNi

(7.9b) ,

(7.9c)

where QNi , Ri and Pi are positive definite matrices. Moreover, the terminal control law is considered to be structured and linear, hence κf (x) = coli∈{1,...,M} (KNi xNi ) .

(7.10)

The first part of this section, i.e. Section 7.4.1, is concerned with distributed synthesis of Pi , KNi and ΓNi , such that conditions (7.2) are satisfied. The second part, i.e. Section 7.4.2, is then concerned with distributed computation of local terminal sets Xf,i (αi ) under polytopic state and input constraints. In order to simplify the notational connection between global and local dimensions, we introduce for each i ∈ {1, . . . , M} lifting matrices Ti ∈ {0, 1}ni ×n and Wi ∈ {0, 1}nNi ×n . These lifting matrices, similar to permuation matrices, have in each row exactly one entry equal to 1, such that xi = Ti x, xNi = Wi x

∀i ∈ {1, . . . , M} .

(7.11)

Remark 7.2. Note that the matrices ΓNi are not required to be definite. Quite the opposite, indefiniteness of these matrices allows for the local cost functions Vf,i (xi ) to increase, while the global cost function Vf (x) still decreases.

7.4.1 Distributed Synthesis of a Structured Terminal Cost PM As discussed in Section 7.3.1, a sufficient condition for Vf (x) = i=1 Vf,i (xi ) to exhibit the required Lyapunov function property is for the local cost terms Vf,i (xi ) to satisfy the conditions (7.2) stated in Theorem 7.1. For structured linear dynamics, as introduced in (3.10), and quadratic cost functions, as introduced in (7.9), conditions (7.2c) and (7.2d) are equivalent to the matrix inequalities (ANi + Bi KNi )T Pf,i (ANi + Bi KNi ) − P¯f,i ≤ T −(Qi +KN Ri KNi ) + Γi , ∀i ∈ {1, . . . , M} , i M X i=1

WiT Γi Wi ≤ 0 ,

(7.12a) (7.12b)

54

7 Nominal Cooperative Distributed MPC for Regulation

where P¯f,i = Wi TiT Pf,i Ti WiT is the matrix Pf,i ⊆ Rni ×ni lifted into the space RnNi ×nNi . The following discussion will demonstrate that conditions (7.12a) and (7.12b) can equivalently be written as a set of coupled LMIs, whereas the coupling structure is the same as the one given in the global system dynamics. Consequently, the synthesis of a structured Lyapunov function according to Theorem 7.1 can be posed as a decomposable convex feasibility problem, which can be solved by distributed optimization methods. Lemma 7.1. Consider for each i ∈ {1, . . . , M} the substitutions Ei = Pf,i−1 , E = diagi∈{1,...,M} (Ei ), ENi = Wi EWiT , YNi = KNi ENi and Fi = ENi Γi ENi . The condition (7.12) is equivalent to the set of LMIs   E¯i + Fi ENi ATNi + YNTi BiT ENi Q1/2 YNTi Ri1/2 i A E + B Y Ei 0 0   Ni Ni  i Ni  ≥0 1/2  Qi E N i 0 InNi 0  0 0 Imi Ri1/2 YNi ∀i ∈{1, . . . , M} , M X

WiT Fi Wi ≤ 0 ,

(7.13a) (7.13b)

i=1

where E¯i = Wi TiT Ei Ti WiT . Proof. We first show that (7.12a) is equivalent to (7.13a) . By multiplying, for each i ∈ {1, . . . , M}, by ENi from both sides, condition (7.12a) can equivalently be written as E¯i + Fi − ENi Qi ENi − YNTi Ri YNi − (ENi ATNi + YNTi BiT )Pf,i (ANi ENi + Bi YNi ) ≥ 0 , ∀i ∈ {1, . . . , M} , (7.14) where E¯i = Wi TiT Ei Ti WiT . By applying the Schur complement, we can equivalently write # " E¯i + Fi − ENi Qi ENi − YNTi Ri YNi (ENi ATNi + YNTi BiT ) ≥ 0 ∀i ∈ {1, . . . , M} . (ANi ENi + Bi YNi ) Ei (7.15) T Separating the term −ENi Qi ENi − YNi Ri YNi from the left hand side matrix in (7.15) and using the Schur complement again, we obtain (7.13a). To see that (7.13b) is equivalent to (7.12b), consider that M X

WiT Γi Wi

≤0⇔

i=1



M X i=1 M X i=1

EWiT Γi Wi E ≤ 0 WiT ENi Γi ENi Wi ≤ 0

7.4 Distributed Synthesis

55



M X

WiT Fi Wi ≤ 0 ,

(7.16)

i=1

where the first equivalence follows from the fact that E has full rank, the second from the fact that E is block-diagonal, and the third from the definition of Fi . Remark 7.3. Matrices ENi , YNi and Fi , that satisfy (7.13), can be found by distributed PM optimization, e.g. by minimizing − log det E = − i=1 log det Ei subject to (7.13), for volume maximization of the ellipsoid {x ∈ Rn |x T Pf x ≤ 1}. (7.13a) directly decomposes into one LMI per system i , which is coupled to neighboring LMIs by ENi . For decomposition of (7.13b), the structure in the sum on the left hand side can be exploited. This includes in particular the fact that nonzero blocks of a matrix WiT Fi Wi only overlap with nonzero blocks of matrices WjT Fj Wj , where either j ∈ Ni or i ∈ Nj . Remark 7.4. Comparing the proposed synthesis problem (7.13) to a synthesis problem for a centralized MPC controller, conservatism enters at several instants. The first and obvious restriction is the choice of the global terminal cost as a quadratic function Vf (x) = x T Pf x, with Pf block-diagonal, as well as the global terminal controller κf (x) = Kf x, with Kf being a sparse matrix. Conservatism also enters when implementing constraint (7.13b) in a distributed fashion, which is not possible in a straight-forward manner due to the sum on the left-hand side. Thus, in order for (7.13b) to be decomposable, restrictions on the matrices Fi have to be taken. They could for instance be chosen to be block-diagonal or even diagonal, in order for (7.13b) to decompose into several matrix inequality constraints, each of which depends on neighboring variables only.

7.4.2 Distributed Synthesis of a Distributed Terminal Set P T Given a separable terminal cost function Vf (x) = x T Pf x = M i=1 xi Pf,i xi fulfilling condition (7.2), any feasible level set thereof can be used as a global set Xfglob = {x ∈ Rn |x T Pf x ≤ α} ⊆ X , as required in Corollary 7.1 to bound the Cartesian product of locally invariant terminal sets. Given such a feasible level set with value α, the local set sizes αi can be P allocated such that M i=1 αi ≤ α. Since the size of the region of attraction of the closedloop system increases with the size of the terminal set, the goal is to find Xfglob as the largest feasible level set of the function Vf (x), given the present state and input constraints. This is achieved by solving the linear program αmax = max α

(7.17a)

α

−1

s.t. ||Pf,i 2 (Gi,j )T ||22 α ≤ (gi,j )2

∀i ∈ {1, . . . , M} ∀j ∈ {1, . . . , lx,i },

(7.17b)

56

7 Nominal Cooperative Distributed MPC for Regulation −1

T ||Pf,N2i KN (Hi,j )T ||22 α ≤ (hi,j )2 i

∀i ∈ {1, . . . , M} ∀j ∈ {1, . . . , lu,i } .

(7.17c)

The constraints (7.17b) and (7.17c) represent conditions on the support function of the α-level set of Vf (x). In particular, (7.17b) ensures that any point x ∈ {x ∈ Rn |x T Pf x ≤ α} fulfills the polytopic state constraints x ∈ X , where Gi,j denotes the jth row of Gi and gi,j the jth entry of gi . Similarly, (7.17c) ensures satisfaction of the polytopic input constraints, i.e. Kf x ∈ U, where Hi,j denotes the jth row of Hi and hi,j the jth entry of hi . The distributed solution to problem (7.17) results in a constrained consensus problem on the shared parameter αmax . For possible solution approaches, the reader is referred to the extensive literature on constrained consensus, e.g. [Olfati-Saber et al., 2007]. A summary of the offline distributed synthesis steps leading to a stabilizing distributed MPC controller is given in Algorithm 7.2. Algorithm 7.2 Offline distributed MPC synthesis Solve system of LMIs (7.13) by distributed optimization to find local terminal cost terms Vf,i (xi ). 2: Solve LP (7.17) by distributed optimization to find the size αmax of the largest feasible level set of the global terminal cost Vf (x). PM 3: Choose initial sizes αi of the local terminal sets, such that i=1 αi ≤ αmax , e.g. by setting αi = α/M for all i ∈ {1, . . . , M}.

1:

7.5 Numerical Example In this section, basic functionality and computational aspects of the proposed cooperative distributed MPC approach are illustrated in a numerical example. In this example, a chain of masses m = 1 kg, connected by springs with spring constants k = 3 N/m and dampers with damping constants b = 3 Ns/m is considered (see Fig. 7.1). The continuous-time ODEs describing the motion of these masses are discretized by the forward Euler method with a sampling time of 0.2 s. The resulting discrete-time system can be written in the

Figure 7.1: Chain of masses, connected by spring and damper elements.

7.5 Numerical Example

57

form (3.8), where xi consists of the position and velocity of mass i and ui is a local force applied to mass i (the disturbance wi is 0). The local states and the input of each system i are constrained as kxi k∞ ≤ 10 and kui k∞ ≤ 1, and the local stage cost matrices QNi and Ri are identity. Furthermore, the MPC prediction horizon is N = 5.

Closed-loop Behavior: Basic Regulation In this example, three control setups are compared regarding their regulation behavior for a chain of M = 5 masses. The first setup is centralized MPC with a terminal set corresponding to the maximum volume feasible ellipsoid, which is characterized by the Hessian of the infinite horizon LQR cost, which is obtained by solving the Riccati equation. The second setup is distributed MPC as proposed in this chapter, and the third setup is distributed MPC

Mass 1 (centr.) Mass 2 (centr.) Mass 1 (distr.) Mass 2 (distr.) Mass 1 ( Mass 2 (

0.25

0.20

0.2

) )

0.0

Force input

Position

0.15

0.10

0.05

0.00

-0.2

-0.4

Mass 1 (centr.) Mass 2 (centr.) Mass 1 (distr.) Mass 2 (distr.) Mass 1 ( Mass 2 (

-0.6

-0.8

-0.05

) )

-1.0

2

4

6

8

10

12

14

16

2

6

4

Simulation step

8

10

12

14

Simulation step

(a) Mass position trajectories.

(b) Input trajectories.

0.88

0.86

0.84

Mass 1 (distr.) Mass 2 (distr.) Mass 5 (distr.)

0.82

0.80

0.78

0.76 2

4

6

8

10

12

14

16

Simulation step

(c) Local terminal set sizes.

Figure 7.2: Trajectories under nominal cooperative distributed MPC for regulation.

58

7 Nominal Cooperative Distributed MPC for Regulation

with a trivial terminal set, which is simply the origin. Consider a zero initial state for all systems except for the first one, which has an initial state of x1 = [0.27 m, 0 m/s]T . The system is simulated over a total of 15 steps, its closed-loop behavior is illustrated in Fig. 7.2. Fig. 7.2(a) depicts the trajectories of the positions of masses 1 and 2 and Fig. 7.2(b) shows the respective input trajectories. Fig. 7.2(c) depicts the terminal set sizes of masses 1, 2 and 5 under distributed MPC as proposed in this chapter. Note that the local terminal set sizes change considerably during convergence of the closed-loop system, and that the trajectories of centralized and distributed MPC, as proposed in this thesis, are quite similar.

Closed-loop Behavior: Region of Attraction (RoA) In this section, we compare the size of the RoA under (i) cooperative distributed MPC as proposed in this chapter, and (ii) cooperative distributed MPC with a trivial terminal set, given by the origin. These RoA sizes are stated relative to the RoA size under centralized MPC with a dense ellipsoidal terminal set, which is expected to be larger. Approximations of the relative RoA sizes are obtained as follows: A total of 100 random initial states are scaled, separately for both (i) and (ii), to the maximum amount under which the distributed MPC problem is feasible. I.e. the boundary of the RoAs of (i) and (ii) is gauged in 100 random directions. Each individual scaling factor is then normalized by the corresponding factor for centralized MPC and the relative RoA sizes for both (i) and (ii) are estimated as the mean of the 100 respective normalized scaling factors. 1.2

relative RoA size

1.0

0.8

0.6

0.4

0.2 distributed terminal set 0.0

2

4

6

8

10

12

14

MPC horizon length

Figure 7.3: RoA under cooperative distributed MPC as proposed in this chapter, compared to a setup with a trivial terminal set.

7.6 Conclusion

59

Fig. 7.3 depicts the estimates of the relative RoA sizes for both considered distributed MPC setups for various prediction horizons. It is apparent that for short prediction horizons, there is a significant gain in RoA-size if non-trivial, locally time-varying terminal sets are used instead of single-point terminal sets. Moreover, it is clear that for long prediction horizons, the RoAs of all controllers, centralized and distributed, converge to the maximum RoA for the given constrained system. Computations: Terminal Cost Synthesis P −1 A structured terminal cost can be synthesized by minimizing − M i=1 log det Pf,i , i.e. by P M maximizing the volume of the ellipsoid {x ∈ Rn | i=1 xiT Pf,i xi ≤ 1}, subject to the LMI conditions (7.13), by distributed optimization. For the given example, we use ADMM (see Section 4.4.3) with ρ = 100 as a distributed optimization method. Furthermore, massspring-damper systems connected according to the network topologies illustrated in Fig. 7.4 are considered, hence (i) chain, (ii) ring, (iii) 2-hop ring and (iv) random graph. As for the random graph, the adjacency matrix G for a network of M systems is generated as follows: 1. Let r ∈ [0, 1] be a prespecified threshold value. 2. Generate a random symmetric matrix G ∈ [0, 1]M×M . 3. For all entries of G: If the entry is less or equal than r set it to 0, else set it to 1. 4. If G represents a connected graph, exit, else increase r and return to step 2. The systems in the obtained random networks had the following average numbers of neighbors: # of systems Average # of neighbors

4 3.5

64 4.625

128 6.05

256 6.90

In Fig. 7.5, the required number of ADMM iterations to reduce the Euclidean norm of the global primal residual of the synthesis problem to a value of less than 2 × 10−3 are depicted. The required number of iterations is observed to increase mildly with the network size for the regular topologies, while it is observed to increase significantly for the random graph topology.

7.6 Conclusion In this chapter, a novel cooperative distributed MPC controller based on a time-varying terminal set was presented. It was shown that both the offline synthesis and the online operation of the controller can be done without central coordination. In a numerical example, the basic functionality of the controller was verified, and it was observed that a controller

60

7 Nominal Cooperative Distributed MPC for Regulation

(a) chain

(b) ring

(c) 2-hop ring

(d) random graph

Figure 7.4: Graph topologies under consideration. 1600 Chain Ring 2-hop ring Random graph

# of it. to convergence

1400

1200

1000

800

600 400

200 0 4

64 128

256

512

1024

# of connected systems

Figure 7.5: Scalability of the proposed terminal cost synthesis method.

using a time-varying terminal set has a significantly larger RoA compared to a controller using the origin as a terminal set.

8 Robust Cooperative Distributed MPC for Regulation 8.1 Introduction In model-based control approaches such as MPC, a common difficulty is the presence of uncertainty in the system model. This uncertainty complicates accurate state predictions and the optimal choice of control inputs. Therefore, several MPC approaches which are able to cope with bounded uncertainty have been developed over the years. For instance, in minmax MPC, e.g. [Scokaert and Mayne, 1998], [Bemporad et al., 2003], the controller aims at finding the best sequence of control inputs under a deliberately counteracting disturbance. Moreover, in tube MPC, e.g. [Chisci et al., 2001] and [Mayne et al., 2005], a nominal MPC problem with tightened constraints is used to define a stabilizing robust MPC control law. In the context of distributed MPC for networks of constrained dynamic systems, robust MPC techniques have mainly been used in non-cooperative methods, in order to decouple the systems in the network from each other. In such an approach, dynamic coupling to neighboring systems is seen as disturbance acting on the local dynamics. This disturbance is accounted for by a locally robust controller design, as done e.g. in [Jia and Krogh, 2002] by local min-max MPC controllers, and in [Farina and Scattolini, 2012] by local tube MPC controllers. Even though, in such a setup, the robust control design is meant to reject dynamic coupling in a distributed nominal system, additional local uncertainty could be handled by the same controller in a straight-forward manner. While robust techniques have been used in several non-cooperative distributed MPC methods, few robust cooperative methods are available to date. In [Richards and How, 2007] and [Trodden and Richards, 2010], robust cooperative methods for networks of uncoupled linear systems, each of which subject to bounded additive uncertainty, were proposed. However, these methods have a sequential nature. In particular, at each time step, only a single system optimizes its control inputs, while the others act according to a previously computed feasible trajectory, that is known to all systems. The closed-loop operation of such a controller is considerably less flexible than the one of a controller based on distributed optimization, in which all systems optimize their inputs in every time step. Therefore, in this

62

8 Robust Cooperative Distributed MPC for Regulation

chapter, robust cooperative distributed MPC methods based on distributed optimization are discussed. Main Contributions Amongst the existing robust MPC approaches, tube MPC has the advantage that the online computational effort is largely the same as that of nominal MPC. This is due to the fact that the optimal control problem solved online in tube MPC is obtained from the nominal MPC problem, where the state and input constraints are tightened. In this chapter, it is shown how existing methodologies for centralized tube MPC can be extended to a cooperative distributed setup. In particular, structural requirements for the two well-established tube MPC schemes proposed in [Mayne et al., 2005], see Section 5.2.1, and [Chisci et al., 2001], see Section 5.2.2, are stated. If these structural requirements are fulfilled, the resulting controllers can be operated by means of distributed optimization techniques in closed-loop. As an additional contribution, corresponding synthesis methods are proposed, which include distributed procedures to design structured ellipsoidal RPI sets and to tighten polytopic state and input constraints. Both the synthesis and the closed-loop operation can be executed in a fully distributed way. Outline In Section 8.2, structural requirements for the robust MPC methods under consideration are stated, under which these methods can be applied in a distributed way. In Section 8.3.1, the distributed synthesis of structured RPI sets is discussed and in Section 8.3.2, distributed constraint tightening is introduced. Furthermore, in Section 8.4, closed-loop operation of the proposed methods is discussed. Section 8.5 provides an illustrative numerical example and Section 8.6 concludes the chapter.

8.2 Problem Statement Tube MPC for centralized LTI systems, as proposed e.g. in [Mayne et al., 2005] and [Chisci et al., 2001], follows a principle which is similar to nominal MPC. In particular, the control inputs are found by online solution of a nominal MPC problem at each time step, where a nominal finite-horizon trajectory is optimized w.r.t. an appropriate cost function. Recursive feasibility under bounded additive disturbances is achieved by using tightened constraints and RPI sets in the MPC problem formulation. The components required for recursive feasibility and closed-loop stability of the two robust MPC approaches under consideration are summarized in Tab. 8.1.

8.2 Problem Statement Robust MPC approach [Mayne et al., 2005]

[Chisci et al., 2001]

63 Stability requirements 1.) RPI set Z 2.) Tightened sets X¯ = X Z and U¯ = U Kt Z 3.) Terminal control law κf (x) = Kt x, terminal cost Vf (·) : Rn → R+ and PI set Xf ⊆ X¯, such that (5.3) is satisfied 1.) Stabilizing control gain Kt for the pair (A, B) 2.) RPI set Xf 3.) Tightened sets as defined in (5.13)

Table 8.1: Components required for closed-loop stability under tube MPC.

In this work, the MPC problems proposed in [Mayne et al., 2005] and [Chisci et al., 2001] are formulated, such that they are amenable to distributed optimization. In order for this to be the case, the components listed in Tab. 8.1 need to exhibit structure. A methodology for finding a combination of a structured control law κf (x) = Kt x = PM coli∈{1,...,M} (KNi xNi ), a structured terminal cost function Vf (x) = i=1 Vf,i (xi ), and a distributed PI set Xf (α1 , . . . , αM ) = Xf,1 (α1 ) × . . . × Xf,M (αM ) was presented in Chapter 7. This chapter is concerned with the structured formulation and synthesis of the remaining components listed in Tab. 8.1, i.e. RPI sets and tightened constraints. We consider networks of constrained LTI systems subject to locally additive disturbances as introduced in (3.8a). In order to simplify the discussion, we exclusively focus on polytopic local state and input constraints of the form Xi = {xi ∈ Rni |Gi xi ≤ gi }, Ui = {ui ∈ Rmi |Hi ui ≤ hi } ∀i ∈ {1, . . . , M} ,

(8.1)

where Gi ∈ Rlx,i ×ni , g ∈ Rlx,i , Hi ∈ Rlu,i ×mi and h ∈ Rlu,i . These local constraint sets can be combined into global sets of the form X = {x ∈ Rn |Gx ≤ g}, U = {u ∈ Rm |Hu ≤ h} ,

(8.2)

where G = diagi∈{1,...,M} (Gi ) ∈ Rlx ×ln , g = coli∈{1,...,M} (gi ) ∈ Rlx , H = diagi∈{1,...,M} (Hi ) ∈ Rlu ×lm and hi = coli∈{1,...,M} (hi ) ∈ Rlu . Furthermore, the local disturbance of each system is assumed to be bounded as Wi = {wi ∈ Rni |wiT wi ≤ vi } ∀i ∈ {1, . . . , M} ,

(8.3)

where vi ∈ R≥0 for each i ∈ {1, . . . , M}. These local bounds can be combined into the global bound W = W1 × . . . × WM ⊆ Rn . (8.4)

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8.3 Distributed Synthesis for Robust Cooperative Distributed MPC 8.3.1 Structured RPI Sets For linear systems, the problem of synthesizing RPI sets has been extensively studied, see e.g. [Blanchini, 1999], [Rakovi´c et al., 2005]. It is well known that the minimum RPI set for a given closed-loop system x + = (A + BKt )x + w , with w ∈ W, is given as ∞ M Zmin = (A + BKt )k W . (8.5) k=0

Definition (8.5) illustrates a fundamental difficulty in the synthesis of RPI sets for dynamically coupled networks of systems: Any RPI set is defined globally, since the infinite horizon in (8.5) couples all systems in the network. Finding an RPI set is therefore generally a global synthesis problem and in this chapter, we formulate the problem such that it can be solved by distributed optimization. For linear systems, most available synthesis methods are either for polyhedral or ellipsoidal RPI sets. In this chapter, only ellipsoidal sets are considered due to the better scalability properties of the corresponding synthesis methods. The main synthesis tool in this regard is the S-procedure, see [Boyd et al., 1994] for details. For instance, given a disturbance set W = {w ∈ Rn |w T w ≤ v }, v ≥ 0, an ellipsoidal RPI set of the form Z = {x ∈ Rn |x T P x ≤ 1} fulfills the implication  xT P x ≤ 1 ⇒ ((A + BKt )x + w )T P ((A + BKt )x + w ) ≤ 1 . (8.6) T w w ≤v The S-procedure can be used to encode this implication by the sufficient matrix inequality  (A + BKt )T P (A + BKt ) (A + BKt )T P  − P (A + BKt ) P 0 0

  0 −P   0 − s0  0 1 0

   0 0 0 0 0    0 0 − s1 0 −I 0 ≥ 0, 0 1 0 0 v (8.7)

where s0 , s1 ≥ 0. By keeping Kt and s0 constant, this matrix inequality can be restricted to be linear in P . Hence, an ellipsoidal RPI set can often be found by LMI techniques. In the following, synthesis approaches for two particularly structured RPI sets are introduced, namely decentralized and distributed ones. 8.3.1.1 Decentralized RPI Sets We define an ellipsoidal decentralized RPI set as the Cartesian product of M local ellipsoids of the form Zi = {xi ∈ Rni |xiT Pi xi ≤ βi }, Pi > 0, βi ∈ R≥0 , each of which is both robust

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65

against the local additive disturbance and the dynamic coupling to neighboring systems. Consequently, each of the local ellipsoids fulfills the condition xjT Pj xj ≤ βj ∀j ∈ Ni wiT wi ≤ vi



⇒ xi+T P xi+ ≤ βi

∀i ∈ {1, . . . .M} ,

(8.8)

where xi+ = (ANi + Bi KNi )xNi + wi for each i . Given condition (8.8), the Cartesian product of the local ellipsoids, i.e. Z = Z1 × . . . × Z M ,

(8.9)

is a global RPI set. Remark 8.1. Condition (8.8) directly translates into a set of M matrix inequality conditions, which are coupled in the same way as the network of dynamic systems. By turning these inequalities into sufficient LMI conditions and combining them with an appropriate separable P cost function, e.g. − M i=1 log det(Pi ) for volume minimization, decentralized RPI sets can be synthesized using standard distributed optimization methods. Note that ellipsoidal RPI sets with minimum volume are interesting in robust MPC as proposed in [Mayne et al., 2005], since in this approach an RPI set Z is used to define an uncertainty tube, the cross-section of which should preferrably be small. Remark 8.2. Another result on decentralized invariance was recently presented in [Bari´c and Borrelli, 2012]. As opposed to the ellipsoidal robust positive invariant sets considered in this chapter, the sets presented in [Bari´c and Borrelli, 2012] are polytopic and robust control invariant. Furthermore, the methodology is restricted to networks of single integrator systems. Decentralized RPI sets can be conservative since they are designed to be robust not only against local disturbances, but also against coupling to neighboring systems. Therefore, in the next section, we will explore distributed RPI sets, which are based on a less conservative structure and do not consider dynamic coupling to be disturbances.

8.3.1.2 Distributed RPI sets We define a distributed ellipsoidal RPI set as n

Z = {x ∈ R |

M X i=1

xNT i PNi xNi ≤ 1} ,

(8.10)

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where PNi ≥ 0 for each i in {1, . . . , M}. For the synthesis of a distributed ellipsoidal RPI set, the global S-procedure encoding the implication  xT P x ≤ 1 ⇒ ((A + BKt )x + w )T P ((A + BKt )x + w ) ≤ 1 , wiT wi ≤ vi ∀i ∈ {1, . . . , M} (8.11) T T T where w = [w1 , . . . , wM ] , can be used. In particular, the matrix inequality, which results from the S-procedure applied to (8.11), is given as   P PM T ¯ ¯ −(A + BKt )T M P (A + BK ) −(A + BK ) P 0 N t t N i i i=1 i=1 P PM   ¯ ¯ − M P (A + BK ) − P 0  N t N i i i=1 i=1 0 0 1   PM   0 0 0 − i=1 P¯Ni 0 0 M    X  (8.12) − s0  si 0 −I¯ni 0  ≥ 0 , 0 0 0 − i=1 0 0 vi 0 0 1 where si ≥ 0 ∀i ∈ {0, . . . , M}, and where P¯Ni and I¯ni are lifted versions of the variables PNi and Ini into the global state space. Specifically, consider lifting matrices Ti ∈ {0, 1}ni ×n and Wi ∈ {0, 1}nNi ×n , which similar to permutation matrices have exactly one entry per row equal to 1, such that xi = Ti x, xNi = Wi x . (8.13) These matrices can then be used to lift the matrices PNi and Ini into Rn×n as P¯Ni = WiT PNi Wi , I¯ni = TiT Ini Ti ,

(8.14)

which enables the formulation of the matrix inequality (8.12). In order to decompose the matrix inequality (8.12), consider the equivalent formulation of the set (8.10) by the M local sets Zi (βi ) = {xNi ∈ RnNi |xNT i PNi xNi ≤ βi } ∀i ∈ {1, . . . , M} ,

(8.15)

P which are coupled by the condition M i=1 βi = 1. Sufficient conditions for inequality (8.12) can be given by the set of matrix inequalities   −(A + BKt )T P¯Ni (A + BKt ) −(A + BKt )T P¯Ni 0   −P¯Ni (A + BKt ) −P¯Ni 0  0 0 βi     0 0 0 −P¯Ni 0 0     − s0  0 0 0  − si 0 −I¯ni 0  ≥ S¯i ∀i ∈ {1, . . . , M} , (8.16) 0 0 βi 0 0 vi

8.3 Distributed Synthesis for Robust Cooperative Distributed MPC

67

in combination with the constraint M X

S¯i ≥ 0 .

(8.17)

i=1

Note that the decomposed conditions (8.16), in combination with (8.17), are equivalent to (8.12), if the matrices S¯i can be chosen arbitrarily. Under arbitrary S¯i , condition (8.17) is however not decomposable. In contrast, if the matrices S¯i are subject to a particular blockdiagonal structural constraints, (8.16) and (8.17) are sufficient for (8.12), but in turn, the constraint (8.17) can be decomposed into a number of constraints, which are coupled in a neighbor-to-neighbor fashion. The only global coupling to remain is in the lower right corner element in (8.17), which states a condition on the global sum over the lower right corner elements of all S¯i . Nevertheless, by using consensus techniques for this global sum, as e.g. proposed in [Schizas et al., 2008], and given block-diagonal matrices S¯i , the matrix inequalities (8.16) and (8.17) can be handled by distributed optimization techniques. Remark 8.3. In order to obtain the desired structured RPI set, a synthesis problem of the form min

PNi ,Si ,βi ,si ∀i∈{1,...,M}

f (PN1 , . . . , PNM )

s.t. (8.16), (8.17) , PNi ≥ 0 ∀i ∈ {1, . . . , M} ,

(8.18a) (8.18b) (8.18c)

can be solved by distributed optimization techniques. In (8.18), constraint (8.18c) ensures that the submatrices are positive semi-definite and the objective function f (PN1 , . . . , PNM ) is a design choice and should be decomposable and enforcing the intended shape and size PM of the resulting RPI set, e.g. − i=1 log det(PNi ) for a heuristic to obtain a small global RPI set. Note that the constraints (8.16) are written in global dimensions. However, the sparsity in the matrices P¯Ni and I¯ni allows to reduce the dimensions of these local matrix inequalities.

8.3.2 Distributed Constraint Tightening In the following, distributed contraint tightening methods for the two robust MPC approaches under consideration are proposed. In Section 8.3.2.1, it is shown how uncoupled local polytopic constraints can be tightened by a globally defined ellipsoid, as required in robust MPC according to [Mayne et al., 2005]. Furthermore in Section 8.3.2.2, it is shown how such constraints can be tightened to account for a k-step bounded disturbance sequence, as required in robust MPC according to [Chisci et al., 2001].

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8.3.2.1 Distributed Tightening of Polytopic Constraints by an Ellipsoidal RPI Set Tightening of general polytopic state and input constraints by a globally defined set Z can be performed using the support function, as introduced in Definition 2.16. Specifically, a tightened state constraint set X¯ = {x ∈ Rn |Gx ≤ g¯}, as well as a tightened input con¯ can be obtained by modifying the polytopic constraints, straint set U¯ = {u ∈ Rm |Hu ≤ h} halfspace by halfspace, as g¯j = gj − σZ (GjT ) ∀j ∈ {1, . . . , lx } , h¯j = hj − σZ (KtT HjT ) ∀j ∈ {1, . . . , lu } ,

(8.19) (8.20)

where Gj and Hj are the jth rows of the matrices G and H respectively, and gj and hj are similarly the jth elements of the vectors g and h. For a decentralized ellipsoidal RPI set as defined in (8.9), the support function evaluation is a purely local operation. In contrast, for a distributed ellipsoidal RPI set, as defined in (8.10), the support function evaulation corresponds to the solution of an optimization problem of the form σZ (a) = max aT x

(8.21a)

x

s.t. xNT i PNi xNi ≤ βi M X

∀i ∈ {1, . . . , M} ,

βi ≤ 1 .

(8.21b) (8.21c)

i=1

If it were not for constraint (8.21c), the constraints in problem (8.21) would be coupled in a neighbor-to-neighbor fashion. However, constraint (8.21c) introduces global coupling into the problem. Since the type of global coupling in (8.21c) is in a scalar variable only, it can be captured within the framework of distributed optimization, e.g. by using the consensus method proposed in [Schizas et al., 2008]. Therefore, the tightening of every single constraint can be computed by solving problem (8.21) by distributed optimization. 8.3.2.2 Distributed Tightening of Polytopic Constraints for a Disturbance Sequence of Length k For robust MPC according to [Chisci et al., 2001], each state and input constraint is tightened by a different amount for every time step in the MPC prediction horizon. However, the tightening operations defined in (5.13) involve Pontryagin differences. These differences contain explicit coupling to non-neighboring systems, which poses a difficulty for distributed computations. Thus, in the following, we propose a new method to perform the constraint tightening in a distributed way, with neighbor-to-neighbor communication only. For this purpose, the k-step support function is introduced.

8.4 Robust Cooperative Distributed MPC: Closed-loop Operation

69

Definition 8.1 (k-Step Support Function). Given a structured linear control law ut (x) = Kt x = coli∈{1,...,M} (KNi xNi ) stabilizing the pair (A, B), the k-step support function is defined as σW k (a) = sup aT y (k)

(8.22a)

w∈W k

s.t. y (0) = 0

(8.22b)

y (l + 1) = (A + BKt )y (l) + w (l) ∀l ∈ {0, . . . , k − 1} , Q where W k denotes the Cartesian product k−1 i=0 W and w is a sequence of k disturbance realizations. The k-step support function indicates the maximum distance a k-step disturbance sequence can drive the system in direction a. A tightened state constraint set X¯k = {x ∈ Rn |Gx ≤ g¯k } as well as a tightened input constraint set U¯k = {u ∈ Rm |Hu ≤ h¯k } can thus be obtained by modifying each half space constraint as g¯k,j = gj − σW k (GjT ) ∀j ∈ {1, . . . , lx } , h¯k,j = hj − σW k (K T HjT ) ∀j ∈ {1, . . . , lu } ,

(8.23a) (8.23b)

where Gj and Hj are the jth rows of the matrices G and H respectively, and g¯k,j and h¯k,j are the jth elements of the vectors g¯k and h¯k . It can easily be shown that the tightened sets obtained by (8.23) are equivalent to those defined in (5.13). Remark 8.4. An evaluation of the k-step support function amounts to the solution of a structured convex optimization problem and can thus be done by distributed optimization. This becomes apparent when looking at the definition of the function, which is essentially an optimization problem with separable cost and structured dynamics constraints.

8.4 Robust Cooperative Distributed MPC: Closed-loop Operation In this section, distributed closed-loop operation of the two robust MPC controllers under consideration is discussed. In particular, it is demonstrated that the MPC problems of both setups can be solved by distributed optimization if the components listed in Tab. 8.1 are structured.

8.4.1 Robust Distributed MPC according to [Mayne et al., 2005] Compared to nominal MPC, the only difference in the closed-loop operation of robust MPC according to [Mayne et al., 2005] is in the initial state constraint (5.5b). If Z is

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chosen to be a decentralized RPI set as defined in Section 8.3.1.1, the coupling structure in the MPC problem proposed by [Mayne et al., 2005] is equal to the one encountered in nominal MPC. Hence, if dynamics and constraints are appropriately structured, the resulting robust MPC controller can be operated in a distributed way using distributed optimization. However, if Z is chosen to be a distributed RPI set, constraint (5.5b) introduces direct global coupling into the MPC problem. Such a constraint, as visible in the set definition (8.10), is imposed on a sum over M local elements and can in principle be handled by distributed optimization techniques. Such techniques, however, come at a cost. For instance, the approach proposed in [Schizas et al., 2008] suggests the use of M additional auxiliary variables in each subproblem, in order to reformulate the direct global coupling into an indirect neighbor-to-neighbor one. This approach is clearly prohibitive if the number M of systems in the network is large. Furthermore, even a modest number of additional auxiliary variables might slow down the optimization software. Therefore, in this section, a sufficient condition for constraint (5.5b) is presented, which is based on local coupling only, even if Z is chosen as a distributed RPI set. In the following explanations, x(t) denotes the system state at time t, x(t + k|t) the kth element of a state trajectory predicted at t, and xr∗1 (t + k|t) the kth element of the optimal state trajectory for problem (5.5) at t. PM Consider a structured RPI set Z = {x ∈ Rn | i=1 xNT i P xNi ≤ 1} and its equivalent reformulation by the sets Zi (t) = {xNi ∈ RnNi |xNT i PNi xNi ≤ βi (t)} ∀i ∈ {1, . . . , M} ,

(8.24)

where the functions βi (·) : N0 → [0, 1] are chosen such that M X

βi (t) ≤ 1 ∀t ∈ N0 .

(8.25)

i=1

A sufficient condition for constraint (5.5b) to hold is that xNi (t|t) ∈ xNi (t) ⊕ Zi (t) ∀i ∈ {1, . . . , M} .

(8.26)

The above constraints offer a way to remove the direct global coupling if appropriate functions βi (t), which fulfill (8.25), can be found. A strategy on how to define these functions, such that additionally recursive feasibility and convergence to Z is guaranteed in closed-loop, is stated in the following theorem. Theorem 8.1. Let for every i ∈ {1, . . . , M} the function βi (·) be defined as (8.27) βi (t) = (xNi (t) − xN∗ i ,r 1 (t|t − 1))T PNi (xNi (t) − xN∗ i ,r 1 (t|t − 1)) , P with βi (0) ≥ 0 and M i=1 βi (0) ≤ 1. Then, problem (5.5) remains feasible ∀t ∈ N0 if (5.5b) is replaced by (8.26).

8.4 Robust Cooperative Distributed MPC: Closed-loop Operation

71

Proof. Recursive feasibility and convergence of the state to Z in closed-loop under robust MPC according to [Mayne et al., 2005] is based on the following argument. Given an optimal state sequence {xr∗1 (t|t), . . . , xr∗1 (t + N|t)} for problem (5.5) at time t, the shifted sequence {xr∗1 (t + 1|t), . . . , (A + BKt )xr∗1 (t + N|t)} is feasible and improving, w.r.t. the cost function (5.5a), at time t +1. In the following, it will be shown that the same properties also hold under the modified constraint (8.26). PM If both constraint (8.26) and i=1 βi (t) ≤ 1 are satisfied at time t, it follows that PM PM ∗ T ∗ i=1 (xNi (t) − xNi ,r 1 (t|t)) PNi (xNi (t) − xNi ,r 1 (t|t)) ≤ i=1 βi (t) ≤ 1, implying x(t) ∈ ∗ xr 1 (t|t) ⊕ Z. Proceeding to time t + 1, since Z is an RPI set it follows that x(t + 1) ∈ xr∗1 (t + 1|t) ⊕ Z. Consequently, it holds that

M X

(x(t + 1) − xr∗1 (t + 1|t))T P (x(t + 1) − xr∗1 (t + 1|t)) =

(8.28)

(xNi (t + 1) − xN∗ i ,r 1 (t + 1|t))T PNi (xNi (t + 1) − xN∗ i ,r 1 (t + 1|t)) =

(8.29)

i=1 M X

βi (t + 1) ≤ 1 ,

(8.30)

i=1

and the shifted trajectory is feasible for constraint (8.26) at t + 1. Feasibility w.r.t. the remaining constraints as well as an improvement in the objective function follow from the original properties of problem (5.5). Thus, both recursive feasibility and convergence are preserved under constraint (8.26).

8.4.2 Robust Distributed MPC according to [Chisci et al., 2001] For distributed closed-loop operation of robust MPC according to [Chisci et al., 2001], the MPC problem (5.12) needs to be solved at every time-step by distributed optimization. Compared to a nominal MPC problem, the only difference in problem (5.12) is the fact that the terminal set is RPI. Closed-loop operation of a decentralized or distributed RPI set constraint was already discussed in the previous section, in connection with robust MPC according to [Mayne et al., 2005]. The only difference in the approach proposed in [Chisci et al., 2001] is the fact that the RPI set poses a constraint for the terminal state instead of the initial one. Nevertheless, the underlying problems and solution techniques are the same. Hence, for a decentralized RPI set, M uncoupled local terminal constraints for the systems in the network are given. For a distributed RPI set, a global coupling constraint is present in the MPC problem. This constraint can be reformulated by a sufficient constraint, which exhibits neighbor-to-neighbor coupling only, by using the approach formalized in Theorem 8.1.

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8.5 Numerical Example Example Setup In this section, the methodology proposed in this chapter is applied to a chain of 5 secondorder systems. The dynamics of the local systems are given in the form (3.8), in particular " # " # " # X 0 0 1 0 0 xi+ = xi + ui + xj + wi ∀i ∈ {1, . . . , M} , (8.31) 1 1 1 0.5 0.5 j∈Ni \i

where N1 = {1, 2}, N5 = {4, 5} and Ni = {i − 1, i, i + 1} for i ∈ {2, 3, 4}. The nominal local state and input constraints are given as Xi = {xi ∈ R2 |kxi k∞ ≤ 10} and Ui = {ui ∈ R|kui k∞ ≤ 10} and for each system the local additive disturbance lies in the ball Wi = {wi ∈ R2 |wiT wi ≤ 0.03}. The prediction horizon at closed-loop operation is N = 3 and the quadratic stage cost matrices are identity for both the state and the input. Closed-loop Behavior: Basic Robust Regulation In the following, a robust regulation scenario for a chain of five systems is considered. The initial state of all systems is zero except for the first one, which has an initial state x1 (0) = [3.2, 0]T . The RPI sets used in the simulations were chosen to be decentralized. In Fig. 8.1, trajectories of the first state component of selected systems are depicted, along with the corresponding input trajectories of the same systems. Note that for both robust MPC approaches considered in this chapter, satisfactory regulation performance is observed. The input magnitude in the first steps of the simulation is larger for the approach based on [Chisci et al., 2001] than for the one based on [Mayne et al., 2005]. This behavior is likely due to the smaller amount of constraint tightening in the former approach, and it results in a faster state regulation towards the origin. Closed-loop Operation: Comparison of Distributed and Decentralized RPI sets An interesting aspect is the comparison of distributed and decentralized RPI sets. In the following, we perform such a comparison for ellipsoidal RPI sets used in robust distributed MPC based on [Mayne et al., 2005]. In this approach, the RPI set corresponds to the cross-section of an uncertainty tube, which contains all possible state trajectory predictions under the given bounded disturbance. Naturally, for a good prediction, this cross-section should be chosen as small as possible. A meaningful synthesis objective is therefore volume minimization, which was done for both a decentralized and a distributed RPI set. The comparison of the volumes of the two sets is not trivial, since the decentralized RPI set is defined as a Cartesian product of ellipsoids in R2 , while the distributed RPI set

8.5 Numerical Example

73

3.5

3.5

3.0

3.0 2.5

First state element

First state element

2.5 2.0 1.5 1.0 0.5 0.0

2.0 1.5 1.0 0.5

-0.5 0.0

-1.0 -1.5

-0.5

0

5

10

15

0

5

Simulation step

10

15

Simulation step

(a) First state component trajectories under [Mayne et al., 2005]

(b) First state component trajectories under [Chisci et al., 2001]

1

1 0

0 -1

-1

input

input

-2

-2

-3 -4

-3 -5

-4 -6 -7

-5 0

2

4

6

8

10

12

14

Simulation step

(c) Input trajectories under [Mayne et al., 2005]

0

2

4

6

8

10

12

14

Simulation step

(d) Input trajectories under [Chisci et al., 2001]

Figure 8.1: Regulation behavior of proposed cooperative distributed robust MPC methods.

is an ellipsoid directly defined in Rn . An ad-hoc volume comparison strategy was chosen as follows: The extensions of the two sets were measured in 2000 random directions, the same ones for each set. The extensions within the distributed RPI set were observed to be 0.776 times smaller than the ones within the decentralized RPI set, which indicates that the distributed set is considerably smaller than the decentralized one. This result makes sense, since a distributed RPI set needs to be robust only against additive disturbance, whereas a decentralized RPI set also needs to be robust against dynamic coupling to neighbors. Thus, the distributed RPI set is less conservative and can be designed with a smaller volume.

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8 Robust Cooperative Distributed MPC for Regulation

Computations: Distributed Constraint Tightening Both robust distributed MPC approaches presented in this chapter rely on constraint tightening. In the approach based on [Mayne et al., 2005], the nominal constraints are tightened by an RPI set, whereas in the approach based in [Chisci et al., 2001], they are tightened in a step-by-step fashion. Numerically, the tightening can be done by a number of distributed evaluations of the support function or the k-step support function, respectively. In the following, numerical properties of these operations are highlighted, where ADMM was used as a distributed optimization method in both cases. The RPI set used for constraint tightening in the approach by [Mayne et al., 2005] is chosen to be distributed, since under a decentralized RPI set constraint tightening is a purely local operation and does not require the use of distributed optimization.

130

140

120

120

Nr. of ADMM iterations

Nr. of ADMM iterations

Given a chain of M systems, the first state constraint of system b M2 c is tightened. In Fig. 8.2(a), the required number of iterations for the distributed evaluation of the support function is depicted for various chain lengths M. Furthermore, in Fig. 8.2(b), the same quantity is depicted for the distributed evaluation of the k-step support function for various values of k. The ADMM stepsize parameter is chosen to be ρ = 7 for the support function and ρ = 0.02 for the k-step support function, the stopping criterion is for the primal variables not to change by more than 5 × 10−3 in five consecutive steps. Note that a saturation in the number of iterations can be observed for long chains of systems in the support function case, and for large values of k in the k-step support function case.

110 100 90 80

100 80 60 40 20

70 2

4

6

8

10

12

14

16

18

20

Chain Length

(a) Distributed evaluation of the support function.

0

0

5

10

15

Prediction step

(b) Distributed evaluation of the k-step support function.

Figure 8.2: Performance of the proposed distributed constraint tightening methods in terms of required ADMM iterations to convergence.

8.6 Conclusion

75

8.6 Conclusion In this chapter, two cooperative distributed robust MPC methods, based on existing centralized robust MPC methods available in the literature, were proposed. It was pointed out that for the use in a cooperative distributed setup, the MPC problems of the two methods under consideration are required to be amenable for distributed optimization. This means in particular, that the RPI sets contained in the problem formulation need to be designed in a structured way. Thus, for both methods, synthesis procedures were presented, which cover the design of structured RPI sets and the tightening of nominal state and input constraints. All synthesis procedures can be performed by distributed computations using distributed optimization. Furthermore, the functionality of the proposed synthesis and closed-loop control methods was verified in numerical examples.

9 Cooperative Distributed MPC for Reference Tracking 9.1 Introduction In this chapter, methodology for cooperative distributed MPC for reference tracking is introduced. In particular, a well-established approach from the literature on centralized tracking MPC is adapted for the use in a distributed system, where the MPC problem is solved online by distributed optimization. In the literature on centralized tracking MPC, mainly two problems have been considered. The first one is offset-free tracking MPC [Muske and Badgwell, 2002], [M¨ader and Morari, 2010], where the main objective is to estimate disturbance signals acting on the dynamics of the system, in order to remove the disturbance induced tracking offset by shifting the reference point. The second problem is tracking MPC under inadmissible output references, hence references, which can not be attained by an equilibrium point under the given constraints. This line of work builds on the theory of output admissible sets [Gilbert and Tan, 1991] and reference governors [Gilbert and Kolmanovsky, 1995]. In the latter, the paradigm is to replace the original reference with the nearest admissible one, for which a feasible equilibrium point exists. A particularly interesting adaptation of this concept was proposed in [Limon et al., 2008], where the reference is filtered directly in the MPC problem. This is in contrast to classical reference governor setups, where two separate optimization problems are solved sequentially, i.e. one for the reference governor and one to evaluate the MPC control law. In spite of the practical relevance of reference tracking, the number of available distributed reference tracking approaches is small. In a non-cooperative setup, tracking of constant references has been considered in [Betti et al., 2012], building on the non-cooperative distributed regulation approach proposed by [Farina and Scattolini, 2012]. Furthermore, in a cooperative setup, a distributed tracking MPC controller was presented in [Ferramosca et al., 2013], combining the work on distributed regulation MPC proposed in [Stewart et al., 2010] with the work on tracking MPC proposed in [Limon et al., 2008]. This controller, however, relies on a globally coupled terminal set in the MPC problem, which complicates the applicability of common distributed optimization methods. Therefore, in this chapter, we

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present a structured cooperative distributed tracking MPC controller. While this controller also builds on the work presented in [Limon et al., 2008], it relies on a structured timevarying distributed invariant set for tracking, which is amenable to standard distributed optimization methods. Main Contributions The methodology presented in this chapter extends the nominal cooperative distributed MPC methodology presented in Chapter 7 to MPC for tracking of piecewise constant output references. In particular, the tracking MPC approach presented in [Limon et al., 2008] is adapted for cooperative distributed MPC. As a first contribution, the distributed invariance concept, which was presented in Chapter 7 for nominal MPC, is extended to distributed invariance for tracking, as required in [Limon et al., 2008]. This allows the formulation of a structured tracking MPC problem, which can be solved by standard distributed optimization methods. As a second contribution, a distributed synthesis procedure for ellipsoidal distributed invariant sets for tracking is proposed. This procedure is particularly designed for networks of linear systems subject to polytopic constraints, and it can be executed directly on the distributed system, no central coordination is required at any point. This aspect is particularly interesting in presence of changing network topologies, where new controllers need to be synthesized on the fly (see the related work in [Zeilinger et al., 2013]). Outline The chapter is structured as follows. In Section 9.2, the problem under consideration is defined and discussed. In Section 9.3, a novel notion of distributed invariance for tracking is introduced. Subsequently, in Section 9.4, a distributed synthesis approach for a cooperative distributed tracking MPC controller is proposed and in Section 9.5, a numerical example is presented. Finally, in Section 9.6, the chapter is concluded.

9.2 Problem Statement The problem considered in this chapter is on the formulation of cooperative distributed tracking MPC controllers. Such a controller is defined through an MPC problem, which is globally defined for the network of systems under consideration, and which is solved at every time step by distributed optimization. In order for distributed optimization to be applicable under the given communication constraints, the global MPC problem is required to be structured. Since well established methodologies for centralized tracking MPC are

9.3 Time-Varying Distributed Invariant Set for Tracking

79

readily available in the literature, the goal is to adapt an existing tracking MPC method, i.e. the one proposed in [Limon et al., 2008], for the use under distributed optimization. Tracking MPC according to [Limon et al., 2008] was briefly summarized in Section 5.3 of this thesis. The components in [Limon et al., 2008], which guarantee stability, are the terminal cost Vf (∆x) and the terminal invariant set for tracking Xtr ∈ R2n+m , which is defined in the space of the state residual ∆x = x − xs , the equilibrium state xs and the equilibrium input us . These components are not naturally structured, even if the system dynamics and constraints are. Therefore, the main challenge is to formulate a terminal cost and a terminal invariant set for tracking in a structured way, without giving up their stabilizing properties, such that the resulting MPC problem is amenable to distributed optimization. Similar to the regulation case presented in Chapter 7, the goal is to formulate and synthesize a structured terminal cost and a structured terminal set for tracking as Vf (∆x) =

M X

Vf,i (∆xi ) ,

(9.1a)

i=1

Xtr (α1 , . . . , αM ) = Xtr,1 (α1 ) × . . . × Xtr,M (αM ) ,

(9.1b)

where Xtr,i (αi ) ⊆ Rni are parametrized local terminal sets, with parameter αi related to the set size. The functionality of the terminal cost is the same as in the nominal regulation case and it is therefore formulated in the same way. Hence, given a suitable terminal control law κf (∆x), Vf (∆x) decreases in every time step, while some Vf (∆xi ) might increase (see Theorem 7.1 for details). Since these terminal cost properties have already been defined in Section 7.3.1, the following Section 9.3 directly focuses on the formulation of a structured invariant set for tracking. Moreover, in Section 9.4, a combined distributed synthesis method for a quadratic terminal cost and an ellipsoidal terminal invariant set for tracking for a network of linear systems, subject to polytopic constraints, is presented.

9.3 Time-Varying Distributed Invariant Set for Tracking As discussed in Section 5.3, an invariant set for tracking, as introduced in [Limon et al., 2008], is defined in the augmented space of z = [∆x T , xsT , usT ]T , where the dynamics     A 0 0 B     + z =  0 In 0  z +  0  (u − us ) = Al z + Bl (u − us ) (9.2) 0 0 Im 0 apply. Therefore, an invariant set for tracking can not, as done in the regulation case, be defined as a level set of the terminal cost function. Instead, in this work, it is chosen as a

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level set of the newly introduced function Vtr (z) = Vf (∆x) + Vx (xs ) + Vu (us ) =

M X i=1

V (∆xi ) + Vx,i (xs,i ) + Vu,i (us,i ) , | f,i {z }

(9.3)

=Vtr,i (zi )

PM T T T ] . Note that Vf (∆x) = i=1 Vf,i (∆xi ) , us,i where for each i ∈ {1, . . . , M}, zi = [∆xiT , xs,i is the global terminal cost in problem (5.16), which in this case fulfills the conditions (7.2), guaranteeing a global decrease even under an increase of some of the local components Vf,i (∆xi ). Furthermore, the functions Vx,i (·) : Rni → R+ and Vu,i (·) : Rmi → R+ are convex and non-negative for all i in {1, . . . , M}. A time-varying distributed invariant set for tracking can be defined using the level sets of the local functions Vtr,i (·), as shown in the following theorem. Theorem 9.1. Let κf (x) = coli∈{1,...,M} (κNi (xNi )) be a structured control law, which stabilizes the system x + = Ax + Bκf (x). Let furthermore Vf,i (·), li (·, ·) and γi (·) be functions for each i ∈ {1, . . . , M}, for which the conditions (7.2) in Theorem 7.1 are T T T satisfied. Moreover, let zi = [∆xiT , xs,i , us,i ] and Xtr,i (αi ) = {zi ∈ R2ni +mi |Vtr,i (zi ) ≤ αi }. Then, under the dynamics z + = Al z + Bl κf (∆x) and the recursion α+ i = αi + γi (∆xNi ), i the time-varying sets Xtr (αi ) are invariant and non-empty at all times, hence (i) zi ∈ Xtr,i (αi ) ⇒ zi+ ∈ Xtr,i (α+ i ) (ii) 0 ≤ αi ⇒ 0 ≤ α+ i Proof. Consider zi ∈ Xtr,i (αi ). By Theorem 7.1 and positivity of li (·, ·), we have 0 ≤ Vf,i (∆xi+ ) ≤ Vf,i (∆xi ) − li (∆xNi , κNi (∆xNi )) + γi (∆xNi ) ≤ Vf,i (∆xi ) + γi (∆xNi ) .

(9.4)

Consequently, we can write + + Vtr,i (zi+ ) = Vf,i (∆xi+ ) + Vx,i (xs,i ) + Vu,i (us,i )

≤ Vf,i (∆xi ) + γi (∆xNi ) + Vx,i (xs,i ) + Vu,i (us,i ) ≤ αi + γi (∆xNi ) = α+ i ,

(9.5)

+ + where the first inequality follows from (9.4) and the fact that xs,i = xs,i and us,i = us,i + under the dynamics z = Al z + Bl κf (∆x). The second inequality follows from the fact that zi ∈ Xtr,i (αi ) is equivalent to Vtr,i (zi ) = Vf,i (∆xi )+Vx,i (xs,i )+Vu,i (us,i ) ≤ αi . Inequality (9.5) proves (i) and, by positivity of Vtr,i (·), also (ii).

9.4 Distributed Synthesis of a Distributed Invariant Set for Tracking

81

Given Theorem 9.1 and a feasible global level set Xtrglob = {z ∈ R2n+m |Vtr (z) ≤ α}, recursive feasibility of the Cartesian product Xtr (α1 , . . . , αM ) = Xtr,1 (α1 ) × . . . × Xtr,M (αM ), PM provided that i=1 αi ≤ α, can be established by the very same argument as for the distributed invariant set for regulation in Corollary 7.1.

9.4 Distributed Synthesis of a Distributed Invariant Set for Tracking In this section, a method for distributed synthesis of a distributed invariant set for tracking is proposed for linear systems subject to polytopic constraints. Not only does the resulting invariant set have a distributed structure as defined in (9.1b), but also can the synthesis procedure itself be executed in a distributed way. The second point is interesting especially for very large-scale systems, where the sheer size of the optimization problem outgrows a centralized computational infrastructure. Furthermore, distributed synthesis is interesting for networks of systems with time-varying connection topology, where for every topology change a new controller has to be synthesized online, and possibly on the network of systems itself. Related work on time-varying network topologies, i.e. for a so-called plug-and-play setup, has recently been presented in [Zeilinger et al., 2013]. Throughout this section, polytopic state and input constraints, as well as quadratic cost functions are considered. In particular, the local constraints are of the form Xi = {xi ∈ Rni |Gi xi ≤ gi } , Ui = {ui ∈ Rmi |Hi ui ≤ hi } ∀i ∈ {1, . . . , M} ,

(9.6)

where Gi ∈ Rlx,i ×ni , gi ∈ Rlx,i , Hi ∈ Rlu,i ×mi and hi ∈ Rlu,i . These local constraint sets can be combined into global sets of the form X = {x ∈ Rn |Gx ≤ g}, U = {u ∈ Rm |Hu ≤ h} ,

(9.7)

where G = diagi∈{1,...,M} (Gi ) ∈ Rlx ×ln , g = coli∈{1,...,M} (gi ) ∈ Rlx , H = diagi∈{1,...,M} (Hi ) ∈ Rlu ×lm and hi = coli∈{1,...,M} (hi ) ∈ Rlu . For the cost function terms, we consider local quadratic stage cost functions li (xNi , ui ) = xNT i QNi xNi + uiT Ri ui

∀i ∈ {1, . . . , M} ,

(9.8)

as well as local quadratic functions Vf,i (∆x) = ∆xiT Pf,i ∆xi Vx,i (xs,i ) = Vu,i (us,i ) =

T xs,i Px,i xs,i T us,i Pu,i us,i

∀i ∈ {1, . . . , M} ,

(9.9)

∀i ∈ {1, . . . , M} ,

(9.10)

∀i ∈ {1, . . . , M} ,

(9.11)

82

9 Cooperative Distributed MPC for Reference Tracking γi (xNi ) = ∆xNT i Γi ∆xNi

∀i ∈ {1, . . . , M} .

(9.12)

Note that the matrices Px,i and Pu,i are positive-semidefinite, Qi , Ri and Pf,i are positive definite, and Γi is allowed to be indefinite. Finally, we consider a linear terminal control law κf (x) = Kf x = coli∈{1,...,M} (KNi xNi ) .

(9.13)

As done in previous chapters, in order to simplify the notational connection between global and local dimensions, we introduce for each i ∈ {1, . . . , M} lifting matrices Ti ∈ {0, 1}ni ×n , Si ∈ {0, 1}mi ×m and Wi ∈ {0, 1}nNi ×n . These lifting matrices, similar to permuation matrices, have in each row exactly one entry equal to 1, such that xi = Ti x , ui = Si u , xNi = Wi x .

(9.14)

Using these matrices, the weighting matrices in the global stage cost function l(x, u) = PM PM x T Qx +u T Ru can be written as Q = i=1 WiT QNi Wi and R = i=1 SiT Ri Si respectively. Similarly for the global functions, Vf (∆x) = ∆x T Pf ∆x, Vtr (z) = ∆x T Pf ∆x + xsT Px xs + PM T PM T T P T , P = usT Pu us and γ(x) = ∆x T Γ∆x, we obtain Pf = f,i i x i i=1 Ti Px,i Ti , i=1 PM PM T T Pu = i=1 Si Pu,i Si and Γ = i=1 Wi Γi Wi . A structured terminal set for tracking Xtr (α1 , . . . , αM ) can be synthesized by techniques similar to those used in the regulation case in Section 7.4. In particular, a global level set Xtrglob = {z ∈ R2n+m |Vtr (z) ≤ α} is feasible and invariant if the conditions (A + BKf )T Pf (A + BKf ) − Pf ≤ −Q − KfT RKf + Γ with Γ ≤ 0 , h i z ∈ Xtr ⇒ Hx Hx 0 z ≤ kx , h i z ∈ Xtr ⇒ Hu Kf 0 Hu z ≤ ku

(9.15a) (9.15b) (9.15c)

are satisfied, where condition (9.15a) additionally enforces the required global Lyapunov property for the terminal cost function Vf (x). The set Xtr (α1 , . . . , αM ) is recursively conPM tained in Xtrglob , if initially it holds that i=1 αi ≤ α. The local function components Vx,i (xs,i ) and Vu,i (us,i ) do not show up in the conditions (9.15) and could in principle be chosen as arbitrary convex functions. However, for the closed-loop performance and the region of attraction under the resulting controller, it is of interest to choose Xtrglob as large as possible. Therefore, we propose a synthesis method, by which the volume of Xtrglob is maximized by simultaneous optimization over the matrices Pf,i , Px,i , Pu,i , Kf,i and Γi ∀i ∈ {1, . . . , M}. This method is detailed in the following lemma. Lemma 9.1. Consider for each i ∈ {1, . . . , M} the substitutions Ei = Pf,i−1 , ENi = −1 −1 diagj∈Ni (Ej ), YNi = KNi ENi , Ex,i = Px,i and Eu,i = Pu,i . The maximum volume ellipglob 2n+m soid Xtr = {z ∈ R |Vtr (z) ≤ 1} satisfying the conditions (9.15) for every point z

9.4 Distributed Synthesis of a Distributed Invariant Set for Tracking

83

contained, is given by the solution of the SDP min E,Ex ,Eu ,Y,F

M X

(− log det Ei − log det Ex,i − log det Eu,i )

(9.16a)

i=1

s.t. (7.13) , (9.16b)   2 gi,j Gi,j Ei Gi,j Ex,i   T Ei 0  ≥ 0 ∀i ∈ {1, . . . , M} ∀j ∈ {1, . . . , lx,i } , (9.16c)  Ei (Gi,j ) Ex,i (Gi,j )T 0 Ex,i   2 Hi,j YNi Hi,j Eu,i hi,j   T ENi 0  ≥ 0 ∀i ∈ {1, . . . , M} ∀j ∈ {1, . . . , lu,i } , (9.16d)  YNi (Hi,j )T Eu,i (Hi,j )T 0 Eu,i where E = diagi∈{1,...,M} (Ei ), Ex = diagi∈{1,...,M} (Ex,i ), Eu = diagi∈{1,...,M} (Eu,i ), Y = PM T PM T i=1 Si YNi Wi , F = i=1 Wi Fi Wi and Fi = ENi Γi ENi . Proof. The objective function (9.16a) makes sure that the determinant of of the inverse of the matrix diag(P, Px , Pu ) is maximized, see Lemma 4.1 for the separability property of the log det operator. This minimizes the determinant of diag(P, Px , Pu ), which in turn maximizes the volume of the ellipsoid Xtrglob = {z ∈ R2n+m |z T diag(P, Px , Pu )z ≤ 1}. Moreover, the constraint that the terminal cost Vf (∆x) decreases while its components Vf,i (∆xi ) might increase (according to Lemma 7.1), is contained in (7.13). The feasibility constraints in (9.15b) and (9.15c) for the set Xtrglob are imposed as constraints on its support function. It is well known that the support function of an ellipsoidal set E = {y ∈ Rq |y T Sy ≤ 1}, S > 0, is given explicitly as 1

σE (h) = max hT y = kS − 2 hk2 , y T Sy ≤1

(9.17)

where h ∈ Rq . The constraint that ellipsoid E is contained in a halfspace hT y ≤ k, k > 0, of a polytope can thus be formulated as 1

hT y ≤ k ∀y ∈ E ⇔ σE (h) ≤ k ⇔ kS − 2 hk22 ≤ k 2 .

(9.18)

Considering the particular structure of Xtrglob , wo obtain the following conditions for the local constraint sets Xi and Ui . h iT 1 2 kdiag(Pf ,i , Px,i )− 2 Gi,j Gi,j k22 ≤ gi,j ∀i ∈ {1, . . . , M} ∀j ∈ {1, . . . , lx,i } , (9.19a) h iT 1 2 kdiag(Pf ,Ni , Pu,i )− 2 Hi,j KNi Hi,j k22 ≤ hi,j ∀i ∈ {1, . . . , M} ∀j ∈ {1, . . . , lu,i } , (9.19b)

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9 Cooperative Distributed MPC for Reference Tracking

where the pair (Gi,j , gi,j ) corresponds to the jth halfspace constraint of Xi and (Hi,j , hi,j ) corresponds to the jth halfspace constraint of Ui . Note that (9.19a) ensures satisfaction of state feasibility (9.15b) and (9.19b) of input feasibility (9.15c). By the Schur complement, (9.19a) and (9.19b) can equivalently be written as (9.16c) and (9.16d) respectively. Remark 9.1. The optimization problem (9.16) has a separable objective function and constraints, which are coupled through the matrices ENi . Thus, it can, similar to the synthesis problem for an invariant set for regulation, be solved by distributed optimization techniques, to obtain a global invariant set for tracking Xtrglob = {z ∈ R2n+m |Vtr (z) ≤ 1}. This value can then be split up into local level set sizes, e.g. by simply assigning a size αi = 1/M to every local terminal set Xtr,i (αi ).

9.5 Numerical Example In this section, basic functionality and computational aspects of the proposed cooperative distributed tracking MPC controller are illustrated by a numerical example. In this example, we consider the same system as in Section 7.5, i.e. a chain of masses m = 1 kg, connected by springs with spring constants k = 3 N/m and dampers with damping constants b = 3 Ns/m, as illustrated in Fig. 9.1. Note that the output yi of each mass i ∈ {1, . . . , M} is defined to be the position of mass i , while no disturbance is acting on the dynamics. The local state and input constraints are chosen as kxi k∞ ≤ 5 and kui k∞ ≤ 5. Moreover, the local stage cost matrices are chosen to be identity for both QNi and Ri and the MPC prediction horizon is of length N = 5.

Figure 9.1: Chain of masses, connected by spring and damper elements.

Closed-loop Behavior: Basic Reference Tracking Consider a chain of five masses which have zero initial states, except for the first one, which has an initial state of [2.8m, 2.8m/s]T . The system is simulated for a total of 120 steps, where the output references for all systems, except system 5, are zero. The output reference of system 5 is 0.5m for simulation steps 1 to 40, −2m for steps 41 to 80 and again 0.5m

9.5 Numerical Example

85

for steps 81 to 120. In Fig. 9.2, the tracking behavior is illustrated, i.e. Fig. 9.2(a) depicts selected output trajectories, Fig. 9.2(b) selected input trajectories and Fig. 9.2(c) selected local terminal set size trajectories. The 0.5m reference is tracked accurately, while the −2m reference is found to be inadmissible and is thus replaced by the controller by an admissible reference, which is approximately −1.2m. In Fig. 9.2(c) it can be observed that during the tracking operation, the local terminal set sizes vary considerably. 4

3 2

3

ref 1

2 0

1

-1 -2

0

-3

-1 -4

-2

-5

0

20

40

60

80

120

100

20

0

40

simulation steps

60

80

100

120

simulation steps

(a) Selected output trajectories.

(b) Selected input trajectories.

0.7 0.6

0.5

0.4

0.3

0.2 0.1

0.0 0

20

40

60

80

100

120

simulation steps

(c) Selected terminal set size trajectories

Figure 9.2: Nominal tracking for a chain of five masses, connected by springs and dampers.

Computations: Tracking Performance under Suboptimal Optimization A practical difficulty in cooperative distributed MPC based on distributed optimization is the fact that the optimization method has to be stopped, at each time step, after a finite number of iterations. This usually means that the available solution to the MPC problem is

86

9 Cooperative Distributed MPC for Reference Tracking

suboptimal to some extent. Therefore, for the same system and tracking scenario which was considered in the previous section, the closed-loop tracking performance under suboptimal distributed optimization is highlighted in this section. Note that ADMM with various fixed numbers of iterations is used as a distributed optimization method. Fig. 9.3 depicts, for chains of 5 and 64 masses, the suboptimality in the 2-norm of the accumulated tracking error. As expected, the performance approaches the optimal one for a large number of iterations, where near optimal performance is already obtained at around 100 iterations. 25

suboptimality in tracking performance [%]

suboptimality in tracking performance [%]

14

12

10

8

6

4

2

0

-2 100

101

102

# of ADMM iterations

(a) Chain of 5 masses.

103

20

15

10

5

0 100

101

102

103

# of ADMM iterations

(b) Chain of 64 masses.

Figure 9.3: Suboptimality in closed-loop tracking due to unfinished distr. optimization.

9.6 Conclusion In this chapter, a novel cooperative distributed tracking MPC controller, based on a centralized method by [Limon et al., 2008], was proposed. It comes with a closed-loop stability guarantee, which is based on a time-varying distributed invariant set for tracking. For linear systems and polytopic constraints, both the controller synthesis and its closed-loop operation can be done by distributed optimization, without central coordination. Basic functionality of the approach was verified in an example, even under mildly suboptimal control inputs.

Part III Computational Aspects of Distributed Optimization in MPC

10 Computational Performance of Distributed Optimization in MPC 10.1 Introduction Distributed MPC approaches can be divided into two categories, namely non-cooperative and cooperative ones. For cooperative distributed MPC, distributed optimization offers a standard framework for iterative negotiation procedures, by which globally optimal control inputs can be found by distributed computations and local communication. Specific distributed optimization methods that have been proposed for MPC include the subgradient method [Wakasa et al., 2008], [Johansson et al., 2008], accelerated gradient methods [Necoara et al., 2008], [Giselsson et al., 2013], the Jacobi method [Venkat et al., 2005], and the alternating direction method of multipliers [Conte et al., 2012b]. For convex MPC problems, all of these methods guarantee convergence to the global optimum.

Main Contribution Despite the prominent position of cooperative distributed MPC in the literature, the performance of distributed optimization in this control approach is poorly understood. It is well known that in some cases distributed optimization methods converge slowly, whereas in contrast e.g. [Boyd et al., 2010] points out that the alternating direction method of multipliers (ADMM), a particular distributed optimization method, often converges to a satisfactory accuracy in a few tens of iterations. The main contribution of this chapter is therefore to provide a systematic computational study on the performance of two well-established distributed optimization methods, namely ADMM and dual decomposition based on fast gradient updates (DDFG), in MPC. In particular, the performance sensitivity of these methods is investigated with respect to a comprehensive but not exhaustive list of aspects, which are fundamental to distributed MPC: (i) the initial state of the problem, (ii) the coupling topology of the network of systems, (iii) the coupling strength among neighboring systems, (iv) the stability of the decoupled systems and (v) the number of systems in the network.

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10 Computational Performance of Distributed Optimization in MPC

Outline In Section 10.2, the setup of the computational study is presented, i.e. the system and controller setup as well as the performance evaluation strategy. Note that details on the distributed optimization methods under consideration can be found in the preliminaries chapter of this thesis, i.e. in Section 4.4.2 for DDFG and in Section 4.4.3 for ADMM. Section 10.3 contains the computational results of the study and the discussions thereof. Finally, Section 10.4 concludes the chapter.

10.2 Setup of the Computational Study The computational results presented in this chapter are regarding the solution of MPC problems for networks of constrained LTI systems by well established distributed optimization methods. In the following, a particular network of coupled constrained LTI systems is introduced in Section 10.2.1, while in Section 10.2.2 it is demonstrated that the resulting MPC problem is amenable to standard distributed optimization methods. Furthermore, in Section 10.2.3, the details of the computational study are introduced, i.e. performance metrics, data illustration and data interpretation strategies.

10.2.1 System Setup We consider networks of nominal LTI systems as introduced in Section 3.2, where xi ∈ R2 and ui ∈ R for each i ∈ {1, . . . , M}. The system dynamics are defined as xi+

=

M X

Aij xj + Bi ui = ANi xNi + Bi ui

∀i ∈ {1, . . . , M} ,

(10.1)

j=1

where the matrices Aij ∈ R2×2 are non-zero only if j ∈ Ni , following Definition 3.1 for neighboring systems. Accordingly, for each i ∈ {1, . . . , M}, the matrix ANi ∈ R2×2|Ni | consists of the blocks Aij for all j ∈ Ni . If Aij is non-zero, it has an upper-triangular structure with equal non-zero entries, hence " # aij aij Aij = , aij > 0 . (10.2) 0 aij Moreover, for all i ∈ {1, . . . , M}, the matrix Bi ∈ R2 has the structure " # 0 Bi = , bi > 0 . bi

(10.3)

10.2 Setup of the Computational Study

91

The non-zero entries in these matrices, i.e. aij and bi , can be adjusted to modify stability and coupling properties of the network of systems. As for the strength of the dynamic coupling, the metric kANi k2 σci = ∀i ∈ {1, . . . , M} (10.4) kAii k2 is used. Furthermore, as for the individual local stability of system i , the metric σsi = λmax (Aii ) = aii

(10.5)

is used, i.e. the largest eigenvalue of the matrix Aii . Note that even if all Aii are stable, the global system might still be unstable due to coupling effects. Furthermore, Assumption 3.2 holds, hence the systems in the network can communicate in a bidirectional way whenever their dynamics are coupled in one way or the other. Finally, the local states and inputs are subject to polytopic constraints of the form xi ∈ Xi = {xi ∈ R2 |kxi k∞ ≤ 10} ∀i ∈ {1, . . . , M} ,

(10.6)

ui ∈ Ui = {ui ∈ R|kui k∞ ≤ 10} ∀i ∈ {1, . . . , M} .

(10.7)

10.2.2 MPC Setup The MPC control law is defined through a global MPC problem for the network of systems (10.1). This global MPC problem is of form (5.1), and its cost function is given as the sum over M local quadratic functions of the form xiT (N)Pf,i xi (N) +

N−1 X

xNT i (k)QNi xNi (k) + ui (k)T Ri ui (k) ,

(10.8)

k=0

where for each i ∈ {1, . . . , M}, the matrices Pf,i , QNi and Ri are chosen to be identity. The MPC prediction horizon is N = 10, and in the MPC problem the dynamics constraints as well as the local state and input constraints are imposed at every prediction step. Note that also the terminal state is constrained to lie in the nominal state constraint set, hence xi (N) ∈ Xi

∀i ∈ {1, . . . , M} .

(10.9)

Remark 10.1. The terminal cost and constraint chosen as above do not guarantee stability of the resulting closed-loop MPC controller. The focus of this chapter, however, is on computational aspects rather than stability. A way to design stabilizing terminal costs and constraints that fit the framework of this computational study is described in Chapter 7.

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The particular structure of the objective function and the constraints in the resulting MPC problem allows for a reformulation of the problem into the equivalent structured form (4.17). Written in this particular form, for each system i in {1, . . . , M}, the vector of global variables zi is composed of system i ’s input and state trajectories. Moreover, the vector of local variables yi is composed of system i ’s input trajectory, as well as the state trajectories of all systems in Ni . Given this copying scheme, all dynamics, state and input constraints can be formulated locally in the vector yi and constraint sets Zi . As pointed out in Chapter 4, optimization problems in form (4.17), and thus also our MPC problem as a special case, can be solved by standard distributed optimization methods, such as DDFG and ADMM.

10.2.3 Performance Evaluation and Interpretation In this section, the methodology to evaluate the performance of the distributed optimization methods under consideration is introduced. Furthermore, the strategy to interpret the measured performance results is explained in detail. 10.2.3.1 Performance Measurement and Data Illustration In all cases considered, the distributed optimization methods are run for a maximum number of 2000 iterations. In ADMM, the step size parameter is held constant at ρ = 120. The performance of the distributed optimization methods is measured as the number of iterations, until two combined conditions are satisfied. The first one is a relative condition on the primal residual kcol{1,...,M} (yi − Ei z)k∞ ≤ 10−3 , (10.10) kzk∞ and the second one is a relative condition on the difference of the primal objective value to the optimum P ∗ | M i=1 Ji (yi ) − Ji (yi )| ≤ 10−3 . (10.11) PM ∗ | i=1 Ji (yi )| As the reference primal optimum in (10.11), the centralized solution provided by the commercial solver CPLEX is used. It is evident that neither condition (10.10) nor condition (10.11) can be evaluated online and in a distributed way. However, the problem of imposing practical distributed stopping criteria is non-trivial and beyond the scope of this chapter. In the present study, the purpose of the stopping criteria introduced above is for a posteriori data analysis. For every test configuration considered, data sets on the performance of the distributed optimization methods for a number of randomly chosen initial conditions are generated.

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The findings are illustrated in box plots, where the upper and lower end of the box denote the 25th and 75th percentile of the data, the median is marked as a dot and the whiskers span the whole range of the data.

10.2.3.2 Interpretation of the Results The study is designed in the spirit of a sensitivity analysis. Hence, one specific test configuration is defined to investigate the impact of exactly one of the system aspects considered. In each of these configurations, the parameters related to the aspect under consideration are varied, while the remaining parameters are held at constant values. The interpretation of observed performance trends is often difficult. However, the norm of the dual optimizer kλ∗ (x0 )k2 can serve as a source for interpretation. Since the optimal dual variables represent prices for specific constraint violations, kλ∗ (x0 )k2 can be viewed as the price of consensus. I.e. the larger this price, the more iterations can be expected to attain a near optimal solution. This interpretation strategy is backed up by the upper bound on the number of iterations for DDFG, which is stated in (4.25). This mathematically rigorous bound is proportional to kλ∗ (x0 )k2 , and even though it is not tight, it can serve as a performance indicator, as the following computational results will show. Therefore, in the presentation of all computational results, the number of iterations to convergence for DDFG is illustrated alongside the price of consensus kλ∗ (x0 )k2 , which is furthermore used to intuitively connect problem aspects, such as system size or coupling strength, with observed performance results. Remark 10.2. A second parameter that affects the bound in (4.25) is the Lipschitz constant L of the gradient of the dual function. According to [Richter et al., 2011], in the problem formulation (4.17), L solely depends on the Hessian of the primal cost function and the constraints (4.17c) related to the copying of neighboring variables. Thus, if the primal cost function is constant as in the present study, L only varies with the structure of the coupling graph. Therefore, in this study, the size of L is only considered when investigating the impact of different coupling topologies on the performance, as done in Section 10.3.2.

10.3 Computational Results and Discussion In this section, the computational results on the performance sensitivity of DDFG and ADMM are presented. The test cases considered are as follows. In Section 10.3.1, the effect of coupling strength and local stability on the performance is investigated and in Section 10.3.2, the same is done for the coupling topology in the global network of systems.

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Furthermore, in Section 10.3.3, the influence of the initial state on the performance is investigated and in Section 10.3.4, the same is done for the network size.

10.3.1 Coupling Strength and Stability Scenario: To analyze the impact of the coupling strength, the dynamics (10.1) of a bidirectional chain of 10 systems are scaled such that for all i ∈ {1, . . . , M}, all σci take the same value in {0.2, 0.4, . . . , 2.0}, while σsi = 1. Likewise, to analyze the impact of local stability, the system dynamics are scaled such that for all i ∈ {1, . . . , M}, all σci = 1 and all σsi take the same value in {0.2, 0.4, . . . , 2.0}. For each combination (σci , σsi ), the resulting MPC problem is solved for 100 random initial conditions. Findings: Figure 10.1 illustrates the number of iterations to convergence for ADMM and DDFG, as well as the magnitude of kλ∗ (x0 )k2 for varying σci , Figure 10.2 illustrates the same quantities for varying σsi . The number of iterations to convergence increases with both of these quantities. ADMM

# of iterations

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# of iterations

DDFG 1500

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500

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Figure 10.1: Performance of ADMM and DDFG for the MPC problem resulting from a network of 10 coupled double-integrators with σsi = 1 and σci ∈ {0.2, 0.4, . . . , 2.0}. The median of kλ∗ (x0 )k2 is depicted in the lower plot.

Interpretation: For either varying σci or σsi , the number of iterations to convergence mainly for DDFG, but also for ADMM, is correlated with kλ∗ (x0 )k2 . This allows the following interpretation: • Coupling Strength: As the coupling strength increases, system i has more incentive to influence the state trajectories of systems in Ni . Therefore, the price of consensus

10.3 Computational Results and Discussion

95

ADMM

# of iterations

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# of iterations

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Figure 10.2: The analog to Figure 10.1 with σci = 1 and σsi ∈ {0.2, 0.4, . . . , 2.0}.

increases and so does the number of iterations. • Stability : As system i becomes more unstable, it becomes more sensitive to its neighbors’ trajectories. Thus, the incentive to influence these increases and, as a consequence, the price of consensus and the number of iterations to convergence also increase.

10.3.2 Coupling Topology Scenario: For 10 systems of form (10.1), the coupling graph is varied according to four topologies (see Fig. 10.3, where the edges in the graphs represent bidirectional coupling): (a) chain, (b) ring, (c) 2-hop ring, (d) 3-hop ring. For each topology, the MPC problem is solved for 100 random initial conditions. The coupling and stability parameters are σci = σsi = 1 for every i ∈ {1, . . . , M}. Findings: Figure 10.4 illustrates the number of iterations to convergence for ADMM and

(a) chain

(b) ring

(c) 2-hop ring

(d) 3-hop ring

Figure 10.3: Graph topologies under consideration.

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# of iterations

# of iterations

DDFG as well as the magnitude of kλ∗ (x0 )k2 . The number of iterations to convergence increases with the graph connectivity and so does the magnitude of kλ∗ (x0 )k2 . The Lipschitz constant L of the gradient of the dual function is found to be 1.5 for chain and ring, 2.5 for the two-hop ring and 3.5 for the three-hop ring. 1000

ADMM

750 500 250 0 1000

chai n

ring

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DDFG

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250

300

0

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Topology

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Figure 10.4: Performance of ADMM and DDFG for the MPC problem resulting from different coupling topologies, the median of the dual optimizer kλ∗ (x0 )k2 is depicted in the lower plot.

Interpretation: While, according to the consensus literature, an improvement in performance with increasing graph connectivity would be expected, our results indicate a decrease in performance, i.e. an increase in the number of iterations. One possible explanation for this behavior is the increase in the dimension of λ∗ (x0 ), which increases the value of kλ∗ (x0 )k2 , i.e. the price of consensus, if the magnitude of the entries of λ∗ (x0 ) remain comparable. A larger kλ∗ (x0 )k2 implies a larger bound (4.25), which suggests an increase in the number of iterations to convergence. Another possible explanation is the observed increase in the Lipschitz constant L of the gradient of the dual function, as the topology is changed from chain, over ring and 2-hop ring, to 3-hop ring. This increase also causes an increase in the bound (4.25).

10.3.3 Initial State Scenario: We consider a bidirectional chain of 10 systems of form (10.1) with σci = σsi = 1 for every i ∈ {1, . . . , M}. 100 random initial states on the boundary of the region of attraction are generated (by bisection). Subsequently, the MPC problem is solved for scaled versions of these initial conditions, whereas the scalar scaling factors lie in {0.2, 0.4, . . . , 1.0}. Findings: Figure 10.5 illustrates an increase in the number of iterations for both DDFG

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97

# of iterations

# of iterations

and ADMM with the initial state approaching the boundary of the feasible set. Furthermore, the spread in the number of iterations (visible by the larger span between the 25th percentile and the 75th percentile of the respective boxes in 10.5) becomes larger and the number of outliers increases. 750

ADMM

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scaling factor for initial state Figure 10.5: Performance of ADMM and DDFG for the MPC problem resulting from a chain of 10 coupled double-integrators with 100 random initial states which were scaled by a factor in {0.2, 0.4, . . . , 1.0}. The median of kλ∗ (x0 )k2 is depicted in the lower plot.

Interpretation: The number of iterations to convergence mainly for DDFG, but also for ADMM, is correlated with the magnitude of kλ∗ (x0 )k2 . This behavior is possibly caused by the fact that, as the initial state approaches the boundary of the feasible set, the range of possible control moves is decreased for every system i ∈ {1, . . . , M}. Such a decrease is likely to increase the price of consensus and thus also the number of iterations.

10.3.4 Number of Connected Systems Scenario: We consider a bidirectional chain of systems of form (10.1), with σci = σsi = 1 ∀i ∈ {1, . . . , M}. The number of connected elements takes values in {3, 50, 100, 150, 200, 250} and for each size, the MPC problem is solved for 10 random initial conditions. Findings: As depicted in Figure 10.6, the number of iterations to convergence increases with the size of the network. The increase is however significantly lower than linear and even seems to settle for a large number of elements. The value of kλ∗ (x0 )k2 correlates with the number of iterations to convergence. Interpretation: The slope of kλ∗ (x0 )k2 closely resembles a square root function. This

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# of subsystems Figure 10.6: Scaling behavior of distributed optimization in MPC for a chain of systems with σc = σs = 1 under ADMM and DDFG. The median of kλ∗ (x0 )k2 is depicted in the lower plot.

behavior can be explained by the fact that, as the length of the chain grows, the entries of the dual optimizer λ∗ (x0 ) do not change in magnitude, but the dimension of λ∗ (x0 ) grows. √ Consequently, the norm kλ∗ (x0 )k2 , grows with order O( M). From this observation and the existence of bound (4.25), we conjecture that the number of iterations, at least for √ DDFG, can be expected to grow according to order O( M) as well. Figure 10.6 suggests that the growth in the number of iterations may be even lower in practice.

10.4 Conclusion In this chapter, the cooperative distributed MPC specific performance of two well-established distributed optimization methods, i.e. DDFG and ADMM, was investigated in a systematic computational study. The results of the study, and the interpretation thereof, offer insight into the connection between fundamental properties of networked dynamic systems and the performance of distributed optimization methods in the corresponding MPC problems. The two attributes, to which the performance of the methods were the most sensitive, were coupling strength and local stability of the systems. The methods however still performed well for coupling strengths which would lead to a very small region of attraction under non-cooperative distributed MPC methods. Under cooperative distributed MPC in contrast, the region of attraction is in principle the same as the one obtained under a cen-

10.4 Conclusion

99

tralized approach. This highlights the fact that the range of applicability of cooperative distributed MPC is much larger than the one of its non-cooperative counterpart. The performance of both DDFG and ADMM was relatively insensitive to varying initial states. Even though initial states close to the boundary of the feasible set result in a slightly larger number of iterations, both methods were still operating relatively reliably. This feature is beneficial for MPC, where the initial state is the only problem parameter that changes during closed-loop operation. Thus, the finding suggests that a distributed optimization method will not undergo dramatic performance variations when used for cooperative distributed MPC in closed-loop operation. Another observation is that iterative distributed MPC scales well with the size of the problem. More specifically for DDFG, as the number of systems increases, the increase in √ the number of iterations is expected to grow by no more than by order O( M). This implies a great potential of iterative distributed MPC for large-scale networks of systems, where any centralized MPC approach eventually becomes intractable. Regarding the methods that were compared, ADMM consistently showed a lower number of iterations to convergence than DDFG. However, depending on the speed of communication in the distributed system, DDFG might still outperform ADMM, since it requires only one communication step per iteration, while ADMM requires two. Note, however, that there are no tuning parameters in DDFG, while the performance of ADMM might still be improved for a specific problem by the tuning parameter ρ.

11 Cooperative Distributed MPC for Wind Farms 11.1 Introduction Wind as a phenomenon is highly variable and hard to predict, which makes the available wind power highly volatile [Burton et al., 2008]. With an ever increasing amount of nominal wind power capacity installed in the grid, the fluctuations in power production that arise when all the available wind power is injected into the grid are difficult to mitigate by classical primary and secondary reserve strategies. Due to the resulting negative impact on stability and operability of power grids, Transmission System Operators (TSOs) are increasingly equipped with authority to restrict the operation of wind farms. In such a restricted operation, the wind farms produce less power than they could potentially harvest under the given wind conditions, i.e. they exhibit a power reserve. This reserve provides a degree of freedom for wind farm control, i.e. as to how the power production is allocated to individual wind turbines. This freedom can be utilized by a wind-farm controller in various ways, e.g. to reduce energy losses in transmission lines [de Almeida et al., 2006] or to reduce the number of farm-wide turbine startups and shutdowns [Moyano and Lopes, 2009]. In this chapter, the power reserve is utilized for the reduction of structural fatigue, which is the accumulation of mechanical load in the turbine components, in particular tower, blades and shaft. The fatigue of long and slender structural components of the wind turbines, i.e. blades, towers and shafts, is typically one of the wind turbine design drivers. That means that the controller design, which aims at reduction of fatigue in those components, can lead to an increase in the wind turbine operating lifetime and thus, effectively, lower the price of the produced energy. Different fatigue reduction strategies, ranging from wind turbine to wind farm level, can be found in the literature. These strategies aim at different phenomena and thus operate in different time scales. At wind turbine control level, a controller with fast sampling rate is typically used to damp the oscillations in structural components at natural frequencies, see e.g. [van Engelen et al., 2007], and on multiples of the rotor frequency, see e.g. [Bossanyi, 2003]. A design of an optimal wind turbine controller based on a spectral

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description of fatigue is proposed in [Hammerum et al., 2007]. At wind farm controller level, a controller is usually designed as a supervisor of wind turbine controllers and therefore has a slower sampling rate and does not have direct access to wind turbine actuators (blade pitch angles and generator torque). However, the wind farm controller design can also be used to reduce fatigue loads of the turbine structure. One of the first contributions that aims at fatigue reduction at wind farm level is found in [Spruce, 1993], where the lifetime profit of a wind farm is optimized by trading off power production against load accumulation at a slow time scale of several minutes. At a faster time scale, a feedback at a 1 Hz sample rate has been suggested in [Spudi´c, 2012], to reduce the farm-wide load accumulation by efficient deployment of local power reserves, which arise due to TSO-enforced power tracking requirements. In order to achieve this fatigue reduction at a fast time scale, the work presented in [Spudi´c, 2012] suggests the use of MPC. The main strength of this methodology lies in its capability to capture practical performance objectives, such as fatigue reduction, as well as plant constraints, such as actuator ranges, both of which is highly useful for the control of wind farms. While the MPC scheme proposed in [Spudi´c, 2012] is based on centralized coordination of the turbines in the farm, it is appealing to extend the methodology to distributed coordination, since, at least at a fast time scale, the state-of-the-art in currently operating wind farms is for the turbines to act largely autonomously. In such a setup, the resulting wind farm controller does not exhibit a single point of failure and can adapt easily to outages of single turbine units. While an unconstrained distributed wind farm controller for fast-scale fatigue reduction has been proposed in [Madjidian et al., 2011], we proposed a distributed MPC controller for the task. Such a controller is additionally capable of taking turbine operational constraints into account, which is expected to result in a more reliable wind farm operation. Since farm-wide fatigue reduction, in particular by reallocating power production among all the turbines in the farm, is an inherently cooperative task, the use of a cooperative distributed MPC scheme, based on distributed optimization, is chosen.

Main Contribution As a first contribution of this chapter, and as an extension of the work by [Spudi´c, 2012], a cooperative distributed MPC approach for fatigue reduction in wind farms is proposed. The fatigue reduction objective, as well as the operational constraints, are captured in a finitehorizon optimal control problem, i.e. an MPC problem, and it is shown that this problem is structured such that standard distributed optimization methods are applicable. As a second contribution, a simulation study is presented, which highlights various computational and

11.2 Control of Wind Farms

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operational aspects of this controller. In particular, the effect of suboptimality in distributed optimization on the fatigue reduction performance is studied, and the impact of different operational conditions on the convergence speed of the distributed optimization methods is investigated. Overall, the simulation results suggest that cooperative distributed MPC is a computationally viable approach for the application under consideration. Outline The chapter is structured as follows: In Section 11.2, the main dynamic phenomena observed in wind farms, as well as common control strategies for wind farms are discussed. In Section 11.3, as a first contribution of this chapter, the application of distributed optimization based cooperative distributed MPC to wind farms is discussed. Section 11.4 contains a detailed explanation of the simulation methodology and setup, as well as the wind scenarios under consideration. Section 11.5 contains the simulation results, which form the main contribution of this chapter. These results are then discussed in Section 11.6 and the chapter is concluded in Section 11.7.

11.2 Control of Wind Farms A wind farm controller governs the active power references of the wind turbines in the farm. Traditionally, this task is of supervisory nature – in cases of abnormal operating states it restricts power production of individual turbines. However, with the new control requirements imposed by grid codes, the wind farm controller needs to take over a new task – the coordination of power production of the turbines. This section aims at giving an overview of the factors relevant to the design and operation of such wind farm control systems. In particular, in Section 11.2.1 and Section 11.2.2, the operational modes of wind farms and turbines are described. In Section 11.2.3, the predominant dynamic phenomena occurrent in wind farms are introduced and a common control architecture to capture them is presented. Finally, in Section 11.2.4, the wind farm model used in the controller under consideration is presented.

11.2.1 Operating Modes for Wind Farms Given the fact that wind is a costless fuel, the most intuitive mode to operate a wind farm is at maximum power output. Since wind is a highly variable phenomenon however, this output fluctuates considerably, which imposes disturbances in the power grid, which need to be compensated for. To this end, primary and secondary reserves need to be assigned, which

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11 Cooperative Distributed MPC for Wind Farms

Delta Production Constraint

Power Gradient Constraint

Power

Absolute Production Constraint

Available Wind Farm Power Wind Farm Power Reference

Time

Production in % of the prod. without freq. ctrl.

is not economically beneficial if the grid-wide share of wind power is large. Furthermore, the inability of the TSOs to influence wind farm active power production can be problematic, especially during outages, congestions or frequency instabilities. For these reasons, modern grid codes typically require wind farm controllers to be able to track prescribed active power profiles, given the wind conditions allow so. As an example, the operating modes required in the Danish grid are depicted in Fig. 11.1. The wind farm power reference, Pwf,ref , (full line on the left plot) is defined relative to the estimated wind farm available power (dotted line on the left plot), or as a function of grid frequency (right plot). These operating modes enable TSOs to limit the wind farm production (or the variability thereof), utilize wind farms as a secondary reserve, or to provide frequency support. Frequency Control

100

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Figure 11.1: Various TSO-imposed operational modes of wind farms [Elkraft System and ELTRA, 2014].

11.2.2 Operation of Wind Turbines as a Part of Wind Farms In order to accomplish the grid code requirements, the primary task of the wind farm controller is to coordinate the power production of individual wind turbines. Hence, it issues power references Pref to individual turbines, which track these references locally using wind turbine controllers. Naturally, not every power reference can be tracked by a wind turbine controller. In particular, the power reference needs to be lower than the power that can be produced by the wind turbine at a given moment. The power that can be produced is the minimum between available wind power, Pav , and nominal wind turbine power, Pnom , see Fig. 11.2. The classical wind turbine controller design, described in [Bossanyi, 2003] and [Spudi´c, 2012], which relies on torque and blade pitch control at a millisecond time scale, ensures excellent tracking performance. In cases when the issued power reference is larger than the currently available power, the wind turbine controller operates in such way that the wind turbine power output is maximized. Therefore, turbine operation with the sole objective of power maximization is equivalent to keeping Pref = Pnom .

11.2 Control of Wind Farms

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Figure 11.2: The available power of a typical wind turbine as a function of the wind speed.

Remark 11.1. One should note that the available power curve depicted in Fig. 11.2 is only valid in a stationary environment. During operation in an inherently dynamic environment, the estimation of the available power is a nontrivial task, which is closely related to the concept of effective wind speed estimation. The issues of this concept, which are described in e.g. [Soltani et al., 2013], lie in the fact that practical wind speed measurements are heavily disturbed, point-based and thus not representable for the whole rotor wind field. Furthermore, the available power directly depends on the dynamic operation of the turbine itself. Remark 11.2. Note that the knee of the available power curve, i.e. the point where Pref = Pav , is not a desirable operating regime for a wind turbine. This regime is characterized by the highest thrust force (for given power reference) and the starting of the pitch mechanism, which is accompanied by undesirable transients and large fatigue accumulation.

11.2.3 Dynamics Relevant to Wind Farms and Wind Farm Control System Architecture In order to model and control the dynamics of wind farms, it is common to introduce a separation of relevant physical phenomena. Typically, a time scale separation in slow quasistatic and fast dynamic time scale is used. A wind farm controller architecture based on such a time scale separation is depicted in Fig. 11.3. It is a hierarchically structured scheme, in which different controllers, operating at different sampling rates, deliver set-points to each other. The components of this scheme, as well as the underlying physical phenomena relevant to their operation, are introduced in the following.

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11 Cooperative Distributed MPC for Wind Farms Wind farm Controller Wind farm Slow-scale controller

Fast-scale controller

Wind turbine controller 1

Wind turbine controller M

Figure 11.3: Wind farm controller architecture. P¯ref denotes the turbine power references generated by the slow-scale controller and Pref denotes the same quantities, modified by the fast-scale controller.

At a quasi-static time scale, the evolution of the mean wind speed, which is typically captured in 10-minute time series, is dominant. Of particular interest is the wake effect, which couples downwind turbines to upwind ones. The wind turbines enveloped by wakes of upwind turbines experience different operating conditions, typically characterized by reduced available power and increased loads. By an appropriate distribution of power production in the wind farm those effects can be mitigated, which motivates the design of a slowscale controller. Such a slow-scale controller can for instance be designed to reduce fatigue [Spruce, 1993], or to provide maximum power reserve in the wind farm [Spudi´c, 2012]. It can also be designed with other slow-scale or stationary phenomena and objectives in mind, e.g. the reduction of transmission losses in power lines [Laks et al., 2011]. On the other end of the dynamic range lies the local interaction of the turbulent wind with the turbines, which is handled by individual wind turbine controllers, which should ensure safe operation of the individual turbines. In this regard, the oscillatory behavior of the turbine components, i.e. rotor, shaft and tower, are relevant. The wind turbine controller ensures as accurate tracking of the wind turbine power reference as possible, given the current operating conditions. This objective can be extended by fatigue reduction techniques, which aim at influencing the operation of the turbines in the vicinity of the natural frequencies of their components. These natural frequencies, following the methodology of modal analysis, typically lie in a range between 0.3 Hz and 2.9 Hz [Jonkman et al., 2009]. Since there is a large gap between the two time scales introduced above, there is additional potential for fatigue reduction at the time scale of seconds, as described in detail in [Spudi´c, 2012]. In particular, at this time scale, the fluctuations of the rotor effective wind speeds at individual turbines are dominant. The rotor effective speed is an artificial quantity describing the influence of the wind field over the entire rotor. It can be used to characterize the aerodynamic torque, as well as the rotor thrust that causes tower bending. Thus, by adapting the turbine power output reference to wind speed fluctuations, tower bending load cycles, which cause fatigue, can be considerably reduced. A controller that accomplishes this task is in this chapter referred to as a fast-scale wind farm controller and its distributed realization is the main goal of this chapter.

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Remark 11.3. Note that at the time scale of seconds, wake effects are not relevant and wind turbine dynamics can be considered decoupled. This is due to the fact that wakes propagate with the mean wind speed. For typical turbine spacings in wind farms, the wind propagation time between two turbines is in the range of several tens of seconds, which is too long for a fast-scale wind farm controller to be relevant.

11.2.4 Control Oriented Wind Farm Modeling As the relevant phenomena for fast-scale fatigue reducing wind turbine controllers are the second-scale fluctuations of the rotor effective wind speeds, a control oriented model is expected to capture the response of wind turbine loads to such fluctuations. Moreover, since the control input is given by the turbine power reference, the effect of changes in that reference should be captured as well. At the second-scale, the dominant turbine dynamics relate to wind turbine controller actions, pitch angle, rotor inertia and rotor effective wind speed fluctuations. Therefore, as suggested in [Spudi´c, 2012], the state vector of each of the M individual turbines in the farm is defined as h iT xi = βi ωr,i ωg,i vi ∈ R4 ∀i ∈ {1, . . . , M} , (11.1) where βi is the pitch angle, ωr,i is the angular speed of the rotor, ωg,i is the filtered angular speed of the generator (which is used in the wind turbine controller) and vi is the rotor effective wind speed. The control input of a turbine is the local power reference ui = Pref,i ∈ R ∀i ∈ {1, . . . , M} ,

(11.2)

and the output quantity is chosen as yi = Ft,i ∈ R ∀i ∈ {1, . . . , M} ,

(11.3)

where Ft,i denotes the rotor thrust force. The dynamics that govern these states, inputs and outputs are fundamentally nonlinear. However, it is shown in [Biegel et al., 2013] and [Spudi´c, 2012] that a linear model of the form xi+ = Ai xi + Bi ui yi = Ci xi

∀i ∈ {1, . . . , M} ,

∀i ∈ {1, . . . , M} ,

(11.4a) (11.4b)

describes the dynamics of the process sufficiently well for feedback controller design. In (11.4), the matrices Ai , Bi and Ci are of appropriate dimension and the sample rate is 1 s. This model is obtained in a phenomenological fashion from the data which is typically known about a wind turbine. For details on the derivation and verification of the discrete-time linear model, the reader is referred to [Spudi´c et al., 2011].

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The wind farm power output is defined as the sum of the individual turbine outputs, hence M X Pwf = Pi , (11.5) i=1

where Pi is the power output of the i -th wind turbine. Remark 11.4. The dynamics of the wind farm power tracking is extremely fast, since it are accomplished by frequency converter control. Therefore, we do not model any power dynamics at the second time scale and assume that Pi = Pref,i . Remark 11.5. Note that in (11.4), the rotor equivalent wind speed is held constant, hence vi+ = vi . This is done in spite of the fact that in the literature a number of contributions on wind prediction models can be found. The accuracy of such models on a second time scale, at least in a deterministic framework, is however not very satisfactory. Thus, considering the relatively short prediction horizon of 4 s used in the simulations, the use of a model such as (11.4) is entirely justified, see e.g. [Knudsen et al., 2011].

11.2.5 Control Objective and Constraints The objective of a fast-scale wind farm controller is (i) to achieve accurate power output tracking and (ii) to reduce structural loads. For this purpose, a quadratic finite-horizon performance objective function of the form Vi (xi , ui ) = (Ci xi,N − y¯i )T Pf,i (Ci xi,N − y¯i )+ +

N−1 X

(Ci xi,k − y¯i )T Qi (Ci xi,k − y¯i ) + (ui,k − u¯i )T Ri (ui,k − u¯i ) ∀i ∈ {1, . . . , M} ,

k=0

(11.6) is minimized, in which y¯i and u¯i are output and input references delivered to turbine i by T T T T T the slow-scale controller and xi = [xi,0 , . . . , xi,N ] and ui = [ui,0 , . . . , ui,N−1 ]T are vectors containing the finite-horizon state and input trajectories. Furthermore, Pf,i ∈ R, Qi ∈ R and Ri ∈ R are positive scalars and N denotes the prediction and control horizon of the finite-time optimal control problem, in this case set to N = 4 steps. Note that we chose this relatively short horizon, since we did not observe a further performance improvement for longer horizons and furthermore, the rotor equivalent wind speed can generally not be predicted far ahead in time. The objective function (11.6) aims at reducing the fatigue in blades, tower and shaft. As for blades and tower, it has been shown by aeroelastic simulations in [Spudi´c, 2012], that a reduction of fatigue damage can be achieved by reducing the second-scale oscillations of the

11.3 Cooperative Distributed MPC for the Control of Wind Farms

109

rotor thrust. This reduction is aimed to be achieved by the terms in (11.6) penalizing the thrust force Ft,i = yi = Ci xi . In particular, these terms give the controller incentive to keep the thrust force close to the output reference, which is expected to reduce the magnitude in possible oscillations. As for the shaft, note that any change in the turbine power reference causes a fast change in the generator torque, which is transmitted to the shaft directly. Therefore, a shaft load reduction is aimed to be achieved by the terms in (11.6) penalizing the power reference Pref,i = ui . These terms give the controller incentive to keep the power output close to the input reference, which is expected to reduce the magnitude of the power output oscillation and thus of the shaft load. The wind farm operation is subject to a number of constraints. The constraints on the power reference of each individual wind turbine i can be written as Pi,min ≤ ui ≤ min {Pi,nom , Pi,av } ∀i ∈ {1, . . . , M} .

(11.7)

In (11.7), the left-hand constraint is the minimum power at which turbine i is operable and the right-hand constraint upper bounds the output power by the minimum among the nominal power Pi,nom and the available power Pi,av of turbine i , which depends on the momentary wind speed. In this work, the available power estimate is based on the rotor effective wind speed estimation, which is done via an extended Kalman filter described in detail in [Knudsen et al., 2011]. As an additional constraint, the wind farm tracking requirement is imposed as M X ui = Pwf,ref , (11.8) i=1

which, considering Remark 11.4, ensures that at each point in time the power output of the farm is equal to the reference Pwf,ref imposed by the TSO. Remark 11.6. Note that the right-hand side inequality in constraint (11.7) is implemented as two separate constraints of the form ui ≤ Pi,nom and ui ≤ Pi,av in this chapter. The latter constraint is furthermore implemented in a soft way, hence it is relaxed as an additional penalty term in the objective function. This is meaningful due to the following reasons: (i) The estimate of Pi,av is highly uncertain, (ii) infeasibility of control inputs is avoided.

11.3 Cooperative Distributed MPC for the Control of Wind Farms This section discusses the application of cooperative distributed MPC to fast-scale control of wind farms, using the model, objective function and constraints introduced in the previous

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section. In Section 11.3.1, the formulation of the global MPC problem and the corresponding MPC control law are introduced. Thereafter, in Section 11.3.2, it is demonstrated that this problem exhibits a structure, which is amenable to standard distributed optimization methods. Before going into the details of distributed MPC, note that from a distributed control perspective the farm-wide power output constraint (11.8) is inconvenient, since it couples the inputs of all wind turbines in the farm at once. In order to obtain a more distributed structure in the MPC problem, we define a new set of input variables as uˆ1 = u1

(11.9a)

uˆ2 = uˆ1 + u2 .. .

(11.9b)

uˆM = uˆM−1 + uM ,

(11.9c)

as suggested in [Madjidian et al., 2011]. The dynamics (11.4), as well as the objective function (11.6) and the constraints (11.7) can be equivalently reformulated in the new input vector. Moreover, constraint (11.8) can be written as a local constraint of turbine M, hence uˆM = Pwf,ref . Note that the local dynamics (11.4), as well as the local objective functions (11.6) and input constraints of all turbines with indices i ≥ 2, exhibit a coupling to the turbine with the previous index, hence they depend on both uˆi and uˆi−1 .

11.3.1 Formulation of the distributed MPC Control Law The MPC control law is defined through a finite-horizon constrained optimal control problem, i.e. the MPC problem, which contains the current state as a parameter and is solved at every point in time. In the cooperative distributed scheme proposed in this chapter, one global MPC problem is formulated for the wind farm, which is solved by distributed optimization. The MPC problem is formulated as the minimization of the cost function (11.6), subject to the dynamics (11.4) as well as the constraints (11.7) and (11.8), hence M  X ∗ V (x) = min (Ci xi,N − y¯i )T Pf,i (Ci xi,N − y¯i )+ u ˆ1 ...,ˆ uM

+

N−1 X

i=1 T

T

(Ci xi,k − y¯i ) Qi (Ci xi,k − y¯i ) + (uˆi,k − uˆi−1,k − u¯i ) Ri (uˆi,k − uˆi−1,k

 − u¯i )

k=0

(11.10a) s.t. xi,0 = xi

∀i ∈ {1, . . , M} ,

(11.10b)

xi,k+1 = Ai xi,k + Bi (uˆi,k − uˆi−1,k ) ∀i ∈ {1, . . , M} ∀k ∈ {0, . . , N − 1} , (11.10c)

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111

Pi,min ≤ uˆi,k − uˆi−1,k ≤ min{Pi,nom , Pi,av } ∀i ∈ {1, . . , M} ∀k ∈ {0, . . , N − 1} , (11.10d) uˆM = Pwf,ref .

(11.10e)

In problem (11.10), we define uˆ0,k = 0 for all k ∈ {0, . . . , N − 1} for the sake of iT h T denotes the current state of the notational simplicity. Furthermore, x = x1T . . . xM wind farm. Note that problem (11.10) is a convex quadratic program, for which there are efficient solution methods. The optimization variables in problem (11.10) are the control T T ]T , which uniquely define state sequences , . . . , uˆi,N−1 inputs in the sequences u ˆi = [uˆi,0 T T T xi = [xi,0 , . . . , xi,N ] by the dynamics constraint (11.10c). The optimal input sequence in ∗T ∗T . . . uˆi,N−1 ]T and the MPC control law the sense of problem (11.10) is denoted as u ˆ∗i = [uˆi,0 for the wind farm is defined as the first element of this optimal input sequence, hence h iT ∗ ∗T ∗T κMP C (x) = uˆ0 = uˆ1,0 . . . uˆM,0 . (11.11) Remark 11.7. Stability in MPC controllers is usually achieved by an appropriate design of a terminal cost as well as a terminal constraint, see [Mayne et al., 2000] for details. Problem (11.10) has no terminal constraint and in the simulations presented in Section 11.5, the terminal cost is set to Pf ,i = Qi . We are aware that this choice does not yield a closed-loop stability guarantee. Stability was however never observed to be an issue in the simulations.

11.3.2 Solution of the MPC Problem by Distributed Optimization Techniques Looking at problem (11.10), it becomes apparent that the problem is highly structured and can be decomposed into M subproblems, where the first one depends on variables of turbine 1 only, and the remaining ones depend on variables of turbines i and i − 1. Wellestablished distributed optimization methods, many of which are described in [Bertsekas and Tsitsiklis, 1989], can be applied to such problems. To elaborate on this assertion, consider an equivalent reformulation of problem (11.10), given as min

(xi ,ζi ) ∀i∈{1,...,M}, z

s.t.

M X

Vˆi (xi , ζi )

(11.12a)

i=1

(xi , ζi ) ∈ Yi (xi ) ∀i ∈ {1, . . . , M} ζi = Ei z

∀i ∈ {1, . . . , M} .

(11.12b) (11.12c)

In problem (11.12), ζ1 = ˆ v1,1 for turbine 1 and ζi = [ˆ vTi,i , ˆ vTi−1,i ]T for any turbine i ∈ {2, . . . , M}, where ˆ vj,i is a copy of variable u ˆj stored at turbine i . The local objective

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functions Vˆi (xi , ζi ) can then be written in local variables of turbine i , hence Vˆi (xi , ζi ) = (Ci xi,N − y¯i )T Pf ,i (Ci xi,N − y¯i )+ +

N−1 X

(Ci xi,k − y¯i )T Qi (Ci xi,k − y¯i ) + (ˆ vi,i,k − vˆi−1,i,k − u¯i )T Ri (ˆ vi,i,k − vˆi−1,i,k − u¯i ) ,

k=0

(11.13) where vˆi,j,k denotes the copy of uˆi,k at turbine j. Similarly, the local constraint sets Yi (xi ) are defined in local variables as  Yi (xi ) = (xi , ζi )|xi,0 = xi , xi,k+1 = Ai xi,k + Bi (ˆ vi,i,k − vˆi−1,i,k ) ∀k ∈ {0, . . . , N − 1} , Pi,min ≤ vˆi,i,k − vˆi−1,i,k ≤ min{Pi,nom , Pi,av } ∀k ∈ {0, . . . , N − 1} , (11.14) where vˆ0,i,k = 0 for all k ∈ {0, . . . , N − 1} and where in the set YM (xM ), the constraint vˆM,M = Pwf,ref is added. The element that couples the local objective functions and constraint sets defined above, is constraint (11.12c) and in particular the matrices Ei , which are chosen to have entries in {0, 1}, with exactly one entry per row equal to 1. Through T T ] , the matrices Ei make sure that for any of vector the consensus vector z = [z1T , . . . , zM T T T ζ = [ζ1 , . . . , ζM ] fulfilling constraint (11.12c), it holds that zi = ˆ vi,i = ˆ vi,i+1 ∀i ∈ {1, . . . , M − 1} .

(11.15)

Note that in form (11.12), as explained in detail in Section 4.4.1, the wind farm control problem exhibits a structure which allows for the application of standard distributed optimization methods. In the following simulation study, we in particular apply the dual decomposition based fast gradient method (DDFG) [Nesterov, 2004], as detailed in Section 4.4.2, and the alternating direction method of multipliers (ADMM) [Boyd et al., 2010], as detailed in Section 4.4.3. These methods do by no means exhaustively cover the range of existing distributed optimization techniques. Nevertheless, they are regarded as an interesting choice, since ADMM is known to be practically performant (see e.g. [Summers and Lygeros, 2012], [Kraning et al., 2013]) and DDFG is equipped with a favorable theoretical guarantee on the convergence rate. However, other distributed optimization methods, such as the subgradient method or the Jacobi method, both explained in [Bertsekas and Tsitsiklis, 1989], could also be used. Remark 11.8. Note that for the application of DDFG, problem (11.12) is made strongly convex by mild smoothing, in order for the dual function to be guaranteed differentiable.

11.4 Setup of the Simulation Study

113

11.4 Setup of the Simulation Study The performance of the controller introduced in the previous section is evaluated in simulations. This section contains a description of the components relevant to these simulations. In particular, the simulation scenarios, as well as the performance evaluation methodology, are described in Section 11.4.1, and the simulation setup is presented in Section 11.4.2.

11.4.1 Simulation Scenarios and Performance Evaluation We consider a wind farm of eight turbines and four wind scenarios of 600 s length each. The scenarios correspond to mean wind speeds of 8 m/s, 11 m/s, 14 m/s and 17 m/s, which cover the most common operating range of wind turbines. Note that the wind farm power references are held constant during these 10-minute scenarios. Furthermore, the wind farm power reference is set to 90% of the average available power at the respective mean wind speed. The wind speeds of 8 and 11 m/s fall below the speed, at which the turbines can produce nominal power. Thus, in these simulations, it is expected that the wind farm power reference will not be attained at all time. Contrary, at wind speeds of 14 and 17 m/s, the available power from the wind is larger than the nominal power. Therefore, the wind farm power tracking constraint should be satisfied most of the time. For all four wind scenarios, two different MPC setups are tested. The prediction horizon of both of these setups is N = 4, and both have a control update period of 1 s. The first setup is given by the original formulation as stated in (11.10), and the second is based on (11.10), where the farm power output constraint (11.10e) has been softened. The rationale behind the softening of the power output constraint is the fact that the TSO requirements on the power output of a wind farm do not typically require strict tracking. Due to large grid inertia, small deviations in the farm-wide power output are acceptable. Therefore, by the soft constrained MPC formulation, we try to investigate whether a further reduction in fatigue might be achievable by not trying to fulfill the power output requirement by all means. The softening is done by introducing a new optimization variable , replacing (11.10e) by the constraints uˆM ≤ Pw f ,r ef +  ,  ≥ 0 ,

(11.16)

and adding a term c with c > 0 to the objective function (11.10a). In terms of controller operation, the main focus is on the performance of the distributed optimization methods, hence the number of iterations required by ADMM and DDFG to solve the MPC problem to a satisfying accuracy. In particular, in every simulation step, a total of 2000 ADMM or DDFG iterations were run and the solution sequences stored. A posteriori, the performance of the respective distributed optimization method was evaluated,

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11 Cooperative Distributed MPC for Wind Farms

which was measured as the number of iterations until the stopping criterion kz − z ∗ k∞ ≤ 10−3 had been satisfied for 5 consecutive iterations. Note that for this analysis, the reference optimal solution z ∗ was found by the commercial solver CPLEX by IBM. This type of analysis aims at measuring the performance of distributed optimization in cooperative MPC for wind farms, it does not suggest a stopping criterion which could be used in practice. Practical stopping criteria for distributed MPC are difficult, since most standard stopping criteria require central coordination, which can not be provided in the distributed setup considered. A simple practical remedy to this is to stop the optimization procedure after a fixed number of iterations. Therefore, in Section 11.5.3, we investigate the closed-loop performance of the approach under various constant iteration numbers. In terms of controller performance, we are interested in two main aspects: (i) The accuracy of the power output tracking and (ii) farm-wide fatigue reduction. The tracking accuracy is measured as the standard deviation from the TSO-imposed output reference over the duration of the simulation. The fatigue is measured as the damage equivalent load (DEL) over the duration of the simulation. The DEL is a measure of the accumulation of load over a particular load history. In particular, it is defined as the magnitude of a sinusoidal load of constant frequency f , which produces the same damage as the original load signal. To obtain the DEL for a particular simulation, the load signals of the turbines in the farm are recorded and then analyzed for load cycles by means of the rainflow-counting algorithm (for details see [Matsuishi and Endo, 1968]). The load cycles are the main cause of fatigue and can be transformed into a DEL by the Palmgren-Miner rule. For details on wind turbine fatigue analysis, the reader is referred to [Sutherland, 1999].

11.4.2 Simulation Setup All simulations are carried out in MATLAB Simulink, where the wind-farm simulation tool SimWindFarm [Grunnet et al., 2010] is used. This tool was developed to be a benchmark simulation setup for rapid testing of fast-scale wind farm controllers. As such it is built to satisfactorily reflect the inherently nonlinear dynamics of a wind farm, while still using models of relatively low complexity. More sophisticated wind farm simulation tools such as SOWFA [Churchfield and Lee, 2012], can not be run without large-scale computation infrastructure. For distributed optimization, the message passing interface (MPI) was used for parallel code execution. For a summary on the choice of the tuning parameters in the MPC objective function (11.10a), the reader is referred to [Spudi´c, 2012]. In ADMM, applied as described in Algorithm 4.2, the parameter choice ρ = 1 is made. In the DDFG method, applied as described in Algorithm 4.1, the Lipschitz constant L of the dual function is computed as

11.5 Simulation Results

115

described in [Richter et al., 2011]. In the distributed optimization methods, the initial value for each dual variable is 0. For the solution of the local optimization problems in the distributed optimization method, the commercial solver CPLEX is used. Remark 11.9. SimWindFarm is designed to simulate turbulent wind representative for the wind farm, which makes this tool suitable for evaluation of fatigue loads. However, SimWindFarm does not support simulations of extreme events that define extreme loads in operational scenarios. Therefore, the influence of the controller on extreme loads can not be evaluated. Note, however, that the design principle described in Section 11.2.5 might easily lead to reduction of extreme loads in operational scenarios.

11.5 Simulation Results This section contains the wind-farm simulation results. In Section 11.5.1, the closed-loop performance, in terms of DEL, of soft and hard constrained MPC is presented and compared to the performance of two simpler control setups. In Section 11.5.2, the performance of both distributed optimization methods, i.e. ADMM and DDFG, is documented in terms of required iterations to convergence at every time step. Finally, in Section 11.5.3, results regarding constant numbers of distributed optimization iterations on the controller performance are presented.

11.5.1 Closed-Loop Performance of Distributed MPC Controllers In this section, the performances of four different controller setups are compared: (i) Constant references: The turbine output power references provided by the slow-scale wind farm controller are not changed by the fast-scale wind farm controller. (ii) PI controller: The TSO-imposed power reference is directly tracked by a fast-scale PI controller. The output of the PI controller is proportionally divided among the individual turbines according to the locally available powers. In order to prevent the integral control from causing actuator saturation in case of a lack of available power, anti-windup is used. For details on the controller design, see [Spudi´c, 2012]. (iii) MPC, hard constraints: The MPC control law defined in (11.11). (iv) MPC, soft constraints: The MPC control law defined in (11.11), where the farm power output constraint (11.10e) is softened. Remark 11.10. Note that for the closed-loop performance considerations presented in this section, the choice of the optimization method in MPC does not matter, since all methods solve the same MPC problem. Thus, the closed-loop performance in terms of DEL would

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11 Cooperative Distributed MPC for Wind Farms

be equal for ADMM, DDFG, or any centralized solution method, provided the methods are allowed to converge to optimality. 2000

Constant references PI controller Hard constrained MPC Soft constrained MPC

1500

Standard deviation of wind farm power 1000 tracking error [kW] 500 0 WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

150

Average damage equivalent loads on the tower base [MNm]

100

50

0

2.5 2

Average damage equivalent loads on the shaft [MNm]

1.5 1 0.5 0

Figure 11.4: Comparison of the performances of the controller setups under consideration.

In Fig. 11.4, an overview of the performances of the controller setups (i) to (iv) is given. Furthermore, in Fig. 11.5, the power output trajectories of the wind farm, as well as of turbine 1, are depicted for the controller setups (i), (ii) and (iii), for a wind speed of 11 m/s. These results are considered valuable for the reader’s understanding of the functionality of the system under the different controllers, and are briefly discussed in Sections 11.5.1.1 and 11.5.1.2. The main contributions of this chapter, which are related to the distributed MPC aspect, are presented in Section 11.5.2 and Section 11.5.3. 11.5.1.1 Comparison of distributed MPC vs. constant references and PI setups. In this section, the performances of the four controllers under consideration are compared for each of the four simulation scenarios. • Wind Speed (WS) 8 m/s. The constant references control setup exhibits some tracking error due to the fact that the slow-scale references P¯i,ref are larger than the

11.5 Simulation Results

117 Constant references PI controller Distr. MPC (hard constr.) Available power estimate

Wind farm power [MW]

45

40

35

30

25

Wind turbine 1 power reference [MW]

8 7 6 5 4 3 2 0

100

200

300

400

500

600

Time [s]

Figure 11.5: Comparison of the trajectories corresponding to the controller setups (i), (ii) and (iii) for a wind speed of 11 m/s.

locally available powers for some turbines and some periods of time. Both PI and MPC controllers remove this tracking error very efficiently by redistributing the power production at the fast-scale level. The MPC controller shows an improvement in terms of the DEL on the tower, which is reduced by approximately 20% percent compared to the PI controller, and 13% percent compared to the constant references setup. The DEL on the shaft increased for both the PI controller and the MPC controller, which is a natural consequence of the power deviations introduced to improve tracking, since they are accomplished by changing the generator torque. • WS 11 m/s. In this scenario, neither the PI controller nor the MPC controller were able to remove the tracking error completely, which suggests that the wind farm power reference was larger than the available wind farm power in some parts of the simulation. This is confirmed by Fig. 11.5, which depicts the trajectories of the wind

118

11 Cooperative Distributed MPC for Wind Farms farm power output for the 11 m/s wind speed scenario, as well as the power reference on wind turbine 1 (as an example). It can be seen that approximately from second 200 to second 350, the available power is lower than the wind farm power reference. In terms of the power tracking standard deviation, PI and MPC deliver approximately the same performance. However, compared to PI, the MPC controller exhibits an improvement by 19% in the tower DEL and by 34% in the shaft DEL. • WS 14 m/s. In this simulation scenario, the available power from the wind is high and the wind turbines are operating relatively far below the available power constraint. The PI and the MPC controller are both improving the wind farm power tracking compared to the constant references setup. The MPC controller accomplishes very similar tracking performance as the PI controller, but with 4% lower tower DEL and 32% lower shaft DEL. • WS 17 m/s. Also in this scenario, the available power is very high and all controllers deliver virtually perfect tracking. For the high wind speeds, no major load reduction can be achieved, since very large deviations of the power references would be required for a reduction, see [Spudi´c, 2012] for a detailed explanation.

11.5.1.2 Comparison of hard constrained and soft constrained MPC. As a side-contribution of this chapter, we compare the MPC formulation in (11.10) with a similar formulation containing a softened farm power output tracking constraint as introduced in (11.16). The comparison is made in order to explore whether the freedom to violate the wind farm power reference can be used to further reduce the DEL of tower and shaft. As expected, the MPC with soft constraints increases the standard deviation of the power tracking error. As can be seen in Fig. 11.4, the standard deviation is similar to that of the constant references setup in all four wind scenarios. In comparison to hard MPC, the tower DEL was not reduced, the shaft DEL however was reduced, which might be interesting for particular wind turbine designs. Note that even though Fig. 11.4 suggests that the constant reference setup outperforms the soft-constrained MPC in terms of wind farm power tracking, such a conclusion should not be generalized. Unlike the constant references setup, the soft-constrained MPC still penalizes the power tracking offset based on fast feedback. Therefore, contrary to the constant references setup, the soft constrained MPC is able to redistribute power references in case of low farm-wide available power and is thus expected to decrease the power tracking error in such cases. Also, unlike the constant references setup, the soft-constrained MPC is capable of tracking a time-varying power reference, like the one required in ”delta mode”

11.5 Simulation Results

119

control (see Fig. 11.1).

11.5.2 Convergence Properties of the Distributed Optimization Methods For the hard constrained MPC setup, the numbers of iterations to the point where the condition kz ∗ − zk∞ ≤ 10−3 has been satisfied for 5 consecutive iterations is illustrated in Fig. 11.6 for both ADMM and DDFG, for each step of the 11 m/s and the 14 m/s wind speed scenarios. Moreover, the average numbers of iterations to solve the MPC problem in all scenarios is illustrated in Fig. 11.7. As a general observation from Fig. 11.7, note that ADMM converges considerably faster than DDFG for the given problem, in all four wind scenarios. Looking at Fig. 11.6, it can be observed that for both ADMM and DDFG, the number of iterations to convergence correlates well with the norm of the Lagrange multiplier vector. This behavior could be expected from the bound (4.25) for DDFG, but it is also present for ADMM. The Lagrange multipliers, i.e. the optimal dual variables, can be interpreted as the price of consensus among the subproblems. Hence, the harder the problem to solve is, the more difficult it is to reach a consensus in the shared variables among the subproblems. Therefore, we expect the magnitude of the Lagrange multipliers to increase whenever the operating conditions make a consensus among the turbines difficult. This is the case, for instance, when there is a lack of available power at individual turbines, which has to be compensated for by other turbines. For the two simulation scenarios illustrated in Fig. 11.6, we observe that the number of iterations is higher in average in the 11 m/s case. Note that in the 11 m/s scenario, as opposed to the 14 m/s one, the farm-wide available power is often not sufficient to meet the farm-wide tracking requirement. This results in a harder task for the optimization method. Hence, not only do the turbine power references have to be redistributed, but also does the softening of the locally available power constraints have to be exploited. In the distributed optimization methods under consideration, both tasks are enforced by Lagrange multipliers of a larger magnitude, which are attained by a higher numbers of iterations. This effect is visible in particular in those simulation steps, where the available wind farm power is not sufficient to meet the farm-wide reference (see Fig. 11.5). The effect can also be seen in Fig. 11.7, where it is visible that the average number of iterations for both ADMM and DDFG is generally lower for the high wind-speed scenarios, where generally less power has to be redistributed due to sufficient local available power.

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11 Cooperative Distributed MPC for Wind Farms

ADMM WS 11 m/s

ADMM WS 14 m/s 80

# of Iter.

# of Iter.

400

200

0

0

400

40 20

600

0

200

400

600

0 0

200

400

600

600

15

400

10

200

5

0

0

200

400

600

Fast gradient WS 11 m/s

1000

0

0

200

400

Fast gradient WS 14 m/s

600

# of Iter.

2000

# of Iter.

200

60

600

600

400 200 0

0

200

400

600

0

200

400

600

20

400 10 200 0

0

200

400

Simulation time [s]

600

0

Simulation time [s]

Figure 11.6: Number of iterations and norm of the Lagrange multiplier vector of both ADMM and DDFG throughout the simulations of the wind speeds 11 m/s and 14 m/s.

11.5.3 Performance of Distributed MPC under Suboptimal Solutions In this section, simulation results concerned with the performance of hard constrained distributed MPC under suboptimal solutions, obtained by constant numbers of ADMM iterations at each time step, are presented. In particular, we consider all wind speed scenarios

11.5 Simulation Results

121

Average number of iterations to convergence

700

ADMM DDFG

600 500 400 300 200 100 0 WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

1500 Optimal solution ADMM 100 iterations ADMM 10 iterations ADMM 5 iterations

1000

500

Damage equivalent loads on the tower base [MNm]

0

150

Damage equivalent loads on the shaft [MNm]

Standard deviation of wind farm power tracking error [kW]

Figure 11.7: Average numbers of iterations for ADMM and DDFG in the four simulation scenarios.

2

WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

WS 8 m/s

WS 11 m/s

WS 14 m/s

WS 17 m/s

100

50

0

1.5 1 0.5 0

Figure 11.8: Performance of distributed MPC under various constant numbers of ADMMiterations.

and compare the performance of the hard-constrained MPC controller under the constant numbers of 5, 10 and 100 ADMM iterations. These iteration numbers qualitatively correspond to various degrees of suboptimality in the MPC control law. The performance results

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11 Cooperative Distributed MPC for Wind Farms

are illustrated in Fig. 11.8 and are discussed in the following. The earlier ADMM is stopped, the larger the standard deviation of the power tracking error becomes. This behavior is most pronounced in the 11 m/s scenario and makes sense since the TSO-imposed power tracking requirement is only satisfied exactly at optimality, where a consensus in the different subproblems is reached. Note furthermore, that the earlier ADMM is stopped, the more of a reduction in shaft fatigue is achieved, especially in the 11 m/s case. Also this behavior makes sense, since local fatigue reduction correlates with the local objective of each turbine, while farm-wide power output tracking corresponds to a global constraint. In the first ADMM iteration for instance, the dual variables in the vector λ are all equal to zero and the local minimizations in step 3 of Algorithm 4.2 are completely decoupled. Only after a number of iterations, once the dual values have reached a certain magnitude, will the consensus start to manifest in the local solutions. Therefore, the less ADMM iterations we allow, the less accurate the global consensus will be and the more emphasis will be put on the turbine’s local fatigue reduction objectives.

11.6 Discussion The simulation results described in Section 11.5 demonstrate the applicability of cooperative distributed MPC to second-scale control of wind farms. With a distributed MPC controller, the same closed-loop performance can be achieved as with a centralized MPC controller. To that end, the advantage of using a distributed control scheme is the fact that there is no single point of failure and the control system can adapt to outages of single turbines. Note that presently installed wind farms are already equipped with communication hardware, as required for a distributed control approach. As the main contribution of this chapter, we investigated the computational performance of the distributed optimization methods ADMM and DDFG in terms of numbers of iterations in closed-loop. It can generally be concluded that in the given setup, ADMM outperforms DDFG significantly. For ADMM, in the scenarios with wind speeds of 17 m/s and 14 m/s, full consensus is reached in average after 31 and 37 iterations for, while the number of iterations for DDFG is approximately 4 and 5 times larger. At lower wind speeds, when the wind farm operates close to its production potential, the magnitude of the Lagrange multipliers increases, which causes a larger number of iterations to convergence for both methods. However, for ADMM, the average number of iterations to convergence is still small, namely 57 and 81 for the simulation scenarios of 8 m/s and 11 m/s, while the average number of iterations required for DDFG is in both cases around 7 times larger. Further on, distributed MPC controllers with constant numbers of ADMM-iterations were considered and their performances were compared to the one of an MPC controller

11.7 Conclusion

123

with perfectly converging ADMM. The results show that a comparably good performance is already given under 10 iterations only. Under 100 ADMM-iterations, the performance is virtually the same as with centralized MPC. This result confirms the viability of the proposed distributed approach for the fast-scale wind farm controller. In particular, given the current state of technology, exchanging information between turbines at a 10 ms rate is possible. Moreover, the results for constant numbers of iterations are interesting also from another perspective, namely for tuning. Specifically, the iteration number could be used as a parameter to actively trade off the local objective of fatigue reduction versus the global objective of farm-wide power output tracking. One thing that needs to be addressed here is the scalability of the distributed MPC controller. In this chapter we use a fairly small number of wind turbines, due to restrictions imposed by the SimWindFarm software. However, the results presented in [Conte et al., 2012b] and [Kraning et al., 2013] suggest that the number of distributed optimization iterations can be expected to increase only moderately with the number of subsystems. In any event, for extremely large wind farms, the division of the turbines into smaller clusters that operate cooperatively in order to reduce fatigue loads is always a viable option. Finally, the idea of softening the power tracking constraint was considered in this chapter. The results suggest that with constraint softening, an increase in power tracking error can be traded off against a decrease in shaft loads. This kind of controller formulation is interesting since the grid codes typically allow some deviation around the wind farm power reference, which can be used to reduce shaft loads. Note that an additional reduction of the tower load in comparison to MPC with hard constraints was not observed.

11.7 Conclusion In this chapter, the application of cooperative distributed MPC to wind farms at a time scale of 1 s was proposed. The simulation results presented imply that this approach yields appealing performance in terms of farm-wide power output tracking and fatigue reduction. Moreover, the results imply that the approach is computationally viable, hence a performance close to the one of centralized MPC can be achieved with only a couple of tens of ADMM iterations in the online solution of the MPC problem. Furthermore, regarding the distributed optimization methods compared, ADMM proved more computationally efficient than DDFG.

Conclusion and Outlook The main goal of this thesis was to contribute to the framework of distributed optimization based cooperative distributed MPC, by designing theoretical tools and providing computational insight. On the theoretical side, we have proposed controller formulations for cooperative nominal, robust and reference tracking MPC. All of the resulting controllers are equipped with rigorous theoretical stability guarantees and can be synthesized and operated in a completely distributed manner by distributed optimization. On the computational side, we have documented the potential of distributed optimization methods in MPC under various system theoretical and operational circumstances. The main conclusions of this thesis can be summarized as follows. • Theory: Established non-trivial stability enforcing techniques, e.g. terminal PI sets, can be customized for the use under communication-constrained distributed optimization. The resulting distributed controllers are considerably less conservative than the ones obtained by trivial techniques, such as single-point terminal sets. • Computations: Distributed optimization methods, when used in cooperative distributed MPC for networked systems, are observed to require the more iterations to convergence, the more interdependent the system’s objectives are. In the following, we elaborate on these main conclusions and discuss possible future research directions based on the results presented in this thesis.

Theory, Thesis Part II Key to a stabilizing cooperative distributed nominal MPC controller is a terminal state constraint, which both guarantees stability and is structured such that distributed optimization methods can be applied under the given communication constraints. While such a constraint can always be trivially stated, i.e. by constraining the terminal state to be zero, it has been shown in this thesis that by choosing a separable terminal state penalty function and time-varying terminal sets, conservatism can be reduced considerably. Similarly for robust and reference tracking MPC, it has been shown that stabilizing MPC controllers can be formulated, such that their structure allows for the application of distributed optimization methods. In particular, for cooperative distributed robust MPC, RPI sets with decentralized

126

Conclusion and Outlook

and distributed structures were proposed, and for cooperative distributed reference tracking MPC, a time-varying distributed terminal invariant set for tracking was presented. For all of these components, distributed synthesis methods were designed. An interesting future research direction would be the application of the proposed distributed synthesis methods to networks with changing topologies. While such setups have recently been studied in context of plug-and-play MPC, i.e. in [Riverso et al., 2012] for a non-cooperative, and in [Zeilinger et al., 2013] for a cooperative setup, a number of interesting open problems remain. In particular, it would be interesting to investigate whether a distributed MPC controller can be synthesized to be robust w.r.t. single unit failures. Furthermore, it would be interesting to investigate regularities in network topologies and coupling properties, under which prespecified cooperative control modules could simply be combined, without running a global synthesis procedure for every incremental topology modification. Another interesting future research direction would be the further development of the distributed invariance concepts proposed in this thesis. It would for example be interesting to investigate if the size of time-varying distributed PI sets could be enlarged by allowing the shape of the local sets to be non-ellipsoidal. Possible alternative shapes are ellipsoids truncated by polytopes [Thibodeau et al., 2009], or convex hulls of ellipsoids and polytopes [Hu and Lin, 2003]. Furthermore, it would be interesting to see if distributed PI or RPI sets could be of use to describe safe operating regions for practical distributed systems, as similarly proposed in [Bari´c and Borrelli, 2012].

Computations, Thesis Part III The computational performance of specific distributed optimization methods in cooperative distributed MPC was investigated in a computational study on abstract networks of linear systems, as well as in an application study on fatigue reducing power output tracking of wind farms. The first study showed that the performance of distributed optimization methods strongly correlates with fundamental properties of coupled networks of dynamic systems, e.g. with local stability or coupling strength. In particular, the number of iterations to convergence for distributed optimization methods was observed to increase the more, the stronger the local system objectives were in conflict with the global one. Such situations commonly manifest in a high price of consensus, i.e. Lagrange multipliers of large magnitude. This observation was confirmed by the results of the wind farm study, where the Lagrange multipliers, along with the number of iterations, were observed to increase in operating conditions, where local fatigue reduction goals have to be sacrificed in order to achieve accurate farm-wide power output tracking.

127 From a computational viewpoint, an interesting research direction would be to investigate the performance of distributed optimization methods in MPC in cases where the communication graph is not equal to the dynamic coupling graph. Distributed optimization algorithms, which can cope with such conditions, exist in the literature, see e.g. [Schizas et al., 2008], [Mota et al., 2013]. In particular, it would be interesting to see how different configurations of available communication links influence the performance of the distributed optimization methods. Another interesting direction would be to find more and better application examples, for which cooperative distributed MPC is a suitable and meaningful control approach. Existing examples are mostly slow applications, such as various types of water distribution systems [Negenborn et al., 2009] or building control [Ma et al., 2012]. It would be interesting to also see the potential of cooperative distributed MPC for faster applications, which operate at a time-scale of seconds rather than minutes. Possible applications are swarms of mobile robots, or further cooperative energy harvesting systems, i.e. farms of wave or kite power units.

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Curriculum Vitae Christian Conte born September 26th , 1983 in Zurich, Switzerland.

2009 – 2014

Doctorate at the Automatic Control Laboratory, ETH Z¨urich, Switzerland

2003 – 2009

Studies in Electrical Engineering, ETH Z¨urich, Switzerland (MSc)

1996 – 2002

Kantonsschule Wiedikon, Zurich, Switzerland (High School, Matura)