Journal of Applied Nonlinear Dynamics

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Mar 1, 2019 - a more profound understating of fractional calculus, as well as the de- ..... SBL method is an analytical solution of characteristic equation, the fitting SBL of the fractional .... The index j in T1j denotes the order of the denominator and “1” in T1j ...... when an impact force is applied at the free end of the beam.
Volume 8 Issue 1 March 2019

ISSN  2164‐6457 (print)  ISSN 2164‐6473 (online) 

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics Editors J. A. Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrical Engineering Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto Portugal Fax:+ 351 22 8321159 Email: [email protected]

Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA Fax: +1 618 650 2555 Email: [email protected]

Associate Editors J. Awrejcewicz Department of Automatics and Biomechanics K-16, The Technical University of Lodz, 1/15 Stefanowski St., 90-924 Lodz, Poland Fax: +48 42 631 2225, Email: [email protected]

Shaofan Li Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, CA 94720-1710, USA Fax : +1 510 643 8928 Email: [email protected]

C. Steve Suh Department of Mechanical Engineering Texas A&M University College Station, Texas 77843-3123 USA Fax:+1 979 845 3081 Email: [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University Balgat, 06530, Ankara, Turkey Fax: +90 312 2868962 Email: [email protected]

Antonio M Lopes

Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics Moscow State University 119991 Moscow, Russia Fax: +7 495 939 0397 Email: [email protected]

Nikolay V. Kuznetsov Mathematics and Mechanics Faculty Saint-Petersburg State University Saint-Petersburg, 198504, Russia Fax:+ 7 812 4286998 Email: [email protected]

Miguel A. F. Sanjuan Miguel A. F. Sanjuan Department of Physics Universidad Rey Juan Carlos 28933 Mostoles, Madrid, Spain Email: [email protected]

UISPA-LAETA/INEGI Faculty of Engineering University of Porto Rua Doutor Roberto Frias, s/n, 4200-465 Porto, Portugal Email:[email protected]

Editorial Board Ahmed Al-Jumaily Institute of Biomedical Technologies Auckland University of Technology Private Bag 92006 Wellesley Campus WD301B Auckland, New Zealand Fax: +64 9 921 9973 Email:[email protected]

Giuseppe Catania Department of Mechanics University of Bologna viale Risorgimento, 2, I-40136 Bologna, Italy Tel: +39 051 2093447 Email: [email protected]

Liming Dai Industrial Systems Engineering University of Regina Regina, Saskatchewan Canada, S4S 0A2 Fax: +1 306 585 4855 Email: [email protected]

Alexey V. Borisov Department of Computational Mechanics Udmurt State University, 1 Universitetskaya str., Izhevsk 426034 Russia Fax: +7 3412 500 295 Email: [email protected]

Riccardo Caponetto DIEEI, (Electric, Electronics and Computer Engineering Department), Engineering Faculty University of Catania Viale A. Doria 6, 95125 Catania, Italy Email: [email protected]

Mark Edelman Yeshiva University 245 Lexington Avenue New York, NY 10016, USA Fax: +1 212 340 7788 Email: [email protected]

Continued on back materials

Journal of Applied Nonlinear Dynamics Volume 8, Issue 1, March 2019

Editors J. A. Tenreiro Machado Albert Chao-Jun Luo

L&H Scientific Publishing, LLC, USA

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Journal of Applied Nonlinear Dynamics 8(1) (2019) 1-3

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Advances in Fractional Order Controller Design and Applications Cosmin Copot1†, Cristina I. Muresan2† , Konrad Andrzej Markowski3† 1

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Department of Electromechanics, Faculty of Applied Engineering, University of Antwerp, Op3Mech, Groenenborgerlaan 171, 2020 Antwerpen, Belgium Dept. of Automation, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Memorandumului Street, no 28, 400114 Cluj-Napoca, Romania Institute of Control and Industrial Electronics, Faculty of Electrical Engineering, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland Submission Info Communicated by J.A.T. Machado Received 1 August 2017 Accepted 30 September 2017 Available online 1 April 2019 Keywords Fractional order systems PI-PD controller Stability boundary locus fitting Standard forms

Abstract Fractional order differentiation is a generalization of classical integer differentiation to real or complex orders. In the last couple of decades, a more profound understating of fractional calculus, as well as the developments in computing technologies combined with the unique advantages of fractional order differ-integrals in capturing closely complex phenomena, lead to ongoing research regarding fractional calculus and to an increasing interest towards using fractional calculus as an optimal tool to describe the dynamics of complex systems and to enhance the performance and robustness of control systems. The research community has managed to bring forward ideas and concepts that justify the importance of fractional calculus for future engineering and science discoveries. Since the emergence of the CRONE controllers and the generalization of the classical PID controller, many researchers have focused on the design problem of fractional order controllers, the optimal tuning, the possible extensions of fractional calculus in advanced control strategies, the problems regarding their implementation, and so on. There are still many issues and open problems left unattended in this area. This special issue aims at presenting some recent developments in the field of fractional order controllers, in order to further raise the interest regarding the increasing tendency of adopting fractional calculus in applications related to modeling and design of control systems. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

It is our strong belief that fractional calculus can help overcoming the obstacles originated by classic theories. The aim of this special issue is directed towards enhancing the idea of using fractional order tools, in order to further stimulate and raise interest regarding the increasing tendency of adopting fractional calculus in applications related to modeling and design of control systems. The main focus of this special issue is directed towards showcasing the latest updates from the applied fractional calculus community. † Corresponding

author. Email address: [email protected] (Cosmin Copot), [email protected] (Cristina I. Muresan), [email protected] (Konrad Andrzej Markowski)

ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.001

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Cosmin Copot et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 1–3

The Special Issue on Advances in Fractional Order Controller Design and Applications of the Journal of Applied Nonlinear Dynamics includes a collection of 7 papers that cover fractional calculus from a wide area of applications, including mathematics, control, modeling, etc. The advantages and utility of using fractional calculus in representing physical phenomena, in achieving improved performance of control systems are strongly highlighted. The wide range of topics addressed in this special issue caters to a large group of readers. The tuning of a fractional order PI(D) controller is presented in [1], where the standard forms for fractional order system is used together with the stability boundary locus fitting method which approximate the fractional order with an integer order transfer function. Fractional calculus is also used in vibration suppression problem in [2] an active wave control of a flexible beam is presented as an example of fractional calculus in vibration control. Fractional calculus helps to represent a system in a more compact manner. Fractional-order models represent a system in more accurate ways and therefore systems of fractional nature can be represented by a fractional order model instead of integer order models. Model based on fractional calculus helps advanced controllers such as predictive control to have a deeper knowledge of the system dynamics and thus to give better results. In [3] the performance of fractional-order system with model predictive control strategy in a constrained environment for non-minimum phase type system is presented. Fractional-order PID controllers have been used in industrial applications and various fields such as mechatronic systems and system identification. In [4], an application of the fractional order controller to robot manipulator joint control is presented. The motion of a manipulator robot is usually given as Cartesian coordinates of the end-effector. However, in order to reach the desired Cartesian trajectory, each joint needs to follow a specific trajectory. Here, a fraction-order PI controller is designed, tested and validated for the inner velocity control loop. The communication between the robot and the controller was establish via the Robotics System Toolbox from Matlab® and the ROS platform. Another application where fractional control can be employed is continuous casting technology [5]. Here, a control strategy based on fractional approach is proposed in order to control the steel slab temperature (a process with time delay) which one of the most important parameters for evaluating the cooling process and inherent material properties. In [6] the body control of a non-stationary vehicle with an active pneumatic suspension under driver inputs was presented. Three main driver inputs, which can modify the center of the mass of the vehicle and thus the equilibrium, are considered. In order to fulfill the desired specification, a robust controller was designed by using fractional order operators. In the field of analysis of dynamic systems one of constant problems is the realisation problem. Most research studies focus on canonical forms of the system, i.e. constant matrix forms, which satisfy the system described by the transfer function. With the use of canonical form we are able to write only one realisation of the system, while in general there exists a set of possible solutions. Another approach is based on digraph representation of dynamic system. In [7], a method that allows finding of (A, B, C, D) matrices directly from obtained digraph structures of one-dimensional SIMO and MISO fractional-order dynamic systems was proposed. Finally, we wish to thank Prof. J.A. Tenreiro Machado and Prof. A. Luo, Editors-in-Chief of the Journal of Applied Nonlinear Dynamics for their encouragement for a special issue on fractional calculus. Each paper went through rigorous peer review process and for that we wish to thank the reviewers for their professional service. References [1] Deniz, F.N., Y¨ uce, A., and Tan, N. (2019), Tuning of PI-PD Controller Based on Standard Forms for Fractional Order Systems, Journal of Applied Nonlinear Dynamics, 8(1), 5-21. [2] Kuroda, M. and Matsubuchi, H. (2019), Active Wave Control of a Flexible Beam Using Fractional Derivative Feedback, Journal of Applied Nonlinear Dynamics, 8(1), 23-33.

Cosmin Copot et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 1–3

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[3] Joshi, M.M. and Vyawahare, V.A. (2019), Constrained Model Predictive Control for Linear Fractional-order Systems with Rational Approximation, Journal of Applied Nonlinear Dynamics, 8(1), 35-53. [4] Copot, C. (2019), An application to robot manipulator joint control by using fractional order approach, Journal of Applied Nonlinear Dynamics, 8(1), 55-66. [5] Copot, D. and Ionescu, C. (2019), A fractional order controller for delay dominant systems. Application to a continuous casting line, Journal of Applied Nonlinear Dynamics, 8(1), 67-78. [6] Bouvin, J.L., Moreau, X., Benine-Neto, A., and Hernette, V. (2019), Alain Oustaloup, Pascal Serrier, CRONE body control with a pneumatic self-leveling suspension system, Journal of Applied Nonlinear Dynamics, 8(1), 79-95. [7] Markowski, K.A. and Hryniow, K. (2019), Method for finding a set of (A, B, C, D) realizations for SingleInput Multiple-Output / Multiple-Input Single-Output one-dimensional continuous-time fractional systems, Journal of Applied Nonlinear Dynamics, 8(1), 97-108.

Journal of Applied Nonlinear Dynamics 8(1) (2019) 5-21

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Tuning of PI-PD Controller Based on Standard Forms for Fractional Order Systems Furkan Nur Deniz†, Ali Y¨ uce, Nusret Tan Department of Electrical and Electronics Engineering, Inonu University, Malatya, Turkey Submission Info Communicated by C. Copot Received 30 July 2017 Accepted 1 October 2017 Available online 1 April 2019 Keywords Fractional order systems PI-PD controller Stability boundary locus fitting Standard forms

Abstract In this paper, a PI-PD controller tuning method is proposed for fractional order systems based on standard forms. SBL fitting integer order approximation method is directly used to obtain appropriate integer order transfer function required in standard forms for the controller design. The controller tuning parameters for approximate transfer function are calculated by using optimization of ISTE integral performance criterion. The obtained tuning parameters are performed for fractional order transfer function. Results give good performance. The results show that the performance of the proposed method is practicable and that the controller parameters for the fractional order models can be tuned by using its integer order approximation transfer function. Also, the results shows that the other methods such as Oustaloup’s and Matsuda’s methods which enable one to obtain integer order approximate transfer functions, cannot be used directly because they do not conform to the standard form. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Fractional calculus has been a challenge for mathematicians for many years. Fractional calculus has attracted the attention of researchers working in the engineering and science with the development in the field of fractional calculus. Fractional order systems have been shown in many studies that have modeled and characterized real systems more accurately [1, 2]. It is important to model and characterize a system accurately, and therefore a lot of studies has been done focusing on fractional order systems [3–5]. Also, in control applications, it has been presented that fractional order controllers perform better than integer order controllers in many studies [6–9]. A fractional order system has infinite dimensional function and it is very difficult to realize and simulate using by conventional software programs [10]. For this reason, different integer order approximation methods such as Matsuda’s method, Oustaloup’s method and SBL fitting method have been developed in order to obtain equivalent integer order models which can characterize fractional order systems [11–13]. Equivalent integer order models of fractional order systems are obtained by using SBL fitting method in this study. Most of these approximation methods intend to approximate a fractional † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.002

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Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

order operator sα , (−1 < α < 1), to an integer order transfer function (such as fourth order, fifth order). In this way, these methods bring out high order approximate equivalent model and model reduction methods require to be used to design a controller. In the present study, a controller design technique for closed loop control system with a fractional order plant is presented for a PI-PD controller using the standard form. As it is known, the PID controller is a very popular controller in the industry due to its easy and practical use [14]. PI-PD controller can be used instead of PID controllers and more satisfactory results may be obtained [15]. Furthermore, using an internal PD feedback loop provides an open loop stable process for an open loop unstable process. PI-PD controller has four tuning parameters for control of processes and these parameters can be calculated using standard forms which provide satisfactory closed loop response by simple algebra [16]. The proposed technique can be briefly summarized as follows; First, SBL fitting method for the fractional order system is used to obtain an equivalent integer order transfer function suitable for the standard form. For this, the SBL method is applied to the fractional order transfer function, not the fractional operator such as the Matsuda’s and Oustaloup’s methods. These approximation methods used to obtain integer order transfer functions of fractional operators allow to obtain equivalent high order integer models. They are very difficult to control in high order systems and to tune the parameters of a controller. However, the SBL fitting method provides directly low order integer model. The closedloop transfer function suitable for the standard form is obtained by using the controller which is desired to estimate its parameters and the equivalent integer order model. In order to identify the controller parameters in the closed-loop transfer function, the optimum values of the standard form transfer function obtained by minimizing the error function in accordance with the ISTE performance criterion as given in [16] are used. These parameters can be changed until the desired performance is achieved. The controller parameters calculated by using the equivalent integer model are then used to control the fractional order transfer function. As can be seen from the simulation results, the proposed controller design technique exhibits a successful performance to control the fractional order transfer function. Preliminary versions of some of the results given in this study was presented in [17].

2 Fractional order systems Fractional order derivative and integrator are treated as an extension of integer order derivative and integrator operators to the case of non-integer orders and it is defined in general form as [18], ⎧ dα ⎪ ⎪ α >0 ⎪ ⎪ dt α ⎨ α (1) α =0 a Dt = ˆ t 1 ⎪ ⎪ ⎪ ⎪ ⎩ (d τ )(−α ) α < 0 a

where, a Dtα denotes non-integer order derivative operator of fractional calculus. Parameters a and t are the lower and upper bounds of integration, and α ∈ R denotes the fractional-order [18]. The Caputo definition of fractional order differentiation is given by ˆ t 1 f n (τ ) α D = dτ , n − 1 ≤ α < n (2) a t Γ(n − α ) a (t − τ )α −n+1 where Γ(.) is Euler’s gamma function [19]. The Laplace transform of the Caputo fractional order derivative is [19]

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

r( t ) + 

C ( s) k p 

ki s

7

N F ( s ) y( t ) DF ( s )

GF ( s )

Fig. 1 The closed loop control system with PI controller for SBL calculation of fractional order plant model.

ˆ

∞ 0

n−1

e−st 0 Dtα f (t)dt = sα F(s) − ∑ sα −k−1 f (k) (0).

(3)

k=0

Considering f (0) = f (1) (0) = f (2) (0) = f (3) (0) =, .., = f (n−1) (0) = 0 fractional order systems is expressed for Laplace transform of fractional order derivative as L {Dα f (t)} = sα F(s). In general, fractional order system is described by the following fractional order differential equation form as [18], vn Dαq y(t) + vn−1 Dαq−1 y(t) + ... + v1 Dα1 y(t) + v0 y(t) = u p Dβ p r(t) + u p−1 Dβ p−1 r(t) + ... + u1 Dβ1 r(t) + u0 r(t). (4) Using L {Dα f (t)} = sα F(s), a general form of transfer function of fractional order systems is expressed as, p N(s) ∑i=0 ui sβi = , (5) T (s) = D(s) ∑qi=0 vi sαi where denominator polynomial coefficients vi and numerator polynomial coefficients ui are polynomial coefficients and fractional orders of the system are denoted by αi ∈ R (i = 0, 1, 2, 3..., q) and βi ∈ R (i = 0, 1, 2, 3..., p). 3 Stability boundary locus (SBL) method This section summarizes calculation of SBL method for fractional order plant transfer functions according to closed loop PI control system given in Fig. 1. Considering a PI controller given by C(s) = k p + ksi and a fractional order transfer function given by GF (s) = DNFF (s) (s) in a closed loop control system, the characteristic equation is written as Δ(s) = 1 +C(s)GF (s) = sDF (s) + (k p s + ki )NF (s)

(6)

=0 where NF (s) and DF (s) are the fractional order numerator and the fractional order denominator polynomials given by p

q

i=0

i=0

NF (s) = ∑ ui sαi and DF (s) = ∑ vi sβi ,

(7)

where, ui ∈ R and vi ∈ R are polynomial coefficients and αi ∈ R and βi ∈ R are fractional orders. After substituting ( jω )α = ω α (cos( π2 α ) + j sin( π2 α )) and solving characteristic equation by equating real and imaginary parts of Δ( jω ) to zero, the following equations are obtained ki NFR (ω ) − k p ω NFI (ω ) − ω DFI (ω ) = 0,

(8)

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Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

ki

S (k p (Z ), ki (Z )) SBL Line Stability Region

kp Fig. 2 A representation of stability region.

ki NFI (ω ) + k p ω NFR (ω ) + ω DFR (ω ) = 0.

(9)

k p (ω ) and ki (ω ) are determined by using the real and imaginary parts which are equal to zero as follows, k p (ω ) = − ki (ω ) =

NFI (ω )DFI (ω ) + NFR (ω )DFR (ω ) , NFI (ω )2 + NFR (ω )2

ω NFR (ω )DFI (ω ) − ω NFI (ω )DFR (ω ) . NFI (ω )2 + NFR (ω )2

(10) (11)

All stabilizing controller parameters k p and ki values locate in stability region as shown in Fig. 2 and one can obtain the stability region by plotting k p (ω ) and ki (ω ) in the frequency range ω ∈ [0, ∞). Also, stability region can be found in a finite frequency range such as ω ∈ [0, ωc ], where ωc is critical frequency [20].

4 Stability boundary locus fitting method SBL fitting method for calculation of the integer order approximate transfer function is given in this section. SBL method is an analytical solution of characteristic equation, the fitting SBL of the fractional order plant transfer function model to SBL of an integer order approximate transfer function can provide an approximation of characteristic polynomials of the systems in a limited frequency range: The fractional order plant transfer function is replaced with an integer order plant transfer function with unknown coefficients in the closed loop control system with PI controller to fit SBL of the systems. The estimated integer transfer function is given by GT (s) =

NT (s) ∑ni=0 ai si an sn + an−1 sn−1 + ... + a1 s + a0 = m = . DT (s) ∑i=0 bi si bm sm + bm−1 sm−1 + ... + b1 s + b0

(12)

Using s = jω and decomposing the numerator and denominator of GT (s) into its real and imaginary parts, we can write (13) in the following form; GT (s) =

NT ( jω ) NT R (ω ) + jNT I (ω ) = . DT ( j ω ) DT R (ω ) + DT I (ω )

(13)

Equating the real and imaginary parts of characteristic equation to zero, the following equations are obtained (14) ki (ω )NT R (ω ) − k p (ω )ω NT I (ω ) − ω DT I (ω ) = 0, ki (ω )NT I (ω ) + k p (ω )ω NT R (ω ) + ω DT R (ω ) = 0.

(15)

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

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These equations give the real part and imaginary part equations, respectively. The numbers of sampling points are selected from SBL of GF (s) so as to be same number of the unknown coefficients in the real part equation for solution. After calculating real part equation, the remaining unknown coefficients are calculated by using imaginary part equation for same sampling points. Thus, all the unknown coefficients of the integer plant transfer function are found in a certain frequency range. To give an example for applying of SBL fitting approximation, m = 4 and n = 1 in (12) is considered and 3 sampling points are requires to solve linear equation systems. Sampling frequencies are {ω1 , ω2 , ω3 } and the sampling points selected from SBL of GF (s) are {(k p1 (ω1 ), ki1 (ω1 )), (k p2 (ω2 ), ki2 (ω2 )), (k p3 (ω3 ), ki3 (ω3 ))}. Equation (10) and (11) are used to calculate these points. The real part equation in matrix form is given by, (16) Ar Br = Cr , where Ar , Br and Cr can be determined using (14), as follows ⎤ ⎡ ⎤⎡ ⎤ b3 −a0 ki1 ω14 −ω12 −(ω12 )k p1 ⎣ ω24 −ω22 −(ω22 )k p2 ⎦ = ⎣ b1 ⎦ ⎣ −a0 ki2 ⎦ . ω34 −ω32 −(ω32 )k p3 a1 −a0 ki3 ⎡

(17)

Solving (17) for any value of coefficient a0 , which any coefficients can be assigned for solution of homogeneous linear equation systems, the parameters b3 , b1 and a1 are found. The imaginary part equation in matrix form is given by (18) Ai Bi = Ci where Ai , Bi and Ci can be obtained using (15), as follows ⎤⎡ ⎤ ⎡ ⎤ ω15 −ω13 ω1 b4 −ω1 a0 k p1 − ω1 a1 ki1 ⎣ ω 5 −ω 3 ω2 ⎦ ⎣ b2 ⎦ = ⎣ −ω2 a0 k p2 − ω2 a1 ki2 ⎦ . 2 2 ω35 −ω33 ω3 b0 −ω3 a0 k p3 − ω3 a1 ki3 ⎡

(19)

The coefficients b4 , b2 and b0 are calculated using (19) and thus all coefficients which provide to fit SBL of GT (s) with SBL of GF (s) can be determined. For illustrative example, consider a fractional order transfer function 2 . (20) GF (s) = 2.1 2s + s0.8 + 2 The first, the last and an appropriate frequency are selected in the frequency range ω ∈ [0.01, 2.35] for this example. Corresponding frequency range is the frequency range in which the stability region is located. The sampling points from SBL of GF (s) are {(-1.0038, 1.934x10−4 ), (1.5964, 0.4324), (4.6349, 0.0023)} at the sampling frequencies {0.01, 1.65, 2.35}. Solving Equation (16) and (18) for a0 = 1, the integer order approximate transfer function for (20) is obtained as, GT 1 (s) =

2.487s + 1 0.1171s4 + 2.755s3 + 1.669s2 + 3.69s + 1.004

(21)

The following integer order transfer function is obtained using Matsuda’s approximation method for ω ∈ [0.01, 100]

G(s)Matsuda =

2s8 + 1272s7 + 1.011 × 105 s6 + 1.368 × 106 s5 + 5.018 × 106 s4 + 4.517 × 106 s3 + 1.173 × 106 s2 + 6.651 × 104 s + 869.2 3.656s10 + 2242s9 + 1.382 × 105 s8 + 1.699 × 106 s7 + 5.789 × 106 s6 + 7.711 × 106 s5 + 8.561 × 106 s4 + 5.401 × 106 s3 + 1.242 × 106 s2 + 6.763 × 104 s + 871

(22)

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Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

Fig. 3 Stability regions of GF (s), GT 1 (s), GOustaloup (s), GMatsuda (s) and SBL sampling points.

The integer order approximation transfer function is obtained using Oustaloup’s approximation method for ω ∈ [0.01, 100],

G(s)Oustaloup =

2s10 + 301.4s9 + 1.366 × 104 s8 + 2.176 × 105 s7 + 1.354 × 106 s6 + 3.237 × 106 s5 + 3.101 × 106 s4 + 1.142 × 106 s3 + 1.642 × 105 s2 + 8301s + 126.2

3.17s12 + 450s11 + 1.86 × 104 s10 + 2.752 × 105 s9 + 1.612 × 106 s8 + 4.079 × 106 s7 + 5.74 × 106 s6 + 5.908 × 106 s5 + 3.997 × 106 s4 + 1.284 × 106 s3 + 1.736 × 105 s2 + 8526s + 127.8 (23) Stability regions for actual system and its approximated models are plotted and SBL fitting performance of the proposed method used the sampling points is shown in Fig. 3. For this example, 3 sampling points are used to obtain directly integer order approximate transfer function. If order of estimated integer order transfer function is increased, the numbers of sampling points must be increased for solution of linear equations system. In this example, Fourier series method is used to obtain the step response of the fractional order transfer function GF (s) [21]. Open loop step responses of GF (s) and integer order approximated models for SBL fitting method, Matsuda’s method and Oustaloup’s method are compared in Fig. 4. The step response obtained for the proposed method is acceptable when it is considered that the order of the approximate transfer function is lower than the other transfer functions obtained Matsuda’s and Oustaloup’s methods. Amplitude and phase responses of GF (s) with proposed approximation method, Matsuda’s method and Oustaloup’s method are compared in Fig. 5. In order to match the stability region of GF (s) with the integer order approximate transfer function, the low frequency range ω ∈ [0.01, 2.35] is used for SBL fitting method. As a result of this situation, frequency response of the fractional order transfer function matched to integer order approximate transfer function obtained the SBL fitting is better for low frequencies. Details of SBL fitting integer order approximation method can be found in [13].

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

11

Fig. 4 Open loop step responses of GF (s) and integer order approximate transfer functions.

Fig. 5 Amplitude and phase responses of GF (s) and integer order approximation plant transfer functions; SBL fitting method for ω ∈ [0.01, 2.35], Matsuda’s method and Oustaloup’s method forω ∈ [0.01, 100].

5 Standard forms Integral performance criterion is used to calculate the controller tuning parameters in accordance with the standard form [16].

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Fig. 6 Optimum values of d1 , d2 and d3 versus c1 for T14 (s) according to ISTE.

A general form of an integral performance criterion is ˆ ∞ [t n e(t)]2 dt, Jn =

(24)

0

where, n = 0, n = 1 and n = 2 corresponds to ISE, ISTE and IST2 E criteria, respectively. A closed loop transfer function for a plant transfer function with no zero and a controller with a zero is determined according to standard form as T1 j =

sj + d

c1 s + 1 j−1 + ... + d1 s + 1 j−1 s

(25)

The index j in T1 j denotes the order of the denominator and “1” in T1 j denotes a zero in the numerator of the standard form. The error function for a unit step is defined as E1 j =

s j−1 + d j−1 s j−1 + ... + (d1 − c1 ) s j + d j−1 s j−1 + ... + d1 s + 1

(26)

The optimum values of the d1 , d2 , and d3 versus c1 are given in Fig. 6 for ISTE criteria. It means that the closed loop transfer function in standard form denoted as T14 (s) provides to minimize E1 j according to the ISTE criteria. This optimum values given in Fig. 6 will be used for tuning of the controllers as shown in the next section.

6 PI-PD controller design for fractional order systems To design the PI-PD controller by using optimum values of d1 , d2 , d3 and c1 for T14 (s) according to ISTE criterion shown in Fig. 6, Equation (16) and (18) are rearranged for b3 = 1 to provide an appropriate transfer function in the standard form. For this, third order plant transfer function with no zero that means m = 3 and n = 0 in (12) for fractional order transfer function in (20) is obtained by using SBL

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

13

Fig. 7 Stability regions of GF (s), GT 2 (s), GOustaloup (s), GMatsuda (s) and SBL sampling points.

fitting approximation method. To apply the proposed method Ar , Br and Cr given in (16) are rearranged as follows,



a0 −ω14 ki1 −ω12 = (27) ki2 −ω22 b1 −ω24 and Ai , Bi and Ci given in (18) are rearranged as follows,



b2 −ω1 a0 k p1 −ω13 ω1 = . −ω23 ω2 b0 −ω2 a0 k p2

(28)

Stability regions of actual systems and its approximate models are demonstrated in Fig. 7. It is to be noted that the stability region of the model obtained by the SBL fitting method is quite different from the real system. For this reason, only two sampling points are used to obtain the model conforming to the standard form. It is seen in the simulations that the PI-PD adjustment is sufficient. Two sampling points are sufficient to solve Equation (27) and (28). We prefer to select the first and the last frequency in a frequency range ω ∈ [0.01, 2.35]. The sampling points from SBL of GF (s) are {(1.0038, 1.1934x10−4 ), (4.6349, 0.0023)}. Solving Equation (27) and (28), the integer order approximate transfer function is found as, GT 2 (s) =

a0 4.629 . = b3 s3 + b2 s2 + b1 s + b0 s3 + 4.726s2 + 5.524s + 4.647

(29)

To observe fitting of time responses of actual systems and it is approximated models, open loop step responses are shown in Fig. 8. It can be seen that approximated model obtained by SBL fitting is different from the others. Similarly, amplitude and phase responses for all of them are demonstrated in Fig. 9. Since SBL fitting method is performed in low frequencies range, frequency response fitting is better in low frequencies. A closed loop control system with PI-PD controller is shown in Fig. 10. Consider PI-PD controller configuration where GF (s) is a fractional order plant transfer function and PI-PD controller is given by CPI (s) = k p (1 + T1i s ) and CPD (s) = (k f + Td s).

14

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Fig. 8 Open loop step responses of GF (s) and integer order approximate transfer functions.

Fig. 9 Amplitude and phase responses of GF (s), GT 2 (s) for ω ∈ [0.01, 2.35], GOustaloup (s), GMatsuda (s) for ω ∈ [0.01, 100].

GF (s) is replaced with GT 2 (s) in the closed loop control system for calculation of PI-PD controller tuning parameters. The closed loop transfer function is obtained as T14 (s) =

a0 k p (sTi + 1) 4 3 2 s Ti + s b2 Ti + s Ti (b1 + a0 Td ) + sTi (a0 k f

+ a0 k p + b0 ) + a0 k p

.

(30)

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

15

Fig. 10 PI-PD controller configuration. Table 1 Optimum values selected from Fig. 6 and PI-PD controller tuning parameters for different values of time scale α . α

d1

d2

d3

c1

Ti

kp

kf

Td

1.681

7.045

7.153

2.811

6.009

3.574

6.168

0.059

3.174

1.7608

6.162

6.504

2.684

5.000

2.839

5.898

0.3665

3.162

2.305

3.243

4.072

2.050

1.813

0.786

4.798

2.781

3.482

2.981

2.414

3.099

1.585

0.221

0.074

1.262

11.558

4.758

Normalized form is obtained by using time scale α = (k p a0 Ti )1/4 and sn = s/α T14 (sn ) =

(sn α Ti + 1) 4 3 −1 2 −2 (sn + sn α b2 + sn α (b1 + a0 Td ) + sn α −3 (a0 k f

+ a0 k p + b0 ) + 1)

.

(31)

Selecting of the time scale α is defined by the values of b2 and d3 . The tuning parameters of the PI-PD controller Ti , k p , k f and Td are determined by the following expressions [16]

α=

b2 , d3

Ti =

c1 , α

kp =

Ti α 4 , a0

kf =

d1 α 3 − a0 k p − b0 , a0

Td =

d2 α 2 − b1 . a0

(32)

The choice of the time scale α depends on the value of b2 if a suitable value of d3 is to be obtained from Fig. 6. Selecting different values of α , gives optimum values as d1 , d2 , d3 and c1 are found from Fig. 6. These values can be used to calculate the tuning parameter of PI-PD controller. The tuning parameters of the PI-PD controller Ti , k p , k f and Td are determined for different values of time scale α as given in Table 1. Also, corresponding optimum values of d1 , d2 , d3 and c1 are given in Table 1. Step responses of the closed loop control system including the PI-PD controller and fractional order plant transfer function GF (s) are shown in Fig. 11 for different values of α . It can be seen from Fig.11 that the PI-PD controller design based on standard form gives good results by using the equivalent integer order transfer function considering the open loop step response of GF (s). As α increases, settling time decreases and rise time increases very little. There has not been a serious difference in overshoot by changing α . It can be said that the proposed design technique is feasible and acceptable considering the obtained results. Step responses of the closed loop transfer function with fractional order transfer function, GF (s) and its equivalent integer order model GT 2 (s) using the tuning parameters of PI-PD controller as Ti = 3.574, k p = 6.168, k f = 0.059 and Td = 3.174 for α = 1.681 are shown in Fig. 12. Closed loop step responses of GF (s) and GT 2 (s) are obtained close to each other for same tuning parameters of PI-PD controller.

16

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

Fig. 11 Closed loop step responses of GF (s) for different values of time scale α .

Fig. 12 Closed loop step responses of GF (s) and GT 2 (s) using same tuning parameters for α = 1.681.

7 PI-PD controller design for a fractional order time delay systems In this section, a PI-PD controller is designed for a fractional order time delay system using SBL fitting approximation method. Consider the fractional order system with time delay parameter L = 1, GF (s) =

1 s1.2 + 1

e−s .

(33)

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

17

Fig. 13 Stability regions of GF (s), GT 3 (s), GOustaloup (s), GMatsuda (s) and SBL sampling points.

The stability region is calculated to be exact solution for fractional order time delay systems. However, the first-order Pade approximation is used for the fractional order time delay system in the time response and frequency response simulations. The first-order Pade approximation of GF (s) is obtained as GFP (s) ≈

−s + 2 s2.2 + 2s1.2 + s + 2

.

(34)

The first and the last frequency are selected in a frequency range ω ∈ [0.01, 1.78] to obtain approximated model. The sampling points selected from SBL of GF (s) are {(-0.998, 1.377x10−4 ), (1.937, -0.035)}. Using the sampling points, the integer order approximation transfer function GT 3 (s) is calculated as GT 3 (s) =

2.281 s3 + 2.115s2 + 3.143s + 2.279

.

(35)

The integer order approximation transfer function is obtained using Matsuda’s approximation method for ω ∈ [0.01, 100], G(s)Matsuda =

s4 + 95s3 + 453.9s2 + 161s + 3.357 e−s . 3.357s5 + 162s4 + 548.9s3 + 548.9s2 + 162s + 3.357

(36)

The integer order approximation transfer function is obtained using Oustaloup’s approximation method for ω ∈ [0.01, 100], G(s)Oustaloup =

s5 + 56.87s4 + 442.3s3 + 531.7s2 + 98.83s + 2.512 e−s 2.512s6 + 99.83s5 + 588.6s4 + 884.5s3 + 588.6s2 + 99.83s + 2.512

(37)

The stability regions of time delay fractional order system GF (s) and its approximation models GT 3 (s), G(s)Matsuda and G(s)Oustaloup are shown in Fig. 13. Open loop step responses of the fractional order time delay system and its approximation models are shown in Fig. 14. The step response of the fractional order system is calculated by FSM method [21]. The step response obtained with SBL fitting is slightly different from the others due to its low

18

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

Fig. 14 Open loop step responses of GFP (s) and integer order approximate transfer functions.

Fig. 15 Amplitude and phase responses of GFP (s), GT 3 (s) for ω ∈ [0.01, 1.78], GOustaloup (s), GMatsuda (s) for ω ∈ [0.01, 100].

order approximate transfer function. Amplitude and phase responses of GF (s), GT 3 (s), GOustaloup (s) and GMatsuda (s) are given in Fig.15. Since the SBL fitting is calculated for low frequencies in which stability region is located, so fitting in lower frequencies is better. Corresponding optimum values and the tuning parameters of the PI-PD controller Ti , k p , k f and Td are given in Table 2 for different values of α . As α decreases, settling time and maximum overshoot decrease. Step responses of the closed loop transfer function with fractional order time delay transfer function GF (s) and its equivalent integer order model GT 3 (s) using the same tuning parameters of PI-PD

Furkan Nur Deniz, Ali Y¨ uce, Nusret Tan / Journal of Applied Nonlinear Dynamics 8(1) (2019) 5–21

19

Fig. 16 Closed loop step responses of GFP (s) for different values of time scale α . Table 2 Optimum values selected from Fig. 6 and PI-PD controller tuning parameters for different values of time scale α . α

d1

d2

d3

c1

Ti

kp

kf

Td

0.703

8.719

8.344

3.010

7.605

10.824

1.156

-0.829

0.428

0.825

5.156

5.760

2.510

4.001

4.749

1.049

-0.696

0.414

1.147

2.739

3.489

1.843

1.013

0.883

0.671

0.143

0.635

1.351

3.109

8.621

1.565

0.167

0.123

0.181

2.183

5.521

controller as Ti = 0.123, k p = 0.181, k f = 2.183 and Td = 5.521 for α = 1.351 are shown in Fig. 17. As can be seen in Fig.17, closed loop step responses of fractional order system and its approximated model obtained by SBL fitting method are almost same for the same tuning parameters of PI-PD controller.

8 Conclusions In this paper, the PI-PD controller design technique is presented for a fractional order systems. The controller design technique is developed based on a standard form that allows simple mathematical calculations. For this, an appropriate integer order approximate transfer function is first obtained by using SBL fitting integer order approximation method for the fractional order transfer function. PIPD controller parameters are calculated by using the parameters of the closed loop transfer function obtained by minimizing the ISTE integral performance criterion and the coefficients of the transfer function obtained by the SBL fitting method. This paper shows that PI-PD controllers with good performance can be designed by using an equivalent integer order model suitable for the standard form structure obtained by the SBL fitting method for a fractional order transfer function. Other methods such as Matsuda’s and Oustaloup’s methods are very difficult to adapt to the standard form structure without using a model reduction method. However, the SBL fitting method allows to obtain the integer order transfer function in accordance with the standard form directly in a fast and easy way.

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Fig. 17 Closed loop step responses of GFP (s) and GT 3 (s) using same tuning parameters for α = 1.351.

Acknowledgement ˙ ¨ IAK) This work is supported by the Scientific and Research Council of Turkey (TUB under Grant no. EEEAG115E388.

References [1] Podlubny, I. (1999), Fractional Differential Equations. San Diego: Academic Press. [2] Ma, C. and Hori, Y. (2007), Fractional-order control: theory and applications in motion control, IEEE Ind. Electron. Mag., vol. Winter, 6-16. [3] Hwang, C. and Cheng, Y.C. (2006), A numerical algorithm for stability testing of fractional delay systems, Automatica, 42(5), 825-831. [4] Petras, I., Vinagre, B.M., Dorˇca´k, L., and Feliu, V. (2002), Fractional digital control of a heat solid: experimental results, in International Carpathian Control Conference ICCC’ 2002, 365-370. [5] Vinagre, B.M., Feliu, V., and Feliu, J.J. (1998), Frequency domain identification of a flexible structure with piezoelectric actuators using irrational transfer function models, in Proceedings of the 37th IEEE Conference on Decision & Control, , no. December, 1278-1280. [6] Oustaloup, A. (1991), The CRONE control, in European Control Conference, 275-283. [7] Xue, D. and Chen, Y. (2002), A Comparative Introduction of Four Fractional Order Controllers, in 4th World Congress on Intelligent Control and Automation, 3228-3235. [8] De-Jin, W.D.J.W. and Xue, C.X.C. (2010), Turning of fractional-order phase-lead compensators: A graphical approach, Control Conf. CCC 2010 29th Chinese, no. 1, 1885-1890. [9] Podlubny, I. (1999), Fractional-order systems and PIλ Dμ -controllers, IEEE Trans. Automat. Contr., 44(1), 208-214. [10] Podlubny, I., Petr´aˇs, I., O’Leary, P., Dorˇca´k, L., and Vinagre, B.M. (2002), Analogue realizations of fractional order controllers, Nonlinear Dyn. Acad. Publ., 29(1), 281-296. [11] Matsuda, K. and Fujii, H. (1993), H(infinity) optimized wave-absorbing control - Analytical and experimental results, J. Guid. Control. Dyn., 16(6), 1146-1153. [12] Oustaloup, A., Levron, F., Mathieu, B., and Nanot, F.M. (2000), Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 47(1), 25-39. [13] Deniz, F.N., Alagoz, B.B., Tan, N., and Atherton, D.P. (2016), An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators, ISA Trans., 62, 154-163.

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[14] [15] [16] [17] [18] [19] [20] [21]

21

˚ Astr¨ om, K.J. and H¨ agglund, T. (2001), The future of PID control, Control Eng. Pract., 9(11), 1163-1175. Tan, N. (2009), Computation of stabilizing PI-PD controllers, Int. J. Control. Autom. Syst., 7(2), 175-184. Atherton, D.P. and Boz, A.F. (1998), Using standard forms for controller design, in UKACC International Conference on Control (CONTROL ’98), 1998, 1066-1071. Deniz, F.N., Y¨ uce, A., and Tan, N. (2016), PI-PD controller Design for Fractional Order Plant Using Standard Forms, in International Conference on Fractional Differentiation and its Applications. Chen, Y.Q., Petras, I., and Xue, D. (2009), Fractional order control-a tutorial, in 2009 American Control Conference, 1397-1411. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., and Feliu, V. (2010), Fractional-order Systems and Controls- Fundamentals and Applications, London: Springer-Verlag. Tan, N., Kaya, I., Yeroglu, C., and Atherton, D.P. (2006), Computation of stabilizing PI and PID controllers using the stability boundary locus, Energy Convers. Manag., 47(18-19), 3045-3058. Atherton, D.P., Tan, N., and Y¨ uce, A. (2015), Methods for computing the time response of fractional-order systems, IET Control Theory Appl., 9(6), 817-830.

Journal of Applied Nonlinear Dynamics 8(1) (2019) 23-33

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Active Wave Control of a Flexible Beam Using Fractional Derivative Feedback Masaharu Kuroda†, Hiroki Matsubuchi Department of Mechanical Engineering, University of Hyogo, Bldg. 6, 2167 Shosha, Himeji, Hyogo 671-2280, Japan Submission Info Communicated by C. I. Muresan Received 30 July 2017 Accepted 1 October 2017 Available online 1 April 2019 Keywords Fractional calculus Vibration Wave Control Flexible structure

Abstract The existence of active wave control has been known in the field of the vibration control of large-size space structures (LSS) since the 1960s. Recently, with the goal of energy and resource conservation, active wave control has come into the spotlight again in the field of the vibration suppression of light and thin members widely used in mechanical structures, including automobiles. Therefore, achieving active wave control is both an old and a new problem. A vibration suppression problem for a thin cantilevered beam is presented as an example for discussion. √ Results clarified that the active wave controller includes √ s and s s terms. Those terms are realized as a 1/2-order derivative and a 3/2-order derivative using fractional calculus. The active wave controller is realized through fractional calculus, which is shown to be an important step in the analysis of this problem. Specifically, the active wave controller can be implemented using fractional derivative feedback. The controller involving the fractional derivatives is realized with a digital signal processor based on definitions of fractional calculus. The vibration suppression effect of active wave control is demonstrated both numerically and experimentally. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Recently, with the goal of energy and resource conservation, light and thin members have come to be widely used in mechanical structures, including automobiles. No machinery can avoid vibrating, and once vibration modes with low damping factors are excited, it takes a long time for those vibrations to be attenuated. Needless to say, such vibrations exert a bad influence on the performance of the machinery. Therefore, many studies have been carried out on the vibration control problem of light and thin mechanical structures. However, most of those studies are of modal control based on a modal analysis of the mechanical structures. As long as a relatively small number of vibration modes are targeted for control, the modal control methodology can work effectively. However, when dozens of vibration modes are targeted for control in light and thin mechanical structures, the effectiveness of modal control is limited. † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.003

24

Masaharu Kuroda, Hiroki Matsubuchi / Journal of Applied Nonlinear Dynamics 8(1) (2019) 23–33

Active wave control can be said to be one approach that overcomes the limits of modal control [1,2]. The energy introduced by a disturbance produces a progressing wave in the mechanical structure that becomes a reflecting wave at the boundary of the mechanical structure. Furthermore, the progressing wave and the reflecting wave interfere with each other to generate a standing wave. Eventually, a vibration mode is excited. Active wave control is a method for suppressing the vibration by eliminating either the progressing wave or the reflecting wave in order to impede the generation of the standing wave [3, 4]. Iwamoto and Tanaka implemented active wave control for a flexible beam using a wave filter which consists of several point-sensors [5, 6]. This paper deals with the relationship between active wave control and fractional calculus. The existence of active wave control has been known in the field of the vibration control of large-size space structures (LSS) since the 1960s. For the reasons mentioned in the first three paragraphs, active wave control has come into the spotlight again recently. However, fractional calculus is absolutely imperative for its realization. Specifically, the active wave controller can be implemented using fractional derivative feedback. The controller involving the fractional derivatives is realized with a digital signal processor based on definitions of fractional calculus. In this study, active wave control was applied to the vibration control problem of a flexible thin cantilevered beam. As a result, the transfer function of the active wave controller contains non-integer √ √ powers of the Laplace operator s such as s and s s that are interpreted as 1/2-order and 3/2-order derivative elements [7]. Therefore, based on definitions of fractional calculus, the active wave controller was realized using the fractional derivative response of the structure. Finally, the vibration suppression effect of active wave control was demonstrated both numerically and experimentally.

2 Theory 2.1

Active wave control

First of all, the vibration suppression problem of a flexible beam is considered. Suppose that we have a slender cantilever beam hanging perpendicularly. The dynamics of this flexible beam can be modeled as eq. (1) according to the Euler-Bernoulli theory. EI

∂ 4u ∂ 2u + ρ A = 0. ∂ x4 ∂ t2

(1)

For simplicity, the terms related to damping and gravity are neglected in the following design process of the controller. Here, u(x,t) expresses the deflection displacement [m], E is the Young’s modulus [N/m2 ], I is the second moment of area [m4 ], ρ is the density [kg/m3 ], and A is the cross-sectional area [m2 ] of the beam. Equation (1) can be converted into an ordinary differential equation through the Laplace transform in which the Laplace operator is s. d4u (2) a2 4 + s2 u = 0. dx Here, a2 = EI/ρ A. In order to avoid additional notation, the converted variables are hereafter described with the same notation as the time-dependent variables. T  For the analysis, the state vector U = u, ˙ θ˙ , m, q is introduced. Here, u˙ = su is the time derivative of the deflection displacement [m/s]; θ˙ = s(du/dx) is the time derivative of the deflection angle [rad/s]; m = (a/EI) M is the bending moment, in which M = EI(d 2 u/dx2 ) [Nm]; and q = (a/EI) Q is the shear force, in which Q = EI(d 3 u/dx3 ) [N]. By use of the state vector U , the converted Euler-Bernoulli equation of the beam can be written as

Masaharu Kuroda, Hiroki Matsubuchi / Journal of Applied Nonlinear Dynamics 8(1) (2019) 23–33

&DQWLOHYHUHG EHDP

b1

a1

b2

a2

25

&RQWUROIRUFHV Fig. 1 Schematic diagram of propagating waves a1 , a2 , b1 , and b2 .

follows.



0 ⎢ U dU ⎢ 0 =⎣ 0 dx −p

1 0 0 0

0 p 0 0

⎤ 0 0 ⎥ ⎥ U = AU . 1 ⎦ 0

(3)

Here, p = s/a [3]. Because the imaginary variables are not exhibited explicitly, eq. (3) is able to be block-diagonalized into Jordan canonical form using the transformation of eq. (5) [8]. ⎡

1 W p 1⎢ dW −1 = ( )2 ⎢ dx 2 ⎣0 0 ⎡√ U=

2p

3/2 1⎢ ⎢ p ⎣ 0 2 −p3/2

1 1 0 0

0 0 −1 −1

⎤ 0 0⎥ ⎥W , 1⎦ −1

√ ⎤ 2p 0 3/2 ⎥ p3/2 −p3/2 p√ ⎥W = V W . √ 2p 0 − 2p⎦ p3/2 p3/2 p3/2

(4)

0

(5)

According to eq. (5), the state variables in the old vector U are converted into the state variables in the new vector W . This transformation can be interpreted as the state variable expression by terms of the propagating waves. This new vector W can be arranged as W = (a1 , a2 , b1 , b2 )T . ⎡√ ⎤ √ 2 1/ p √0 a1 √ ⎢ ⎢ a2 ⎥ 2 ⎢ ⎥ = 1 ⎢√0 1/ √p ⎣b1 ⎦ 2p ⎣ 2 −1/ p 0 √ √ b2 0 1/ p − 2 ⎡

√ ⎤⎡ ⎤ −1/ p u˙ √ ⎥⎢ ˙ ⎥ 1/ p ⎥ ⎢ θ ⎥ . √ 1/ p ⎦ ⎣m ⎦ √ q 1/ p

(6)

Each component of this new state vector W is the amplitude of a propagating wave mode as shown in Fig. 1. The components a1 and a2 are the amplitudes of the wave modes which are progressing toward the fixed-end side of the beam from the free-end side of the beam, whereas the components b1 and b2 are the amplitudes of the wave modes which are coming from the fixed-end of the beam and going toward the free-end of the beam. Next, the active wave controller is derived. In order to eliminate the waves coming from the fixed end of the beam, the components b1 and b2 in W in eq. (6) are each set to zero at the control actuator location. Consequently, the active wave controller can be designed. As a result, the control force (mc , qc )T can be obtained as follows [3].





2as u −s u ˙ 2a/s −1 mc = = ˙ qc θ θ 1 − 2s/a −s 2s/a s

(7)

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Masaharu Kuroda, Hiroki Matsubuchi / Journal of Applied Nonlinear Dynamics 8(1) (2019) 23–33

Of course, the vibration caused by the remaining waves a1 and a2 propagating from the free end of the beam exists. However, standing waves cannot be generated from interference between waves. This is because the waves b1 and b2 have already been eliminated, and so intensifying interference never occurs between the incoming waves (a1 and a2 ) and the outgoing waves (b1 and b2 ). Eventually, the characteristic of a semi-infinite long beam stabilize in the finite long beam. Therefore, the excitation of the vibration modes is suppressed. It is worthwhile to note that information obtained through a modal analysis of the structure, such as eigen-frequencies and eigenmodes, is not required at all in the design procedure of the active wave controller. Therefore, a controller which has a broad frequency range can be designed free from the limitations of the modal-analysis-based method. √ √ Instead, terms such as s and s s appear in the controller. Because these terms can be interpreted as a 1/2-order derivative element and a 3/2-order derivative element, the concept of the fractional calculus is essential. 2.2

Fractional calculus

In order to implement fractional derivative elements with a digital signal processor, first, an algorithm based on the Gr¨ unwald-Letnikov definition is derived [9]. G q a Dt f

n−1

(t) =

f (k) (a) ∑ Γ (k − q + 1) (t − a)k−q + k=0

ˆ

t

a

(t − τ )n−q−1 (n) f (τ ) d τ . Γ (n − q)

(8)

In this definition, D is a differentiation operator, G denotes Gr¨ unwald-Letnikov, q is the differentiation order, a is the lower limit of the integral, and t is the variable by which f (t) is differentiated. In the case that 0 < q < 1 and n = 1, eq. (8) can be rewritten as follows. G q a Dt f

(t) =

1 (t − a)−q f (a) + Γ (1 − q) Γ (1 − q)

ˆ a

t

(t − τ )−q

d f (τ ) dτ . dτ

(9)

In addition, the condition T = t − a = const. is adopted. This so-called “short memory principle” elicits the L1 algorithm. In other words, the discretized version of the Gr¨ unwald-Letnikov definition of the fractional derivative can be expressed as follows [10]. G q a Dt f (t)

=

( j + 1)T T −q N q (1 − q) f (t − T ) N−1 jT [ ))(( j + 1)1−q − j1−q )}]. + ∑ {( f (t − ) − f (t − q Γ(2 − q) N N N j=1

(10)

In eq. (10), N is the number of histories and T is the history time. Practically, N must be chosen based on the performance of the digital signal processor (DSP). As shown in Fig. 2, the L1 algorithm expressed by eq. (10) is converted into a digital program using MATLAB/Simulink. First of all, the input signal f (t) is split and sent to two blocks. One signal is kept as f (t) because it is only multiplied by the gain of one. The other is sent through a unit delay block. This unit delay block is an element which holds the inputted signal for one sampling period before outputting the signal. Therefore, the second signal is outputted as f (t − NT ). Next, the difference signal between the two outputs is obtained. Similarly, N unit delay elements are connected in series, and the difference signals are outputted in matrix form. Moreover, f (t − T ) appearing during this process is outputted separately. Furthermore, after this matrix signal is multiplied by the corresponding coefficient matrix and f (t − T ) is multiplied by the corresponding coefficient, those two outputs are added to create one signal. Finally, the signal is multiplied by the final coefficient, and Dq f (t) can be obtained. √ Figure 3 shows the transfer function of s realized with fractional calculus. The 3/2-order derivative can be obtained by the 1/2-order derivative of the first-order derivative of the signal. Figure 4 shows

Masaharu Kuroda, Hiroki Matsubuchi / Journal of Applied Nonlinear Dynamics 8(1) (2019) 23–33

27

 Fig. 2 MATLAB/Simulink program for the L1 algorithm.

 √ Fig. 3 Transfer function of s realized with fractional calculus (N = 402, T/N = 0.002 s).

 √ Fig. 4 Transfer function of s s realized with fractional calculus (N = 402, T/N = 0.002 s).

√ √ the transfer function of s s realized with fractional calculus. The transfer function of s has a gain characteristic with a slope of 10 dB/decade and a phase characteristic of 45 degrees. On the other √ hand, the transfer function of s s has a gain characteristic with a slope of 30 dB/decade and a phase characteristic of 135 degrees. As shown in Figs. 3 and 4, these characteristics are approximately realized except in the low frequency range.

3 Numerical simulation Next, the mathematical model for the numerical simulation is explained. As shown in Fig. 5, the disturbance force fd acts at xd . As the control forces, the shear force and the bending moment are supposed to be applied. In this research, the colocation case in which the deflection and the deflection angle can be measured at the location where the control forces act is considered. The shear force qc is

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Fig. 5 Locations of disturbance, control forces, and sensors.

supposed to act at xc and the bending-moment actuator is supposed to be attached from x1 to x2 . Utilizing the Dirac delta function δ and the Heaviside step function H, the equation of motion of the beam with control can be written as follows [11]. EI

∂ 4u ∂ 2u ∂2 + ρ A = f δ (x − x ) + q δ (x − x ) + m {H (x − x2 ) − H (x − x1 )} . d d c c c ∂ x4 ∂ t2 ∂ x2

(11)

On the right-hand side, the first term is the disturbance, the second term is the shear force, and the third term is the bending moment. The deflection of the beam can be expressed with the modal expansion method as follows. u (x, t) = ∑ am (t) φm (x).

(12)

m

Furthermore, each eigenmode function is normalized and orthogonalized. ˆ l φm φn dx = δmn 0

(13)

Here, l is the length of the beam and δmn is the Kronecher delta. The shear force and the bending moment have been obtained as the control forces by eq. (7). √ 1 ∂u ∂u mc = Gm {− (xc ) + 2aDt2 (xc )}, ∂t ∂x

2 3 2 2 ∂ u D u(xc ) + (xc )}. qc = Gq {− a t ∂ t∂ x In eqs. (14) and (15), Gm and Gq are the respective feedback gains. Substituting eqs. (12), (14), and (15) into eq. (11) yields the following equation. d 4 φm d 2 am + ρ A φ ∑ 2 m dx4 m m dt

3 2 dam d φm (xc )}δ (x − xc ) = fd δ (x − xd ) + Gq {− Dt2 am φm (xc ) + ∑ ∑ a m m dt dx EI ∑ am

(14) (15)

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+ Gm {− ∑ m

1 √ dam ∂ φm ∂2 (xc )} 2 {H(x − x2 ) − H(x − x1)}. φm (xc ) + 2a ∑ Dt2 am dt ∂x ∂x m

29

(16)

Both sides of eq. (16) are multiplied by the eigenmode function φn and integrated from 0 to l with respect to x. Utilizing also the relationships expressed in eqs. (13), (17), and (18) [11], we obtain eq. (19).

∂2 ∂ {δ (x − x2 ) − δ (x − x1 )} , {H (x − x2 ) − H (x − x1 )} = 2 ∂x ∂x ˆ

l 0

d 2 an ωn2 ρ Aan + ρ A 2 dt

φn

d φn d φn ∂ {δ (x − x2 ) − δ (x − x1 )} = − (x2 ) + (x1 ) , ∂x dx dx

(17)

(18)

3 2 dam d φm (xc )}φn (xc ) Dt2 am φm (xc ) + ∑ ∑ a m m dt dx 1 √ d φn d φn dam ∂ φm (xc )}{− (x2 ) + (x1 )} φm (xc ) + 2a ∑ Dt2 am +Gm {− ∑ ∂x dx dx m dt m

= fd φn (xd ) + Gq {−

(19)

Based on eq. (19), impulse responses and frequency responses are simulated numerically. In practice, some damping effect is added to eq. (19). The results herein are depicted as impulse response and frequency response at the sensor location when an impact force is applied at the free end of the beam. Those for numerical simulations of the case without control are shown in Figs. 6 and 7, respectively. As shown in Fig. 6, the influence of the second mode is clearly recognizable. As shown in Fig. 7, the first vibration mode is a very steep 30.4 dB and some higher modes are also prominent. Figures 8 and 9 show the respective simulation results with active wave control in which both a shear-force actuator and a bending-moment actuator are utilized. Comparing Fig. 8 with Fig. 6, the amplitude of the first wave is attenuated from 1.26 × 10−4 m to 0.480 × 10−4 m. As shown in Fig. 8, the transient wave converges in five seconds. As shown in Fig. 9, not only the resonant peak of the first mode but also that of the second mode has disappeared. Therefore, vibration suppression by active wave control is very effective not only in the time domain but also in the frequency domain. However, a contact-type shear-force actuator such as an electrodynamic shaker cannot be used in the experiments. This is because the mass of the moving part of the electrodynamic shaker affects the light and thin cantilevered beam so that the dynamic characteristics of the cantilever beam itself may be altered strongly. Therefore, the vibration suppression effects with active wave control in which only the bending-moment actuator is applied were simulated, shown in Figs. 10 and 11. As shown in Fig. 11, the resonant peak of the first mode is lessened by 2.0 dB. The higher modes are still prominent. Although vibration suppression effects can be confirmed in comparison with the no-control case, the control effects are substantially reduced. Without the shear-force actuator, the controller can no longer realize perfect active wave control.

4 Experiment Photographs of the experimental setup and the cantilever beam are shown in Fig. 12. A thin slender steel beam (length: 875 mm, width: 20 mm, thickness: 0.4 mm, right photograph) is suspended vertically as a cantilever (left photograph). A schematic diagram of the experimental setup is shown in Fig. 13. The control system consists of two laser displacement sensors (IL-S100 [measurement range 70 to 130 mm, repeatability 4 mm,

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 Fig. 6 tion).

Impulse response (without control, simula-

 Fig. 8 Impulse response (active wave control with the shear-force and bending-moment actuators, simulation).

 Fig. 10 Impulse response (active wave control with only the bending-moment actuator, simulation).

 Fig. 7 Frequency response (without control, simulation).

 Fig. 9 Frequency response (active wave control with the shear-force and bending-moment actuators, simulation).

 Fig. 11 Frequency response (active wave control with only the bending-moment actuator, simulation).

sampling rate 0.33/1/2/5 ms]; Keyence Co.) for sensing, a piezoelectric actuator (Material code D [dielectric constant ε33 /ε0 = 4500, ε11 /ε0 = 4700, dielectric loss tangent tan δ = 2.0(%)], 58 mm × 18 mm × 0.18 mm; Nihon Ceratec Co. Ltd.) for actuation, and a DSP (DSP6085SH [600 MHz, Max 4.2 GFLOPS]; MTT Corp.) for the controller. The deflection of the beam can be measured as the average of the two sensor outputs. The deflection angle of the beam can be obtained as the difference of the two sensor outputs divided by the distance between the two sensors. Therefore, the deflection and deflection angle can be utilized as the feedback signals. A bending-moment actuator based on a piezo element is utilized as the control actuator. The bending-moment actuator is attached at 437.5 mm from the fixed end of the beam and the actuator length is 60 mm.

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 (a) Experimental setup Material SUS430

Length 875 mm

(b) Cantilever beam Width 20 mm

Thickness 0.4 mm

Fig. 12 Photographs of (a) the experimental setup and (b) the cantilever.

 Fig. 13 Outline of the experimental setup.

31

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 Fig. 14 Impulse response (without control, experiment).

 Fig. 16 Impulse response (active wave control with only the bending-moment actuator, experiment).

Fig. 15 Frequency response (without control, experiment).

 Fig. 17 Frequency response (active wave control with only the bending-moment actuator, experiment).

For the DSP acting as the controller, an L1 algorithm programmed using MATLAB/Simulink is downloaded into the DSP through Real-Time Workshop, and then the fractional derivative controller is implemented. Figures 14 and 15 show the experimental results without control (as previously for the simulation results, these are in the form of, respectively, impulse responses and frequency responses at the sensor location when an impact force is applied at the free end of the beam). As shown in Fig. 14, the transient wave continues over 50 seconds. As shown in Fig. 15, the resonant peak of the first mode is very steep. The effect of the first mode stands out and other modes are unremarkable. Figures 16 and 17 are the results with active wave control in which only the bending-moment actuator is used. As shown in Fig. 16, the transient wave converges in 50 seconds. The amplitude of the first wave is hardly changed from the no-control case, 5.53 mm. As shown in Fig. 17, the resonant peak of the first mode is attenuated by 4.2 dB. Because of the lack of a shear-force actuator, active wave control cannot be achieved sufficiently. Although the vibration suppression effects are less than what they should be, control effects can be confirmed to some extent.

5 Conclusions Active wave control as a vibration suppression method for a flexible structure was investigated. The √ √ active wave controller contained s and s s terms. Because these terms can be interpreted as a 1/2order derivative element and a 3/2-order derivative element, the active wave controller could be realized using fractional derivative feedbacks. Both numerical simulation and experiment results demonstrated vibration suppression effects with active wave control for a light and thin cantilevered beam. The vibration suppression by active wave control with both a shear-force actuator and a bending-

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33

moment actuator is very effective not only in the time domain but also in the frequency domain. However, the vibration suppression effects by active wave control with only the bending-moment actuator are substantially reduced. Without the shear-force actuator, the controller can no longer realize perfect active wave control. In the future, a perfect active wave control should be realized by additional use of a non-contact-type shear-force actuator such as an electromagnet. Furthermore, objects to which the active wave control is applied should be extended from one-dimensional structures such as thin beams to two-dimensional structures such as thin plates.

References [1] MacMartin, D.G. and Hall, S.R. (1991), Control of Uncertain Structures Using an H∞ Power Flow Approach, J. Guidance, 14(3), 521-530. [2] Miller, D.W., Hall, S.R., and von Flotow, A.H. (1990), Optimal Control of Power Flow at Structural Junctions, J. Sound and Vibration, 140(3), 475-497. [3] Vaughan, D.R. (1968), Application of Distributed Parameter Concepts to Dynamic Analysis and Control of Bending Vibrations, Trans. ASME J. Basic Engineering, 157-166. [4] von Flotow, A.H. and Sch¨ afer, B. (1985), Experimental Comparison of Wave-Absorbing and Modal-Based Low-Authority Controllers for a Flexible Beam, AIAA Guidance, Navigation and Control Conference, 851922, 443-452. [5] Iwamoto, H. and Tanaka, N. (2003), Active Wave Feedforward Control of a Flexible Beam Using Wave Filter Constructed with Point Sensors, Transaction of the Japan Society of Mechanical Engineers Series C, 69(685), 2233-2239. [6] Iwamoto, H. and Tanaka, N. (2004), Active Wave Feedback Control of a Flexible Beam Using Wave Filter: Theoretical Verification of Basic Properties, Transaction of the Japan Society of Mechanical Engineers Series C, 70(689), 46-53. [7] Kuroda, M. (2007), Active Wave Control for Flexible Structures using Fractional Calculus, In J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Eds.) Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 435-448. [8] Matsuda, K. and Fujii, H. (1993), H∞ Optimized Wave-Absorbing Control: Analytical and Experimental Results, J. Guidance, Control, and Dynamics, 16(6), 1146-1153. [9] Podlubny, I. (1999), Fractional Differential Equations, Academic Press: San Diego. [10] Oldham, K. B. and Spanier, J. (2006), The Fractional Calculus, Dover: Mineola. [11] Fuller, C. R., Elliot, S. J. and Nelson, P. A. (1997), Active Control of Vibration, Academic Press: San Diego.

Journal of Applied Nonlinear Dynamics 8(1) (2019) 35-53

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Constrained Model Predictive Control for Linear Fractional-order Systems with Rational Approximation Mandar M. Joshi1 , Vishwesh A. Vyawahare2† 1 2

Electrical Engineering Department, Dar-al-Handasah, Pune, India, Department of Electronics Engineering, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai, India Submission Info Communicated by K.A. Markoski Received 30 July 2017 Accepted 2 October 2017 Available online 1 April 2019 Keywords Model predictive control Fractional-order systems Integer order approximation Internal model control

Abstract This work deals with the design of model predictive control (MPC) strategy for linear fractional-order (FO) systems. Two FO systems with different characteristics, highly oscillatory response and nonminimum phase type, are considered to test the performance of the MPC design. Conventionally, a system with FO dynamics is represented using a finite memory integer-order model. In order to evaluate the performance of MPC for such a situation, the integer-order rational approximation (Oustaloup’s recursive approximation) of the FO system is considered as the model, whereas the output of FO plant is calculated analytically by solving the linear FO differential equation at each sampling instant. The MPC methodology is applied in a constrained environment with limitations on the control input magnitude and its rate. The results confirm that designed MPC strategy works satisfactorily for the FO systems. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Model predictive control (MPC) is one of the few control strategies that are popular amongst both academia and industry. It is now a mature field with applications ranging from chemical, mechanical, power electronics, biological systems, and many more [1–3]. MPC provides the way to work near the limits with optimized performance [4]. The problems of actuator limits, saturation and measurement of different states of the system which generally arise in other control methodologies are inherently dealt in MPC. Fractional calculus, which deals with the derivatives and integrals of arbitrary non-integer order (real or complex), helps to represent a system in a more compact manner. Fractional-order (FO) models represent a system in more accurate ways [5] and hence systems with FO dynamics can be represented by a FO model more faithfully as compared to the conventional integer-order (IO) model. In this perspective, it is interesting to analyze the performance of MPC in presence of FO Models. Hence, utilization of fractional theory to develop model in MPC and its effects are worth exploring. † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.004

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The theory of linear FO systems is very well developed and various notions for analysis like stability, root locus, state-space analysis, controllability, observability, etc., have been established in a clear-cut way. The design of FO control (FO PID, optimal, robust controller) is also becoming a mature field [6]. Literature survey reveals that there are more than one strategies available for utilization of model predictive control for FO systems. Reference [7] uses Grunwald-Letnikov’s definition for replacing the non integer derivation operator. The performance is illustrated with practical results on a thermal system. Reference [8] represents a unique approach towards the predictive control methodology for FO system by developing a new fractional order predictive functional control. Hence, model predictive control for FO system is a contemporary research area. This paper presents the performance of MPC for FO systems if the system under consideration is represented using rational approximations. It is a general practice to use the state-space models for the implementation of MPC. An FO transfer function (FOTF) results in a pseudo state-space model which is not suitable for MPC. This warrants the use of a rational approximation, that is, an IO equivalent (transfer function) which can represent that FOTF and also can be converted to state-space model. An approximation for a fractional operator sα or s−α with 0 < α < 1 limits the approximation scope and defines the boundary for which approximation methods provide satisfactory representation. In this paper, MPC is applied to two FO systems: the standard FOTF given in [9] and a nonminimum phase type FO system using Oustaloup’s recursive approximation [6] to obtain the model for prediction. Comparison of the frequency and step response of the original and approximated FOTFs has been carried out to validate the replacement of original FO system with its approximated counterpart. In order to make MPC realistic, analytically calculated output equation for the FO system has been utilized to represent process model. Model predictive control strategy is tested on the FO system in the presence of constraints in the plant input parameters. Furthermore, the performance of the designed MPC is also tested in the presence of bounded disturbance and parametric uncertainty (see [10, 11]). The paper is organized as follows. Section 2 introduces important definitions in fractional calculus, dynamics of FO systems and Oustaloup’s recursive approximation method. Philosophy of model predictive control is given in brief in section 3. Section 4 discusses the details of the two Fractional-order systems used in this work. The results are presented in section 5 and section 6 concludes the work.

2 Fractional calculus The mathematics involving the study of derivatives and integrals of arbitrary non-integer order is more commonly known as the Fractional Calculus (FC). Various eminent mathematicians like L’hospital, Euler, Riemann, etc., and engineers and inventors like Heaviside, Caputo, have made profound contributions to this interesting branch of mathematics [12,13]. The topic is marked by various peculiarities, viz., non-unique definition of the derivative operator, non applicability of common laws of differentiation (semigroup property, chain rule, etc.), lack of an easy and straightforward geometrical and physical interpretation for the operators ([14, 15]), and many others. Also see [12, 16–18]. In recent years, the differential equations involving fractional derivatives and integrals (Fractional Differential Equations-FDEs) have found growing involvement in the modeling of real-world systems and control theory ([19–21]). One of the major applications of FDEs is in the modeling of anomalous diffusion occurring in complex systems [22]. There are now ample number of books, monographs, conference proceedings, available on fractional calculus and its applications [23–32] For a detailed history and bibliography of fractional calculus, see [13]. This section discusses about the important definitions and terms related to fractional calculus, and a brief introduction to the fractional-order systems.

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2.1

37

Fractional-order integrals and derivatives: definitions

A fractional integral or derivative operator a Dtα , where a and t are the limits of the operation and α ∈ R is the generalized integro-differential operator and popularly known as differintegral [17, 33]. It is defined as ⎧ α ⎪ d : α > 0, ⎪ ⎪ ⎪ dt α ⎪ ⎨ α 1 : α = 0, (1) a Dt = ⎪ ⎪ ˆ t ⎪ ⎪ ⎪ ⎩ (d τ )−α : α < 0. a

An FO integral of order α is an extension of the Cauchy’s formula for repeated integrals, which gives a closed-form convolution representation for the n successive integrations of a function f (t): ˆ t 1 n (t − τ )n−1 f (τ )d τ , n ∈ Z+ . (2) a Jt f (t) = (n − 1)! a Using the result (n − 1)! = Γ(n), and replacing n by a positive real number α , the Riemann-Liouville FO integral is defined as ˆ t 1 α (t − τ )α −1 f (τ )d τ , α ∈ R+ . (3) a Jt f (t) = Γ(α ) a The three definitions, most frequently used for the general fractional derivatives (FD) are the Grunwald-Letnikov, the Riemann- Liouville and the Caputo definition [12]. In all these definitions, the function f (t) is assumed to be sufficiently smooth and locally integrable. 1. The Grunwald-Letnikov (GL) definition: Using Podlubny’s short memory principle [16], it is defined as α a Dt

−α

f (t) = lim h h→0

[ t−a h ] ∑ (−1) j

α

C j f (t − jh),

(4)

j=0

where [x] means the integer part of x and α C j is the binomial coefficient. This definition is generally used for the numerical calculations. 2. The Riemann-Liouville (RL) definition: It is obtained using the Riemann-Liouville FO integral (3) and is given as ˆ dn t 1 f (τ ) α D f (t) = dτ , a t Γ(n − α ) dt n a (t − τ )α −n+1

(5)

for n − 1 < α < n, n ∈ Z+ and Γ(·) is the Gamma function. 3. The Caputo definition: It is defined as α a Dt

1 f (t) = Γ(n − α )

ˆ a

t

f n (τ ) dτ , (t − τ )α −n+1

(6)

for n − 1 < α < n, n ∈ Z+ , where f n (τ ) is the nth -order derivative of the function f (t). The lower limit frozen to a = 0 if the function is causal. It is seen that the Caputo definition is more restrictive than the RL. Nevertheless, it is preferred by the engineers and physicists because the FDEs with Caputo derivatives have the same initial conditions as that for the integer order differential equations . Note that the FDs calculated using these three definitions coincide for an initially relaxed function (that is, f (t)|t=0 = 0).

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2.2

Fractional-order system dynamics

FO systems can be represented by IO models as well as FO models. Any system with fractional dynamics is better represented by fractional order models [34]. Theoretically, fractional order systems are of infinite order. A fractional-order system can be mathematically represented by following equation: an Dαn y(t) + an−1 Dαn−1 y(t) + · · · + a1 Dα1 y(t) + a0 y(t) = bm Dβm u(t) + bm−1 Dβm−1 u(t) + · · · + b1 Dβ1 u(t) + b0 u(t). where Dαi and Dβi represents Caputo or Riemann Liouville Fractional Derivative (RLFD). 2.3

Oustaloup’s recursive approximation

In this technique, frequency domain identification is used to obtain a rational function (integer-order function) of a given fractional operator s±α . This can be written as an optimization problem, where the cost function will be of the form [35] ˆ 2 ˆ dw, (7) J = W (w)|G(w) − G(w)| where W (w) is the weighting function and G(w) represents the frequency response of original function ˆ represents the frequency response of IO approximation of which in this case is s±α . The symbol G(w) α s . The cost function is minimized in order to achieve the best approximation [35]. Frequency domain identification is used to obtain the rational function of a given fractional operator. This approximation is given by following equations [35]: N s + w k , (8) sα ≈ K ∏ s + w k k=1 where the poles, zeros and gain can be evaluated as [6]: (2k−1−α )/N

wk = wl wu K=

wαh

(2k−1+α )/N

wk = wl wu  wu = wh /wl ,

,

where wl and wh are lower and higher frequency values. Simpler approximations can be obtained by using low values of N, but it will also cause the appearance of the ripple in both gain and phase behaviors. To practically eliminate such a ripple, N has to be increased. A higher value of N makes the computation heavier [5]. Although this paper presents results obtained by using Oustaloup recursive approximation (ORA) method, there are other methods of approximation that can be utilized to obtain the approximation of an FO system. A study on the use of discrete approximation methods for fractional order system was carried out to test their feasibility [36]. ORA has been used for designing and implementation of fractional controllers. Reference [37] shows the tuning of fractional PID controllers using ORA. Reference [38] presents the implementation of non-integer order controller for air heating process trainer with the help of ORA. Efficient rules for determining the suitable value of parameters used for ORA in case of FOPID controllers is presented in [39]. All this literature survey confirms that ORA is one of the powerful approximation method for FO operators and hence is used in this work.

3 Model predictive control Model Predictive Control (MPC) is an optimal control strategy based on numerical optimization. It is a computational technique for improving control performance in applications within the process and

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39

due to this predictive control has become arguably the most widespread advanced control methodology currently in use in industry [1–3, 40, 41]. 3.1

Principle of model predictive control

For a system to be controlled by MPC, the defined reference trajectory r(k) is the ideal trajectory along which a plant should return to the set-point trajectory w(k), at any instant k. The current error between the output signal y(k) and the set-point is represented as: err(k) = w(k) − y(k).

(9)

The reference trajectory is chosen such that if the output followed it exactly, then the error after i steps would be (10) err(k + i) = exp−iTs /Tre f err(k), where Tre f is the time constant of exponential assuming that reference trajectory approaches the setpoint exponentially and Ts is the sampling interval. Therefore, the reference trajectory is given as: r(k + i|k) = w(k + i) − y(k + i).

(11)

The notation r(k + i|k) represents that the reference trajectory depends on the conditions at time k [5, 42]. Once a future trajectory has been obtained, the first element of that trajectory is chosen to be applied as the input to the plant. The notation uˆ represents the predicted value input at time k + i at instant k; the actual input u(k + i) will probably be different than u. ˆ Hence u(k) becomes u(k|k) ˆ and the whole sequence of prediction, output measurement and evaluation of input trajectory is repeated for every sampling instance. This is known as receding horizon strategy, since prediction horizon slides along the sampling time scale by one sampling instance at each step while its length remains the same. The control law for the unconstrained case is used to evaluate optimal control effort u∗ to be given to the plant/process and to the model (for details refer [4]). At each instant the optimal control effort u∗ is updated by evaluating the control law by considering revised process output yk , model output yˆk and the states x. 3.2

Constrained model predictive control

Although MPC was basically used for multivariable systems and was used extensively in large oil refineries and chemical process, it has also been used for very focused applications on small/large scale. Park in his paper [43] develops MPC for spacecraft rendezvous and docking of the spacecraft with a rotating platform and also for avoiding debris, making MPC utilizable for precise operations like spacecraft maneuvering. Dunbar uses MPC for coordinating multi-vehicle formation [44]. Falcone, et al., have developed predictive active steering control for autonomous vehicle systems [45]. Apart from automotive/moving systems MPC was also used for energy efficient climate control of a building [46]. MPC has also been used for control of power converters [47] and industrial turbo-diesel engine [48], making it a control method adopted on a large scale as well as small scale and on many different applications. MPC offers application of constraints to the system for evaluation of optimal results, which depicts the real world limitations on the system. Furthermore, physical limitations of system can be accounted using constraints. Once the constraints are fixed then it is desired that the optimal predictions should not violate those constraints. Although, it is desired that constraints should not be violated, still to find feasible solution soft constraints can be relaxed. Hence, relaxing the soft constraints to find feasible solution is inevitable unless optimal predictions within the limits of constraints get feasible solution. In this paper input and input rate were used as constraints to simulate the actuator limitations in the real scenario [49–51].

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Online computation r(k)

u*

Optimizer

FO Plant

Model states xm

y(k)

y(k)

y(k)

Fig. 1 Block Diagram for MPC [4].

4 Fractional-order systems Two different FO systems (single-input-single-output) were used to simulate MPC strategies. The details are given below. 4.1

Podlubny system

This FO system is the well-known Podlubny system [16] and is represented as: G1(s) =

1 Y (s) = . 2.2 U (s) 0.8s + 0.5s0.9 + 1

(12)

The FOTF (12) corresponds in the time domain to the three term fractional order differential equation. The system in underdamped in nature with oscillatory step response. It has two poles in the principal Riemann sheet close to the imaginary axis. 4.2

Non-minimum phase type fractional-order systems

This FO system is a non-minimum phase (NMP) type system where α , β , a, b ∈ R+ and is represented as below: sβ − b . (13) G2(s) = α s +a The output of an NMP system to a step input shows an initial inverse response and hence it is difficult to control [52]. The operators sα or sβ can be represented in the form of sv sδ for α , β > 1 where v is an integer and δ is between zero and one, which limits the scope of approximation. Hence for simulations, the assumption of 0 < α , β < 1 should suffice to represent the range of this type of system (13) for approximation. The block diagram of MPC strategy for a fractional-order plant is shown in Fig. 1. Here yk is process output, yˆk is model output, xm represent model states and u∗ is optimal control effort. The block plant model is the rational approximation of the FO system which gives the future states whereas the block FO Plant represents the original FO system the output of which is calculated analytically.

5 Results In order to make the simulations more realistic, the output states of the plant were evaluated separately using the original plant. In the real world, output states will be measured from the plant/process and

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41

will be used with the predicted states for MPC. Following subsection presents the derivation for process output equation of the FO plant. 5.1

Process output evaluation

When MPC is applied to a real process/plant, process output yk is measured in real time at every instant for evaluation of optimal control effort. In case of simulations, generally the model of the process/plant is used to evaluate the process output. Hence, the bias [yk − yˆk ] is redundant as yk is equal to yˆk at each instant. In this work, the process output yk is evaluated by formulating an output equation in time domain for the input given at each instant to the FO plant/process for respective α values (in case of NMP FO system). The expression of yk in time domain was obtained by applying inverse Laplace transformation [6] to the FOTF (12, 13) for different α values. Laplace transform of Caputo fractional derivative and Mittag-Leffler Function (MLF) are used to evaluate the expression of process output in time domain. For the evaluation of process output expression, definition of 1parameter MLF, 2-parameter MLF and that of Laplace transform of Caputo Derivative is required and are as follows: Definition for Laplace transform of Caputo fractional differentiation is [53] L[C0 Dtα f (t)] = sα F(s) −

m−1

dk

∑ sα −k−1( dt k f (t))t=0 ,

(14)

k=0

where 0 < α < 1, m − 1 < α ≤ m and m ∈ N. Definition for 1-parameter MLF is [54] ∞

Eα (t) =

tk

∑ Γ(α k + 1) ,

(15)

k=0

where α ∈ R+ . Definition for 2-parameter MLF is [54] ∞

Eα ,β (t) =

tk

∑ Γ(α k + β ) ,

(16)

k=0

where α , β ∈ R+ . Analytical expression for process output yk is obtained by using inverse Laplace transformation on the expression by rewriting transfer function in terms of Y (s), U (s) using definition of Caputo fractional differentiation and using Laplace transform of Caputo fractional differentiation (14). Process output equations for the FO systems mentioned at section 5 of this paper are presented below: 5.2

Podlubny system

1 Y (s) = . 2.2 U (s) 0.8s + 0.5s0.9 + 1

(17a)

above expression can be rearranged as shown below (0.8s2.2 + 0.5s0.9 + 1)Y (s) = U (s).

(17b)

replacing differential operator sα with Caputo differentiation C0 Dtα to account initial/current conditions (0.8C0 Dt2.2 + 0.5C0 Dt0.9 + 1)Y (t) = U (t).

(17c)

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for 0 < α < 1 , using (14) for Laplace transform of Caputo differentiation, (17c) becomes d Y (t)t=0 ] + 0.5[s0.9Y (s) −Y (0)] +Y (s) = U (s). dt 0.8s0.2 ( dtd Y (t)t=0 ) Y (0)(0.8s1.2 + 0.5) U (s) + + . Y (s) = 0.8s2.2 + 0.5s0.9 + 1 0.8s2.2 + 0.5s0.9 + 1 0.8s2.2 + 0.5s0.9 + 1

0.8[s2.2Y (s) − s1.2Y (0) − s0.2

(17d) (17e)

U(t) is a constant value at any K th instant and hence U (s) = uk−1 s where uk−1 is the constant value obtained as the optimal control effort, dtd Y (t)t=0 can be replaced by [Y (k − 1) − Y (k − 2)]/K and again that will be a constant value and hence its Laplace transform will be a constant value divided by s. Substituting the above said changes, equation (17e) becomes: Y (s) =

Y (0)(0.8s1.2 + 0.5) 0.8s0.2 ([Y (k − 1) −Y (k − 2)]/K) U (k − 1) + + . s(0.8s2.2 + 0.5s0.9 + 1) 0.8s2.2 + 0.5s0.9 + 1 s(0.8s2.2 + 0.5s0.9 + 1)

(17f)

This expression is simplified to obtain Laplace inverse. The expression (17f) is split into three parts to obtain residues. Y1 (s) =

1 (0.8s2.2 + 0.5s0.9 + 1)

.

0.8s1.2 + 0.5 . (0.8s2.2 + 0.5s0.9 + 1) 0.8s0.2 . Y3 (s) = (0.8s2.2 + 0.5s0.9 + 1)

Y2 (s) =

(17g) (17h) (17i)

substitute s0.1 = w 1 . (0.8w22 + 0.5w9 + 1) 0.8w12 + 0.5 . Y2 (s0.1 ) = Y2 (w) = (0.8w22 + 0.5w9 + 1) 0.8w02 . Y3 (s0.1 ) = Y3 (w) = (0.8w22 + 0.5w9 + 1) Y1 (s0.1 ) = Y1 (w) =

(17j) (17k) (17l)

Now by using ‘residue’ command of MATLAB [55] residues for each expression obtained, for example R2 R3 Rn R1 + + + ··· + . w − P1 w − P2 w − P3 w − Pn

(17m)

R2 R3 Rn R1 + + + · · · + 0.1 . s0.1 − P1 s0.1 − P2 s0.1 − P3 s − Pn

(17n)

Y1 (w) = replacing w = s0.1 Y1 (s) =

Similarly residues for Y2 (s) and Y3 (s) were obtained and by using following expressions evaluation of Laplace inverse becomes possible. L−1 (

R1 sa − P1

(a−1)

) = (tk

)(R1)(Ea,a (P1(tka )));

(17o)

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L−1 (

R1 s(sa − P1)

) = (tka )(R1)(Ea,a+1 (P1(tka ))),

43

(17p)

where tk is the kth instant of time period t. Finally an expression for yk can be obtained for Podlubny system which takes into account initial/current conditions of plant. yk = s1k uk−1 + s2k yk−1 + s3k [(yk−1 − yk−2 )/k],

(18)

where s1k , s2k , s3k are constant values obtained at instant k through MLF of respective terms. 5.3

Non-minimum phase type system

sβ − b Y (s) = α . U (s) s + a

(19a)

sα Y (s) + aY (s) = sβ U (s) − bU (s).

(19b)

replacing differential operator sα with Caputo differentiation C0 Dtα to account initial/current conditions C α 0 Dt Y (t) + aY (t)

β

=C0 Dt U (t) − bU (s).

(19c)

for 0 < α , β < 1 , using (14) for Laplace transform of Caputo differentiation, (19c) becomes sα Y (s) −Y (0) + aY (s) = sβ U (s) −U (0) − bU (s). Y (s) =

Y (0) U (0) U (s)(sβ − b) + α − . sα + a s + a sα + a

(19d) (19e)

taking Laplace inverse term by term L−1 (

β U (s)(sβ − b) −1 (s − b) ) = L ) ∗ L−1 (U (s)). ( sα + a sα + a

(19f)

U (t) is a constant value at any K th instant and hence U (s) = uk−1 s where uk−1 is the constant value obtained as the optimal control effort, using [6] for Laplace inverse, (19f) becomes β buk−1 U (s)(sβ − b) −1 s uk−1 ) = L − ). ( sα + a s(sα + a) s(sα + a)

(19g)

U (s)(sβ − b) b ) = uk−1 [t α −β Eα ,(α −β +1) (−at α )] − ( ) × uk−1 [1 − Eα (−at α )]. sα + a a

(19h)

L−1 ( L−1 (

Again U (0) is a constant value at any K th instant and hence U (0) = uk−2 s where uk−2 is the constant value obtained as the optimal control effort, using [6] for Laplace inverse of second term of (19e) L−1 (

Y (0) ) = yk−1 × [(t α −1 )Eα ,α (−at α )]. sα + a

(19i)

Again using [6] for Laplace inverse of third term of (19e) L−1 (

1 U (0) ) = × uk−2 [1 − Eα (−at α )]. α s +a a

(19j)

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44

hence, yk is obtained by using (19h), (19i) and (19j) as: b yk = uk−1 [(t α −β Eα ,(α −β +1)(−at α )) − ( )(1 − Eα (−at α ))] a 1 +yk−1 [tkα −1 Eα ,α (−atkα )] − ( ) × uk−2 [1 − Eα (−at α )]. a Now, by substituting a = 1 and b = 1, an expression for yk can be obtained for the FO plant described by (13) with a = b = 1 and this expression takes into account initial/present conditions of plant. yk = uk−1 [(t α −β Eα ,(α −β +1) (−t α )) − (1 − Eα (−t α ))] + yk−1 [tkα −1 Eα ,α (−tkα )] − ×uk−2 [1 − Eα (−t α )].

(21)

Thus, the process output equations are derived for both the FO systems. These equations will provide real plant output states. This evaluation makes the simulation more realistic and signifies the difference in real output and model output which is similar for output states in case of a real working plant. In the next subsection, steps for the formulation of MPC for FO system using rational approximation are given. 5.4

Model predictive control of FO system - an example

1. FO Plant The NMP type system (13) with a = 1, b = 1, β = 0.5 and α = 0.5 is considered. G2(s) =

s0.5 − 1 . s0.5 + 1

(22)

2. Frequency range of operation We will consider the frequency range from 10−2 to 10−6 rad/sec. The results in this paper were obtained for the above said frequency range. 3. Oustaloup Approximation Using Oustaloup approximation method, we will evaluate rational approximation for (22). We will limit the iterations to two numbers only for this example. G2(s) =

999s4 + 1.008e8 s3 + 9.102e10 s2 + 8.101e11 s − 9e11 . 1001s4 + 1.012e8 s3 + 1.11e11 s2 + 1.21e12 s − 1.1e12

(23)

The accuracy of rationalization is very limited in this case due to small number of iterations. 4. Cost Function Choice of the cost function is a bit critical problem. It needs both engineering and theoretical knowledge to select a proper cost function. The cost function should be as simple as one can get away with for the desired performance. Choice of cost function affects the complexity of the implied optimization and hence it should taken into consideration that the cost function should be simple enough to have a straightforward optimization [4]. For this reason, 2-norm measures were used in this work as a cost function. ny

nu −1

i=1 ny

i=0

nu −1

i=1

i=0

J = ∑ rk+i − yk+i 22 + λ = ∑ ek+i 22 + λ

∑ Δuk+i 22

∑ Δuk+i 22,

(24a) (24b)

Magnitude in decibels

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45

FOTF Oustaloup

0 −100 −200 −300 −2 10

0

10 Frequency in rad/s

0

10 Frequency in rad/s

10

2

10

4

10

2

10

6

4

10

Phase in degrees

0 −50 −100 −150 −200 −2 10

10

Fig. 2 Bode Plot for G1(s) =

6

1 . 0.8s2.2 +0.5s0.9 +1

where λ is the weighting scalar; first term represents the sum of squares of the predicted tracking errors from an initial horizon to an output horizon ny and the second term represents the sum of squares of the control over the control horizon nu . It is assumed that control increments are zero beyond the control horizon, that is Δuk+i|k = 0, i ≥ nu . 5. Formulation of MPC strategy for FO system Optimization of cost function provides control law and MPC strategy is formulated. Once the results are obtained, it is necessary to check the rationalization of FO system to validate its application. The next subsection presents the same. 5.5

Comparison of frequency and step response

Any FO system (13) is basically an infinite dimensional system. The varying α and β represent each individual system of this type. The maximum range of transfer functions represented by fractionalorder system (13) was endeavored to encompass in the following results by considering extreme and middle values of α and β in the range of 0 < α , β < 1. In the plots, few transfer functions of FO system type (13) and Podlubny system (12) were compared with its obtained approximation via frequency as well as step response. FOTF represents the original FO transfer function and Oustaloup represents approximation obtained for the same FO transfer function. Fig. 2 and Fig. 3 show the frequency and step response for system (12) and Fig. 5, Fig. 6 depict the frequency and step response for system (13) with a = 1, b = 1, β = 0.5 and α = 0.5, while Fig. 8, Fig. 9, Fig. 11, and Fig. 12 show the same for system (13) with a = 1, b = 1, β = 0.5, α = 0.9 and a = 1, b = 1, β = 0.1, α = 0.9 respectively. The prediction equations were developed for strictly proper state space models. However, to make the state space model for the FO system of type (13) to be strictly proper in nature, a stable pole was added at a very high frequency [ −(e+15 )]). Discretization of the obtained state space model was carried out with sampling time T = 1s. Hence, the state space models used in this paper are controllable and

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1.6

FO Approxiamtion

1.4

Output in units

1.2 1 0.8 0.6 0.4 0.2 0 0

10

20

30 40 Time in seconds

Fig. 3 Step Response for G1(s) =

50

60

70

1 . 0.8s2.2 +0.5s0.9 +1

1

Process Reference Model

0.5

0 0

10

20 30 40 Output plot (Constrained)

50

60

10

20 30 40 Control Effort (Constrained)

50

60

1

0.5

0 0

Fig. 4 Output and control effort for G1(s) =

1 . 0.8s2.2 +0.5s0.9 +1

observable in nature. These six plots show that the approximated function captures all the dynamics of the original function, even in the presence of added pole. Hence, the approximated function can be used to obtain the state space model which will be used for prediction purpose in model predictive control strategy. Moreover, these results suggests that the effect of approximation of the plant and addition of pole preserves the nature of original system.

Magnitude in decibels

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47

0 −2 −4 FOTF Oustaloup

−6 −8 −2 10

0

10 Frequency in rad/s

0

10 Frequency in rad/s

10

2

10

4

10

2

10

6

4

10

Phase in degrees

200 150 100 50 0 −2 10

10

Fig. 5 Bode Plot for G2(s) =

6

s0.5 −1 . s0.5 +1

1

FOTF Oustaloup

0.8 0.6

Output in units

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

10

20

30 40 Time in seconds

50

Fig. 6 Step Response for G2(s) =

5.6

60

70

s0.5 −1 . s0.5 +1

Model predictive control of Podlubny system

Model predictive control with input rate constraint Δu  ±0.05 was applied on FO plant defined at (12). Results were obtained with input constraint −2 ≤ U ≤ +2. Fig. 3 shows reference, process output yk and model output yˆk for respective FO plants. Subsequent plot shows the optimal control effort u∗ at each instant for each plant. A sampling time of 1 second is used for these results. Here the simulation results with T = 1s are presented. It was observed that the simulation of MPC for this FO plant is also possible for other step sizes.

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Process Reference Model

1 0.8 0.6 0.4 0.2 0 0

50 100 Output plot (Constrained)

150

50 100 Control Effort (Constrained)

150

0 −0.2 −0.4 −0.6 0

Magnitude in decibels

Fig. 7 Output and control effort for G2(s) =

s0.5 −1 . s0.5 +1

0 −10 −20 −30 −40 −50 −2 10

FOTF Oustaloup 0

10 Frequency in rad/s

0

10 Frequency in rad/s

10

2

10

4

10

2

10

6

4

10

Phase in degrees

200 150 100 50 0 −50 −2 10

10

Fig. 8 Bode Plot for G2(s) =

5.7

6

s0.5 −1 . s0.9 +1

Model predictive control of non-minimum phase type FO system

Model predictive control strategy is applied to the FO system (13). Discrete state space model with T = 1s was used for the prediction and was obtained using ORA method (8). Process output equation derived above was used to deliver the output state. The output and control effort plots are shown in Fig.7, 10 and 13, for the tracking of reference signal. Output plot shows reference, process output yk and model output yˆk for respective FO plants. Subsequent plot shows the optimal control effort u∗ at each instant for respective plants. Simulations were carried out to obtain output and input values for

Mandar M. Joshi, Vishwesh A. Vyawahare / Journal of Applied Nonlinear Dynamics 8(1) (2019) 35–53

0.4

49

FOTF Oustaloup

0.2

Output in units

0 −0.2 −0.4 −0.6 −0.8

−1 0

10

20

30 40 Time in seconds

50

Fig. 9 Step Response for G2(s) =

60

70

s0.5 −1 . s0.9 +1

Process Reference Model

1 0.8 0.6 0.4 0.2 0 0

50 100 Output plot (Constrained)

150

50 100 Control Effort (Constrained)

150

0 −0.2 −0.4 −0.6 0

Fig. 10 Output and control effort for G2(s) =

s0.5 −1 . s0.9 +1

constrained condition with input rate constrain Δu  ±0.05 and input constrain −2 ≤ U ≤ +2. After some trial runs for selection of input horizon, it was observed that input increments were similar for the selected range for input horizon of 2. Selection of output horizon was a trade-off between optimization of performance index for tracking error signal and control input. Selecting output horizon to be 10, gave the results which are presented in this paper. It has been observed that for both the FO systems, the target trajectory was closely followed by the process output and time required to follow the changing target trajectory was due to the presence of constraints. In unconstrained case, target trajectory was followed without any delays. It was

Mandar M. Joshi, Vishwesh A. Vyawahare / Journal of Applied Nonlinear Dynamics 8(1) (2019) 35–53

Magnitude in decibels

50

0 −20 −40 −60 −80

FOTF Oustaloup

−100 −2 10

0

10 Frequency in rad/s

0

10 Frequency in rad/s

10

2

10

4

10

2

10

6

4

10

Phase in degrees

200 100 0 −100 −2 10

10

Fig. 11 Bode Plot for G2(s) =

6

s0.1 −1 . s0.9 +1

0.05

FOTF Oustaloup

0 −0.05

Output in units

−0.1 −0.15 −0.2 −0.25 −0.3 −0.35 −0.4 0

10

20

30 40 Time in seconds

50

Fig. 12 Step Response for G2(s) =

60

70

s0.1 −1 . s0.9 +1

further observed that the hard constraints make the process output sluggish due to the limitations imposed on the control input to the system. However, the model predictive control tries to follow the trajectory within the given limits. These results demonstrate that model predictive control strategy works successfully for these two linear FO systems.

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51

Process Reference Model

1 0.5 0 0

50 100 Output plot (Constrained)

150

50 100 Control Effort (Constrained)

150

0 −0.2 −0.4 −0.6 −0.8 0

Fig. 13 Output and control effort for G2(s) =

s0.1 −1 . s0.9 +1

6 Conclusion The work presented in this paper addresses the development and analysis of model predictive control strategy for linear fractional-order systems using approximation method in the presence of input and input rate constraints. In order to simulate a realistic situation, the integer-order rational approximation of the FO system was considered as the model for FO system. From the results, it can be in general concluded that MPC can be successfully applied to linear FO systems. The process output follows the reference trajectory very closely. The reference signal used for the simulation contained both positive and negative steps and hence it can be inferred that MPC for the FO systems can work for all types of limiting reference signals. The results presented in this paper can be extended to design the MPC strategy for nonlinear FO systems.

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Journal of Applied Nonlinear Dynamics 8(1) (2019) 55-66

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

An Application to Robot Manipulator Joint Control by Using Fractional Order Approach Cosmin Copot† University of Antwerp, Department of Electromechanics, Op3Mech Groenenborgerlaan 171, 2020 Antwerp, Belgium Submission Info Communicated by C.I. Muresan Received 27 July 2017 Accepted 2 October 2017 Available online 1 April 2019 Keywords Fractional-order control Joint control Robotics PID control Manipulator arm

Abstract This paper presents the application of fractional-order control strategy for the velocity control of a manipulator robot joint. A wellknown approach for robot joint control is to use an independent controller for each joint axis with a nested structure: an outer loop which computes the desired joint velocity in order to minimize the position error; and an inner loop which ensures the required velocity of the joint. A fractional-order controller was designed, tested and implemented on a manipulator robot. The communication between the robot and the controller was established via the Robotics System Toolbox from Matlab® and the ROS (Robot Operating System) platform. The experimental results indicate that the flexibilities of the fractional-order PI controller allows it to outperform the conventional integer-order PI controller. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Manipulator robots can be described as automated electromechanical systems with flexible functionalities (i.e. programming according to the environmental conditions in which they operate). The main role of the manipulator robot is to help the human operators in fulfilling repetitive and/or tasks with increased risk in industrial environment. From the perspective of robot control focus is put on the structure and functionality of the robot command unit. Based on the dynamic and geometrical model and the tasks which need to be performed, appropriate commands are established. In order to provide these commands to the actuators, the hardware and the software, it is required to use the feedback signals obtained from the sensory system unit [1]. Due to the complexity of the manipulator robot, the control architecture has a hierarchical structure. The upper level takes the decision with respect to the actions that need to be taken (e.g. simple test based on “if - then” actions), while the lower level carries out the control of the joints. The typical structure of the control system consists of a computer on the upper level and a system with one or more microcontrollers to drive the actuators of the joints on the lower level. † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.005

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Mechatronic systems, and in special robots, are very popular for control application due to their interdisciplinary nature [2–4]. For linear mechatronic systems, the proportional-integral-derivative (PID) controller has been widely used given its simple structure and robustness [5]. Usually, the design of conventional integer-order PID controllers is based on the model of the system. However, with the development of the fractional-calculus, researchers in the engineering field become to realize that many real processes are more adequate to be described by fractional-order state equations [6, 7]. Thus, the conventional integer-order PID controller becomes less suitable for the control of the fractional-order ‘reality’. A suitable way to improve the control performance is to use a controller of a similar ‘nature’ as the ‘reality’ [8], i.e. a fractional-order PID controller. Fractional calculus has been used (relatively) recently in modeling and control applications [9–11]. The attractiveness of the fractional order PID controllers resides in their potential to increase the closed loop performance and robustness of the closed loop system, due to the extra tuning parameters available, as compared to the conventional controller. With fractional order controllers, the order of differentiation and integration may be used as supplementary tuning parameters and thus more specifications can be fulfilled at the same time, including the robustness to plant uncertainties, such as gain and time constant changes [10,12,13]. Fractional-order controllers are described by fractional-order differential equations. Fractional-order PID controllers have been used in industrial applications [14] and various fields such as mechatronic systems [15, 16], and system identification [17]. In this paper a fractional calculus based control strategy for speed control of a joint within a manipulator robot is presented. In this research study a KUKA KR16 robot with a KRC2 controller is considered. The motion of a manipulator robot is usually given as Cartesian coordinates of the endeffector. However, in order to reach the desired Cartesian trajectory, each joint needs to follow a specific trajectory. Here, a fractional-order PI controller is designed, tested and validated for the inner velocity control loop. The communication between the robot and the controller was establish via the Robotics System Toolbox from Matlab® and the ROS platform. The robustness of the fractional-order PI controller and its performances are compared against an integer-order PI controller. The performances of both classical integer-order approach and fractional-order approach are analyzed through simulations and real-time experiments. The experimental results revealed better performances of the fractional approach in comparison with the classical one. The paper is organized as follows: Section 2 is dedicated to the robot manipulator descriptor. In Section 3, a brief introduction on controller design i.e. the integer-order and the fractional-order controller is given. In Section 4, the simulation and the experimental results of the designed controllers are presented. Finally, Section 5 addresses the concluding remarks and indicates future perspectives.

2 Robot manipulator description A robot manipulator, depicted in Figure 1, can be defined as a kinematic chain consisting of multiple rigid bodies which are interconnected by joints. Each joint has one degree of freedom (translational or rotational) and can move its outward neighbouring link with respect to its inward neighbour. In the work of Spong et.al [1] is mention that a simple representation of a robot manipulator joint can be modeled as a spring with linear constant stiffness. Based on the Lagrangian formulation, the dynamics of a global manipulator robot with n revolute joints is given by [18]: Q = M(q)q¨ +C(q, q) ˙ q˙ + F(q) ˙ + G(q) + J(q)T g

(1)

where q, q, ˙ q¨ ∈ ℜn represents the joint variables (the generalized joint coordinates, the velocity and the ˙ is the n × 1 vector of the Coriolis and centrifugal acceleration), M(q) ∈ ℜn×n is the inertia matrix, C(q, q) forces, G(q) is the gravity force. The n × m matrix J(q)T is the transpose of the Jacobian matrix of the robot while g is the joint force vector applied at the robot end-effector. Starting from the assumption

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Fig. 1 Schematic overview of a KUKA KR16 manipulator robot. Gearbox

Jm 00

Load Inertia

Motor Motor Inertia

Jl

Fig. 2 Schematic overview of a classical robot joint drivetrain.

that the pose, the velocity and the acceleration are known and based on equation (1), it is possible to calculate the required joint forces. Thus, the differential equation which describes the manipulator rigid-body dynamics is known as the inverse dynamics. Nowadays, a large number of the industrial robots are driven by brushless servo motors. A schematic overview of a classical robot joint is illustrated in Figure 2. Several studies in the area of control of manipulator robots considers as control variables the Cartesian coordinates. However, in order to obtain the desired Cartesian trajectory for the end-effector attached to the robot, each joint axis must follow a specific trajectory. The common approach to control the robot joints is based on the decentralized techniques which consider an independent controller for each joint. As was already mention, the industrial robot considered for our experiments is electrically actuated. If the robot joint drivetrain is driven by the current: (2) i = Ka u then, the torque generated by the motor is proportional to the current:

τ = KM i

(3)

where Ka is the transconductance of the amplifier, Km is the motor torque constant and u is the applied

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58

disturbance

+_

Position Control

+_

Speed Control

Robot Joint

Fig. 3 The structure of the nested control architecture.

control voltage. Based on this assumption, the dynamics of a motor attach to a joint j can be defined as: Jm ω˙ + Bω + τc (ω ) = Km Ka u

(4)

with τc the Columbus friction torque, B the viscous friction and Jm the total inertia seen by the motor for joint j computed as: 1 (5) Jm = Jm j + 2 M j j . Gj Since on real life disturbance torques such as gravity and friction can act on the joints, the common approach where independent control systems are considered to control the robot joints is no-longer efficient. The control performance is decreasing leading to high overshoot and large steady-state error. A classical approach to overcome these shortcomings is to use a nested control structure composed of two loops: one outer loop and one inner loop, as depicted in Figure 3. The outer loop is used to control the position and to provide the velocity of the joints in order to minimize the position error, while the inner loop is used to minimize the error between the actual velocity of the joint and the velocity demanded by the outer loop. For simplification reason, if the Columbus friction from equation (4) is ignored, the equivalent Laplace transform of (4) is given by: sJΩ(s) + BΩ(s) = Km KaU (s)

(6)

where Ω(s) and U (s) are the Laplace transform of their corresponding signal from time domain ω and u. Thus, the transfer function of a motor drivetrain attached to a robot joint can be written as: Km Ka Ω(s) = . U (s) Js + B

(7)

Typically, a proportional controller is used to drive the robot joint from the actual velocity to its demanded velocity: ˙ u∗ = K p (q˙∗ − q).

(8)

Since the inertia matrix is a function of the manipulator pose, the components of the interaction matrix are varying with respect to the robot configuration, and due to the disturbances introduced by the system, the proportional controller cannot satisfy the specifications. Thus, in order to increase the performance of the system, a proportional-integral controller can be considered:   Ki (q˙∗ − q). ˙ u∗ = K p + s

(9)

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3 Controller design and comparative study In the last years, fractional calculus has been increasingly employed in control engineering area of research due to several advantages. In fact, the fractional order PI λ Dμ controller represents a generalized structure for the integer-order PID controller defined by ˆ t d (10) e(τ )d τ + Kd e(t), u(t) = K p e(t) + Ki dt 0 where K p , Ki and Kd represent the controller parameters and e(t) is the error between the current output and the desired setpoint. The transfer function of the PI λ Dμ is given by: CFO−PID (s) = K p +

Ki + Kd sμ , sλ

(11)

with λ ∈ (0, 1] and μ ∈ (0, 1] being the fractional-order of the integral and derivative components. It can be noticed that a fractional-order controller provides more flexibility in the controller design because it has five parameters to select in order to fulfill the desired specifications. However, this also implies that the tuning procedure will increase in complexity. 3.1

Integer-order control approach

An integer-order PI controller is implemented on the system for comparison purposes. This controller is tuned with FRtool (A Frequency Response Tool for CACSD in Matlab® ) [19] designed at Ghent University, a tool compatible with the Matlab® platform. This computer-aided-design software uses Nichols charts of the system in order to tune the controller (Figure 4). An important feature of the FRtool controller design tool is the possibility to define practicallymeaningful design specifications -which will guide the designer in the tuning process. These specifications have to be converted to graphical restrictions to make the designer’s task easier. Some of the traditional design specifications are gain margin (GM) and phase margin (PM). However, these specifications have not necessarily a clear physical meaning to a potential user (unless this user is, for example, a control engineer) since these are based on mathematical insight and system theory. Therefore, more practical specifications - which can be easily interpreted by any user - are settling-time (small red circle) and overshoot (%OS) of the closed-loop time response, and of course robustness (Ro) of the design. The design specifications can be introduced using the options denoted in the lower right part in the main FRtool interface; e.g. the overshoot and the robustness specifications are visible in the chart (Figure 4-left). Closed-loop performance can be evaluated with the options in the lower left part of the main FRtool interface (Figure 4-left). In order to design a controller with FRtool, the user does not need a detailed knowledge of the frequency-response background theory, i.e. adding zeros and poles in the design window of the FRtool (Figure 4-right). 3.2

Fractional-order control approach

Fractional calculus represents the generalization of the integration and differentiation to an arbitrary order. The Laplace transform of the fractional-order operators for the fractional-order integral is defined as [10]: (12) L Iα f (t) = s−α F(s) while for the fractional-order derivative, the equation is given by: L Dα f (t) = sα F(s)

(13)

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PHC Gain

Pole %OS

x

Ro

Zero PM GM Real Part

Imaginary Part

Matlab workspace

Fig. 4 Graphical interface of FRtool. After the system has been imported from Matlab workspace (see window in the lower right part), it appears as a curve in the Nichols chart corresponding to the loop frequency response (PHC).

with α ∈ (0, 1]. The general transfer function of a fractional-order PI (FO-PI) controller is given as: CFO−PI (s) = K p (1 +

Ki ). sλ

(14)

The design of the FO-PI controller is usually based on a phase margin and a gain crossover specification, to which a third criteria may be added in order to uniquely determine the three controller parameters. In order to tune the fractional-order PI controller, the open-loop transfer function needs to be computed as: (15) Gopen−loop (s) = CFO−PI (s)GP (s) where GP (s) is the process to be controlled. The tuning of the FO-PI controller implies the computation of the three parameters K p , Ki and λ according to three performance specifications imposed: 1. Gain crossover frequency ωcg specification: |Gopen−loop ( jωcg )| = 1.

(16)

  ϕm = arg Gopen−loop ( jωcg ) + π .

(17)

2. Phase margin ϕm specification:

3. Robustness to variations in the gain of the plant:     d arg Gopen−loop ( jωcg )    dω

ω =ωcg

= 0.

(18)

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Table 1 Motor parameters. Parameter

Value

Unit

Torque constant

1.38

Nm/A

Rotor moment of inertia

20 × 10−6

Kgm2

Shaft torsional stiffness

109000

Nm/rad

Maximum torque

24.70

Nm

Max. permissible speed (mech.)

6000

rpm

Gear ratio

16/1

Maximum inverter current

18

A

The performance specifications given in equations (16) and (17) can be re-written as: |CFO−PI ( jωgc )| =

1 , |GP ( jωgc )|

arg [CFO−PI ( jωgc )] = −π + ϕm − arg [GP ( jωgc )] which can be further detailed as:      1 πλ πλ  −λ K p 1 + Kiωgc − j sin cos  = |G p ( jωgc )| ,  2 2 

 πλ πλ −λ − j sin = −π + ϕm − arg [GP ( jωgc )] . cos arg 1 + Kiωgc 2 2

(19) (20)

(21) (22)

The third specification that refers to the robustness against gain uncertainties can also be written as:

 −λ cos πλ − j sin πλ d arg 1 + Ki ωgc 2 2 d (arg [GP ( jωgc )]) =− . (23) d ωgc d ωgc Since the FO-PI controller has three independent parameters, these can be adequately tuned to meet three performance specifications. By solving the above equations, one can get λ , Ki , and K p . From the above analysis it can be observed that λ and Ki can be obtained jointly. Thus, the procedure to tune the FO-PI controller is as follows [10]: 1. Given ωgc , the gain crossover frequency; 2. Given ϕm , the desired phase margin; 3. Plot curve 1, corresponding to Ki with respect to λ , according to (22); 4. Plot curve 2 corresponding to Ki with respect to λ , according to (23); 5. Obtain λ and Ki from the intersection point between curve 1 and 2; 6. Calculate K p from (21). 4 Simulation and experimental results In this section, the simulation and experimental results that were conducted in order to validate the fractional order controller are presented. The manipulator robot considered in this study is a KUKA KR16 with a KRC-2 controller, Figure 5. The speed control was developed and applied to joint 2 of the manipulator robot. The parameters of the selected joint are given in Table 1. The transfer function of the motor drivetrain corresponding to joint 2 is given by: GP (z) =

1 . (1.5s + 2.3) 10−3

(24)

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Fig. 5 The KUKA manipulator robot.

The performance specifications regarding the gain crossover frequency and phase margin are: ωcg = 1.2 and ϕm = 66◦ . Using graphical methods, two curves for the Ki parameter as a function of the fractional order λ are plotted and the intersection of the two curves lead us to the next solution for Ki and λ as Ki = 11.8 and λ = 0.93. These values are further used to compute the value for the third parameter K p using (21): K p = 2.3. Thus, the FO-PI controller was obtained as: CFO PI (s) =

2.3s0.93 + 11.8 . s0.93

(25)

The integer-order PI controller was designed using the FRtool and the corresponding PI parameters are given as: K p = 3.1 and Ki = 14.7. In order to test the controllers prior to the actual implementation, a Simulink benchmark was created (following the robotics toolbox descried in [18]) in which the previously determined FO-PI controller and integer-order PI controller were implemented. A first simulation was performed considering only a proportional controller and the results are illustrated in Figure 6. It can be notice that if the variance of the gravity (which acts as a disturbance) is not taken into account, then the proportional controller give satisfactory performance, Figure 6(a). However, if we add a disturbance at the load, the controller has a large steady state error, Figure 6(b) and therefore a PI controller was considered, Figure 7(a). The simulation results for fractional order feedback loop by using the Simulink model are depicted in Figure 7(b). A closer view of the output is illustrated in Figure 8. The fractional order PI controller is compared with the integer order PI controller and the obtained results indicate that the fractional controller outperforms the classical integer-order controller.

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(a)

63

(b)

Fig. 6 The output of the inner velocity loop with a sawtooth demand and a proportional control: (a) no disturbance are considered; (b) with gravity disturbance.

(a)

(b)

Fig. 7 The output of the inner velocity loop with gravity disturbance: (a) PI control; (b) FO-PI control.

(a)

(b)

Fig. 8 The output zoom-out of the inner velocity loop: (a) for the PI control; (b) for the FO-PI control.

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Fig. 9 Visualization of the ROS computation graph.

Fig. 10 The output of the inner velocity loop.

The performance of the PI controllers has also been evaluated in the real time application using the setup depicted in Figure 5. In order to communicate with the KUKA robot the Robotics System Toolbox of Matlab® and the ROS platform were used. The running nodes and topics within ROS platform related to this case study are illustrated in Figure 9. Again, the experimental results from Figure 10 show that the fractional order controller outperforms the integer order PI controller. If for the simulation results the superiority of the fractional controller is slightly visible, in case of real time the fractional order PI is clearly superior to the classical controller. The simulation and experimental results reveal that fractional-order controller obtain better performances when dealing with this type of processes in comparison with classical PI controller. To further

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Table 2 Performance index for output error in the different control strategies. Controller

IAE

P - without gravity dist

58.9949

P - with gravity dist

551.8461

PI

163.6640

FO-PI

145.0029

prove the effectiveness of the proposed methodology, the performance of the controllers is evaluated using the well known performance index, i.e. integral of the absolute magnitude of the error (IAE) with the mathematical definition given by equation (26) and the results are presented in Table 2: ˆ IAE =

|error| dt.

(26)

5 Conclusions In this paper, the application of fractional-order control for the robot joint velocity was presented. The performance of the fractional-order PI controller is analyzed and compared with integer-order PI controller. The experimental results indicate that the fractional-order controller outperforms the classical integer-order controller. Due to the flexibility of the fractional controller introduced by the extra parameter, the system can fulfill more specification but its implementation requires additional effort in comparison with classical PI controller.

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[14] Vinagre, B.M., Monje, C.A., Calderon, A.J., and Suarez, J.I. (2007), Fractional PID controllers for industry application: a brief introduction, Journal of Vibration and Control, 13, 1419-1429. [15] Fan, H., Sun, Y., and Zhang, X. (2007), Research on fractional order controller in servo press control system, Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, Harbin, China, pp. 934-938. [16] Ma, C. and Hori, Y. (2004), Fractional order control and its application of controller for robust two-inertia speed control, Proceedings of the 4th International Power electronics and Motion Control Conference, 3, China, pp. 1477-1482. [17] Schlegel, M. and Cech, M. (2004), Fractal system identification for robust control, the moment approach, Proceedings of 5th International Carpathian Control Conference, Krakow, pp. 1-6. [18] Corke, P.I. (2011), Robotics, Vision & Control, Springer, ISBN 978-3-642-20143-1. [19] De Keyser R. and Ionescu C.M. (2006), FRtool: A frequency response tool for CACSD in Matlab, IEEE International Symposium on Computer Aided Control Systems Design, Munich, pp. 2275–2280.

Journal of Applied Nonlinear Dynamics 8(1) (2019) 67-78

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

A Fractional Order Controller for Delay Dominant Systems. Application to a Continuous Casting Line Dana Copot†, Clara Ionescu Ghent University, Department of Electrical Energy, Metals, Mechanical Constructions and Systems Research group on Dynamical Systems and Control Technologiepark 914, 9052 Ghent, Belgium Submission Info Communicated by K. A. Markoski Received 27 July 2017 Accepted 3 October 2017 Available online 1 April 2019 Keywords Fractional order control Continuous casting line Delay system

Abstract Continuous casting technology implies an exotherm process from liquid steel to solid slab. During this process, the temperature at the surface of the slab is one of the most important parameters for evaluating the cooling process and inherent material properties. To ensure a specific temperature gradient, several control elements need to be evaluated; e.g. to optimize casting speed, to determine intensity of the second cold and determine liquid depth. In this paper a fractional order control strategy is proposed to control the steel slab temperature, governed by delay dominant dynamics. The reference tracking for deisred temperature at end of casting line is evaluated by means of performance metrics. The results obtained indicate that the proposed methodology outperforms other strategies used for comparison. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Continuous casting (CC) technology originated almost 60 years ago, nowadays utilized in more than 95% of the world steel production. To ensure competitiveness through the years, the product quality, production efficiency, operating safety, casting of special steels and alloys have increased considerably. These developments were allowed by equipment revamps, updates in installation setups, computer process, post-process control, better problem understanding by long-term experiments, computer simulations, etc. The work of Nortmanton et al. shows that there is still a strong need in the effort for optimization of the casting process in order to achieve better quality, high productivity in the environmental friendly operation [1]. In the past decade, worldwide steel companies have focused in increasing energy efficiency of the production processes [2]. In steel industry the slab reheating furnaces are considered one of the processes which requires the most energy [3]. These are the sub-sequent manufacturing steps in the processing of slab resulted from the casting line to the hot strip mill sub-process to obtain the steel rolls. Continuous casting (CC) represents the process of converting the molten steel to a slab. It is a very complex process which involves many variables such as: casting speed, mold oscillation characteristics, steel † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.006

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grade, mold dimensions and metal flow. Production quality is considerably influenced by temperature variations (e.g. problems like cracks might occur which require expensive surface-reconditioning or even rejection of the product) during cooling [4]. However, slab surface temperature is not only related to the cooling intensity but also depends on the slab cross-section dimensions, casting speed (e.g. due to the high casting speeds and short response times, setting the spray water flow rates becomes for operators increasingly difficult), shell thickness etc. Another very important factor which may affect the surface quality is the mold level in the continuous casting machine. In order to ensure a good surface quality of the slab the variation in mold level should be of the order of only few millimeters. Several steelmakers control the CC process by conventional PID control systems, but PID control should not be used for nonlinear systems and systems with fast changes of the processing parameters. Furtmueller and del Re discussed challenges and control issues in the CC and analyzed limitations of PID control loops [5]. Fractional operators have been often applied in the last years by different authors [6–8] to model and control processes with challenging dynamics. An interesting feature of fractional-order controllers is that they exhibit some advantages when designing robust control systems in the frequency domain for processes whose parameters vary in a large range. Generic heat diffusion systems were controlled using a fractional order PID controller plus a Smith predictor [9]. Recently, few articles have appeared about the application of fractional controllers to industrial furnaces [10, 11]. These describe different behaviours that can be achieved by changing the fractional order of the differential/integral operators. However, nor tuning methodology, neither robustness analysis were provided. In this paper we propose a fractional order controller for the continuous casting line which can be represented as a first order plus dead time (FOPDT) system. Performance of the controller is investigated via a comparison to traditional integer order controllers using performance metrics such as relative mean square (MSE), the integral square error (ISE), the integral absolute error (IAE) and the integral time weighted absolute error (ITAE). This paper is organized as follows. Section 2 elaborates on the proposed controller, followed by a comparative study with other established controllers. In section 4 a short description of the continuous casting line and approximated control scheme for simulation purposes is given. In section 5 the simulation results and their discussion is presented. A final section provides the conclusion of this research and points to future work.

2 Fractional order controller The benefits of fractional order control over integer order control have been long proven on a manifold of systems [12], from process industry [13] to mechatronics [14–16]. In this paper we propose a variation of fractional order control tuned based on plant’s dynamic properties, i.e. the process time constant and time delay values. This controller is also able to adapt its tuning to LPV (Linear parameter-varying) systems, by providing at all times the updates on the time constant and/or time delay values. This controller can thus be used in manufacturing applications where LPV is an inherent property. In essence, any process dynamics can be reduced to the generic form: K e−Ls (1) Ts+1 with K the static gain of the system, L the time delay and T the time constant. Usually, these are extracted easily from the step response of the process in open loop [17, 18], or from simple sinusoid experiments [19]. The processes are then classified in three groups, depending on the relative dynamic of the system, computed as: L (2) τ= L+T G(s) =

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whereas considered to be lag dominant if τ < 0.5, balanced for τ ≈ 0.5 and delay dominant for τ > 0.5. whereas considered to be lag dominant if τ < 0.5, balanced for τ ≈ 0.5 and delay dominant for τ > 0.5. A fractional order PI controller can be represented by the following transfer function: C1 (s) = Ki · (

1 + T s1−α ) sα

(3)

and the controller is a combination of fractional integral and fractional derivative elements. The open loop transfer function of controller and process is then given by C(s)G(s) = Ki K

e−L sα

(4)

with corresponding gain: C( jω )G( jω ) =

Ki K ωα

(5)

and phase

π (6) φ = −Lω − α . 2 The controller can be further designed by means of Ziegler-Nichols methods [20,21] or relay feedback autotuning methods, as described in [22–24]. However, for the purpose of this work, a model based tuning is necessary for adapting controller parameters during operation of the process without necessity of additional tests (e.g. relay, relay+delay etc). Assume that the user defines a desired gain margin Gm and a desired phase margin φm . This is not unusual, since both specifications relate to time performance indexes as overshoot, settling time, rise time and to frequency domain indexes as robustness [25, 26]. We have that 1 Gm φm = π + ∠C( jωg )G( jωg )

C( jω p )G( jω p ) = −

(7)

with ω p is the phase crossover frequency and ωg is the gain crossover frequency of the open-loop system and controller. It follows that the gain is: KKi =1 ωgα ω pα = Gm KKi

(8)

and the phase is:

φm = π − Lωg − α π =α

π + Lω p . 2

π 2

(9)

From these relations one can extract the system

ωp α ) , ωg π − α π2 ωp = . ωg π − φm − π π2

Gm = (

(10)

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In these relations the unknown is then the fractional order α which can be extracted using numerical optimization from the relation: π − α π2 α ) (11) Gm = ( π − φ− π π2 With the value of α the following controller parameters can be determined analytically:

π − φm − α π2 , L α − α π2 , ωp = L ω pα ωgα Ki = , = KGm K Kd = T Ki . ωg =

(12)

Although the above method has its advantages as it converges to an optimal solution, it requires nonlinear optimization to be executed online. Instead, we propose to make use of the approximated process parameter information to simplify the tuning procedure. Taking into account the intrinsic relation between integral action and specific dynamics of the process one may extract the following rules of thumb [27, 28]: • if τ < 0.05, then α = 0.7; • if 0.05 ≤ τ ≤ 0.1, then α = τ + 0.6; • if 0.1 ≤ τ ≤ 0.2, then α = τ + 0.7; • if 0.2 ≤ τ ≤ 0.4, then α = 0.9; • if 0.4 ≤ τ ≤ 0.6, then α = τ + 0.5; • if τ ≥ 0.6, then α = 1.1. In this way, the parameter α is determined from the process characteristics, without requiring optimization. For given phase margin, then controller parameters can be directly obtained as described above. Notice that the values of τ may change if delay estimation techniques are introduced in the algorithm. Depending on the application, such delay estimation methods are not difficult to implement and thus can be exploited to increase robustness of controller to variations in delay values in larger operating range and maintain performance despite great fluctuations in these parameters.

3 Comparative study To better illustrate the benefits of the proposed controller, a comparison to other established controllers has been introduced. This section describes briefly the tuning rules used to determine the controllers for the comparative study. Most of the controllers presented in [27] are designed taking into account the process description

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from (1). Two controllers are selected, with the following tuning rules: Kp 1 , with Ti s 0.2978 1 ) Kp = ( K τ + 0.000307 0.8578 ) Ti = T ( 2 τ − 3.402τ + 2.405

(13)

Kp 1 , with Ti s (0.7303 + 0.5307T /L)(T + 0.5L)Ti , Kp = K(T + L) Ti = T + 0.5L.

(14)

C2 (s) = K p +

and C3 (s) = K p +

Another well established set of controllers for processes with time delay are designed in [29]. Two controllers are used, namely: C4 (s) = K p +

Kp 1 Ti s

T , (L + T 0)K T + 0.5L ; K p = kc T Ti = T + 0.5L; kc =

(15)

with T0 a tunning parameter affecting the speed of the closed loop. Kp 1 Ti s L + 2T ; K p = 0.35 KL Ti = T + L/2;

C5 (s) = K p +

(16)

Finally, important robustness issues have been addressed in a tuning methodology based on internal model control principles in [30]. Specifications as maximum sensitivity are allowed for optimal tuning. The controller structure has been selected as: Ti s + 1 Ti s 1 T Kp = K T0+L Ti = min(T, 4(T 0 + L));

C6 (s) = K p

(17)

Notice that these controllers are also designed to allow adaptation to LPV system dynamics, such that a fair comparison is done. These controllers have been tested and compared in simulation and the results presented hereafter.

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Fig. 1 Illustration of the continuous casting process.

4 Process description In the CC process, depicted in Figure 1, molten steel flows from a ladle, through a tundish into the mold. Once in the mold, the molten steel freezes against the water-cooled copper mold walls to form a solid shell. Drive rolls lower in the machine continuously withdraw the shell from the mold at a rate or ”casting speed” that matches the flow of incoming metal, so the process ideally runs in steady state. Below mold exit, the solidifying steel shell acts as a container to support the remaining liquid. In the secondary cooling zone water is sprayed to cool the surface of the strand between the rolls in order to maintain its surface temperature until the molten core is solid. The most significant phenomena which govern the CC process and determine the quality of final product are given hereafter. Liquid metal flows into the mold cavity through a submerged entry nozzle and is directed by the angle and geometry of the nozzle ports. The direction of the steel jet controls turbulent fluid flow in the liquid cavity, which affects delivery of superheat to the solid/liquid interface of the growing shell. The liquid steel solidifies against the four walls of the water-cooled copper mold, while it is continuously withdrawn downward at the casting speed. Mold powder added to the free surface of the liquid steel melts and flows between the steel shell and the mold wall to act as a lubricant, so long as it remains liquid. The resolidified mold powder, or slag, adjacent to the mold wall cools and greatly increases in viscosity, thus acting like a solid. This relatively solid slag layer often remains stuck to the mold wall, although it is sometimes dragged intermittently downward at an average speed less than the casting speed. Heat conduction across the slag depends on the thickness and conductivity of its layers, which, in turn, depends on their velocity profile, crystallization temperature, viscosity, and state (glassy, crystalline, or liquid). Finally, the flow of cooling water through vertical slots in the copper mold withdraws the heat and controls the temperature of the copper mold walls. After exiting the mold, the steel shell moves between successive sets of alternating support rolls and spray nozzles in the spray zones. Improvements into the mixing of the steel via electromagnetic actuators are given in [31]. A block diagram of the process is given in Figure 2. Based on baseline dominant dynamic characteristics, a simplified structure of the control loop has been elaborated hereafter. The values are not coming from a real life data, but they are able to capture the main dynamical properties necessary to illustrate the feasibility of the controlled loop in a realistic

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Fig. 2 Schematic representation of the continuous casting process.

environment. The transfer function for the hydraulic actuator which moves the stopper in the tundish, is given by: 39.48 HA(s) = 2 . (18) s + 8.796s + 39.48 A PD type controller transfer function has the following form: PD = 0.3(s + 5)2 .

(19)

The dynamics of the stopper are given by: Stopper(s) =

0.8 . s+1

(20)

The actual stopper position is measured with a position sensor which has the following dynamics (the output of the sensor is expressed in V): Sensor(s) =

1 . 0.1s + 1

(21)

The next element represents the effect of a variation of the stopper position on the flow speed of the steel. The electro-magnetic stirrer is used to change the speed of the flow of steel in order to maintain the speed within a certain interval. As long as the speed of the steel flow is within these limits, the quality of the steel slab will be sufficient. The transfer function representing these dynamics is given by: 1 . (22) EMS(s) = 2 s + 2s + 1 Temperature gradient loss is represented by a first order dynamics, followed by a time delay indicating the measurement of temperature at the end of the CC line. The transfer function capturing these effects is given by: −0.8 −30s e . (23) DL(s) = 10s + 1 In practice, all these time constants and delay values may vary slightly due to variations in the level of steel in the tundish and stopper position variations, effects of disturbances (e.g. new laddle supply, noise, vibrations, etc).

5 Simulation study results and discussion The proposed fractional order controller is applied on a simulator of the real system in Matlab/Simulink based on Figure 2. The results obtained from the simulation are depicted in Figure 3. It can be noticed

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Fig. 3 Comparison between the proposed fractional order controller and other established controllers.

Fig. 4 Comparison between the proposed fractional order controller and other established controllers (the controllers with a satisfactory performance).

that the proposed fractional order controller (C1) has a very good closed loop performance (e.g. no overshoot). The controllers used for comparison performed similarly, except C2. The performance of the controllers is evaluated using several performance metrics such as: the relative mean square error (MSE), the integral squared error (ISE), the integral absolute error (IAE)

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Table 1 Controllers Parameters. Controller

Kp

Ki

K

L

kc

T0

C1

0.642

0.024

0.7

30

-

-

C2

0.642

0.007

0.7

30

-

-

C3

0.638

0.024

0.7

30

-

-

C4

0.779

0.024

0.7

30

0.3896

25

C5

0.875

0.024

0.7

30

-

1.5

C6

0.389

0.024

0.7

30

-

25

and teh integral time weighted absolute error (ITAE) which are defined as: 1 n (Re f − Tout )2 ∑ n i=1 U2

MSE =

(24)

n

ISE = ∑ (Re f − Tout )2 T s

(25)

i=1

n

IAE = ∑ abs(Re f − Tout )2 T s

(26)

i=1 n

ITAE = ∑ t · abs(Re f − Tout )2 T s

(27)

i=1

where Re f is the reference temperature value of the slab expected at the end of the CC process, Tout is the current slab temperature and n is the length of the time vector t. The resulting metrics for each controller are graphically represented in Figure 6. Notice that controller C2 is not able to control the system with a reasonable performance. For sake of clarity, an additional figure depicts the results of the selected best performance controllers, i.e. Figure 4. The percentage overshoot (OS%) and the settling time (ST) are calculated and presented in Figure 6 along with the performance metrics. In Table ?? the controllers parameters are given.

6 Conclusions In this paper a fractional order control strategy for temperature control in a continuous casting line has been proposed. The temperature control (i.e. temperature gradients in the continuous casting process) is related to the quality of the end-product. A tuning method for the fractional order controller parameters has been proposed. The method allows for LPV process dynamics. A comparative study has been performed between the proposed controller and other five controllers from literature. The closed loop performance has been evaluated based on four different metrics: i.e. mean square error, the internal squared error, the integral absolute error and the integral time weighted absolute error. Additionally, the overshoot and the settling time of the closed loop system supported our claim that the proposed controller performs well.

References [1] Northmanton, A., Ludlow, V., and Smith, A. (2015), Improving quality of continuously cast semis by an understanding of shell development and growth, Technical Steel Research Series ,1-351.

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(a)

(b)

(c)

Fig. 5 Performance comparison in terms of (a) ISE (b) IAE (c) ITAE. [2] Weng, Y., Dong, H., and Gan, Y. (2011), Advanced steels. The recent scenario in steel science and technology, Springer, Beijing. [3] Mullinger, P. and Jenkins, B. (2008), Industrial and process furnaces, Principles, design and operation, Elsevier, Oxford. [4] Stankovski, M., Dimirovski, G., Dinibutun, T., and Gacovski, Z. (2001), Iterative design of a two level control and supervision system for industrial furnaces, in: Proceedings of 9th Mediterranean Conference on Control and Automation, Dubrovnik, Croatia.

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(a)

(b)

(c)

Fig. 6 Performance comparison in terms of (a) MSE, (b) settling time ST and (c) percentage overshoot OS%. [5] Furtmueller, C. and del Re, L., Control issues in continuous casting steel, 17th World Congress The International Federation of Automatic Control. [6] Podlubny, I. (1999), Fractional order systems and PIλ Dμ controllers, IEEE Transactions on Automatic and Control, 44(1), 208-214. [7] Oustaloup, A., Cois, O., Lanusse, P., Melchior, P., Moreau, X., and Sabatier, J. (2006), The crone approach: theoretical developments and major applications, in: Proceedings of the 2nd Workshop of the International Federation of Automatic Control on Fractional Differentiation and its Applications, Porto, Portugal.

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[8] Monje, C., Vinagre, B., Feliu, V., and Chen, Y. (2008), Tuning and auto-tuning of fractional order controllers for industry applications, Control Engineering Practice, 16(9), 798-812. [9] Jesus, I. and Machado, J.T. (2008), Fractional control of heat diffusion systems, Nonlinear Dynamics, 54, 263-282. [10] Isfer, L.D., Lenzi, E.K., Teixeira, G.M., and Lenzi, M.K. (2010), Fractional control of an industrial furnace, Acta Scientiarum Technology, 32(2), 279-285. [11] Mora, I., Rivas-Perez, R., Feliu-Batlle, V., and Castillo-Garcia, F., Identification and fractional order control of temperature in steel slab furnace. [12] Machado, J., Kiryakova, V., and Mainardi, F. (2011), Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 16(3), 1140-1153. [13] Petras, I. and Vinagre, B. (2002), Practical application of digital fractional order controller to temperature control, Acta Montanistica Slovaka, 7(2), 131-137. [14] Muresan, C., Ionescu, C., Folea, S., and De Keyser, R. (2015), Fractional order control of unstable processes: the magnetic levitation study case, Nonlinear Dynamics, 80, 1761-1772. [15] Copot, C., Burlacu, A., Ionescu, C., Lazar, C., and De Keyser, R. (2013), A fractional order control strategy for visual servoing systems, Mechatronics, 23(7), 848-855. [16] Copot, C., Zhong, Y., Ionescu, C., and De Keyser, R. (2013), Tuning fractional pid controllers for a steward platform based on frequency domain and artificial intelligence methods, Central European Journal of Physics, 11(6), 702-713. [17] Liu, T., Wang, Q. G., and Huang, H.P. (2013), A tutorial review on process identification from step or relay feedback test, Journal of Process Control, 23(10), 1597-1623, DOI:10.1016/j.jprocont.2013.08.003. [18] Tan, K., Lee, T., and Jiang, X. (2001), On-line relay identification, assessment and tuning of pid controller, Journal of Process Control, 11(5), 483-496, DOI:10.1016/S0959-1524(00)00012-3. [19] De Keyser, R., Muresan, C., and Ionescu, C. (2016), A novel auto-tuning method for fractional order pi/pd controllers, ISA Transactions, 62, 268-275. [20] Hang, C., ˚ Astr¨ om, K., and Ho, W. (1991), Refinements of the ziegler-nichols tuning formula, Control Theory and Applications, 138(2), 111-118, DOI:0143-7054. [21] ˚ Astr¨ om, K. and H¨ agglund, T. (2004), Revisiting the ziegler-nichols step response method for pid control, Journal of Process Control, 14(1), 635-650, doi:10.1016/j.jprocont.2004.01.002. [22] Ionescu, C. and De Keyser, R. (2012), The next generation of relay-based pid autotuners (part1): Some insights on the performance of simple relay-based pid autotuners, in: IFAC Conf. on Advances in PID Control (PID’12), International Federation of Automatic Control, Brescia, Italy, iSSN: 1474-6670. [23] De Keyser, R., Joita, O., andIonescu, C. (2012), The next generation of relay-based pid autotuners (part 2): A simple relay based pid autotuner with specified modulus margin, in: IFAC Conf. on Advances in PID Control, International Federation of Automatic Control, Brescia, Italy, 10-3182/20120328-3-IT-3014.00022. [24] De Keyser, R., Dutta, A., Hernandez, A., and Ionescu, C. (2012), A specification based pid autotuner, in: 2012 IEEE International Conference on Control Applications, IEEE, Dubrovnik, Croatia, 1621-1626. [25] De Keyser, R. and Ionescu, C. (2010), A comparative study of three relay-based pid autotuners, in: Proc. of the IASTED Conf. On Modelling, Identification and Control, ACTA Press, Phuket, Thailand, 2010. [26] ˚ Astr¨ om, K. and H¨ agglund, T. (2006), Advanced PID Control, ISA, Berlin Heidelberg. [27] Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., and Feliu-Batlle, V. (2010), Fractional-order systems and controls: fundamentals and applications, Springer Science Business Media, London. [28] Vilanova, A.V.R. (2012), PID control in the third millenium, Springer, Berlin Heidelberg. [29] Normey-Rico, J. and Camacho, E. (2010), Control of dead-time processes, Springer, Springer Verlag London. [30] Skogestad, S. (2003), Simple analytic rules for model reduction and pid controller tunning, Journal of Process Control, 13, 291-309. [31] Dekemele, K., Ionescu, C., De Doncker, M. and De Keyser, R. (2016), Closed loop control of an electromagnetic stirrer in the continuous casting process, in: European Control Conference, Aalborg, Denmark.

Journal of Applied Nonlinear Dynamics 8(1) (2019) 79-95

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

CRONE Body Control with a Pneumatic Self-leveling Suspension System Jean-Louis Bouvin1,3†, Xavier Moreau1,3 , Andr´e Benine-Neto1,3 , Vincent Hernette2,3 , Pascal Serrier1,3 , Alain Oustaloup1,3 1 2 3

IMS, Univ. Bordeaux, CNRS, Bordeaux INP, 351 cours de la Lib´eration, 33405 Talence cedex, France PSA Groupe, 2 route de Gisy, 78943 V´elizy-Villacoublay, France OpenLab PSA Groupe - IMS Bordeaux Submission Info Communicated by C. Copot Received 30 July 2017 Accepted 3 October 2017 Available online 1 April 2019 Keywords CRONE control Body control Robust control Suspension

Abstract This paper deals with vehicle body control under driver inputs through a pneumatic system using the CRONE approach and follows the objective of vehicle level control through the same system. A comparison between this active system and a passive metallic suspension equipping a Citro¨en C4 Picasso is presented after modeling the pneumatic (leveling) system of a quarter vehicle model with two degrees-of-freedom. Results show an excellent body control as well as, by using the CRONE approach, a robustness of the stability-degree.

©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Vehicle suspension systems can be divided into two types according to its architecture. Indeed, the first type of suspension, called “traditional” architecture, is only composed of a passive spring-damper unit. Then, all other systems are classified as “complex” architecture [1]. The leaf spring or the MacPherson suspension are some of the most common examples of so-called traditional suspensions. Another example, in another technology, is the passive monospheric hydropneumatic suspension developed by Paul Mag`es in 1954 [2]. The complex architectures appeared with the advent of mechatronics in order to improve the comfort and safety aiming to resolve the dilemmas inherent in traditional architectures [3]. Bose suspension [4], Active Wheel [5] but also Hydractive suspension are some examples of so-called complex suspensions. The role of typical automotive suspension system can also be divided into two main aspects as seen in Fig. 1. On the one hand, its role in dynamics and, on the other hand, its role in statics by supporting the vehicle load and allowing, or not, to maintain the ride height independently of load by using active systems [6]. † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.007

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Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

Fig. 1 Roles of the automotive suspension system.

Self-leveling systems already exist in different technologies: hydro-pneumatic systems with the Hydractive CRONE suspension [7–13], hydro-mechanical systems with the Superlift system from Magneti Marelli, electro-mechanical systems with the electrically-driven ball screw drive-actuator from Schaeffler [14] and pneumatic systems, as used in the Citro¨en C4 Picasso (see Fig. 2) and trucks [15–19]. In this paper, we focus on the body control of a traveling vehicle under driver inputs. This functionality is complementary to the one of vehicle level control through the pneumatic system [20] which only operates when the vehicle is parked. Indeed, the three different forms of driver inputs consist of acting on accelerator or brake pedals as well as on the steering wheel. All of these actions induce load transfers which tend to modify the center of mass, and thus the equilibrium, but they only exist with a traveling vehicle. On the contrary, the objective of vehicle level control responds to an user request, when the vehicle is parked, and has an interest for people of reduced mobility (it facilitates the access to driver or passenger compartments) or for loading/unloading the trunk. Section 2 presents the modeling of the pneumatic leveling system while section 3 deals with the method of body control based on a CRONE control strategy, which synthesis methodology is completely developed, enabling the design of a robust controller by using fractional-order operators. In section 4 results between the active pneumatic system will be compared to those obtained with a passive metallic suspension of the Citro¨en C4 Picasso. Finally, conclusions and perspectives are presented in section 5.

2 System modeling The vehicle considered in this paper is the Citro¨en C4 Picasso, in which an air suspension has already been integrated. As seen in Fig. 2, the suspension system is composed of a shock absorber and a pneumatic cushion, enabling control of the vehicle height for a parked vehicle and the body control under driver inputs. It also acts as the spring when none of the latter active functions are ongoing. The quarter vehicle model with two degrees-of-freedom of the studied system is represented in Fig. 3, m2 and m1 being, respectively, the sprung and the unsprung masses (kg), b2 the viscous coefficient of the damper (Ns/m), k1 and b1 designating, respectively, the equivalent vertical stiffness (N/m) and viscous coefficient of the tire (Ns/m), v2 (t) and v1 (t) representing, respectively, the vertical velocities of m2 and m1 (m/s), f0 (t) modeling the load transfer (N), v0 (t) the velocity of the road vertical excitation (m/s) and Ua (t) being the force generated by the pneumatic actuator. In the first instance, the pneumatic system considered here is not equipped with an air reservoir.

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81

Fig. 2 Pneumatic rear suspension of the Citro¨en C4 Picasso (1. Chassis, 2. Shock absorber, 3. Sensor, 4. Sensor lever, 5. Link, 6. Suspension arm, 7. Pneumatic cushion, 8. Wheel spindle).

Fig. 3 Quarter vehicle model with two degrees-of-freedom with the pneumatic leveling system (1. Pneumatic cushion, 2. Leveling valve, 3. Air compressor, 4. Piston, 5. Link).

This way, air is directly pumped from outside into the system via the leveling valve for inflation and pumped out for deflation. Thereafter, a reservoir, in which air has already been pumped from outside and compressed, may be added enabling a faster inflation whilst avoiding the outside air compression step that could result, in some instances, to a large temperature increase into the system. 2.1

Validation model

Several models of pneumatic actuators already exist in the literature, but these models either represent linear pneumatic cylinder [21–24] or air springs without considering the leveling system [25] or using a reservoir [26]. As modeled in [20], the dynamic behavior of the two degrees-of-freedom system is defined by the generalized equation:

vi (t) =

1 mi

ˆ

t 0

FΣi (τ )d τ + vi (0),

(1)

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Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

Fig. 4 Validation model of the pneumatic actuator.

Fig. 5 Single block diagram representation of the validation model.

with i = 1 or i = 2, vi (0) = 0 chosen as an initial state afterwards and FΣi (t) described as follows:  FΣ1 (t) = Fd (t) −Ua (t) + Ft (t) − m1 g, FΣ2 (t) = f0 (t) − Fd (t) +Ua (t) − m2 g,

(2)

with Fd (t) being the force generated by the damper on the mass m2 (with NLd a nonlinear function of (v2 (t) − v1 (t))) and Ft (t) the force generated by the tire on the mass m1 defined by the following equations: ˆ t (3) Ft (t) = k1 (v0 (τ ) − v1 (τ ))d τ + b1 (v0 (t) − v1 (t)), 0

Fd (t) = NLd (v2 (t) − v1 (t)).

(4)

The force generated by the pneumatic actuator Ua (t) is defined as follows: Ua (t) = Sv (Pv (t) − pa ) ,

(5)

where pa = 1.105 Pa being the absolute atmospheric pressure, Sv the effective area of the air spring (m2 ) and, ˆ t 0 1 0 Vv (1 + 0 0 Pv (τ )qu (τ )d τ ), (6) Pv (t) = Pv Vv (t) Pv Vv 0 ˆ t ˆ t qd (τ )d τ = Sv (v2 (τ ) − v1 (τ ))d τ , (7) vv (t) = 0

0

Pv (t) = Pv0 + pv (t) being the absolute pressure into the air spring (Pa ) with Pv0 the chosen equilibrium static absolute pressure, Vv (t) = Vv0 + vv (t) the volume of the air spring (m3 ) with Vv0 the equilibrium static volume, qu (t) the control flow (m3 /s) and qd (t) being the displacement flow (m3 /s). Thus, the validation model of the pneumatic actuator is developed in Fig. 4. 0 Finally, the validation model of the plant ´ t is represented in Fig. 5 with Z2 (t) = Z2 + z2 (t) being the suspended mass height (m) where zi (t) = 0 vi (τ )d τ + zi (0).

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

2.2

83

Analysis model

In order to design robust control-systems in frequency domain, the previous model is linearized around its static equilibrium point (Pve = Pv0 , Vve = Vv0 , qeu = 0 and qed = 0). Equations (4) and (6) become: Fd (t) = b2 (v2 (t) − v1 (t)), ˆ t 1 Pv (t) = Pv0 + (qu (τ ) − qd (τ ))d τ , κ C2 0

(8) (9)

V0

with C2 = κ1 Pv0 the pneumatic capacity of the air spring (m5 /N), κ (1 ≤ κ ≤ γ ) the polytropic coefficient v of the pneumatic transformation and γ = 1.4 the adiabatic index or ratio of specific heats at constant pressure and at constant volume. Finally, using (1) - (9) (around the operation point), the analysis model can be developed. 2.3

Synthesis model

After linearizing the validation model around the static equilibrium point, the principle of superposition is applied in order to define the synthesis model, which is the model used for robust control-systems in frequency domain synthesis. Thus, this synthesis model can be represented with the following transfer function of the plant:  K Z2 (s)  , (10) = P(s) = Qu (s)  F0 (s)=0 s(1 + 2ξ ωs0 + ( ωs0 )2 ) V0 (s)=0

with Z2 (s) = LT {z2 (t)}, Qu (s) = LT {qu (t)}, F0 (s) = LT { f0 (t)}, V0 (s) = LT {v0 (t)} and where LT denotes the Laplace Transform. The obtained transfer function of the plant P(s) is an uncertain third order. Its gain K, its natural frequency ω0 and its damping factor ξ depend on the values of the system parameters and uncertainties. The dynamics of the distributor is assumed to be negligible compared to that of the regulation loop. In this study, the parameters and uncertainties are considered to be bounded according to: ⎧ m2 ∈[270.7 kg, 513.2 kg] , ⎪ ⎪ ⎪ ⎪ ∈[1, γ ] , ⎨κ Hz, 1.15 Hz] ftarget ∈[0.95 (11)  , ⎪ 2 , 49 cm2 , ⎪ ∈ 40 cm S ⎪ ⎪ ⎩ v b2 ∈[1515 Ns/m, 1851 Ns/m] , with ftarget the given natural frequency with the nominal sprung mass m2nom . Taking the extreme values and knowing that: ⎧ 1 ⎪ ⎪ K = ⎪ ⎪ Sv ⎪

⎪ ⎨ (2π ftarget )2 , ω = 0 ⎪ ⎪ κ ⎪ ⎪ ⎪ b2 ⎪ ⎩ξ = 2m2 ω0

(12)

these uncertainties lead to gain K, natural frequency ω0 and damping factor ξ variations such that: ⎧  ⎨ K ∈ 205 s/m2 , 251 s/m2 , (13) ω ∈ [5.97 rad/s, 6.11 rad/s] , ⎩ 0 ξ ∈ [0.30, 0.47] .

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Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

Magnitude (dB)

60 40 20 0 -20 -40

Phase (deg)

-90 Pextr1 Pnom Pextr2

-135 -180 -225 -270 10 -1

10 0

10 1

10 2

Frequency (rad/s)

Fig. 6 Bode diagrams of the nominal transfer function Pnom (s) (green), and of the extreme transfer functions Pextr1 (s) (blue) and Pextr2 (s) (red).

Fig. 7 Feedback system.

Thus, three transfer functions can be defined for the controller design as presented in Fig. 6: a nominal one, defined with the average values of parameters, and two extreme ones, namely: ⎧ ⎨ Pnom (s) =P(s, m2nom , κnom , ftargetnom , Svnom , b2nom ), Pextr1 (s)=P(s, m2extr1 , κextr1 , ftargetextr1 , Svextr1 , b2extr1 ), ⎩ Pextr2 (s)=P(s, m2extr2 , κextr2 , ftargetextr2 , Svextr2 , b2extr2 ).

(14)

3 Controller design In order to design the controller enabling the body control under driver inputs, the feedback system presented in Fig. 7 is used, where C(s) is the transfer function of the controller, Hcons (s) the reference signal, U (s) = Qu (s) the control signal, Du (s) the input disturbance signal, N(s) the sensor noise and Hmes (s) the measured signal. In order to control the body under driver inputs, a second generation CRONE controller is designed. Moreover an illustration of its contribution will be done comparing this active system with the passive metallic version of the Citro¨en Grand C4 Picasso.

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

3.1

85

Specification requirements

The requirements of the system are the following: • control of the vehicle body under driver inputs; • a crossover frequency ωcg = 2π 10 rad/s; • a robust resonant peak QT = Mr = 3dB. 3.2

Second generation CRONE controller design

The CRONE control system design (CSD) methodology [27] is a frequency-domain approach (loop shaping), which has been developed since the 1980s [28–32]. It offers the best choice in comparison with other fractional order controller, in terms of readiness for real application [33]. It presents three approaches, related to the three generations of CRONE control. The first two approaches ensure robustness of the stability-degree with respect to the parametric plant perturbation, whereas the third generation minimizes the variations of the stability-degree. The application of these three strategies is also possible using the CRONE toolbox [34, 35], developed by the CRONE research group. The first generation CRONE strategy is particularly appropriate when the desired open-loop gain crossover frequency ωcg is within a frequency range where the plant frequency response is asymptotic. It proposes to use a controller without phase variation around ωcg . When ωcg is within a frequency range where the plant uncertainties are only gain-like, the second generation CRONE approach enables to produce a constant open-loop phase for which the Nichols locus is a vertical straight line, or frequency template, which ensures the robustness of phase and modulus margins. Finally, the third generation CRONE strategy must be used when the plant frequency uncertainty domains are of various types (not only gain-like). It replaces the vertical template by a generalized template which is optimized by minimizing the stability degree variations. As can be seen in Fig. 6, ωcg is within a frequency range where the plant uncertainties are only gainlike. Therefore, the second generation CRONE strategy can be used. It defines the open-loop transfer function (in the frequency range [ωA , ωB ] corresponding to the range where frequency ωcg varies) by that of a fractional integrator:

β (s) = (

ωcg n ) , with n ∈ IR and n ∈ [1, 2] . s

Moreover, the sensitivity functions S(s), T (s), CS(s) and GS(s) can be defined as following: ⎧ 1 ⎪ ⎪ , S(s) = ⎪ ⎪ 1 + β (s) ⎪ ⎪ ⎪ ⎪ β (s) ⎪ ⎪ ⎪ ⎨ T (s) = 1 + β (s) , C(s) ⎪ ⎪ ⎪ , CS(s)= ⎪ ⎪ 1 + β (s) ⎪ ⎪ ⎪ ⎪ P(s) ⎪ ⎪ ⎩ GS(s)= , 1 + β (s)

(15)

(16)

with β (s) = C(s)P(s) the open-loop transfer function, S(s) the sensitivity function, T (s) the complementary sensitivity function, CS(s) the input sensitivity function and GS(s) the input disturbance sensitivity function. As presented in Fig. 8, the Nichols locus of β (s) is a vertical straight line around ωcg . Its phase location is determined by order n only and being the desired shape of the open-loop frequency response Nichols locus (called frequency template).

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Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

Fig. 8 Frequency template in the Nichols plane.

The vertical template thus defined slides on its own axis with the parameters variations ensuring: • a robust phase margin MΦ equals (2 − n)π /2; • a robust resonant peak Mr = QT =

sup|T ( jω )| ω

|T ( j0)|

=

1 sin(nπ /2) ;

• a robust modulus margin Mm = inf|β ( jω ) + 1| = (sup|S( jω )|)−1 = sin(nπ /2); ω

ω

• a robust damping ratio ζ = cosθ = cos(π − πn ) = − cos(π /n) directly deduced from the central half-angle θ formed by the two closed-loop complex poles thus introducing the notion of robust oscillatory mode.

Then, the fractional open-loop transfer function has to be band-limited and complexified by including integral and low-pass effects as defined in Eq. (17):

β (s) = β0 (

1 + ωsl s ωl

nl

) (

1 + ωsh 1+

s ωl

)n

1 (1 + ωsh )nh

= C(s)P(s),

(17)

where ωl and ωh are, respectively, the transitional low and high frequencies, n is the non-integer order around the crossover frequency ωcg , nl and nh are, respectively, the integer orders at low and high frequencies and where β0 is a gain defined as following:

β0 = (

ωcg nl ωcg 2 (n−nl ) ωcg 2 (nh −n) ) (1 + ( ) ) 2 (1 + ( ) ) 2 . ωl ωl ωh

(18)

The asymptotic Bode diagrams of the fractional open-loop transfer function is presented in Fig. 9, with ωcgnom being the crossover frequency for the nominal plant Pnom (s). In this way, the fractional controller transfer function can be expressed as: C(s) = β (s)P−1 (s),

(19)

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

87

Fig. 9 Asymptotic Bode diagrams of the fractional open-loop transfer function β ( jω ).

or, with P(s) = Pnom (s) the nominal plant: Pnom (s) =

s

s(( ω0

nom

Knom 2 ) + 2ξnom ω0s

+ 1)

nom

,

(20)

with Knom the gain of the nominal plant, ξnom the damping factor and ω0nom the natural frequency. The expression of the fractional controller transfer function can be developed as: C(s) = β0 (

1 + ωsl s ωl

nl

) (

1 + ωsh 1 + ωsl

s(( ω0s )2 + 2ξnom ω0s + 1) 1 nom nom ) . (1 + ωsh )nh Knom n

Moreover, considering that n = 2 − m with 0 < m < 1 and C0 = C(s) = C0 s((

s

ω0nom

)2 + 2ξnom

s

ω0nom

+ 1)(

or, C(s) = C0 s((

s

ω0nom

2

) + 2ξnom

s

ω0nom

+ 1)(

1 + ωsl

1 + ωsl s ωl

s ωl

nl

)

)nl (

β0 , Eq. (21) can be rewritten with: Knom

1 + ωsh 1 + ωsl

1

(21)

)2−m

1 , (1 + ωsh )nh

1 + ωsh

2

1 + ωsh

( ) ( )−m . (1 + ωsh )nh 1 + ωsl 1 + ωsl

(22)

(23)

Finally, the fractional controller transfer function can be expressed with: C(s) = CI (s)CNI (s) = C0 s( with CI (s) = C0 s(

( ω0s )2 + 2ξnom ω0s nom

nom

( ωsl )nl

( ω0s )2 + 2ξnom ω0s nom

nom

( ωsl )nl

+ 1 (1 + ωs )nl −2 1 + ωs l h −m ) ( ) , (1 + ωsh )nh −2 1 + ωsl + 1 (1 + ωs )nl −2 l ) (1 + ωsh )nh −2

(24)

(25)

being an expression of integer order, and CNI (s) = (

1 + ωsh 1 + ωsl

)−m

(26)

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

88

Fig. 10 Asymptotic gain diagram of the fractional open-loop transfer function β ( jω ).

being a phase lead cell of non-integer order m (with 0 < m < 1). In a second step, the low, medium and high frequencies orders, respectively nl , n and nh , must be determined according to the specification requirements. To begin, the low frequency order nl determines the precision at steady-state. In order to ensure a zero following error, nl = 2. Then, to ensure a robust resonant peak Mr = 3dB, or a robust phase margin MΦ = 45◦ , n = 1.5 (and m = 2 − n = 0.5) and, finally, in order to limit the sensitivity of the process input, we choose nh = n p + 1 = 4 where n p represents the plant order at high frequencies. Consequently, the final expression of the controller is: C(s) = CI (s)CNI (s), with CI (s) = C0 s(

( ω0s )2 + 2ξnom ω0s nom

nom

+1

( ωsl )2 (1 + ωsh )

) and CNI (s) = (

(27) 1 + ωsl

1 + ωsh

)0.5 .

(28)

In a third part, the transitional low and high frequencies (ωl and ωh ) are determined. First of all, we know that ∀ωcg ∈ [ωcgmin ; ωcgmax ], ωA ≤ ωcg ≤ ωB so that ωA ≤ ωcgmin ≤ ωcg ≤ ωcgmax ≤ ωB . Then, ωh √ ωA B choosing that ωl ωh = ωcgnom and r = ω ωA > 1, ωl = a and ωh = bωB are set, hence ω = abr. Finally, l ωl and ωh can be expressed as following: ⎧ ⎨ ω = ωcgnom √ , l a r √ (29) ⎩ ω =ω b r, h

1

cgnom

1

with r = Δβ n = ΔP n representing the gain variations around ωcg , as can be seen in Fig. 10. 1

In the present case ωcgnom = 62.8 rad/s, 20 log10 ΔP = 1.3dB so that r = (ΔP) 1.5 = 1.11 and, with a = b = 10, ⎧ ⎨ ω = ωcgnom √ = 5.98 rad/s l 10 r √ . (30) ⎩ ω =ω 10 r = 660.48 rad/s h cgnom It should be noted that the following method enables to design, in the best possible manner, the length of the template according to the gain variations. In practice, we start with a = b = 10 and then increase these values if the template is not long enough to reach the asymptotic behavior in the frequency range [ωA , ωB ]. Finally, a and b can have different values. Thus, using a = 50 and b = 10, the following transitional frequencies are found: ⎧ ⎨ ωl = ωcgnom √ = 1.19 rad/s 10 r , (31) √ ⎩ ωh =ωcgnom 10 r = 660.48 rad/s

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

and β0 can be determined using: ωcg nl ωcg 2 (n−nl ) ωcg 2 (nh −n) β0 = ( ) (1 + ( ) ) 2 (1 + ( ) ) 2 = 384.17. ωl ωl ωh In a fourth step, the gain C0 can be determined knowing β0 and Knom = C0 =

β0 = 2.43 m2 . Knom

1 Svnom

89

(32) = 225.63 m−2 : (33)

Thus, the optimal parameters of the open-loop transfer function and, a posteriori, of the second generation CRONE controller are the following: ⎧ n =1.5 , m =0.5; ⎪ ⎪ ⎪ ⎪ , nh =4; nl =2 ⎪ ⎪ ⎨ r =1.11 , β0 =384.17 (34) , C0 =2.43 Knom =225.63 ⎪ ⎪ ⎪ ⎪ ⎪ a =50 , b =10; ⎪ ⎩ ωl =1.19 rad/s, ωh =660.48 rad/s. Finally, in the last step, the rational form Crat (s) of the controller transfer function C(s) can be determined using the CRONE approximation defined by Oustaloup [36]. For that, the rational form CR (s) of the phase-lead cell of non-integer order CNI (s), whose expressions are mentioned below, needs to be identified. N 1 + s 1 + ωsl m ωi (35) CNI (s) = ( s ) and CR (s) = ∏( s ). 1 + ωh i=1 1 + ωi Thus, using the four parameters m, ωl , ωh and N, CR can be defined using the following equations: ωh 1 αη = ( ) N so that α = (αη )m and η = (αη )1−m (36) ωl 1

1

ω1 = ωl η 2 and ωN = ωh η − 2 ,

(37)

 ωi+1 = αηωi and ωi+1 = αηωi .

(38)

Then, using Eq. (34) and N = 8 in the present case, the values of CR can be determined (see Eq. (39)). As described by Oustaloup [36] and by Malti, Melchior, Lanusse and Oustaloup [34], the bigger N the more precise the approximation of the fractional differentiator in the frequency band [ωA , ωB ]. As a rule of thumb, two of the three corner frequencies (in the numerator and the denominator) are used per decade. ⎧ α =1.49 , η =1.48; ⎪ ⎪ ⎪  =1.46 rad/s , ω =2.16 rad/s; ⎪ ω 1 ⎪ 1 ⎪ ⎪  =3.21 rad/s , ω =4.76 rad/s; ⎪ ω ⎪ 2 2 ⎪ ⎪  ⎪ ⎨ ω3 =7.06 rad/s , ω3 =10.48 rad/s; (39) ω4 =15.54 rad/s , ω4 =23.07 rad/s; ⎪  ⎪ ⎪ ⎪ ω5 =34.22 rad/s , ω5 =50.81 rad/s; ⎪ ⎪ ⎪ ω6 =75.35 rad/s , ω6 =111.87 rad/s; ⎪ ⎪ ⎪ ⎪ ω  =165.90 rad/s, ω7 =246.32 rad/s; ⎪ ⎩ 7 ω8 =365.29 rad/s, ω8 =542.36 rad/s. At last, the rational form CR (s) of the phase advance cell of non-integer order CNI (s) can be defined and, using Eq. (28) and Eq. (35), the rational form Crat (s) of the CRONE controller can be expressed as following: ( ω0s )2 + 2ξnom ω0s + 1 N 1 + ωs  nom i ) ∏( (40) Crat (s) = CI (s)CR (s) = C0 s( nom s 2 s ). ( ωl ) (1 + ωsh ) 1 + i=1 ωi

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

90

Gain (dB)

60 40 20 C(s) C (s)

0

rat

-20 10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 3

10 4

10 5

Frequency (rad/s)

Phase (deg.)

200 100 0 -100 10 -1

10 0

10 1

10 2

Frequency (rad/s)

Fig. 11 Bode diagrams of the fractional form C(s) and of the rational form Crat (s) of the second generation CRONE controller.

40

100

20

Open-Loop Gain (dB)

Magnitude (dB)

0 dB 30

0

-100

Phase (deg)

-200 -90 Pextr1 Pnom Pextr2

-180

3 dB 6 dB

0 -10 -20

-270

-360 10 -1

1.5 dB 10

-30

10 0

10 1

10 2

10 3

Frequency (rad/s)

(a) Bode diagrams

10 4

10 5

-40 -270

-225

-180

-135

-90

-45

0

Open-Loop Phase (deg)

(b) Nichols diagram

Fig. 12 Open-loop Bode and Nichols diagrams for Pextr1 (s) (blue), Pnom (s) (green) and Pextr2 (s) (red).

The Bode diagrams of C(s) and Crat (s) are presented in Fig. 11, showing that such approximation is adequate.

4 Performance analysis Bode and Black-Nichols diagrams of the open-loop with the CRONE controller previously designed, considering nominal Pnom (s) and extreme plants Pextr1 (s) and Pextr2 (s) are plotted in Fig. 12. Moreover, the sensitivity functions S(s), T (s), CS(s) and GS(s), defined in (16), are also presented in Fig. 13. Thus, it can be deduced from these figures that the crossover frequency ωcg and the highest resonant peak QT vary, according to the plant variations, respectively from 59.9 to 66.9 rad/s and from 2.99 dB to 3.13 dB.

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

91

5 0

Pextr1 Pnom Pextr2

-5

-10

-10

Magnitude (dB)

Magnitude (dB)

Pextr1 Pnom Pextr2

0

-20

-30

-40

-15 -20 -25 -30 -35 -40

-50

-45 -60 10 0

10 1

10 2

-50 10 0

10 3

Frequency (rad/s)

10 2

10 3

Frequency (rad/s)

(a) Sensitivity function S(s)

(b) Complementary sensitivity function T (s)

60

40

10 1

10 Pextr1 Pnom Pextr2

Pextr1 Pnom Pextr2

0 -10

Magnitude (dB)

Magnitude (dB)

-20 20

0

-20

-30 -40 -50 -60 -70 -80

-40

-90 -60 10 0

10 1

10 2

10 3

10 4

-100 10 0

Frequency (rad/s)

10 1

10 2

10 3

Frequency (rad/s)

(c) Input sensitivity function CS(s)

(d) Input disturbance sensitivity function GS(s)

Fig. 13 Sensitivity functions for Pextr1 (s) (blue), Pnom (s) (green) and Pextr2 (s) (red).

Fig. 14 Time-domain feedback system.

For the time domain performances, and in order to illustrate the performances in terms of body control under driver solicitations, the load transfer f0 (t) = 500 N (corresponding to a braking condition for a rear axle suspension) will be applied with the validation model whose feedback system is presented in Fig. 14. The results will be compared to those of the passive metallic version of the Citro¨en Grand C4 Picasso. Results of the simulations using the validation models (active pneumatic and passive metallic) and considering the three parametric states previously defined (Pnom (s), Pextr1 (s) and Pextr2 (s)) are shown in Fig. 15. Furthermore, the validation model of the pneumatic system used here is a more complete version of the one presented in Fig. 4 as it takes account of the temperature (and its variation) inside

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

92

0.045

0.045 Pextr1 Pnom Pextr2

0.04

0.035

0.03

0.03

0.025

0.025

Height (m)

Height (m)

0.035

0.02 0.015

0.02 0.015

0.01

0.01

0.005

0.005

0

0

-0.005

0

0.5

1

1.5

2

2.5

3

3.5

Pextr1 Pnom Pextr2

0.04

4

-0.005

0

0.5

1

1.5

Time (s)

2

2.5

3

3.5

4

Time (s)

(a) Output z2 (t) (Active pneumatic suspension)

(b) Output z2 (t) (Passive metallic suspension)

600

600 F0 Fb2extr1

400

400

Fb2ref Fb2

200

200

extr2

extr1

Ua ref

0

Force (N)

Force (N)

Ua

Ua extr2

-200

0 -200

-400

-400

-600

-600

-800

-800

F0 Fb2extr1 Fb2ref Fb2extr2 Fk2 extr1 Fk2

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

(c) Load transfer f0 (t), actuator force Ua (t) (around Ua0 ) and damping force fb2 (t) (Active pneumatic suspension)

ref

Fk2 extr2

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

(d) Load transfer f0 (t), spring force Fk2 (t) and damping force fb2 (t) (Passive metallic suspension)

Fig. 15 Output z2 (t) and forces deployed by the different elements of the suspension after a load transfer of amplitude f0 = 500 N, with the validation models the active pneumatic and the passive metallic suspensions, for Pextr1 (s) (blue), Pnom (s) (green) and Pextr2 (s) (red).

the pneumatic cushion. This model will be developed in a future contribution. Finally, the implement of the control loop with the CRONE controller enables an excellent body stability compared to the passive metallic suspension of the Citro¨en Grand C4 Picasso which, as expected, endures the load transfer. Indeed, whereas the steady-state value of the ride height z2 (t) (representing Z2 (t) around the equilibrium Z20 ) is constant, according to the plant variations, at 3.25 cm for the passive metallic system, it stays constant at zero for the active pneumatic system. Thus, as the output Z2 (t) is regulated, for the active system, at its equilibrium value Z20 , it can be noted that z2 (t) is indeed annulled. On the contrary, for the passive system, and as the ride height is not regulated, it can be observed that the suspension is only enduring the load transfer, which translates into an output Z2 (t) different of its equilibrium value Z20 (or z2 (t) = 0), without acting to annul this variation. In this way, the body stability ensured by the active system can be characterized by the regulation of the ride height at its equilibrium but above all by its settling time, which enables an improved roll and pitch control permitting a greater security under driver solicitations.

Jean-Louis Bouvin et al. / Journal of Applied Nonlinear Dynamics 8(1) (2019) 79–95

93

Fig. 16 Time-domain system with feedback and feedforward control.

5 Conclusion In this article, the modeling of an active pneumatic suspension system based on the air spring, and enabling the body control of the vehicle under driver inputs, has been developed. A methodology allowing the synthesis of a second generation CRONE controller has thus been proposed in order to meet the required specifications which are a crossover frequency ωcg = 2π 10 rad/s and a robust resonant peak QT = 3dB. Due to the system characteristics and the desired specifications, the CRONE controller offers an excellent body control compared to the passive metallic suspension that just endures the load transfer without regulation of the ride height. It would also be interesting to compare the performances offered with the CRONE controller to those obtained with a PID or another robust controller. Future work will be oriented on a strategy involving two CRONE controllers: one designed for the height regulation for a parked vehicle as in [20] and another for the body control of the vehicle under driver inputs for a traveling vehicle. Indeed, the levels of control effort and dynamics asked (a decade below the natural frequency of the car body for the height regulation to ensure appropriate safety and a decade above for the body control) to respond to these issues are not the same, so that the requirement specifications, which implies the use of two different controllers. Moreover, and in the future, this feed-back system will be paired with a feed-forward control system enabling the relaxation of the imposed constraints for the design of the CRONE controller, as seen in Fig. 16.

Acknowledgements This work took place in the framework of the OpenLab ‘Electronics and Systems for Automotive’ combining IMS laboratory and PSA Groupe company.

References [1] L´et´ev´e, A. (2014), Etude de l’influence des suspensions de v´ehicule de tourisme sur le confort vibratoire, le comportement routier et les limites de fonctionnement : l’approche CRONE en mati`ere de formalisation, d’analyse et de synth`ese, Universit´e Bordeaux I. ´ [2] Brioult, R. (1987), Citro¨en, l’histoire et les secrets de son bureau d’´etudes, Edifree. [3] Jezequel, L. (1995), Active Control in Mechanical Engineering, International Symposiums Series, Hermes,

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84. [4] Brown, S.N. (2010), Active suspension system for vehicle. [5] Laurent, D., Sebe, M. and Walser, D. (2000), Assembly comprising a wheel and a suspension integrated into the wheel. [6] Tseng H.E. and Hrovat, D. (2015), State of the art survey: active and semi-active suspension control, Vehicle System Dynamics, 53/7, 1034-1062. [7] Oustaloup, A., Moreau, X. and Nouillant, M. (1997), From fractal robustness to non-integer approach in vibration insulation: the CRONE suspension, Decision and Control, 1997., Proceedings of the 36th IEEE Conference on, 5, 4979-4984. [8] Altet, O., Nouillant, C., Moreau, X. and Oustaloup, A. (2003), La suspension CRONE Hydractive : mod´elisation et stabilit´e : Automatismes : les syst`emes hybrides, REE. Revue de l’´electricit´e et de l’´electronique ISSN 1265-6534 s, 9, 84-93. [9] Altet, O., Moreau, X., Moze, M., Lanusse, P. and Oustaloup, A. (2004), Principles and Synthesis of Hydractive CRONE Suspension, Nonlinear Dynamics, 38, 453-459. [10] Rizzo, A., Moreau, X., Hernette, V. and Oustaloup, A. (2011), Suspension CRONE Hydractive : du concept au v´ehicule d´emonstrateur, Groupe de Travail ” Automatique et Automobile ” - Journ´ees d’´etude Automatique et Automobile -GTAA/JAA 2011 : Grenoble, France. [11] Rizzo, A. (2012), L’approche CRONE dans le domaine des architectures complexes des suspensions de v´ehicules automobiles : la suspension CRONE Hydractive, Universit´e Bordeaux I [12] L´et´ev´e, A., Moreau, X., Hernette, V. and Rizzo, A. (2013), La suspension CRONE Hydractive. Analyse et synth`ese, Revue des syst`emes, 47/4-8, 519-545. [13] L´et´ev´e, A., Moreau, X., Hernette, V. and Rizzo, A. (2013), De la suspension Hydractive de s´erie a` la suspension CRONE Hydractive implant´ee sur v´ehicule Citro¨en C5, JDMACS 2013 : Strasbourg, France. [14] Baeuml, M., Dobre, F., Hochmuth, H., Kraus, M., Krehmer, H.,Langer, R. and Reif, D. (2014), The Chassis of the Future, Schaeffler Seminar, Schaeffler, 392-411, 2016-10-18. [15] Yuexia, C., Long, C., Ruochen, W., Xing, X., Yujie, S. and Yanling, L. (2016), Modeling and test on height adjustment system of electrically-controlled air suspension for agricultural vehicles, International Journal of Agricultural and Biological Engineering, 9/2, 40. [16] Xu, X., Chen, L., Sun, L. and Sun, X. (2013), Dynamic ride height adjusting controller of ECAS vehicle with random road disturbances, Mathematical Problems in Engineering, Hindawi Publishing Corporation. [17] Kim, H., Lee, H., and Kim, H. (2007), Asynchronous and synchronous load leveling compensation algorithm in airspring suspension, Control, Automation and Systems, 2007. ICCAS’07. International Conference on, IEEE, 367-372. [18] Kim, H. and Lee, H. (2011), Height and leveling control of automotive air suspension system using sliding mode approach, IEEE Transactions on Vehicular Technology, IEEE, 60/5, 2027-2041. [19] Yokote, M., Ito, H, Kawagoe, K. and Kawabata, K. (1988), Height control system for automotive vehicle with vehicular profile regulating feature. [20] Bouvin, J.-L., Moreau, X., Benine-Neto, A., Oustaloup, A., Serrier, P. and Hernette, V. (2017), CRONE control of a pneumatic self-leveling suspension system, IFAC 2017 World Congress: Toulouse, France. [21] Brun, X.(1999), Commandes lin´eaires et non lin´eaires en ´electropneumatique. M´ethodologies et Applications, INSA de Lyon. [22] Brun, X., Belgharbi, M., Sesmat, S., Thomasset, D. and Scavarda, S. (1999), Control of an electropneumatic actuator: comparison between some linear and non-linear control laws, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, SAGE Publications, 213/I5, 387-406. [23] Turki, K. (2010), New approaches for the synthesis of nonlinear robust control laws. Application to an electro pneumatic actuator and proposal a solution to the ”stick slip” phenomenon, INSA de Lyon. [24] Girin, A. (2007), Contribution to nonlinear control of electropneumatic system with a new test bench for aeronautics application, Ecole Centrale de Nantes (ECN). [25] Presthus, M. (2002), Derivation of air spring model parameters for train simulation, Lule˚ a University of Technology, Sweden. [26] Robinson, W. D. (2012), A pneumatic semi-active control methodology for vibration control of air spring based suspension systems, Digital Repository@ Iowa State University. [27] Lanusse, P., Malti, R. and Melchior, P. (2013), CRONE control system design toolbox for the control engineering community: tutorial and case study, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 371/1990. [28] Oustaloup, A. (1983), Syst`emes asservis lin´eaires d’ordre fractionnaire, Masson: Paris, France. [29] Oustaloup, A. (1991), La commande CRONE, Herm`es Editor: Paris, France. [30] Oustaloup, A., Mathieu, B. and Lanusse, P. (1995), The CRONE control of resonant plants: application to

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a flexible transmission, European Journal of control, Elsevier, 1/2, 113-121. [31] Lanusse, P. (1994), De la commande CRONE de premi`ere g´en´eration `a la commande CRONE de troisi`eme g´en´eration, Universit´e Bordeaux I. [32] ˚ Astr¨ om, K. J. (2000), Model uncertainty and robust control, Lecture Notes on Iterative Identification and Control Design, Systems Engineering and Control Department of the Universidad Polit´ecnica de Valencia, 63-100. [33] Xue, D. and Chen, Y. (2002). A comparative introduction of four fractional order controllers. In Intelligent Control and Automation, 2002. Proceedings of the 4th World Congress on (Vol. 4, pp. 3228-3235). IEEE. [34] Malti, R., Melchior, P., Lanusse, P. and Oustaloup, A. (2011), Towards an object oriented CRONE toolbox for fractional differential systems, IFAC Proceedings Volumes, Elsevier, 44/1, 10830-10835. [35] Oustaloup, A., Melchior, P., Lanusse, P., Cois, O. and Dancla, F. (2000), The CRONE toolbox for Matlab, Computer-Aided Control System Design, 2000. CACSD 2000. IEEE International Symposium on, IEEE, 190-195. [36] Oustaloup, A. (1995), La d´erivation non-enti`ere, Herm`es Editor: Paris, France.

Journal of Applied Nonlinear Dynamics 8(1) (2019) 97-108

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Method for Finding a set of (A, B, C, D) Realizations for Single-Input Multiple-Output / Multiple-Input Single-Output One-dimensional Continuous-time Fractional Systems Konrad Andrzej Markowski†, Krzysztof Hryni´ow Warsaw University of Technology, Faculty of Electrical Engineering Institute of Control and Industrial Electronics, Koszykowa Street No 75, 00-662 Warsaw, Poland Submission Info Communicated by C. I. Muresan Received 1 August 2017 Accepted 4 October 2017 Available online 1 April 2019 Keywords Realization Fractional systems MISO and SIMO Digraphs Algorithm

Abstract In the paper presented is a method allowing for determination of a set of (A, B, C, D) realizations for fractional-order dynamic systems. Proposed method is an extension of previously proposed algorithm that was used to determine realizations of fractional-order 1D single-input single-output (SISO) dynamic systems for both single-input multipleoutput (SIMO) and multiple-input single-output (MISO) fractional dynamic systems. The main advantage of the method over canonical forms is that the algorithm finds a set of realizations, not just a single realization. Also, the solutions found tend to be minimal in terms of size of state matrix A. Additionally, the method allows for the possibility of obtaining a set of state matrices directly from digraph form of the system and can be efficiently used as GPGPU computer algorithm. Proposed method is presented in pseudo-code and illustrated with example. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In the field of analysis of dynamic systems one of constant problems is the realization problem. Most research studies focus on canonical forms of the system, i.e. constant matrix forms, which satisfy the system described by the transfer function [1, 2]. With the use of canonical form we are able to write only one realization of the system, while in general there exists a set of possible solutions. Another approach is based on digraph representation of dynamic system and was first introduced in [3] and [4]. The first definition of the fractional derivative was introduced by Liouville and Riemann at the end of the 19th century [5]. Mathematical fundamentals of fractional calculus are given in [5–8]. Some others applications of fractional-order systems can be found in [9–16]. Most existing solutions for the problem of determination of entries of the state matrix A for a given characteristic polynomial are based on canonical forms of the system [2]. Canonical forms are constant matrix form, which satisfy the system described by the transfer function. All of the state-of-the-art † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2019.03.008

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methods based on canonical forms [17–21] are capable of giving one of the possible realizations of the characteristic polynomial. Due to the complexity of the problem there were no classical methods of determination of the whole set of possible realizations for a given characteristic polynomial. Such solution was proposed for non-fractional 1-D and 2-D single-input single-output (SISO) systems in [22, 23] and for fractional SISO systems in [24]. The method is based on the multi-dimensional digraphs theory and is extension of basic method proposed in [3] and [4]. Method of constructing digraphs and their properties can be seen for example in [25]. In addition to finding a set of possible solutions for the given characteristic polynomial, proposed method is also able to find solutions that are minimal in rank of A matrix. The parallel algorithm proposed as practical implementation of the proposed method for SISO system was shown in [26] as superior to methods proposed in [17–21] in both number of obtained realizations and the minimality of matrix size. Computer algorithm for SISO systems (both standard and dynamic) was presented in details in [27]. This paper presents an extension of the parallel algorithm described in [26–28], that allows for determination of a set of (A, B, C, D) realizations for fractional-order one-dimensional (1-D) SIMO and MISO systems. Algorithm finds not only A matrices, but also B and C matrices from digraph representation of the system. Realisations obtained using the proposed method are at this stage are equal – having the set of possible realizations allows for additional selection (for example when searching for reachable solution or one that is observable) and opens the possibilities of more in-depth analysis of given dynamical system. This work has been organised as follows: Section 2 presents the main results of the paper – proposition how to find (A, B, C, D) matrices directly from constructed digraph and pseudo-code of parallel numerical method, while Section 3 illustrates the working of the method with an example. In Section 4 concluding remarks and future work are presented. One-Dimensional Fractional-Order SIMO and MISO System. time fractional linear system described by equations: α 0 Dt x(t)

= Ax(t) + Bu(t),

Let be given the continuous-

0 < α  1,

(1)

y(t) = Cx(t) + Du(t), where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ R p are the state, input and output vectors, respectively and A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n and D ∈ R p×m . The following Caputo definition of the fractional derivative will be used: ˆ t dα 1 f (n) (τ ) C α D = = dτ , (2) a t dt α Γ(n − α ) a (t − τ )α +1−n where α ∈ R is the order of a fractional derivative, f (n) (τ ) = function.

d n f (τ ) dτ n

and Γ(x) =

´∞ 0

e−t t x−1 dt is the gamma

Theorem 1. The Laplace transform of the derivative-integral (2) has the form L

C

α 0 Dt



n

= sα F(s) − ∑ sα −k f (k−1) (0+ ). k=1

The proof of the Theorem 1 is given in [6]. After using the Laplace transform to (1), Theorem 1 and taking into account ˆ ∞ x(t)e−st dt, X (s) = L [x(t)] = 0

(3)

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99

L [Dα x(t)] = sα X (s) − sα −1 x0 we obtain:

  X (s) = [In sα − A]−1 sα −1 x0 + BU (s) , Y (s) = CX (s) + DU (s), U (s) = L [u(t)] .

(4)

After using (4) we can determine the transfer matrix of the system in the following form: T(s) = C [In sα − A]−1 B + D ∈ R p×m (s).

(5)

Matrices A, B, C and D are called a realization of the transfer matrix T(s) if they satisfy the equality (5). The realization (A, B, C, D) is called minimal if the dimension of the state matrix A is minimal among all possible realizations of T(s). 2 Main Results The essence of the proposed method for determining a minimal realization for one-dimen-sional fractionalorder continuous-time system described by the model (1) will be presented for single-input multipleoutput (SIMO) and multiple-input single-output (MISO) dynamic systems. We will be assume that commensurate fractional-order continuous-time system will be considered. Therefore, transfer matrix can be considered as a pseudo-rational function of the variable λ = sα in the form: T(λ ) = C [In λ − A]−1 B + D ∈ R p×m (λ ). 2.1

(6)

SIMO systems

The transfer matrix (6) for SIMO system can be written in the following form: ⎤ ⎤ ⎡ ⎡ n1,1 (λ ) T1 (λ ) ⎥ ⎢ T2 (λ ) ⎥ N(λ ) 1 ⎢ ⎥ ⎢ n2,1 (λ ) ⎥ ⎢ = T(λ ) = ⎢ . ⎥ = ⎥, ⎢ . .. ⎦ d(λ ) ⎣ ⎣ .. ⎦ d(λ ) T p (λ ) n p,1 (λ )

(7)

where: d(λ ) = λ n + an−1 λ n−1 + an−2 λ n−2 + . . . + a1 λ + a0 , ni,1 (λ ) = b(i,n) λ n + b(i,n−1) λ n−1 + . . . + b(i,1) λ + b(i,0) ,

(8)

i = 1, 2, . . . p.

(9)

From (6) we have: T(∞) = lim T(λ ) = lim {C [In λ − A]−1 B + D} = D λ →∞

λ →∞

since limλ →∞ [In λ − A]−1 = 0. Using (7) – (9) we obtain: ⎤ ⎡ ⎤ ⎡ ⎡ b(1,n) T1 (∞) T1 (λ ) ⎢ T2 (λ ) ⎥ ⎢ T2 (∞) ⎥ ⎢ b(2,n) ⎥ ⎢ ⎥ ⎢ ⎢ T(∞) = lim T(λ ) = lim ⎢ . ⎥ = ⎢ . ⎥ = ⎢ . .. ⎦ ⎣ .. ⎦ ⎣ .. λ →∞ λ →∞ ⎣ b(p,n) T p (λ ) T p (∞)

⎤ ⎥ ⎥ ⎥ = D. ⎦

(10)

100 Konrad Andrzej Markowski, Krzysztof Hryni´ ow / Journal of Applied Nonlinear Dynamics 8(1) (2019) 97–108

The strictly proper transfer function is given by the equation ⎡ Tsp (λ ) = T(λ ) − D = C [In λ − A]−1 B =

1 ⎢ ⎢ ⎢ d(λ ) ⎣

n 1 (λ ) n 2 (λ ) .. .

⎤ ⎥ ⎥ ⎥, ⎦

(11)

n p (λ ) where d(λ ) is given by the equation (8) and b(i,1) s +

b(i,0) , n (s) =

b(i,n−1) sn−1 + . . . +

(12)

where:

b(i,n− j) = b(i,n− j) − b(i,n) · an− j for i = 1, 2, . . . , p; j = 1, 2, . . . , n. 2.2

MISO systems

The transfer matrix (6) for MISO system can be written in the following form:   N(λ )  1  = n1,1 (λ ) n1,2 (λ ) . . . n1,m (λ ) T(λ ) = T1 (λ ) T2 (λ ) . . . Tm (λ ) = d(λ ) d(λ )

(13)

d(λ ) = λ n + an−1 λ n−1 + an−2 λ n−2 + . . . + a1 λ + a0 ,

(14)

where:

n1,i (λ ) = b(i,n) λ n + b(i,n−1) λ n−1 + . . . + b(i,1) λ + b(i,0) ,

i = 1, 2, . . . m.

(15)

From (6) we have: T(∞) = lim T(λ ) = lim {C [In λ − A]−1 B + D} = D λ →∞

λ →∞

since limλ →∞ [In λ − A]−1 = 0. Using (13) – (15) we obtain:     T(∞) = lim T(λ ) = lim T1 (λ ) T2 (λ ) . . . Tm (λ ) = T1 (∞) T2 (∞) . . . Tm (∞) = λ →∞ λ →∞   = b(1,n) b(2,n) . . . b(m,n) = D.

(16)

The strictly proper transfer function is given by the equation Tsp (λ ) = T(λ ) − D = B [In λ − A]−1 C =

 1  n 1 (λ ) n 2 (λ ) . . . n m (λ ) , d(λ )

(17)

where d(λ ) is given by the equation (14) and b(i,0) , n( ¯ λ ) = b¯ (i,n−1) λ n−1 + . . . + b¯ (i,1) λ +

where b¯ (i,n− j) = b(i,n− j) − b(i,n) · an− j for i = 1, 2, . . . , m; j = 1, 2, . . . , n.

(18)

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101

Algorithm 1 DetermineStateMatrices() 1: Determine number of cycles in characteristic polynomial - it is equal to number of terms in characteristic polynomial (and equal to number of binomials formed); 2: for binomial = 1 to cycles do 3: Determine digraph D for every binomial; 4: BinomialRealisation1D(binomial); 5: Determine number of variants for each binomial (except the first) and put it into array variants. Variants mean different placement positions of the sub-digraph on the final digraph; 6: end for 7: Create all possible combinations of variants of all binomials; 8: Remove all redundant variant sets by checking their control sums; 9: Determine digraph for each variant as a combination of the digraph binomial representations; 10: for all variant do 11: Determine PolynomialRealisation1D(variant) 12: Check number of cycles in digraph, if differs from cycles remove solution as improper; 13: Record as proper realization of characteristic polynomial – in this moment we have the structure of matrix A; 14: end for 15: for all realizations do 16: Each realization gives as a different structure of A matrix that can be filled by a set of possible weights; 17: Now the structure of B and C matrices need to be determined from constructed digraph; 18: [B,C] = FullRealisation1D(realization) 19: Determine weights of the arcs in digraph and write state matrices A, B, C; 20: end for 2.3

Algorithm for determination matrices A, B and C

Creation of digraph structure that allows to determine the A matrices was presented in [28] and detailed computer algorithm for SISO systems was presented in [27]. Digraph structure build is based on the characteristic polynomial of the transfer function. Matrices B and C can be obtained from digraph by adding two additional types of vertices - input vertex s and output vertex y. Wages of arcs between vertices vi and s correspond to wages of input matrix B and wages of arcs between vertices vi and y corresponding to wages of output matrix C. Intersection vertices are vertices that belong to the intersection set of all digraph representations of binomials consisting of given characteristic polynomial as presented in [26]. Due to the complexity of the problem in this article we are discussing the case where all arcs from single input vertex lead to intersection vertices and all arcs to every output vertex start in intersection vertices – such case was earlier discussed for SISO systems in [29] and [30]. In [31] three classes of digraph structures were proposed and all such solutions are belonging to defined there class K1 – digraph structures where all cycles have common vertex. Such structures have been found to always create valid numerical solutions (for non-positive systems) that can always be computer numerically in feasible time with use of the dedicated computer algorithm. To determine B and C matrices from digraph we create all the possible paths from input vertex (or verices in case of MISO system) to output vertex (or vertices in case of SIMO system) and multiply all arc wages on the given path to form a binomial form. Then, we add all the created binomials. Details are presented in Section 3. In other case, when not all arcs from input vertex lead to intersection vertices and/or not all arcs to output vertex start in intersection vertices observed digraph structure can also

102 Konrad Andrzej Markowski, Krzysztof Hryni´ ow / Journal of Applied Nonlinear Dynamics 8(1) (2019) 97–108

Algorithm 2 BinomialRealisation1D(binomial) 1: size equals size of the binomial (sum of powers of all the variables); 2: Create digraph D for given binomial; 3: for node = 1 to size do 4: if node == size then 5: asize,1 = 1; 6: else 7: asize−1,size = 1; 8: end if 9: end for belong to class K1 if there exists such intersection vertex or vertices or to much more complicated classes K2 or K3 . In this case there exist then additional complications when determining the B and C matrices from digraph and there is need for tracing all the paths on digraph that start in input vertex and end in output vertex, creation of sub-graphs for each cycle by removal of certain vertices and multiplication of wages on sub-graphs by (−1). Such method worked for SISO systems, but will be more time-consuming and complicated for SIMO and MISO systems. Proposed method starts with creating digraphs for all binomials that form the characteristic polynomial, then joins them by the use of composition relative to vertices to create all possible variants of digraphs representing polynomial realization – which represents the characteristic polynomial given. All parts of the algorithm can be paralleled, which is important for developing fast computer algorithm. Algorithm 1 presents in pseudo-code the basics of the proposed method. Parts of the algorithm are presented in simplified form and some functions are only mentioned in comment and their working is not presented in detail – interested reader should look into [27] for more presentation of implemented algorithm’s code for SISO systems on which proposed method is based. Algorithm 2 is much simpler from algorithm constructing binomial realizations presented earlier in [27]. It was proven in [23] that for 1D binomials both growth and prune steps can be omitted and only one solution will be created with different variants representing placement of the solution on different nodes in final digraph. Algorithm 3 is addition to algorithm presented in [26,28], allowing for creation of B and C matrices directly from digraph for SIMO or MISO dynamic systems. Remark 1. It should be noted that, algorithm is divided into blocks, each realising different part of the operation. Each block is a function and can be switched to another block to deal with different problem efficiently without the need to change whole algorithm and without performing unnecessary operations for simpler problems – such solution allowed for modification of only some parts of tried and tested algorithm for SISO systems to form a basis for experimental method for fractional-order SIMO and MISO systems. Some functions can be run in different order – for example: we can determine weights of A matrix first and then construct matrices B and C that match given matrix A, or we can determine the structure of all three matrices and then at the end fill the weights in all of them.

3 Example Problem: Find a minimal realization of the transfer matrix given in the form: T(s) =

1 s0.7 + 3 s1.4 + 3 s1.4 + 4s0.7 + 4 s0.7 + 1 s0.7 + 2

.

(19)

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103

Algorithm 3 FullRealisation1D(realization) 1: if SIMO then 2: for all Sinks Sc j , j = 1, . . . , p do 3: Create all paths from Sb (source) to Sc j (sinks); 4: for all paths do 5: Multiply all wages along the path to form binomial; 6: end for 7: end for 8: else 9: for all Sources Sb j , j = 1, . . . , m do 10: Create all paths from Sb j (sources) to Sc (sink); 11: for all paths do 12: Multiply all wages along the path to form binomial; 13: end for 14: end for 15: end if 16: Add all binomials created from paths to create a polynomial; 17: Distribute binomials based on b to form a set of equations from which B and C matrices can be created; Solution: Transfer matrix can be considered as a pseudo-rational function of the variable λ = s0.7 in the form: 2 λ +3 1 λ +3 1 λ2 +3 λ +3 = . (20) T(λ ) = (λ + 2)2 λ + 1 λ + 2 λ 2 + 4λ + 4 λ + 1 λ + 2 In the first step we must determine matrix D. For this purpose, we should rewrite the transfer matrix (20) into the following form:

 ≡(λ 2 +3)(λ +1) ≡(λ +2)2 ≡(λ +3)(λ +2)(λ +1)          λ 3 + λ 2 + 3λ + 3 λ 2 + 4λ + 4 λ 3 + 6λ 2 + 11λ + 6 . (21) T(λ ) = 3 2 λ + 5 λ + 8 λ + 4    ≡(λ +2)2 (λ +1)

By the use equation (10) and (21) we obtain:   D = lim T(λ ) = 1 0 1 .

(22)

λ →∞

From (17) and (22) we have: Tsp (λ ) = T(λ ) − D =



−4λ 2 − 5λ − 1

λ 2 + 4λ + 4 λ 3 + 5λ 2 + 8λ + 4

λ 2 + 3λ + 2

 .

(23)

In the nest step we must determine state matrix A. Multiplying the denominator of the (23) by λ −3 we obtain the characteristic polynomial d(λ ) = 1 + 5λ −1 + 8λ −2 + 4λ −3 ,

(24)

which can be written as a sum of binomials in the following form: d(λ ) = (1 + 5λ −1 ) ∪ (1 + 8λ −2 ) ∪ (1 + 4s−3 ) = B2 ∪ B1 ∪ B0 ,          1−a2 λ −1

1−a1 λ −2

1−a0 λ −3

(25)

104 Konrad Andrzej Markowski, Krzysztof Hryni´ ow / Journal of Applied Nonlinear Dynamics 8(1) (2019) 97–108

v1

v2

v3

v1

v2

(a)

v1

v2

v2

(i)

v1

v2

(b)

v3

v1

(e)

v1

v3

v1

v1

(c)

v2

v3

v1

v3

v1

(g)

v2

v1

v3

v3

v2

v3

(h)

v2

(j)

v2

(d)

v2

(f)

v3

v3

v3

v1

(k)

v2

v3

(l)

Fig. 1 All possible realizations of the characteristic polynomial (24). w(v1 ,v3 )λ −1 w(v1 ,v1 )λ −1 v1

w(v2 ,v1 )λ −1

w(v1 ,v2 )λ −1

v2

w(v3 ,v2 )λ −1

v3

Fig. 2 One of the possible realizations of the characteristic polynomial (24).

where ∪ is digraph operation on vertices called composition relative to vertices. In details this kind of relation is presented in paper [30]. For each simple binomial B2 , B1 and B0 we create digraph representations. In the next step we determine all the possible digraph structures corresponding to the characteristic polynomial (24) by the use of algorithm presented in the paper [27]. Remark 2. It should be noted that, for considered polynomial (24), we have 18 possible digraphs structures: 12 proper realizations in class K1 ; 6 realizations which must be rejected. A half of realization presented in Figure 1(a)–Figure 1(f). We will obtain the next 6 realizations by the change of all arcs directions in digraph, as presented in Figure 1(g)–Figure 1(l). Each digraph-structure must satisfy two conditions. The first condition relates to the existence the common part in digraph (intersection vertex, presented as vertex in blue on Figure 1). The second condition relates to non-existence of additional cycle in the digraph. Then, from all the potential proper realizations (Figure 1), we choose the realization presented in Figure 2 to present in detail. From obtained digraph structure we can write the state matrix in general form: ⎤ ⎡ 0 w(v1 , v1 ) w(v2 , v1 ) (26) 0 w(v3 , v2 ) ⎦ , A = ⎣ w(v1 , v2 ) 0 0 w(v1 , v3 ) where: a2 = w(v1 , v1 ) = −5, a1 = w(v1 , v2 ) · w(v2 , v1 ) = −8,

(27)

a0 = w(v1 , v3 ) · w(v3 , v2 ) · w(v2 , v1 ) = −4. After determination of digraph weights we get: ⎡

⎤ −5 1 0 A = ⎣ −8 0 1 ⎦ . −4 0 0

(28)

Konrad Andrzej Markowski, Krzysztof Hryni´ ow / Journal of Applied Nonlinear Dynamics 8(1) (2019) 97–108 w(v1 ,y1 )

v1

w(v2 ,v1 )s−1

w(v3 ,v2 )s−1

v2

105

v3

y1 w(s1 ,v1 )

w(s1 ,v2 )

w(s1 ,v3 )

s1

Fig. 3 Realisation of the polynomial n11 (λ ). w(v1 ,y1 )

v1

w(v2 ,v1 )s−1

v2

w(v3 ,v2 )s−1

v3

y1 w(s2 ,v1 )

w(s2 ,v2 )

w(s2 ,v3 )

s2

Fig. 4 Realisation of the polynomial n12 (λ ).

In the next step we must determine matrices B and C. Multiplying nominator of the strictly proper transfer function (23) by λ −2 we obtain the following polynomial in the form which is needed to draw the digraph:   (29) N(λ ) = n11 (λ ) n12 (λ ) n13 (λ ) =   −1 −2 −1 −2 −1 −2 1 + 4λ + 4λ 1 + 3λ + 2λ . = −4 − 5λ − λ To the digraph presented in Figure 2 we add source vertex s and output vertex y and connect them. In this example we will assume that source is connected with the vertex belonging to the intersection set (vertex v1 in Figure 2 marked in blue) of digraph corresponding to binomials B2 , B1 and B0 . Assuming that matrix C contains one non-zero entry, we obtain the following three subtasks: Subtask 1: Polynomial n11 (λ ) = −4 − 5λ −1 − λ −2 . The source vertex s1 is connect with vertices v1 , v2 and v3 . The vertex v1 is connect with output vertex y1 . We obtain a digraph presented in Figure 3. Then, using the digraph presented in Figure 3 we can write the set of the equations ⎧ w(s1 , v1 ) · w(v1 , y1 ) = −4 ⎨ λ −1 w(s1 , v2 ) · w(v1 , y1 ) = −5 (30) ⎩ −2 λ w(s1 , v3 ) · w(v1 , y1 ) = −1 and after solving them we can write matrices in the following form: ⎤ ⎡ ⎤ ⎡ −4 w(s1 , v1 )   1 ⎣ −5 ⎦ , C = w(v1 , y1 ) 0 0 . B11 = ⎣ w(s1 , v2 ) ⎦ = w(v1 , y1 ) −1 w(s1 , v3 )

(31)

Subtask 2: Polynomial n12 (λ ) = 1 + 4λ −1 + 4λ −2 . The source vertex s1 is connect with vertices v1 , v2 and v3 . The vertex v1 is connect with output vertex y1 . Following in the same way as in Subtask 1 we receive digraph presented in Figure 4 and the set of the equation ⎧ w(s2 , v1 ) · w(v1 , y1 ) = 1 ⎨ −1 λ w(s2 , v2 ) · w(v1 , y1 ) = 4 (32) ⎩ −2 λ w(s2 , v3 ) · w(v1 , y1 ) = 4 After solving (32) we obtain matrix in the following form:

106 Konrad Andrzej Markowski, Krzysztof Hryni´ ow / Journal of Applied Nonlinear Dynamics 8(1) (2019) 97–108 w(v1 ,y1 )

v1

w(v2 ,v1 )s−1

v2

w(v3 ,v2 )s−1

v3

y1 w(s3 ,v1 )

w(s3 ,v2 )

w(s3 ,v3 )

s1

Fig. 5 Realisation of the polynomial n13 (λ ).



⎤ ⎡ ⎤ w(s2 , v1 ) 1 1 ⎣ ⎦ ⎣ B12 = w(s2 , v2 ) = 4 ⎦, w(v1 , y1 ) w(s2 , v3 ) 4

  C = w(v1 , y1 ) 0 0 .

(33)

Subtask 3: Polynomial n13 (λ ) = 1 + 3λ −1 + 2λ −2 . The source vertex s1 is connect with vertices v1 , v2 and v3 . The vertex v1 is connect with output vertex y1 . Following in the same way as in Subtask 1 we receive digraph presented in Figure 5 and the set of equations ⎧ w(s3 , v1 ) · w(v1 , y1 ) = 1 ⎨ −1 λ w(s3 , v2 ) · w(v1 , y1 ) = 3 (34) ⎩ −2 λ w(s3 , v3 ) · w(v1 , y1 ) = 2 After solving (32) we obtain matrix in the following form: ⎤ ⎡ ⎤ ⎡ 1 w(s3 , v1 )   1 ⎣ 3 ⎦, C = w(v1 , y1 ) 0 0 . B13 = ⎣ w(s3 , v2 ) ⎦ = w(v1 , y1 ) 2 w(s3 , v3 )

(35)

Remark 3. It should be noted that output matrix C in all subtasks has the same structure and the input matrix B depends on weight of arc w(v1 , y1 ). Because we divided our task into three independent subtasks, then input matrix three cases and have the following form: ⎤ ⎡ ⎡ −4 1 w(s1 , v1 ) w(s2 , v1 ) w(s3 , v1 )   1 ⎦ ⎣ ⎣ −5 4 B = B11 B12 B13 = w(s1 , v2 ) w(s2 , v2 ) w(s3 , v2 ) = w(v1 , y1 ) −1 4 w(s1 , v3 ) w(s2 , v3 ) w(s3 , v3 )

B is submission of ⎤ 1 3 ⎦. 2

The output matrix C have the following form:   C = w(v1 , y1 ) 0 0

(36)

(37)

for w(s1 , v1 ) ∈ R. The desired minimal realization (A, B, C, D) of the (19) is given by (28), (36), (37) and (22). 4 Concluding Remarks In this paper was presented modified method based on multi-dimensional digraphs theory that allows for finding (A, B, C, D) matrices directly from obtained digraph structures of one-dimensional SIMO and MISO fractional-order dynamic systems. Proposed method is an extension of previously published

Konrad Andrzej Markowski, Krzysztof Hryni´ ow / Journal of Applied Nonlinear Dynamics 8(1) (2019) 97–108

107

algorithm that allowed finding a set of solutions only for non-fractional 1-D and 2-D single-input singleoutput (SISO) dynamic systems. Presented algorithm finds a set of possible realizations instead of just a few, as was in the case of canonical forms approach. Moreover, proposed solutions are minimal in terms of rank of A matrix. Algorithm presented in the paper is possible to implement numerically in both time and memory efficient way due to it’s construction that allows parallelism and implementation of GPU (Graphics Processing Units). Described method was presented along with pseudo-code algorithm and illustrated in the article with example. At the moment the main limitation of the method is that it only works for some digraph structures belonging to specified class that need to have intersection vertex (or vertices) common to every cycle in digraph. In addition every arc from input vertices must lead to one of intersection vertices and every arc to output vertices must start in one of intersection vertices. Further work includes overcoming both of those limitations and extending the algorithm for 2-D fractional-order dynamic systems and 1-D and 2-D multiple input multiple output (MIMO) fractional-order dynamic systems.

References [1] Luenberger, D.G. (1979), Introduction to Dynamic Systems: Theory, Models, and Applications, chap. Positive linear systems, Wiley. [2] Benvenuti, L. and Farina, L. (2004), A tutorial on the positive realization problem, IEEE Transactions on Automatic Control , 49, 651-664. [3] Fornasini, E. and Valcher, M.E. (1997), Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs, Linear Algebra and Its Applications, 263, 275-310. [4] Fornasini, E. and Valcher, M.E. (2005), Controllability and reachability of 2D positive systems: a graph theoretic approach, IEEE Transaction on Circuits and Systems I , pp. 576-585. [5] Nishimoto, K. (1984), Fractional Calculus, Decartess Press. [6] Podlubny, I. (1999), Fractional Differential Equations, Academic Press. [7] Das, S. (2011), Functional Fractional Calculus, Springer Berlin Heidelberg. [8] Ortigueira, M.D. (2011), Fractional Calculus for Scientists and Engineers, Academic Press. [9] Petras, I., Sierociuk, D., and Podlubny, I. (2012), Identification of parameters of a half-order system, Signal Processing, IEEE Transactions on, 60, 5561-5566. [10] Podlubny, I., Skovranek, T., and Datsko, B. (2014), Recent advances in numerical methods for partial fractional differential equations. Control Conference (ICCC), 2014 15th International Carpathian, pp. 454-457, IEEE. [11] Martynyuk, V. and Ortigueira, M. (2015), Fractional model of an electrochemical capacitor. Signal Processing, 107, 355-360. [12] Machado, J. and Lopes, A.M. (2015), Fractional state space analysis of temperature time series. Fractional Calculus and Applied Analysis, 18, 1518-1536. [13] Machado, J., Mata, M.E., and Lopes, A.M. (2015), Fractional state space analysis of economic systems. Entropy, 17, 5402-5421. [14] Muresan, C.I., Dulf, E.H., and Prodan, O. (2016), A fractional order controller for seismic mitigation of structures equipped with viscoelastic mass dampers. Journal of Vibration and Control , 22, 1980-1992, DOI: 10.1177/1077546314557553. [15] Muresan, C.I., Dutta, A., Dulf, E.H., Pinar, Z., Maxim, A., and Ionescu, C.M. (2016), Tuning algorithms for fractional order internal model controllers for time delay processes. International Journal of Control , 89, 579-593, DOI: 10.1080/00207179.2015.1086027. [16] Markowski, K.A. (2017), Fractional kinetics of compartmental systems. First approach with use digraph-based method, Proc. SPIE , 10445, (in press). [17] Kaczorek, T. (1987), Realization problem for general model of two-dimensional linear systems, Bulletin of the Polish Academy of Sciences, Technical Sciences, 35, 633-637. [18] Bisiacco, M., Fornasini, E., and Marchesini, G. (1989), Dynamic regulation of 2D systems: A state-space approach, Linear Algebra and Its Applications, 122-124, 195-218. [19] Xu, L., Wu, Q., Lin, Z., Xiao, Y., and Anazawa, Y. (2004), Futher results on realisation of 2D filters by Fornasini-Marchesini model, 8th International Conference on Control, Automation, Robotics and Vision, Kunming, China, 6-9th December , pp. 1460-1464. [20] Xu, L., Wu, L., Wu, Q., Lin, Z., and Xiao, Y. (2005), On realization of 2D discrete systems by Fornasini-

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Marchesini model, International Journal of Control, Automation, and Systems, 4, 631-639. [21] Kaczorek, T. (2007), Positive realization of 2D general model, Logistyka, nr 3, 1-13. [22] Hryni´ow, K. and Markowski, K.A. (2014), Parallel digraphs-building algorithm for polynomial realisations. Proceedings of 2014 15th International Carpathian Control Conference (ICCC), pp. 174-179. [23] Hryni´ow, K. and Markowski, K.A. (2015), Optimisation of digraphs-based realisations for polynomials of one and two variables. Szewczyk, R., Zieli´ nski, C., and Kaliczy´ nska, M. (eds.), Progress in Automation, Robotics and Measuring Techniques, vol. 350 of Advances in Intelligent Systems and Computing, pp. 73-83, Springer International Publishing. [24] Markowski, K.A. and Hryni´ ow, K. (2017), Finding a Set of (A, B, C, D) Realisations for Fractional OneDimensional Systems with Digraph-Based Algorithm, vol. 407, pp. 357-368, Springer International Publishing. [25] Bang-Jensen, J. and Gutin, G. (2009), Digraphs: Theory, Algorithms and Applications (2nd Edition), Springer-Verlag. [26] Hryni´ow, K. and Markowski, K.A. (2015), Optimisation of digraphs creation for parallel algorithm for finding a complete set of solutions of characteristic polynomial. 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 2015 , pp. 1139-1144. [27] Hryni´ow, K. and Markowski, K.A. (2016), Parallel digraphs-building computer algorithm for finding a set of characteristic polynomial realisations of dynamic system, Journal of Automation, Mobile Robotics & Intelligent Systems (JAMRIS), 10, 38-51. [28] Hryni´ow, K. and Markowski, K.A. (2015), Digraphs minimal positive stable realisations for fractional onedimensional systems, Domek, S. and Dworak, P. (eds.), Theoretical Developments and Applications of NonInteger Order Systems, vol. 357 of Lecture Notes in Electrical Engineering, pp. 105-118, Springer International Publishing. [29] Markowski, K.A. (2016), Digraphs structures corresponding to minimal realisation of fractional continuoustime linear systems with all-pole and all-zero transfer function, 2016 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR). [30] Markowski, K.A. (2017), Two cases of digraph structures corresponding to minimal positive realisation of fractional continuous-time linear systems of commensurate order. Journal of Applied Nonlinear Dynamics, 6, 265-282, DOI: 10.5890/JAND.2017.06.011. [31] Hryni´ow, K. and Markowski, K.A. (2015), Classes of digraph structures corresponding to characteristic polynomials. Szewczyk, R., Zieli´ nski, C., and Kaliczy´ nska, M. (eds.), Challenges in Automation, Robotics and Measurement Techniques, vol. 440 of Advances in Intelligent Systems and Computing, pp. 329-339, Springer International Publishing.

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Aims and Scope The interdisciplinary journal publishes original and new research results on applied nonlinear dynamics in science and engineering. The aim of the journal is to stimulate more research interest and attention for nonlinear dynamics and application. The manuscripts in complicated dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in nonlinear dynamics and engineering nonlinearity. Topics of interest include but not limited to • • • • • • • • • • • •

Complex dynamics in engineering Nonlinear vibration and control Nonlinear dynamical systems and control Fractional dynamics and control Dynamical systems in chemical and bio-systems Economic dynamics and predictions Dynamical systems synchronization Bio-mechanical systems Nonlinear structural dynamics Nonlinear multi-body dynamics Multiscale wave propagation in materials Nonlinear rotor dynamics

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Yuefang Wang Department of Engineering Mechanics Dalian University of Technology Dalian, Liaoning, 116024, China Email: [email protected]

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Indexed by Scopus and zbMATH

Journal of Applied Nonlinear Dynamics Volume 8, Issue 1

March 2019

Contents Advances in Fractional Order Controller Design and Applications Cosmin Copot, Cristina I. Muresan, Konrad Andrzej Markowski….……..……....

1-3

Tuning of PI-PD Controller Based on Standard Forms for Fractional Order Systems Furkan Nur Deniz, Ali Yüce, Nusret Tan……………...…………………………..

5-21

Active Wave Control of a Flexible Beam Using Fractional Derivative Feedback Masaharu Kuroda†, Hiroki Matsubuchi..................................................................

23-33

Constrained Model Predictive Control for Linear Fractional-order Systems with Rational Approximation Mandar M. Joshi, Vishwesh A. Vyawahare...………...….....................………..…

35-53

An Application to Robot Manipulator Joint Control by Using Fractional Order Approach Cosmin Copot.....................................................................................................…

55-66

A Fractional Order Controller for Delay Dominant Systems. Application to a Continuous Casting Line Dana Copot, Clara Ionescu…………………………………………………….....

67-78

CRONE Body Control with a Pneumatic Self-leveling Suspension System Jean-Louis Bouvin, Xavier Moreau, Andr´e Benine-Neto, Vincent Hernette, Pascal Serrier, Alain Oustaloup…………………………………………...…......

79-95

Method for Finding a set of (A,B,C,D) Realizations for Single-Input MultipleOutput / Multiple-Input Single-Output One-dimensional Continuous-time Fractional Systems Konrad Andrzej Markowski, Krzysztof Hryniów……..…......................…….…...

97-108

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