AUTOMATION OF LINEAR AND NONLINEAR CONTROL SYSTEMS DESIGN BY EVOLUTIONARY COMPUTATION Yun Li, Kim Chwee Ng, Kay Chen Tan, Gary J. Gray, Euan W. McGookin, David J. Murray-Smith and Ken C. Sharman Centre for Systems and Control & Department of Electronics and Electrical Engineering University of Glasgow, Glasgow G12 8LT, United Kingdom E-Mail:
[email protected], Fax: +44 141 330 4907
Abstract: This paper develops a uniform definition of control system design problems using a vector space. Design difficulties by conventional analytical and numerical means are discussed. Associated problem-classification is presented, together with a possible automation technique that transforms unsolvable design problems to solvable analysis and non-NP design problems. Practically realisable design automation methodology is then developed, using genetic algorithm based evolution programs to transform the non-NP problems to NP-complete problems. This methodology is applied to the establishment of uniform linear control systems. Design automation of linear and nonlinear control systems is illustrated through two examples. It is shown that this methodology leads to globally optimised control systems, in addition to the ease of design achieved by the computerised automation. Keywords: Control system design, Genetic algorithms, Intelligent control, Computer-aided control systems design, Evolutionary computing, Evolution programs, Linear control systems, Nonlinear control systems
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neers. They accommodate, however, few direct or automated design facilities. These deficiencies have contributed to the failure of many advanced control schemes to be widely accepted by practising engineers.
INTRODUCTION
In the process of designing a linear or nonlinear control system, it is usually necessary to obtain a number of parameters of the controller in order to define a “good design” that meets a number of performance requirements under certain practical constraints. This design problem is equivalent to a multi-dimensional optimisation problem in an, almost certainly, multimodal space. If the objective function (or cost function) is differentiable under practical constraints in the multi-dimensional space, the design problem may be solved easily by setting its vector derivative to zero. Finding the parameter sets that result in zero first-order derivatives and that satisfy the second-order derivative conditions would reveal all local optima. Then comparing the values of the performance index of all the local optima, together with those of all boundary parameter sets, would lead to the global supremum. The corresponding parameter set would thus represent the best design.
This paper attempts to solved these problems in a uniform way. The following section develops a uniform definition of a control system design problem. Design difficulties encountered in conventional numerical methods are discussed. Then associated problem-classification and a possible automation technique are presented. Practically realisable design automation methodology is developed in Section 3. In Section 4, this methodology is applied to the design of uniform linear control and nonlinear control systems. Finally, conclusions are outlined in Section 5. 2.
THE DESIGN PROBLEM
2.1 Problem formation
However, the multi-dimensional objective function in a control system design problem is usually not differentiable in practice. This is mainly due to the explicit and implicit constraints of the physical system, such as actuator saturation and bandwidth limits. On contrast to a design problem, an analysis problem of a practical control system is always solvable with the assistance of numerical algorithms. Existing computer-aided control system design (CACSD) packages are just developed to carry out this task. These provide a passive simulation tool for control engi-
In control system design practice, the structure of a controller is usually determined by the control scheme or control law that the design engineer opts to apply. The design task is thus to optimise the parameters of the controller that best meet the design objective. This means that a parameter set of the controller represents a design candidate of the system. DEFINITION 1 In the context of design, a candidate control system, Pi, is defined by a uniform vector representation given by: -1-
{
}
Pi = p1 , ..., p n ∈ R n
NP = {Problems that cannot be solved in polynomial time
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but can be solved otherwise}
where i stands for the ith design candidate, n the number of parameters required by the control law, pj ∈ R the jth parameter of the ith design candidate with j ∈ {1, ..., n}, and Rn the ndimensional real Euclidean space.
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DEFINITION 2 The solution space of a control system design problem is defined by
S=
{ P , ∀i i
}
p j ∈ R and j ∈{1,..., n} ⊆ R n .
(2)
DEFINITION 3 The fitness of a control system design, Pi, is defined by a function f(Pi): Rn→R+ which represents the performance index of the control system with respect to the design requirements or specifications, where R+ is the non-negative real space.
Fig. 1. Classification of numerical problems. Not directly shown in Fig. 1, the following definitions also exist: P
The performance index needs to reflect the following design criteria in the presence of practical system constrains:
∪ NP-COMPLETE = NP = {Problems that can be solved in polynomial time}; and NP-COMPLETE = NP-HARD = {Problems that cannot be solved by any deterministic algorithms in polynomial time} = {Problems that are at least as hard as an NP problem}.
NP ∪
(1) An excellent transient response in terms of rise-time, overshoots and settling-time; (2) An excellent steady-state response in terms of steady-state error; (3) Acceptable stability margins; (4) Robustness in terms of disturbance rejection; and (5) Robustness in terms of parameter sensitivity.
It should be noted that the hypotheses:
COMPLETE = ∅;
NPP
DEFINITION 4 A control system design problem is defined as the problem of finding a design given by:
Po = Po ∈ S
f ( Po ) = sup f ( Pi ) . ∀ Pi ∈S
P
⊇ NP ; or = NP
have never been proven untrue and still remain a mystery in computer science today, despite a great deal of research effort has been made during the past several decades. The above classification has, however, encountered no controversial cases so far and is widely accepted in computer science and algorithm engineering. It is reported that the majority of science and engineering problems belong to the category of NP-hard problems (Sedgewick 1988).
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In the previous section, it is highlighted that many practical control system design problems are unsolvable problems in the analytical domain, but practical control system analysis problems are solvable problems in the numerical domain. The following question is, however, left unanswered: Are practical control system design problems solvable in the numerical domain?
2.3 Practical control system design problems are unsolvable by conventional numerical means
2.2 Problem classification
The numerical means used in most existing CACSD packages are conventional calculus-based methods. They can perform very well in computer-assisted design, if they are incorporated with a numerical optimisation tool for quadratic objective/fitness functions. Such a tool is based on conventional gradient-guidance or hill-climbing techniques. Some control system design problems could be transformed by these techniques to a P problem, easily resulting in polynomial design time. Such scenario does almost not, however, exist in practical systems. In dealing with practical design problems, there exist the following drawbacks of conventional techniques:
Before answering the above question, it is desirable to review the problem-classification (Sedgewick 1988) used in computer science and algorithm engineering. This is depicted in Fig. 1. The clear area represents the set of unsolvable problems and the shaded areas represent solvable problems. The solvable problems are further divided into three categories as follow: = {Problems that can be solved by a deterministic algorithm in polynomial time}; NP-COMPLETE = {Problems that cannot be solved by any deterministic algorithms in polynomial time but can be solved by a non-deterministic algorithm in polynomial time}; and P
(1) Existence Problem: Gradient guidance passively adjusts Pi using ∇f(Pi) or well-defined smooth slopes of the objectives (Goldberg 1989); (2) Practical Problem: The method is impossible to work with
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hard constraint conditions (Michalewicz 1992), such as domain constraints, domain and control signal equalities and inequalities. Nor does it work properly in the noisy search space (Goldberg 1989) in practical applications; (3) Multi-modal Problem: The sequentially guiding process usually leads to a local optimum (Goldberg 1989) and is difficult to evaluate Pi at the boundary of S, although optimisation at parallel points may overcome this problem to a certain extent; (4) A-Priori Problem: It is difficult to incorporate knowledge and experience that a designer may have on the design.
2.4 Practical control system design problems can be solvable by non-NP numerical means Since the analysis problem is solvable and encounters no difficulties as those highlighted by the conventional drawbacks in the numerical domain, one approach to achieving a solvable and possibly automated design could be to exhaustively evaluate in S all the possible design choices Pi ∀i,. To illustrate this method, let n = 8 and suppose in S each parameter has 10 possible values. Then there are a total of: max (i) = 108
Using a CACSD package based on these techniques for design, a design engineer needs to solve these problems by heuristic simulations. He/She has first to input certain a-priori controller parameters, such as those obtained from some preliminary analysis, and should then undertake simulations and evaluations using the package. If the simulated performance of the “designed” control system does not meet the specification, the designer would modify the values of the parameters randomly or by his/her realtime gained experience. The engineer would then run the simulations repeatedly until a “satisfactory” design emerges. Clearly, such a design process is neither automated nor easily carried out, since mutual interactions among parameters are hard to predict (multi-dimensional problem).
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permutations of design choices. Every candidate design could then be encoded by a string of 8 decimal digits. By enumerating all digits of the string one by one could span the entire quantised design space S. Now suppose that each evaluation by numerical simulation takes 0.1 second on an extremely fast computer. Then the entire design process would take 0.1 second x 108 = 3.8 months to complete. This is unacceptable in practice. Although such a search scheme does transform an unsolvable problem to a non-NP problem, the search time is in the order of O(pn), where p is the quantisation dimension of the parameters. Even the highly regarded exhaustive/enumerative scheme dynamic programming breaks down on problems of “moderate” dimensionality and complexity (Bellman 1957; Goldberg 1989). Some specialised, or problem-dependent, numerical schemes work much more efficiently than the exhaustive search, but they are confined to a very narrow problem domain.
A possible alternative to this manual approach is to incorporate an approximation algorithm (Sedgewick 1988) or a random-walk technique (Goldberg 1989) in the optimisation process to achieve a computerised design. The performance may be further improved by using the simulated annealing (SA) technique, which allows some inferior neighbouring positions to replace the current one for possible correct directions leading to the global optimum. Convergence towards the correct directions or towards the global optimum is not, however, guaranteed (Michalewicz 1992). Using these techniques, it is easy to answer the question:
The exponential search time required by an exhaustive search mechanism could, however, be largely reduced, if a-priori experience of the designer could be incorporated in the search and the interim results of the evaluations could be used to guide the search intelligently. The following section presents techniques that achieve these and allow an non-NP problem to be transformed to an NP problem.
Is there a design such that f(Pi) < fo ? but difficult to answer:
3.
Is there a design such that f(Pi) ≥ fo ? where fo∈R+. Clearly, the resulting “satisfactory” design from these techniques does not necessarily offer the best or near-best performance.
DESIGN AUTOMATION BY EVOLUTION
3.1 The methodology Sedgewick (1988) pointed out that one way to extend the power of a digital computer (or the Turing machine) is to endow it with the power of intelligent nondeterminism: to assert that when an algorithm is faced with a choice of search options, it has the power intelligently to “guess” the right one. Artificially emulating Darwin’s evolutionary principle of “survival-of-the-fittest” in natural selection and genetics, evolution programs (EPs) (Michalewicz 1992) based on evolutionary programming, evolution strategies, genetic algorithms (GAs), and genetic programming search the solution space intelligently in a nondeterministic way. They have been found very powerful and robust in searching poorly understood, irregular and complex spaces for optimisation and machine learning (Goldberg 1989). Here the genetic programming is an extension of the GA, where arithmetic and logic operators are encoded together with parameters, and the execution of the program itself leads to evolution of the solution space. Since the architecture of a genetic algorithm includes those of evolution strategies and evolutionary programming, an evolution program is thus generally a GA with data structure
In addition, a modern paradigm of CACSD should also meet the open environment (Barker 1995) and other design challenges as listed below: (1) (2) (3) (4)
Complexity of practical systems; Required high quality and accuracy of design; Speed of design; Competition with available design tools (in terms of ease of use, for example); and (5) Robustness, reliability and safety arising from the design. It is found many CACSD systems do not yet meet these challenges easily, due to the drawbacks of conventional techniques listed earlier. In the view point of optimal designs, the control system design problems by conventional means are thus marked as unsolvable.
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(Michalewicz 1992). This approach is depicted by Fig. 2.
procedure evolution program begin construct f(Pi) t := 0 initialise P(t) evaluate P(t) while (not terminate-condition) do begin t := t+1 survive P(t) ⊆ P(t-1) genetically operate P(t) evaluate P(t) end end
Data A non-NP
Structure
Problem
Genetic Algorithm An NP-complete Problem
Evolution Program
Fig. 2. EP = GA + data structure. Fig. 3. The structure of an evolution design algorithm. Such an algorithm may be applied in virtually any (non-NP) problem (Goldberg 1989), no matter how complex the system is, since most such systems are analysable by numerical means. An EP transforms a non-NP problem to an NP problem, i.e., this nondeterministic algorithm requires a search time bounded by a polynomial of n (instead of an exponential of n as resulting from the exhaustive search). This methodology can thus also be used in control system design, meeting the speed and competition challenges.
3.2 Limitations From the above discussions, it can be concluded that a design problem of control systems, as well as other decision making systems, can always be solved by an evolution program under the following conditions: (1) The solution space to design, S, is finite or can be represented by a finite quantisation; (2) The system is analysable, i.e., the performance of candidate designs, f(Pi), can be evaluated; and (3) The performance index, f(Pi), has values with more information than a simple Yes-or-No answer. This information is needed to guided the nondeterministic evolution.
Since the EP simultaneously evaluates f(Pi) at multiple points in the solution space S, it easily overcomes the multi-modal drawback. These points form a population of candidate designs as defined by:
{
}
P ( t ) = Pi ( t ), ∀i ∈{1,..., z} Pi ( t ) = { p1t ,..., pnt } ⊆ S , (5)
Further, for the computer-assisted design automation, reasonable encoding schemes (Sedgewick 1988) must be used. An example of such a scheme is the binary coding, where the number of bits used to represent the decimal number 12 should be equal to log2(12) = 4 and not equal to 12 itself. Otherwise the dimension of the solution space is unduly amplified exponentially.
where z is the size of the population. The designer’s expertise or known controllers can be encoded and incorporated easily in the initial population, which will usually lead to a faster convergence and will thus overcome the a-priori drawback. Further, all the design criteria and practical constraints can be included by the fitness function, since this function needs not to be differentiated. The structure (Michalewicz 1992) of an evolution program is shown in Fig. 3. There, in the reproduction process, the survivalof-the-fittest principle results in a subset of P(t), which has the same size as P(t-1), with more offspring obtained from fitter individuals. The two probabilistic genetic operators are crossover cj(Pi): ∏S→S and mutation mj(Pi): ∑S→S. A third operator, inversion invj(Pi): ∑S∪∏S→S, can be derived from these two operators and is thus usually neglected. For details of genetic and evolution programs and their operators, refer to Goldberg (1989) and Michalewicz (1992).
4.
DESIGN EXAMPLES
4.1 Linear controllers can be unified by an EP In the design of linear control systems, such as the proportional plus integral plus derivative (PID), phase-lead/lag, linear quadratic, H∞ or µ-synthesis based controllers, their transfer functions are of the uniform given by: G( s) =
The termination-condition is met if the specified number of generations have been evolved or the standard deviation of fitness within the generation is narrowed to a certain range. If, however, the fitness does not improve after a certain number of generations, new individuals can be randomly generated to invade the population. Due to the nondeterministic global searching feature, an EP may not arrive at the exact supremum. This can however be improved by incorporating the SA (Li et al 1995d) or conventional optimisation (Michalewicz 1992) techniques to fine-tune the local search.
L
p n s n + + p m+ 2 s + p m+1 V (s) = E( s ) pm s m + p m−1 s m−1 + + p2 s + p1
L
(6)
where m p4 , s > 0 2 & 0 p 6 , − es > 0 p8 , − es
Consider a DC servo-mechanism for velocity control as an example. This system is described by a differential equation: KT RB JR + LB & && ( t ) + ω ω ( t ) = v (t ) ω (t ) + LJ LJ LJ
(7)
where ω(t) the angular velocity, KT=13.5 NmA-1 the torque constant, R=9.2 Ω the resistance of the winding, L=0.25 H the inductance, and J=0.001 kgm2 the moment of inertia of the motor shaft and load. There is a time-delay of 60 ms in the system.
are the switching gains. The switching condition is determined by the sliding surface given by: s(e, e& ) = {( e, e& ) | p9 e + e& = 0}
∑(
)
(12)
in order to achieve robust nonlinear control.
A simple fitness function that reflects the design criteria can be an exponentially amplified time-weighted L1 norm given by: N f = exp −α e i + ∆e i i i=1
(11)
The convergence of the fitness is shown in Fig. 5. Since an existing manually tuned controller is available, it was later incorporated in the initial design population. This has led to a faster convergence as shown in Fig. 6. The EP automated process yields the following parameter set:
(8)
where i is the time-index in the simulation and ∆e the amount of change of error used to suppress oscillations. In (8), N=100 is the total time steps chosen to ensure that the simulations reach a steady-state for a sampling period of 10 ms. A practical implicit constraint in the form of control action inequalities given by v(t)∈[-5, +5] is conveniently incorporated in the simulations and is used in evaluating (8).
., s 0 3.2, − es > 0 .03, s > 0 & 0
Running the reusable EP (which is fine-tuned by an SA in this paper) automates the design. The population size is 20 with each parameter being coded by 4 decimal digits. On a 50 MHz Intel 80486 processor running Pascal, it took 45 minutes for evolving 50 generations of design. A globally optimised third-order controller is given below: Po = {0.0, 29.2, 23.8, 1.0, 77.7, 89.9, 24.3, 0.9}
(9)
The performance of this controller is shown in Fig. 4 and is compared with a best-tuned PID controller. The ripples occurred are due to the sudden artificial changes of the friction of and the command to the servo-system. Fig. 5. Curves of fitness without experience incorporated.
Fig. 4. Responses of the uniform and PID controllers. Fig. 6. Curves of fitness with a-priori experience.
4.2 Automation of sliding mode controller design
The performances of this and the manually design controllers are compared in Fig. 7. It can be seen that the EP designed controller also offers better performance, in addition to the convenience obtained from the automated design. Both nonlinear controllers are shown to have superior robustness to the linear controllers as shown in Fig. 4.
To illustrate the applicability of the EP based design automation methodology to nonlinear control, this sub-section presents a design example of sliding mode control for the system given by (7). A generalised controller is given by (Li et al 1995e):
v ( t ) = -φ + ϕ 0 ∫ e( t ) dt + ϕ 1 e( t ) + ϕ 2 e&( t )
(10)
where
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Barker, H.A.. (1995). Open environments and object-oriented methods for computer-aided control system design, Control Eng. Practice, 3, (3), 347-356. Bellman, R. (1957). Dynamic Programming. Princeton University Press, Princeton, NJ. Goldberg, D. (1989). Genetic algorithms in searching, optimisation and machine learning. Addison-Wesley, Reading, MA. Gray, G.J., Y. Li, D.J. Murray-Smith and K.C. Sharman (1995). Specification of a control system fitness function using constraints for genetic algorithm based design methods, Proc. First IEE/IEEE Int. Conf. on GA in Eng. Syst.: Innovations and Appl., Univ. of Sheffield, U.K. Häußler, A., Y. Li, D.J. Murray-Smith and K.C. Sharman (1995). Neurocontrollers designed by a genetic algorithm, Proc. First IEE/IEEE Int. Conf. on Genetic Algorithms in Eng. Syst.: Innovations and Appl., Univ. of Sheffield, U.K. Li, Y. (1995a). Modern information technology for control systems design and implementation, Proc. 2nd Asia-Pacific Conf. Control and Measurement, ChongQing, China, 17-22. Li, Y., K.C. Ng, A. Häußler, V.C.W. Chow and V.A Muscatelli (1995b). Macroeconomics modelling on UK GDP growth by neural computing, Pre-prints of IFAC/IFIP/IFORS/ SEDC Symp. Modelling and Control of National and Regional Economies, Australia. Li, Y. and K.C. Ng (1995c). Genetic algorithm based techniques for design automation of three-term fuzzy systems (invited paper), Proc. 6th Int. Fuzzy Sys. Asso. World Cong., São Paulo, Brazil. Li, Y., K.C. Tan, K.C. Ng and D.J. Murray-Smith (1995d). Performance based linear control system design by genetic evolution with simulated annealing, Proc. 34th IEEE Conf. Decision and Control, New Orleans, LA,. Li, Y., K.C. Ng, D.J. Murray-Smith, G.J. Gray and K.C. Sharman (1995e). Genetic algorithm automated approach to design of sliding mode control systems. Int. J. Control, (to appear). Ng, K.C., and Y. Li (1994). Design of sophisticated fuzzy logic controllers using genetic algorithms. Proc. 3rd IEEE Int. Conf. on Fuzzy Systems, IEEE World Congress on Computational Intelligence, Orlando, FL, 3, 1708-1712. Ng, K.C., Y. Li, D.J. Murray-Smith and K.C. Sharman (1995). Genetic algorithms applied to fuzzy sliding mode controller design, Proc. First IEE/IEEE Int. Conf. on GA in Eng. Syst.: Innovations and Appl., Univ. of Sheffield, U.K. McGookin, E.W, D.J. Murray-Smith and Y. Li (1995). Segmented simulated annealing applied to sliding mode controller design, IFAC World Congress, San Francisco, CA, June 1996 (submitted). Michalewicz, Z. (1992). Genetic Algorithms + Data Structure = Evolutionary Programs. Springer-Verlag, Berlin. Sedgewick, R. (1988). Algorithms. Addison-Wesley, Reading, MA., 2nd Ed. Sharman, K.C., A.E. Esparcia-Alcazar and Y. Li (1995). Evolving digital signal processing algorithms by genetic programming, Proc. First IEE/IEEE Int. Conf. on GA in Eng. Syst.: Innovations and Appl., Univ. of Sheffield, U.K. Tan, K.C., Y. Li, D.J. Murray-Smith and K.C. Sharman (1995). System identification and linearisation using genetic algorithms with simulated annealing, Proc. First IEE/IEEE Int. Conf. on GA in Eng. Syst.: Innovations and Appl., Univ. of Sheffield, U.K.
Fig. 7. Performances of sliding mode controllers designed manually and by the EP. 4.3 Other applications The EP based methodology has also been applied at University of Glasgow to the following problems: (1) PID based control of a lift drive system with response shape and industrial design specification based fitness (Gray et al 1995); (2) Performance based linear control of nonlinear systems (Li et al 1995d); (3) Sliding mode control of multivariable, coupled, asymmetrical nonlinear liquid-level and submarine control systems (Li et al 1995e; McGookin et al 1995); (4) Fuzzy logic control (Ng and Li 1994; Li and Ng 1995c); (5) Fuzzy sliding mode control (Ng et al 1995); (6) Neural computing (Li et al 1995b; Sharman et al 1995) and neurocontrol (Häußler et al 1995); (7) Linear system identification and its parallelism (Li 1995a); and (8) Nonlinear identification and linearisation (Tan et al 1995). 5.
CONCLUSIONS AND FURTHER WORK
This paper has developed a uniform definition of control system design problems. Unsolvable control system design problems by conventional optimisation means are transformed to solvable problems by using evolution programs. This methodology enables design automation in polynomial time. It overcomes the conventional drawbacks, satisfies the design criteria and meets the design challenges by trading off precision slightly for improved tractability, robustness and ease of design. In addition, superior performances offered by this methodology are illustrated by two design examples. Progress on related research made by the Evolutionary Computing and Control Group at Glasgow is also highlighted.
Acknowledgement: This research is supported in part by the UK Engineering and Physical Sciences Research Council under grant GR/K24987 (“Evolutionary Programming for Nonlinear Control”) funded to Dr. Li, Prof. Murray-Smith and Dr. Sharman. Dr. Gray and Mr. McGookin is grateful to the EPSRC, and Mr. Ng and Mr. Tan to the University and CVCP, for their financial support. REFERENCES
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