Cooperative Particle Swarm Optimization for Robust Control System Design Renato A. Krohling1 Universidade Federal do Espírito Santo Departamento de Engenharia Elétrica Campus de Goiabeiras, C.P. 01-9001 29060-970, Vitória, ES, Brazil E-mail:
[email protected]
Leandro dos S. Coelho Pontificia Universidade Católica do Paraná Laboratório de Automação e Sistemas Rua Imaculada Conceição, 1155 80215-030, Curitiba, PR, Brazil E-mail:
[email protected]
Abstract In this paper, a novel design method for robust controllers based on Cooperative Particle Swarm Optimization (PSO) is proposed. The design is formulated as a constrained optimization problem, i.e., the minimization of a nominal H 2 performance index subject to a H ∞ robust stability constraint. The method focuses on two (PSOs): One for minimizing the performance index, and the other for maximizing the robust stability constraint. Simulation results are given to illustrate the effectiveness and validity of the approach.
Keywords: Particle Swarm Optimization, Robust Controller, H 2 Performance Index, H ∞ -Robust Stability Constraint. 1 Introduction The robust control system design treated here is concerning the mixed H 2 /H ∞ optimal control problem which consists of the design of a controller which internally stabilizes the plant and satisfies the robust stability constraint on the H ∞ norm. The mixed H 2 /H ∞ problem has received a great deal of attention from the theoretical viewpoint of design methods (Bernstein and Haddad, 1994), (Doyle et al., 1994), (Snaizer, 1995) etc. In the last few years some research interest has been devoted to develop practical design method for this relevant problem (Chen et al., 1995), (Lo Bianco, and Piazzi, 1997), (Krohling, 1998), and (Krohling et al., 1999).
Yuhui Shi EDS Embedded Systems Group 1401 E. Hoffer Street Kokomo, IN 46902 USA Email:
[email protected]
subject to a H ∞ robust stability constraint. For the solution of that problem a hybrid method consisting of a GA with binary codification for minimization of the ISE performance index together with a numerical algorithm for evaluating the H ∞ robust stability constraint was proposed. Another solution method for the problem is proposed in Lo Bianco, and Piazzi (1997) which is based on a genetic/interval approach, consisting of a GA for minimization of the ISE performance index and interval methods for evaluating the H ∞ robust stability constraint. In this paper, based on previous work by Krohling (1998), and Krohling et al (1999) we consider the design of robust controllers for uncertain plants formulated as constrained optimization by using PSO. Firstly, the synthesis of mixed H 2 /H ∞ optimal fixed-structure controller is formulated as the minimization of a nominal H 2 performance index subject to a H ∞ robust stability constraint. Next, for the solution of the highly non-linear constrained minimization problem for which a closed solution can not be obtained, a new method based on two PSOs is proposed: One for minimizing the ISE performance index, and the other for maximizing the robust stability constraint. Simulation results are presented to show the effectiveness and validity of the novel approach. The rest of this paper is organized as follows: section 2 describes the problem; section 3 is devoted to an explanation of particle swarm optimization; in section 4 the design method using two PSOs is presented; in section 5 a design example is given and section 6 presents some conclusions.
In Chen et al (1995), the mixed H 2 /H ∞ optimal PID control design problem was formulated as a minimization of the integral of squared-error (ISE) performance index 1
Renato A. Krohling was sponsored by the Brazilian Research Council (CNPq) under Grant 301009/99-6
1
R ( s) E ( s )
U ( s) C ( s, k )
-
α z (w, k ) = α n (w, k ) (C(jw, k )G0 (jw)Wm (jw))(C(− jw, k )G0 (− jw) Wm (− jw)) = (1 + C(jw, k )G0 (jw))(1 + C (− jw, k )G0 (− jw))
Y ( s)
α (w, k ) =
G ( s)
Thus the condition for robust stability in the frequency domain is represented by:
Figure 1: Feedback control system.
max (α ( w , k )) 0.5 < 1
2 Problem Description In the context of single-input single-output linear timeinvariant systems (Doyle et al., 1992), consider the feedback control system shown in figure 1. The fixedstructure controller is described by a rational transfer
function C(s,k) where k = [k1, k2, ...,k m ] stands for the vector of the controller parameters. It is assumed that the plant to be controlled is described by the nominal transfer function G (s ) and undergoes a perturbation described by ∆G (s ), which is assumed to be stable and limited by
The function α ( w , k ) can also be expressed in the following form: p
T
∆G( jw ) < W ( jw ) , ∀w ∈ [0 , ∞ )
where the function W(s) is stable and known. In this paper is considered that the plant is described by the multiplicative uncertainty model G given by G := {G ( s , ∆ ) = G ( s ) (1 + ∆G ( s ) ) } .
It is required that ∆G (s ) does not cancel unstable poles of G (s ) in forming G ( s , ∆ ). The Condition for robust stability is stated, as follows (Doyle et al., 1992): If the nominal control system (∆( s ) = 0) is stable with the controller C(s, k) , then the controller C(s,k) guarantees robust stability of the control system, if and only if the following condition is satisfied: C(s, k )G(s)∆G(s) 1 if max (a(w, k i ))0.5 < 1 0 , (13)
If the individual k i does not satisfy the stability test applied to the characteristic equation of the system, then k i is an unstable individual and it is penalized with a very large positive constant M 2. In case that the particle k i satisfies the stability test, but not the robust stability constraint, it is an infeasible individual and is penalized with M 1.max α(w, k i ), where M 1 is a positive constant. Otherwise, the particle k i is feasible and is not penalized. The method can be summarized as follows: Given the plant with transfer function G0(s), the controller with a fixed-structure, a transfer function C (s , k ) , and a weighting function W(s) determines the error signal E(s) and the robust stability constraint α ( w , k ). Specifying the lower and upper bounds of the controller parameters it is suitable to describe the method in the form of an algorithm given in the following list: List 2:
1) Initialize the two populations PSO_1 k i (i = 1,...,µ1 ) and PSO_2 w j (j = 1,...,µ 2 ) within the corresponding ranges. 2) For each particle k i of the PSO_1, run the second PSO_2 for a number maximum of iterations, iter_ 2 max in order to calculate the maximum
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value of α ( w , k i ) given by g 2best . If no particle of PSO_1 satisfy the constraint max (α ( w , k i )) 0.5 < 1 then a feasible solution is assumed to be non-existent and the algorithm stops. In this case, a new controller structure has to be assumed. 3) Run PSO_2 for each particle w a number maximum of iterations, iter _ 2 max in order to calculate the maximum value of F2(wj) according to the equation (12) given by g 2 _ best . 4) Run PSO_1 for each particle k i , in order to calculate the value of F1(k i ) according to the equations (11)and (13). 5) If the maximum number of iterations of PSO_1 iter _ 1max is not reached, then set iter _ 1 = iter _ 1 + 1 go to step 3, otherwise stop, and the solution of the optimization given by g1 _ best .
problem
k*
is
5 Design Example The plant is described by the transfer function (Lo Bianco and Piazzi, 1997): G ( s) =
18 . s ( s + 2) 2
The plant uncertainty is expressed by
∆G ( s) = W ( s) =
01 . s + 01 . s + 10 2
The fixed-structure controller is given by following transfer function C(s , k)= k1
s 2 + 2k 4 k 5 s + k 52 (s + k 2)( . s + k 3)
For the control system as shown in Figure 1, the error transfer function E(s), the performance index J 5 , and the robust stability constraint are given in the appendix. The parameters of the controller are searched within the following bounds: k1 = [1, 1000]; k 2 = [1, 100]; k 3 = [1, 100]; k 4 = [0, 1]; k 5 = [0.1, 100].
PSO parameters setting: Population size of PSO_1 µ1 = 50 , population size of PSO_2 µ 2 = 30 , penalty constant M 1 = 100 , M 2 = 100000.
Nominal plant Plant with perturbation
Figure 3: The closed-loop step response. The method using PSOs described in the previous section, has been applied to the design of the robust controller. The best controller parameters for this example obtained by the minimization of the performance index J 5(k) subject to the robust stability constraint max (α ( w , k )) 0.5 < 1 using the proposed method is given by k * = [1000; 14.418; 14.569; 1; 0.541].
The minimal value of the performance index after 100 generations is J5(k* ) = 0.1506. The results obtained by the proposed method is equivalent to those reported in (Lo Bianco and Piazzi, 1997; Krohling, 1998 and Krohling et al, 1999). In order to test the controller obtained by the proposed method, a unit step signal is applied to the system as shown in Figure 1. The closed loop response for both cases: a) nominal plant and b) with plant uncertainty ∆G ( s) is shown in Figure 3. It shows that the controller obtained by the proposed method can control the perturbed plant G ( s , ∆ ) successfully. 6 Conclusions
This paper presents a method for synthesis of robust controllers with fixed-structure. The problem is formulated as an optimization problem with a constraint of type H ∞ norm. A method based on two PSOs is presented: One PSO is used to minimize the nominal H 2 performance index and the other to maximize the H ∞ robust stability constraint. The validity of the novel method is demonstrated by a design example and the simulation result is compared with those reported in the literature. The results obtained so far demonstrate clearly the potential of intelligent search methods based on swarm intelligence for synthesis of robust controllers. Particle swarm optimization can also be applied to design other kinds of controllers
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formulated as constrained optimization problems. Future work in this direction will be dedicated.
Kennedy, J., Eberhart, R.C. and Shi, Y., Swarm Intelligence. Morgan Kaufmann Publishers, San Francisco 2001.
References
Potter, M. A. and De Jong, K. “A Cooperative Coevolutionary Approach to Function Optimization.” in Proc. of the third Parallel Problem Solving from Nature. Springer-Verlag, New York, pp. 257, 1994.
Bernstein, B.S. and Haddad, W.M. “LQG Control with a H ∞ Performance Bound: A Riccati Equation Approach,” IEEE Trans. Automatic Control. vol. 34, no. 3, pp. 293-305, 1994. Doyle, J.C., Zhou, K., and Glover, K. Boddenheimer, B.: “Mixed H 2 and H ∞ Performance Objectives II: Optimal Control,” IEEE Trans. Automatic Control. vol. 39, no. 8, pp. 1575-1587, 1994. Doyle, J.C. and Francis, B.A. and Tannenbaum, A. R. Feedback Control Theory. Macmillan Publishing, New York, USA, 1992. Snaizer, M. “An Exact Solution to General SISO Mixed H 2 / H ∞ Problems via Convex Optimization,” IEEE Trans. Automatic Control. vol. 39, no. 12, pp. 2511-2517, 1994. Newton, G.C., Gould, L.A and Kaiser, J.F. Analytic Design of Linear Feedback Controls. John Wiley & Sons, New York, 1957. Jury, E.I. Inners and the Stability of Dynamic Systems. John Wiley, USA, 1974. Chen, B.-S.; Cheng, Y.-M.; Lee, C.-H. “A Genetic Approach to Mixed H 2 / H ∞ Optimal PID Control,” IEEE Control Systems Magazine, pp. 51-60, 1995. Lo Bianco, C.G. and Piazzi, A. “Mixed H 2 / H ∞ FixedStructure Control via Semi-Infinite Optimization,” in Proc. of the 7th IFAC International Symposium on CACSD, Gent, Belgium, pp. 329-334, 1997. Krohling, R.A. “Genetic Algorithms for Synthesis of Mixed H 2 / H ∞ Fixed-Structure Controller,” in Proc. of the 13th IEEE International Symposium on Intelligent Control, ISIC’98, Gaithersburg, USA, September, pp. 30-35, 1998. Krohling, R.A., Coelho, L.S., and Coelho, A.A.R. “Evolution Strategies for Synthesis of Mixed H 2 / H ∞ Fixed-Structure Controllers,” in Proceedings of the IFAC World Congress, Beijing, China, July, pp. 471-476, 1999. Eberhart, R.C. and Kennedy, J. “A new optimizer using particle swarm theory,” in Proc. of the 6th. Int. Symposium on Micro Machine and Human Science, Nagoya, Japan, pp. 39-43. IEEE Service Center, Piscataway, NJ, 1995. Kennedy, J. and Eberhart, R.C. “Particle swarm optimization,” in Proc. of the IEEE Int. Conf. on Neural Networks IV, IEEE Service Center, Piscataway, NJ, pp.1942–1948, 1995.
Potter, M. A. “The Design and Analysis of a Computational Model of Cooperative Coevolution”. Doctoral dissertation, George Mason University, 1997.
Appendix The error transfer function is given by:
E( s) =
c4 s 4 + c3s 3 + c2 s 2 + c1s + c0 d5s + d4 s 4 + d3s 3 + d2 s 2 + d1 s + d0 5
with c0 = 0, c1 = 2 k2 k3 , c2 = 2( k2 + k3 ) + k2 k3 , c3 = 2 + k2 + k3 , c4 = 1 d0 = 1.8k1k52 , d1 = 3.6k1k4 k5 , d2 = 2 k2 k3 + 1.8k1, d3 = 2( k2 + k3 ) + k2 k3 , d4 = 2 + k2 + k3 , d5 = 1 The performance index J 5 is given by Newton et al. (1957): J5 =
A 2 m5
with A = c42 m0 + (c32 - 2 c2 c4 ) m1 + ( c22 - 2 c1c3 + 2 c0 c4 ) m2 + + (c12 - 2 c0 c2 ) m3 + c02 m4 , m1 = - d0 d3 + d1d2 , m2 = - d0 d5 + d1d 4 , m0 = ( d3 m1 - d1m2 ) / d5 , m3 = ( d2 m2 - d 4 m1 ) / d0 , m4 = ( d2 m3 - d 4 m2 ) / d0 , m5 = d0 ( d1m4 - d3 m3 + d5 m2 ).
The robust stability constraint α ( w , k ) is given by
α(w,k) =
αz (w,k) αn (w,k)
with
α z ( w , k ) = 0.0324k12 k 54 + (-0.0648k12 k 52 + + 0.1296k12 k 42 k 52 ) w 2 + (0.0324k12 ) w 4
(
αn (w, k ) = 100 + (-20 + 4k12k42k52 )w2 + w4 ) * ( 3.24k12k54 + (-6.48k12k52 - 7.2k1k2k3k52 + + 12.96k12k42k52 )w2 + (3.24k12 + 7.2k1k2k3 +
Shi, Y. and Eberhart, R.C., “Parameter selection in particle swarm optimization,” in Evolutionary Programming VII: Proc. EP98, Springer-Verlag, New York, pp. 591-600, 1998.
+ 4k22k32 - 14.4k1k2k4k5 - 14.4k1k3k4k5 -
Shi, Y. and Eberhart, R.C., “A modified particle swarm optimizer,” in Proc. of the IEEE International Conference on Evolutionary Computation, Piscataway, NJ, IEEE Press, pp. 6973, 1998.
+ 3.6k1k3k52 ) w4 + (-7.2k1 - 3.6k1k2 + 4k22 -
- 7.2k1k2k3k4k5 + 7.2k1k52 + 3.6k1k2k52 + - 3.6k1k3 + 4k32 + k22k32 + 7.2k1k4k5 )w6 + + (4k22k32 )w8 + w10
)
Fan, H.-Y. and Shi, Y., Study of Vmax of the particle swarm optimization algorithm“ in Proc. of the Workshop on Particle Swarm Optimization. Purdue School of Engineering and Technology, IUPUI, Indianapolis, IN, 2001.
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