MIXED H2/HINF-BASED PID CONTROL USING GENETIC ALGORITHMS: EXPERIMENTAL EVALUATION A. Rangel-Merino, R. Lagunas, J. C. Martinez-Garcia, A. Soria

Departamento de Control Automatico CINVESTAV-IPN A.P. 14-740, 07300 Mexico D.F., Mexico E-mail: [armer,rlagunas,martinez,soria]@ctrl.cinvestav.mx

Abstract: In this paper we propose an experimental evaluation of a PID mixed H2 /H∞ control methodology based on the application of genetic algorithms. We consider an experimental hydraulic servo system as the plant that is affected by disturbances acting on its output; the performance objective corresponds to the control of the position of the hydraulic cylinder.

1. INTRODUCTION

d(t) W(s)

As is pointed out in [2], the so-called mixed H2 /H∞ control designs are quite useful for robust performance design for systems under parameter perturbation and uncertain disturbance. However, as is also pointed out in [2], the conventional output feedback designs of mixed H2 /H∞ optimal control are complicated and not easily implemented. By these reasons, it is quite common to Þx the structure of the controller in order to express mixed H2 /H∞ control in terms of a tractable numerical optimization problem in the parameter vector space. The real parameter vector obtained as a solution to the optimization problem corresponds then to a particular Þxed-structure controller which satisÞes the speciÞed control problem. The Proportional Integral Derivative control (PID control) law is a very succesful industry-oriented Þxedstructure controller. As far as numerical optimization techniques are concerned, evolutionary computing [7] offers some powerful tools. In particular, Genetic Algorithms, initially inspired from the processes of natural selection and evolutionary genetics, have been succesfuly applied in control and signal processing design (see for instance

r(t)

e(t) +

yd(t) C(s)

P(s)

+

-

Fig. 1. Control system with disturbance. [8]). We are interested here in the experimental evaluation of a PID mixed H2 /H∞ control methodology based on the application of a standard genetic algorithm. With this o bjective in mind, we follow the procedure described in [2] to obtain the gains of a PID controller which solves a positioning control problem. The concerned plant is an experimental DC servosystem affected by a disturbance acting on the output. The paper is organized as follows: Þrst of all, we discuss in Section 2 the problem statement, i.e., the PID mixed H2 /H∞ control methodology applied to the positioning control problem. Section 3 is dedicated to a brief description of Genetic Algorithms. We also recall in Section 3 the PID mixed H2 /H∞ control methodology based on the application of Genetic Algoritms. For our particular application we apply

a comercial Genetic Algorithms software [1], which implements a general purpose algorithm. We present in section 4 the results obtained when applying the discussed methodology to a experimental servosystem. We conclude with some Þnal remarks in Section 5

2. PROBLEM STATEMENT 2.1 Mixed H2 /H∞ control Consider the feedback control scheme shown in Figure 1, where: r denotes the reference input signal; yd denotes the output signal; d denotes the disturbance signal; e denotes the tracking error input signal; P denotes a Linear Time-Invariant (LTI) Single-Input Single-Output (SISO) plant; C denotes a LTI SISO controller, and W denotes the so-called frequency proÞle of the disturbance signal d. The control problem is then deÞned as follows:

Chosing the energy as the measure of a given signal, OTCP can be reformulated in formal terms as follows: DeÞnition 2. Let the SISO transfer functions P (s) and W (s) be given. Let also a real disturbance attenuation level γ > 0 be given. Find a controller C (s) such that: min J, with J := C

∞

e2 (t)dt,

(1)

0

and:

kyd (t)k2 sup = d(t)$L2 kd(t)k2 ° ° ° ° W (s) ° ° ° 1 + P (s)C(s) ° ≤ γ,

Remark 3. The problem considered in DeÞnition 2 is a typical mixed H2 /H∞ control one. In fact the problem corresponds to the minimization of the H2 -norm of a signal (1), with a H∞ -norm constraint (2). 2.2 PID mixed H2 /H∞ control

DeÞnition 1. Optimal Tracking Control Problem (OTCP): taking into account the tracking control scheme shown in Figure 1, Þnd a controller C which minimizes the tracking error signal e for a speciÞc reference signal r, while insuring both disturbance attenuation and closed-loop internal stability.

Z

Fig. 2. Hydraulic Servo System

(2)

Let us now Þx the structure of the controller to a PID one, i.e, C (s) = k1 + ks2 + k3 s,where: k1 , k2 and k3 denote the Proportional, the Integral and the Derivative gains of the controller, respectively. If we suppose that the parameter domain of {k1 , k2 , k3 } guarantees the stability of the closed-loop system (such parameter domain can be characterized by the Routh-Hurwitz criterion), and applying the Parseval’s theorem (see for instance [9]), we have that (1) can be rewritten as follows: Z

∞

e2 (t)dt = Z j∞ 1 e (−s) e (s) ds min k1 ,k2 ,k3 2πj −j∞ Z j∞ 1 B(s)B(−s) ds, = min k1 ,k2 ,k3 2πj −j∞ A(s)A(−s) J :=

0

where A(s) and B(s) are Hurwitz polynomials. Let us denote m the degree of A (s), and let us assume that the degree of B(s) is equal to m − 1. Thus the polynomials A(s) and B(s) are given by:

(3)

∞

while insuring closed-loop internal stability. L2 stands for the space of all real valued Lebesgue integrable functions.

A(s) =

(4)

In the previous deÞnition k·k∞ stands for the H∞ norm of the transfer function · (see for instance [4]).

(6)

i=0

m−1 P k=0

Please note that:

m m X Y ak sk = am (s − zi ) k=0

and B(s) = r(s) . e(s) = 1 + P (s)C(s)

(5)

bk sk ,where {z1 , z2 , . . . , zm } is the set

of zeros of A (s). It is also assumed that a0 6= 0 and am > 0. Remark 4. In what follows we shall change J in (5) by Jm (k1 , k2 , k3 ) to strictly indicate its dependence on m, k1 , k2 , and k3 .

As is pointed out in [2], the optimization problem (5) can be solved in this case by the residue theorem (see [10]and [3] for the details). Indeed: N um Jm (k1 , k2 , k3 )= Den Num = · ¸ dm−2 Qm−2 − dm−3 Qm−3 a0 dm−1 Qm−1 + · · · + (−1)m−1 d1 Q1 m−1 + (−1) am d0 Q0 Den = m−1 (−1) 2am a0 ¸ · am−1 Qm−1 −am (am−2 Qm−2 − am−5 Qm−3 + am−7 Qm−4 + · · · ) (7) Pm−1 i where: dl = i,j=0 (−1) bi bj , with i + j = 2l. The Qi , i = 1, 2, , m − 2 are formed from |Ω|, with:   am−1 am−3 am−5 · · · ··· 0 0 . .    am am−2 am−4 · · · · · · .. ..     .. ..  0 am−1 am−3 am−5 · · · . .     .. .. ..  , Ω :=  . am am−2 am−4 · · · . .    . .. ..   .. 0 am−1 am−3 · · · . .     . .. .. .. ..  .. . a1 0  . . . 0 0 ··· ··· · · · a2 a0 by deleting the Þrst, (m−1)-th and m-th columns and the Þrst (i + 1)-th and m-th rows, and Q0 and Qm−1 are given by: Q0 = a2 Q1 − a4 Q2 + a6 Q3 − a8 Q4 + · · ·

Q1 = am−2 Qm−2 − am−4 Qm−3 + am−6 Qm−4 − · · ·

As far as the attenuation disturbance constraint (2) is concerned, we have that: s ° ° ° ° W (s) β(ω) ° = ° ≤ γ, (8) sup ° 1 + P (s)C(s) ° ω∈[0,∞) α(ω) ∞ where β(ω) and α(ω) are some appropiate polynomials of ω. Since the peaks of β(ω)/α(ω) occur at the point which satisfy: dα(ω) α(ω) dβ(ω) d β(ω) dω − β(ω) dω = = 0, dω α(ω) α2 (ω)

d β(ω) dω α(ω)

(ω) : = α(ω) =

n Y

i=1

need be found.

dα(ω) dβ(ω) − β(ω) (9) dω dω

(ω − λi ) = 0

d β(ω) dω α(ω)

(ω).

Thus, the attenuation disturbance constraint (2) is equivalent to: v u u t

sup

λri ∈{λr1 , λr2 ,..., λrl

β(λri ) ≤ γ. α(λ ri ) }

(10)

Summarizing, the PID mixed H2 /H∞ Control Problem (PIDH2 /H∞ CP) is deÞned as follows: DeÞnition 6. PIDH2 /H∞ CP: let the transfer functions P (s) and W (s) be given. Consider a disturbance attenuation level γ also be given. Find a PID controller C (s) = k1 + ks2 + k3 s, assuming that the parameter domain of {k1 , k2 , k3 } guarantees the stability of the closed-loop q system, such that Jm (k1 , k2 , k3 ) is minimized and supλri (β(λri )/α(λri )) < γ is satisÞed. Jm (k1 , k2 , k3 ), β(ω), α(ω), and λri , i = 1, . . . , m, are deÞned as in (7), (8), and (9), respectively. 2.3 An illustrative example In order to illustrate the PID mixed H2 /H∞ control methodology we have identiÞed the model of an experimental hydraulic servosystem shown in Þgure ??. This servosystem is included in a Client-Server experimental setup, mainly oriented to the evaluation of control laws and model identiÞcation algorithms. The Client is allocated in a PentiumT M based computer running at 450 MHz. The sampling rate was set to 1 KHz. The position is measured using a variable resistence that is read using the data acquisition card is the ServotoGo model 1. This I/O card has 12 bits digital to analog converters with an output range [10 volts, +10 volts]. The model identiÞcation algorithm, in fact a well known least-squares algorithm (see for instance [13]), was implemented using the MatLabT M /SimulinkT M software running under the WINCONT M environment. The Server is installed in a PentiumT M based computer running at 900 MHz. The model obtained from the identiÞcation process is the following one: y (s) 1965 = , (11) s(s + 56) u (s) where: u (s) denotes the control signal and y (s) denote the output signal. The output (angular position) is measured in number of turns, while the input signal is given in volts. We take as the reference signal the following one r (s) = 1s .We take as the frequency proÞle of the disturbance the transfer function W (s) = 1 s+1 . P (s) =

only the real roots of: num

Remark 5. In what follows we shall denote {λr1 , λr2 , . . . , λrl } the set of real roots of num

2.3.1. The internal stability constraint First of all, we have that e (s) = r (s) − yd (s) = Te (s) r (s) ,where: µ

¶ 1 1 + P (s) C (s) s2 (s + 56) . = 3 s + (56 + 1965k3 ) s2 + 1965k1 s + 1965k2

Te (s) : =

Consequently, the characteristic polynomial, say pc (s), is given by: pc (s) = s3 +(56 + 1965k3 ) s2 +1965k1 s+1965k2 . (12) Now, the Routh-Hurtwitz criterium let us to conclude that the parameter domain of {k1 , k2 , k3 } insuring internal stability (of the closed-loop system) is characterized by: k2 > 0, k3 > −56/1965, (13) k1 > k2 / (1965k3 + 56) . 2.3.2. The H2 optimization problem Taking into R j∞ account (5) and (7) we have −j∞ e (−s) e (s) ds = R j∞ B(s)B(−s) −j∞ A(s)A(−s) ds, where:

B (s) := b0 +b1 s+b2 s2 and A (s) := a0 +a1 s+a2 s2 +a3 s3 ,

with:

and num

d β(ω) dω α(ω)

(ω)

= 0.2 × 10−7 ω 13 + 0.12544 × 10−3 ω 11 +(0.19 + 13.79k3 + 242.09k32 −0.24k1 + 0.44 × 10−2 k2 −1 2 9 +0.15k 2 k3 − 0.77 × 10 k1 )ω ¢ ¡ 2 + ¡0.30k2 k3 − 0.15k2 + 0.88 × 10−2 k2 − 0.15k12 ω 7¢ + 484.35k2 k3 − 242.17k12 + 13.8k2 − 242077055k22 ω 5 −484.35k22 ω3 (16) PIDH2 /H∞ CP Summarizing, for our current example PIDH2 /H∞ CP is deÞned as follows (we Þx γ = 0.1): Problem 7. Let: 1 1965 , W (s) = , λ = 0.1, s(s + 56) s+1 1 r (s) = , s

P (s) =

be given. Consider the parameter stability domain of {k1 , k2 , k3 } (13): k2 > 0, k3 > −56/1965, k1 > k2 / (1965k3 + 56) .

a0 = 1965k2 , a1 = 1965k1 a2 = (56 + 1965k3 ) , a3 = 1 b0 = 0, b1 = 56, b2 = 1 Remark that for our current example m = 3. Consequently: b2 a0 a1 + (b21 − 2b0 b2 )a0 a3 + b20 a2 a3 J3 (k1 , k2 , k3 ) = 2 µ 2a0 a3 (−a0 a3 + a1 a2 ) ¶ 1965k1 + 3136 = 0.00002544529262 −k2 + 56k1 + 1965k1 k3 (14) 2.3.3. The H∞ optimization problem As far as the disturbance attenuation constraint is concerned, we have that: W (s) 1 + P (s)C(s) s2 (s + 56) = (s + 1) (s3 + (56 + 1965k3 ) s2 + 1965k1 s + 1965k2 ) β (ω) = (1/10000)ω 6 + 31.36ω 4 , α (ω) = 0.1 × 10−3 ω 8 + (22k3 + 386.12k32 −0.39k1 + 0.31)ω6 +(−22k2 + 386.12k32 − 0.39k1 + 386.12k12 4 +22.00k 3 − 772.245k2 k3 + 0.3136)ω + ¢ ¡ 2 2 386.12k2 + 386.12k1 − 22k2 − 772.24k2 k3 ω 2 +386.1225k22 (15)

Find a set {k1 , k2 , k3 } satisfying the previous inequality constraints and such that : min J3 (k1 , k2 , k3 ) {k1 ,k2 ,k3 } µ µ min

{k1 ,k2 ,k3 }

0.00025

=

1965k1 + 3136 −k2 + 56k1 + 1965k1 k3

¶¶

(17)

and (see (10) and (15)): v à ! u u 0.0001λ6ri + 0.31λ4ri t sup < 0.1, α (λri ) λri ∈{λr1i ,λr2 ,...,λrl } with: α (λri ) = 0.1 × 10−3 λ8ri +(22k3 + 386.12k32 − 0.39k1 + 0.31)λ6ri +(−22k2 + 386.12k32 − 0.39k1 + 22k3 2 4 −7722.24k 2 k3 + 0.31 + 386.12k1 )λri ¢ ¡ 2 2 + 386.12k2 + 386.12k1 − 22k2 − 772.24k2 k3 λ2ri +386.12k22 . (18) Where λri , for i = 1, ..., l, denote the ri -th real root of :

= 0.2 × 10−7 ω 13 + 0.12 × 10−3 ω11 + (0.19 + 13.79k3 +242.09k32 − 0.24k1 +0.44 × 10−2 k2 + 0.15k2 k3 − 0.77 × 10−1 k12 )ω 9 2 −2 +0.30k k2 − 0.15k12 ω 7 ¢ 2 k3 − 0.15k2 + 0.88 × 10 ¡ 2 + 484.35k2 k3 − 242.17k1 + 13.8k2 − 242077055k22 ω 5 −484.35k22 ω 3 (19) Problem 8. In what follows we shall denote {k1∗ , k2∗ , k3∗ } the solution to Problem 7. We can at this level proceed to solve the proposed problem. Fig. 3. PID Control 3. THE GENETIC ALGORITHMS APPROACH There exists a huge quantity of publications concerning Genetic Algorithms and its applications in Automatic Control and Signal Processing (see for instance [1], [8],[11], and [12]). As is pointed out in [12], Genetic Algorithms (as a class of stochastic optimization techniques) can be interpreted as one particular implementation of a Monte Carlo optimization technique and can be applied to arbitrary optimization problems (like the one speciÞed by DeÞnition 6). Motivated by the mechanisms of natural selection and evolutionary genetics, a typical Genetic Algorithm behaves as speciÞed by the following genetic dialect (see [12]): “We start out with a randomely chosen qualitative genetic pool. We evaluate the quality of the entire genetic pool. We rank the genetic strings according to their quality. We deÞne the Þtness of a genetic string as:Þtness= total1.0error .We then add up the Þtnesses of all genetic strings in the genetic pool and deÞne the relative Þtness of a genetic string as: relative Þtness= sum overÞtness all Þtnesses .We then replace the entire genetic pool by a new pool in which each genetic string is represented never, once, or multiple times proportional to its relative Þtness. Poor genetic strings are removed, while excellent genetic strings are duplicated many times. We then pair the genetic strings up arbitrarely. Each pair produces exactly two offspring, one consisting of the head of the Þrst string concatenated with the tail of the second and the other consisting of the head of the second string concatenated with the tail of the Þrst. We then let the old generation die and replace the entire genetic pool by the new generation. The algorithm is repeated until convergence”. There exist several implementations of the genetic computing strategy schematized above. For our purposes we chose a MATLABT M -based tool called FlexToolsT M [1]. This MATLABT M toolbox implements a general-purpose Genetic Algorithm which constraint each genetic string to be coded as a binary pattern. Both the Þtness function (i.e., Jm (k1 , k2 , k3 ) in our case) and the disturbance attenuation constraint function are coded as MATLABT M procedures, each

functions receiving as their argument the vector {k1 , k2 , k3 }. It is quite obvious that the initial genetic pool belongs to the parameter domain characterized by (13). As far as the set of Genetic Algorithm descriptors, we must choose the following one: Number of generations, Population size, Crossover probability, Mutation probability and Selection operator.

3.1 The computing procedure In order to implement the solution of PIDH2 /H∞ CP through the Genetic Algorithm Approach we follow the following sequential procedure: a)We Þx the Genetic Algorithms descriptors mentioned above. b) We randomely choose a genetic pool of PID gains {k1 , k2 , k3 } (coded in binary terms) constrained to belong to the stability parameter domain. In fact, we Þx range values for k1 , k2 , and k3 .c)We compute the Þtnesses of all genetic strings, taking directly Jm (k1 , k2 , k3 ) as the Þtness function.d)We apply the roulette wheel selection technique to choose the best subset of the population of PID gains.e)We proceed to pair the genetic strings (and to apply mutation) in order to obtain a new population. f) We verify the disturbance attenuation constraint for each member of the chosen population. We decrease the Þtness of the members which do not satisfy the disturbance attenuation constraint and e) we repeat the procedure step c) to step f) until the Þxed number of generations is attained. The Þnal binary result is Þnally decoded to obtain the PID gains. Both the crossover probability and the mutation probability are chosen following what is indicated in [7]. Experimental Results Let us now continue with our illustrative example (see Subsection 2.3): Consider Problem 7. The set of Genetic Algorithms descriptors is Þxed as follows: Number of generations = 57, Population size = 47, Crossover probability = 0.6, Mutation probability = 0.077 and Roulette wheel selection. In order to compute {k1∗ , k2∗ , k3∗ } we take a

quite obvious that the remarked difference is due to the nonlinear nature of the real system.

Fig. 4. Control Signal subset of the parameter stability domain charcaterized by k1 ∈ [0, 0.7], k2 ∈ [0, 1], and k3 ∈ [0, 0.06]. We apply the procedure speciÞed in Subsection 3.1. For our current example we obtain the following results: k1∗ s

= 0.667

k2∗

= 0.00009873, µ

supλr ∈{λr ,λr ,...,λr } 2 i 1i l

k3∗

= 0.06

0.0001λ6r +0.3136λ4r i i α(λri )

= 0.0669 < 0.1.

¶

The computed solution shows some particularities: k1∗ and k3∗ take the extreme values of theirs corresponding ranges, while k2∗ tends to be equal to zero. Because of the nature of J3 (k1 , k2 , k3 ), k1∗ and k3∗ must be small to guarantee a small value of J3 (k1 , k2 , k3 ). As far as k2∗ is concerned, the obtained result conÞrms that integral control is not necessary, since the model of the hydraulic servosystem includes an integral action. It is necessary to say that a big value of the proportional gain gives rise to response overshoots. This undesired behaviour can be corrected injecting damping, i.e., increasing the derivative gain. Figure 3 shows the behaviour of the real system corresponding to the PID gains, and Þgure 4 presents the control signal.The noise in the control signal is due primarlly to the position sensor that in our case is a variable resistance. It should be remarked that the obtained values of the PID gains dont saturate the actuator, a proportional hydraulic servo-valve, that saturates in [−2, 2].

4. CONCLUDING REMARKS. Concerning our experimental evaluation, we compare the behaviour obtained when applying the computed controller to the real system with the behaviour obtained when applying the computed controller in a simulated control scheme. Both behaviours are very close, but the simulated control scheme does not present an appreciable effect of the disturbance. It is

In general, we can conclude that the Genetic Algorithms Approach to solve mixed H2 /H∞ control problems (considering PID controllers) is a good choice when the dynamics of the real plant are close to the dynamics of a linear system. In our study we apply a standard Genetic Algorithm which does not proÞt from both the nature of the plant and the controller. The nature of the control scheme must be coded in the optimization procedure, e.g., the constraint limiting the values gains in our illustrative example can be coded in terms of the speciÞed range for the gain. In the optimization proces it would be interesting to investigate the minimization of the control signal that would also limit the gains in the PID controller.

Bibliography [1] FlexTools(Ga) M 2.1 Flexible Intelligence Group. June 1995. [2] Bor-Sen Chen, Yu-Min Cheng, and Ching-Hsiang Lee: “A Genetic Approach to Mixed H2 /H∞ Optimal PID Control”, IEEE Control Systems, October 1995, pp. 51-59. [3] E.I. Jury and A.G. Dewey, “A General Formulation of the total Square Integrals for Continuos Systems”. IEEE Trans. Autom. Control, AC-10, pp. 119-120, Jan.1965. [4] B.A. Francis, “A Course in H ∞ Control Theory”, SpringerVerlag, Berlin, 1986. [5] M. Vidyasagar, “Control Systems Synthesis: A Factorization Approach”, MIT Press, Cambridge, MA, 1985. [6] J.H. Holland, “Adaptive in Natural and ArtiÞcial Systems”, Ann Harbor, MI: Univ. Mich. Press, 1975. [7] Bäck Thomas, “Evolutionary Algorithms in Teory and Practice”, Informatik Centrum Dortmund, Germany Oxford University Press, 1996. [8] K. F. Man, k. S. Tang, S. Kwong and W. A. Halang, “Genetic Algorithms for Control and Signal Processing”. Springer-Verlag, Advances in Industrial Control, London, UK, 1997. [9] J. Doyle, K. Glover, and A. Tannenbaum, “Feedback Control Theory”, Macmillan, New york, 1992. [10] P. V. O’Neil, “Advanced Engineering Mathematics”, Brook/Cole Publishing Company, CA, USA, 1995. [11] D. E. Goldberg, “Genetic Algorithms in Search, Optimization, and Machine Learning”. Reading, MA: addison wesley, 1989. [12] F. E. Cellier, “Continuous System Modeling”. SpringerVerlag, New York, USA, 1991. [13] K. J. Astrom and B. Wittenmark, “Computer Controlled System: Theory and Design”, Second Edition, Prentice Hall, Englewood Cliffs, New Jersey, USA, 1990.