genetic algorithms for optimal feedback control design

Pergamon Engng Applic. Art~[: Intell. Vol. 9, No. 4, pp. 403411, 1996

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved

Pll: S0952-1976(96)00034-6 o952-1976/96 $15.oo+0,0o

Contributed Paper

Genetic Algorithms for Optimal Feedback Control Design

SOURAV KUNDU Kanazawa University, Japan

SEIICHI KAWATA Tokyo Metropolitan University, Japan

(Received October 1995; in revised form April 1996)

This paper presents a technique for optimal feedback control design which combines a relatively recent artificial intelligence (AI) method, the genetic algorithm ( GA ), and the more traditional methods of control system design, achieved via a new problem formulation. The performance function of a control system is generally formulated as a linear combination of xT Qx and ur Ru, where Q is the state weighting matrix and R is the control weighting matrix. These matrices are difficult to ascertain in real-worM cases. The approach outlined here formulates the optimal feedback control design as a multiple-criteria problem, thereby avoiding use of the weighting matrices. It is shown that using the proposed problem formulation, a non-linear state feedback can also be implemented, which expands the search space for the design. A numerical example is computed to show the efficacy of such a method.

Copyright © 1996 Elsevier Science Ltd

Keywords: Artificial intelligence, Pareto sets, genetic algorithms, multicriteria optimization.

1. INTRODUCTION

Most of the applications of artificial intelligence (AI) methods to control-system design are formulated either as a parameter optimization task or control structure learning task) The genetic algorithm (GA) model, 2,3 which is studied within the realm of AI, has been used for state feedback control design by Porter: Huang and Fogarty 5 and Goldberg 6 and others, but the selection mechanisms of the GA in these applications work on a single-valued scalar fitness function. In most methods, the fitness function for the GA is generally taken as the performance index function for the control system. In the control-system design problem, the dynamic processes that are considered are characterized by the state-space equation:

~(t) = Ax(t) + Bu(t), (1)

Correspondence should be addressed to Sourav Kundu, Department of Human and Mechanical Systems Engineering, Faculty of En- gineering, Kanazawa University, 2-40-20 Kodatsuno, Kanazawa 920, Japan. Email: [email protected].

where x is the process state, u is the control input and A and B are known matrices. Design of a linear state feedback:

u(t) = -kx(t) (2)

is sought, where k is a suitable gain matrix. Here, instead of seeking a gain matrix to achieve specified closed-loop pole locations, a gain to minimize a specified performance criterion J (cost function), expressed as the integral of a quadratic form in the state x plus a second quadratic form in the control u, is sought. The performance criterion is characterized by the integral:

J = [xr(t)Qx(t) + ur(t)Ru(t)]dt o

(3)

where both Q and R are positive definite and symmetric matrices. The lower limit of the integral to is considered as the present time and the upper limit tf is the terminal or final time. The matrices Q and R are called the state weighting matrix and the control weighting matrix, respectively. It is possible to find the control gain matrix in terms of these weighting matrices Q and R. The

403

404 SOURAV KUNDU and SEIICHI KAWATA: OPTIMAL FEEDBACK CONTROL

matrices A and B of equation (1) can be used along with Q and R to find k of equation (2) directly. 7

The issue of considerable concern to the control- system designer is the selection of the weighting matrices Q and R. In practical cases, the minimization of the quadratic integral of equation (3) is often not a true design objective. The problem, however, is that the true design objective can seldom be expressed in mathematical terms. In some instances when it can be described in mathematical terms, it is usually impossible to solve for an optimal control law. Expression of the design objective in the form of a quadratic integral is a prudent compromise between formulating the real problem, that cannot be solved, and formulating an artificial problem that can be solved. The quadratic form x r ( t ) Q x ( t ) , in the performance index in equation (3), represents a penalty on the deviation of the state x from the origin, and the term u r ( t ) R u ( t ) attempts to limit the magnitude of the control signal u.

It is intuitive to the understanding here that if it is possible to re-formulate equation (3) in such a way as to avoid the implicit necessity of using weighting matrices Q and R, that is, to consider the problem as one of multiple criteria, one can arrive at a control design technique which eliminates the difficulties outlined in the choice of the Q and R. By using the multiple-criteria formulation and applying the Pareto 8 set analysis to obtain optimal design solutions, the versatility of Q and R is retained, without actually using them to obtain the control design solution. This new technique is presented in this paper.

The research presented here covers three main aspects of the optimal feedback control design technique:

(i) A mathematical re-formulation of the traditional control-system design formulation.

(ii) A new algorithm to solve multiple-criteria design problems using Pareto sets and the genetic algorithm.

(iii) By the use of (i) and (ii) it is shown that a non-linear state feedback can be considered, which expands the search spaces for the design, and that this search algorithm produces better design solutions compared to the linear state feedback approach, which works on a constricted search space.

2. PROBLEM FORMULATION

2.1. The optimal feedback control problem

Consider a multiple-input linear time-invariant system described by the state space equations:

k(t) = Ax(t) + Bu(t) (4)

y(t) = Cx(t) (5)

with given initial conditions:

x(t0) = x0 (6)

where x c ?~n, u ~ 'Y~", A ~ ~ m x n , B ~ Nnx,- and C Nm×,, and the pair A B is controllable, given the initial and terminal state and the performance index, x(t) is an n-dimensional state vector, u(t) is an r-dimensional control input, and x0 is a constant n-dimensional vector. Associated with this system is a performance index, the minimization of which is the goal of the control-system design task. The conventional description of the performance index is:

I; J (u ( t ) ) = [ x r ( t ) Q x ( t ) + ur ( t )Ru ( t ) ]d t , (7)

subject to equation (4), where Q and R are both positive definite and symmetric matrices. The control-system design problem is to find a control input u(t) such that the given performance index in equation (7) is minimized. Deducing Q and R depends on the experience of the control-system designer, and a number of trial-and-error strategies are generally required for satisfactory deduc- tion. According to linear control theory, the algebraic Riccati equation can be expressed as:

A r P + PA - P B R - 1 B r P + Q = 0, (8)

the solution of equation (8) being the matrix P. Using P, the optimal control input can be calculated as:

u(t) = - R - I B r P x (9)

and its minimal, J (u ( t ) ) , is:

J ( u ( t ) ) m i n = xgPx0. (10)

2.2. Mathematical reformulation

Equation (7) is resolved for implementing a multicriteria design strategy. This changes the search spaces for design. Reformulation of equation (7) is proposed, where Jl (u(t)) and J2(u(t)) are defined as follows:

I; Jl (u(t)) = [ x r ( t ) Q x ( t ) ] d t (11)

J2 (u(t)) = [ur ( t )Ru( t ) ]d t . (12)

If Q ~ I and R ~ I (identity matrix), the meaning of equations (11) and (12) still remains the same, provided the design problem is treated as multicriteria problem by avoiding the use of the scalar weighting matrices Q and R. The goal of the control system design task is thus redefined as finding a control input u(t) such that the set of Pareto-optimal solutions in the space of Jl(u(t)) and J2(u(t)) minimizes both the performance functions of equations (11) and (12) simultaneously.

From the theory of dynamic control systems, the control input u(t) for the optimal control system design using linear state feedback is described as:

u(t) = r + k tx t + k2x2 + ... + knxn, (13)

where the gain matrixk = ki, (i = 1, 2 ..... n) ~ ~r×n, x(t) is an n-dimensional state vector, u(t) is an r-dimensional

SOURAV KUNDU and SEIICHI KAWATA: OPTIMAL FEEDBACK CONTROL 405

control input, and r is the reference input vector. By the use of the reformulation described in equations (11) and (12) it is now possible to construct a non-linear state feedback with combination of linear and non-linear terms for the optimal control input u(t) as:

u(t) = r + k l l x l + kl2X2 +...

- x T K 2 1 x "

xT K22x + klnxn + . , (14)

xT g2n x .

where matrix K = K2~, (i = 1, 2 ..... n) ~ ~ × ~ and matrix k = kl /, ( j = 1, 2 .. . . . n) ~ ~"×~. The introduction of the matrix K expands the search space for the design. 9 This expanded search space helps to find much better design solutions, which are exemplified by the numerical example in Section 4. The algebraic Riccati equation shown in equation (8) cannot be applied in this case of non-linear state feedback formulation, and the proposed multicriteria design method with the GA solution technique is then an efficacious control-system design method for the solution of equation (14).

3. GENETIC ALGORITHM SOLUTION TECHNIQUE

Genetic algorithms (GAs) 2 are distinguished by their parallel investigation of several areas of a search space simultaneously by manipulating a population, members of which are coded problem solutions. The task environ- ment for these applications is modeled as an exclusive evaluation function which, in most cases, is called a "fitness function", that maps an individual of the population into a real scalar. The motivational idea behind GAs is natural selection. Genetic operators like selection, crossover and mutation are implemented to emulate the process of natural evolution. A population of "or- ganisms" (usually represented as bit strings) is modified by the probabilistic application of the genetic operators from one generation to the next. GAs have a potential for multi-dimensional optimization, as they work with a population of solutions rather than a single solution. A more detailed explanation of the theory and working of the GA can be found in the literature. 3

Genetic algorithms have two distinguishing features that make them very attractive when considering multicriteria optimization:

(1) They can be used to solve all classes of non- linear programming models, i.e. continuous, integer, discrete, and mixed models. They can be especially effective when solving highly non-linear and highly constrained models. On the other hand they can also be used to provide quick, near-optimal solutions for other models like large linear integer programming models and some classes of networks.

(2) Since genetic algorithms work with a population of candidate solutions, a whole set (population) of Pareto solutions (nondominated solutions) can be generated in a single GA generation. This means that even for a very complicated non-linear programming problem one can obtain the whole set of Pareto-optimal solutions after a single running of the program, and the decision making process can be based on this set. This implies that there is no need to repeat the calculations in order to get another Pareto solution, as it is for most classes of inter- active methods. Thus, the optimal feedback control design process will be less time consuming, and will be based on a knowledge of all possible alternatives.

Previous research on use of GAs for multicriteria analysis can be found in the literature. 1°- 17 The genetic algorithm solution technique that has been used here for optimal feedback control design, utilizing multicriteria analysis, is presented in this section. The assignment of a fitness measure to a design solution is based on reference to the dynamically updated Pareto optimal set generated during progressive GA runs. 18-19 The Pareto-opt ima! set of control design solutions is defined as:

"Let X be a set of feasible control design solutions. The Pareto optimum is defined as: a design solution (control input) u* (t) ~ U, with N performance indices, is Pareto optimal if and only if there exists no u(t) ~ U s u c h t h a t J i ( u ( t ) ) <_ Ji(u*(t)) f o r / = 1,2 ..... N with Ji(u(t)) < Ji(u*(t)) for at least one i. Here U ~ ~n is the universe of admis- sible control inputs. This definition is based upon the intuitive assumption that the control input, u* (t) is chosen as the optimal one if none of the performance indices, Ji (U ( t)) ,

can be improved without worsening at least one other performance index. ''8

The Pareto set is plotted to find the set of the optimal solutions that are generated, considering all the N performance criteria simultaneously without any relative importance (weights) given to any of them. With reference to this Pareto set, a fitness value is awarded to each control design solution. The GA uses this fitness measure to perform the selection operation. The technique proposed here generates control inputs, and compares the control output considering this Pareto optimality con- dition. At the end of the GA run, the control-system designer chooses one of the solutions in the final Pareto- optimal set to suit his or her design goals and requirements.

The fitness is the prox im i t y value of a feasible design solution, added to a shared common fitness value of the Pareto-optimal solutions produced during the previous fitness function evaluations. The fitness is a measure of how f a r the new design solution is from most recent

406 SOURAV KUNDU and SEIICHI KAWATA: OPTIMAL FEEDBACK CONTROL

Pareto Solution l

Pareto Frontier

/' n-- c of outo

d:ll/ ~ Pareto \ III Solution 2 Negative Pareto Space

\ d 2 / - ~ Pareto . . i d 3 ~O Solution 3

I ~ _ _ ~ "~--Solution 4 New Solution ~ ~ , , , ~

n ~ d ~ ' ~ - ' - - - - ~ Pareto Positive Pareto Space u5 ~ o l u t i o n 5

0,0 "~ df fitness for one-dimensional GA

• 2.-

L Performance Index 2 (J2)

Solution for l-d GA

Fig. 1. Fitness calculation methods in GA-based design in control.

Pareto frontier. This Pareto frontier gets updated dynamically during the successive GA runs. An explanation of the notion of proximity in the solution space would help to understand the process better. Every feasible design solution occupies a definite position in the n-dimensional universal space of solutions. This position will have some metrical proximity value (distance) measured from each

of the Pareto solution vectors, found in the previous fitness function evaluations. Each of these proximity values are measured. The value returned as the fitness of a design solution is the least one of all these different proximity values. Analogous to an ordinary scalar GA, where the fitness value of a solution is measured as its metrical distance (straight line distance) from zero on a linear scale, this method is equivalent, remembering that the shortest distance between any two given points is a straight line.

3.1. "Proximity" values in the solution space

Figure 1 shows the equivalent ways in which an ordinary GA and the multiple-criteria GA assign fitness to a new solution. In a single-criterion GA, the fitness dr is the shortest distance (a straight line joining two given points) between the solution and some base measure ([0,0] in Fig. 1). For a multiple criteria problem the fitness is the mini-

mum of dl, d2, d3, d4 and d5 in Fig. 1. The minimum is d3 in this case. The reference measure for di, d2, &, d4 and d5 is the cumulative Pareto set of all the solutions found till the immediately preceding fitness function evaluation. Thus, the fitness is essentially equivalent to that of the working of a one-dimensional simple OA. A solution will always have a proximity value, regardless of where it lies, whether in the negative or positive spaces of the given Pareto frontier. When it lies in the positive Pareto space

(see Fig. 1), this proximity value should be greater than when it lies in the negative Pareto space, as the former is a better solution. This is taken care of by changing the signs in the proximity values. In the case when a solution lies in the negative Pareto space of the present Pareto frontier, the method to assign the fitness is different from that used when it lies in the positive Pareto space. This method is described in detail in Sections 3.2 and 3.3.

3.2. Updating of the Pareto set

At each fitness-evaluation step, a set of Pareto-optimal solutions is maintained, all of which "share" the fitness value that is the highest among all the fitness values of the Pareto solutions at that particular fitness evaluation step of the algorithm. All the members of a Pareto set have this same shared common fitness value. A new solution in a certain generation can fall in any one of the three, and only three, following categories:

(a) It is a new Pareto-optimal solution, and it dominates some (or all) of the Pareto-optimal solutions found up until the immediately preceding fitness function evaluation.

(b) Although it is a new Pareto-optimal solution it does not actually dominate any of the Pareto-optimal solutions found up until the immediately preceding fitness function evaluation. (It lies in the present Pareto frontier.)

(c) It is not a Pareto-optimal solution.

For every new solution in a certain generation, a proximity value is first assigned to it. Then the three different categories mentioned above are dealt with in three sepa- rate ways given below, whereby a fitness value is returned


for the GA selection mechanism to work. Note that for category (c) solutions above the solutions are not re- moved from the Pareto set as in Belegundu's work, 16 but are kept in the population, although with a lower fitness value. This is done as a measure to induce some form of atavism in the evolutionary process, and to insure against the loss of any otherwise useful genetic material.

(i) For category (a) solutions the new fitness value is set to the proximity value added to the previous shared common fitness value of the Pareto-optimal solutions, and the Pareto set is updated by remov- ing those old solutions that this new Pareto solution dominates.

(ii) For category (b) solutions the new fitness value is set to the proximity value added to the previous shared common fitness value of the Pareto-optimal solutions, and the Pareto set is updated by adding this new Pareto solution to the old Pareto set.

(iii) For category (c) solutions, the proximity value sub- tracted from the shared common fitness value of the Pareto-optimal solutions previously found.

3.3. Description of the algorithm The main idea of the algorithm is contained in evalu-

ating the fitness for each solution that the GA generates. This fitness has a greater value if the solution is farther away from the existing Pareto set.

Each Pareto solution has a value which will be called thef i tness potential, and which will be denoted by p / for l = 1, 2 ..... lp, where lp is the number of existing Pareto solutions. Let f/ = I f l/, f2/ ..... J3/] 7" be a vector of objective functions for the lth Pareto solution. For each new solution which is the gain matrix k [as in equation (2)] that is generated by the GA, the system is simulated. The values of the cost-performance indices shown in equations (11) and (12) for that newly generated gain matrix, are calculated, and the solution is plotted in the space of cost performances Jl and J2 given in equations (11) and (12). The relative distances of this solution from all of the most recent Pareto solutions are calculated using the following formula:

i=t ~/~ / f o r / = 1,2 ..... lp. (15)

where qbi is the transformed objective function after penalizing with constraint violation factors. The minimum proximity for a solution is taken into consideration when calculating its fitness value. The complete algorithm is described below. A solution referred to by the notation k [as in equation (2)] means the gain matrix that is generated by the GA. The cost performance in the space of Ji and J2, after the system is simulated, by using that gain matrix k.

(i) Let t = 1 and j = 1, where t is the index of the GA generation, and j is the index of a solution generated within the tth GA generation.

(ii) An initial random population is generated. The first member of this initial population is taken as a Pareto-optimal solution with a "fitness potential" value equal to pl, where pl is an arbitrarily chosen value called the starting fitness potential.

(iii) Substitute j = j + 1 and check if j _< J, where J is the assumed number of solutions generated within the tth generation. If this is true go to step (iv), and if not, go to step (viii).

(iv) For the initial generation the solution matrix k is generated at random. But for the second and subse- quent generations the solution matrix k is generated on the basis of the fitness F calculated in step (vii).

(v) For each solution which is generated by the GA, the relative proximities from all existing Pareto solutions using the formula (15) is calculated, i.e. di(k) for l = 1,2 ..... lp.

(vi) The minimum value from the set {all(k)}, using the following formula:

dr. = min {d/(k)} (16)

where the index l* indicates which of the existing Pareto solutions is nearest to the newly generated solution.

(vii) The newly generated solution k is compared with the existing set of Pareto solutions in the following way.

(a) If it is a new Pareto solution which dominates some (or at least one) of the existing Pareto solutions, then calculate the fitness value using the following formula:

F = Pmax + dl*(k) (17)

and substitute Pmax = F. Update the set of Pareto solutions, i.e. remove any solutions that are dominated by the newly generated solution, and substitute the value of F for the fitness potential value of this solution. Then go to step (iii).

(b) If it is a new Pareto solution which is added to the existing Pareto set, calculate the fitness value using the following formula:

F = PI* + d l , ( k ) (18)

and add this solution to the existing set with the fitness potential value equal to F. If F > Pm~x, substitute Pm~x = F, and go to step (iii). Otherwise go directly to step (iii).

(c) If it is not a new Pareto solution, calculate the fitness value using the following formula:

F = PI* - dr, (k). (19)

If F >__ 0 go to step (iii). If not, then substitute F = 0 and go to step (iii). This substitution is

408 SOURAV KUNDU and SEIrCH1 KAWATA: OPTIMAL FEEDBACK CONTROL

dictated by the fact that the GA operates only on positive fitness values.

(viii) Substitute Pmax for all existing Pareto solutions:

p/ = Pmax for l = 1,2 ..... lp (20)

where lp is the number of existing Pareto solutions. This is where Pmax is "shared" among all Pareto solutions.

(ix) Substitute t = t + 1. If t > T, then terminate calculations, where T is the preassigned number of generations to be considered. If t _< T, check if there is an improvement through the last t* generations, where t* is also a user-preassigned number. If there is any improvement, substitute j = 0 and go to step (iii). Otherwise terminate the GA runs and subse- quent calculations. Such a situation means that the GA has not been able to find a new Pareto-optimal solution through the last t* generations.

Here, two types of termination criteria are used. The termination criterion which indicates that there is absolutely no improvement in the fitness after running t* number of generations, is the more important criterion. Alternatively, the calculations can be stopped just after running the preassigned number of generations. This second criterion refers to the computational time for gen- erating the Pareto solutions.

4. NUMERICAL EXAMPLE

This section presents results of a control problem taken from Ref. 20, p. 610. The problem has been solved using the proposed problem formulation for optimal feedback control design (Section 2.2) and the multiple-criteria genetic algorithm described in Section 3. The control system considered here can be represented by state-space equation:

~(t) = Ax(t) + Bu(t) (21)

and a state feedback controller can be selected:

• For a linear state feedback the u(t) is a linear function of the measured state variable x:

u(t) = r + k l x l + k2x2 + ... + knxn. (22)

• For a non-linear state feedback the u(t) is a combination of linear and non-linear terms in the function of the measured state variable x:

u(t) = r + klxl + k2x2 + ...

Xr Kl X ]

x r Knx ]

(23) + knxn +

Consider a simple system with two state variables and one input variable:

dE ,] [001 ] x,] d---t x2 0 x2 + u(t) (24)

by setting:

[Ool J [o] A = 0 , B = a n d x = x2

in equation (21). First, a linear state feedback design is chosen such that:

U(t) = - k l x l - k2x2. (25)

Secondly, a non-linear state feedback design is chosen such that:

[k3k4][Xl]k5 k6 x2 (26) u/ t ) = -k ,x~ - k2x2 + [L xl x2 ]

with the control u(t) as a combination of the linear and non-linear terms of the two state variables. For non-linear state feedback, equation (24) produces:

£t = x2 (27)

2C2 = - k l X l - k2x2 + [1_ xl x2 ]j

By application of the reformulation in equations (11) and (12), the two following performance indices are chosen, both of which are to be mimmized simultaneously.

Jl (u(t)) = (Xl 2 + x~) dt (29)

J? J2(u(t)) = (u 2) dt. (30)

For using linear state feedback, the goal of the control- system design task is to find the optimal value of the decision variables kl, k2 [see equation (25)], such that when the system is simulated by using those values, equations (29) and (30) are minimized simultaneously. The eigenvalues of the system are taken as -+_j, and the upper and lower limits of kl, k2 as _+20.47. The GA genotype bit length is 24, which takes into account the values of k~ (12 bits) and k2 (12 bits). Note here that (211 - 1)/100 = 20.47, and the 12th bit is for handling the negative values of kl and k2.

For using non-linear state feedback the goal of the control-system design task is to find the optimal value of the decision variables kl, k2, k3, /%4, k5 and k6 [see equation (26)], such that when the system is simulated by using those optimal values, equations (29) and (30) are minimized simultaneously. The eigenvalues of the system are taken as _+j, and the upper and lower limits ofkl , k2, k3, k4, k5 and k6, as +_20.47. The GA genotype bit length is 72, which takes into account the values of kt (12 bits) through k6 (12 bits). Note here that (211 - 1 ) / 100 = 20.47, and the 12th bit is for handling the negative values of kj, k2, k3, k4, k5 and k6.


Starting System Response Plot 1 0.51 ,

0'8 t 0

0.6 k l = + 8 . 2 1 0

I [ k2 = + 2 . 3 3 0

°-'I/11 -° I I / ,2=+2.3,,,,0

o .21 / xl(to)=+t

15

- 2 i 1 14 0 2 4

T i m e - - >

Plot of State variable x2 (starting)

x2(to) = o

I i 6 8 1'o

Time -->

Fig. 2. Starting solution for GA generation number 1 by linear state feedback.

14

0

- 0 . 1

-0.2 ? - 0 . 3

m

~ -0.4

-0.5

-0.6

-0.'

Plot of State variable x2 (Optimal)

x2( t0) = 0

. . . . 'o '2 2 4 6 8 1 1 14 T i m e - - >

0

- 0 .1

- 0 . 2

? ~ -0,3

~-0 .4 N N

-0.5

-o.6! i

- 0 . 7

Plot of State variable x2 (Optimal)

x2(t0) = 0

i i ~ i 1 |

2 4 6 8 10 12 14 T i m e - - >

Fig. 3. Finishing solution for GA generation number 50 by linear state feedback.

4.1. Explanation of results

Figures 2 and 3 show the results from the computer simulation of the system described by equation (24), using linear state feedback. The design goal is to stabilize the system in minimum time, with minimum overshoot. The starting values of xi and x2 are 1 and 0 respectively. At generation 1 of the GA, there is a large amount of overshoot of both the Xl and x2 variables, before stabilizing in about 6 sec. Figure 3 plots the error response at the 50th generation of the GA, when the final design solution was achieved using linear state feedback. Both the state variables stabilize in about 4.6 sec. Figure 2 was plotted using one of the only two Pareto solutions found in generation 1 of the GA, whereas Fig. 3 was plotted using one of the 27 Pareto solutions found in the 50th generation of the GA.

Results of the simulation using non-linear state feedback are presented in Figs 4 and 5. Comparison of these results with those of Figs 2 and 3 gives a visualization of what has been achieved by expanding the design search space via utilization of equation (14), in the place of equation (13). Figure 4 presents the error response of one of the only two Pareto solutions found in the GA generation number 1. The system stabilizes in 3.2 sec which is better than the linear state feedback approach. Fig- ure 5 presents the error response of one of the 14 Pareto solutions found in the 50th generation of the GA. It is clear that, with absolutely no overshoot, the state variables smoothly stabilize in about 2.2 sec.

Figure 6 shows theprogress of the Pareto-optimal frontier during the 50 generations of the GA. In generation 1 there are only two solutions. In generation 50, 14 Pareto

410 SOURAV KUNDU and SEIICHl KAWATA: OPTIMAL FEEDBACK CONTROL

1

0.8

0.6

? 0.4

R 8

0.2

0

-0 ,2 0

Start ing System R e s p o n s e Plot

k l = 18.410 k2 = 6.900

k3 = -15 .390 k4 = 1.200

k5 = 0.600 k6 = -18 .290

J1 = 969.35672089 J2 = 1,37724787

xl(tO) = +1

Plot of State var iable x2 (starting) 0.21

° L -0 .2

-0 .4

-0 .6

-0.8~

-1 I- x2(t0) = 0

-1.2~-

-1.41-

- 1 6 F

i i i / i _1.81 i i i i 4 6 8 10 12 14 0 2 4 6 8

T i m e - - > T i m e - - >

Fig. 4. Starting solution for GA generation number 1 by non-linear state feedback.

i i 10 12 14

Opt im ized Sys tem R e s p o n s e Plot 0.05 Plot of State variable x2 (Opt imal )

0.8

0.6

t " 0 4

g_

8 0.2

0

-0 .2

L

k l = 13.530 k2 = 7.6000

k3 = 16.230 k4 = -12.770

k5 = -18 .070 k6 = -12.240

J1 = 3.13142640 J2 = 0.28371573

xl(tO) = +1

0 0 5 f

0.1

0.15~

-o.2 t

0.25 t

0.3 L 0 2 4 6 8 i i h i i i

2 4 6 8 10 12 14 Time - - > T i m e - - >

Fig. 5. Finishing solution tbr GA generation number 50 by non-linear state feedback.

10 12 14

solutions have been found by the GA. The designer can now choose any of these 14 Pareto solutions according to his or her preference, and the design considerations and requirements. One of these 14 solutions was used to plot Fig. 5. This solution is marked by "*" in Fig. 6 to show its position in the final Pareto frontier. For these simulations, the G A population size was 70. The GA crossover probability was 0. 5. The GA mutation probability was 0.09. The number of generations required to produce all the results presented was 50.

5. CONCLUSIONS

The engineering goal of AI is to solve real-world engineering problems using concepts from AI as an embodiment of ideas about representing and using the

domain knowledge. Control-design techniques use this domain knowledge for the control application to devise a control system that typically minimizes a weighted linear quadratic performance index. Using any of the methods based on heuristic weighting of control performance criteria, one can obtain a single best solution which clearly reflects the choice of weights. Using the proposed method, a reasonably large set of Pareto-optimal solutions is obtained, which are well distributed all along the Pareto frontier. From this set the control-system designer can choose a design solution according to his or her design focus and requirements. The simulations show that non-linear state feedback produces better results by expanding the design search space. The multicriteria design formulation, along with the genetic algorithm solution technique, is essentially required to implement this non-linear state feedback approach, as no mathe-


1.8 PROGRESS IN PARETO FRONT DURING GA RUNS

i r i i i i , 1 1

1.6

CM

~x l .4 +

< 1.2 x l i

o4 1

g $ o.8

~o.6

~ 0.4 0..

0.2

'~- Pareto Frontier at Generation 1

GA Population = 70

Mutation Rate = 0.09

Crossover Rate = 0.5

# of Pareto Soln. in 1st GA generation =2 - Marked with "o"

- #o f ParetoSoln. in 50th GA generation =14 - Marked with "+"

Pareto Frontier at Generat on 50 --~>--" '~

0 I _ _ I I i I I I i I

1 2 3 4 5 6 7 8 9 Performance Criterion 1 (J1 = u^2)

10

Fig. 6. Pareto frontier progression as GA generations proceed.

matical apparatus exists to solve this. Use of inequality constraints under this formulation is the direction of future research.

Acknowledgement~ Sourav Kundu acknowledges the Ministry of Education, Japan for financial support. The research support of Prol: Andrzej Osyczka of Tokyo Metropolitan University, and numerous ideas and suggestions by Pro1: David E. Goldberg of the department of General Engineering of University of Illinois at Urbana Champaign, are acknowledged.

REFERENCES

1. Passino K. M. Intelligent control lbr autonomous systems. IEEE Spectrum, pp. 55--62. IEEE (1995).

2. Holland J. H. Adaptation in Natural and Artificial Systems. Uni- versity ot" Michigan Press, Ann Arbor (1975).

3. Goldberg D. E. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA (1989).

4. Porter B. Genetic design of control systems. Trans. Soc. Instrument and Control Engineers, Vol. 34(5), pp. 393~,02 (1995).

5. Huang R. and Fogarty 32 C. Learning prototype control rules for combustion control with genetic algorithm. J Modeling, Mea- surement and Control, C 38(4), 55-64 (1992).

6. Goldberg D. E. Genetic algorithms and rule-learning in dynamic system control. In Proc. First Int. Conf. on Genetic Algorithms and Their Applications, Pittsburg, PA (Edited by Grefenstette J.), pp. 8-15. Lawrence Erlbaum Associates, Hillsdale, NJ (1985).

7. Friedland B. Control System Design." An Introduction to State- Space Methods. McGraw-Hill, Singapore (1987).

8. Pareto V. Manuale di Economia Politica, Societa Editrice Libraria, Milan (1906). Translated into English: A. S. Schwier. Manual of Political Economy. MacMillan ( 1971 ).

9. Gero J. S., Louis S. J. and Kundu S. Evolutionary learning of novel grammars lbr design improvement. Artificial Intelligence in Engineering Design, Analysis and Manufacturing ( AIEDAM), Vol. 8(2), pp. 83 94 (1994).

10. Schaffer J. D. Some experiments in machine learning using vector evaluated genetic algorithms. Unpublished doctoral dissertation, Vanderbilt University, Nashville (1984).

11. Schaffer J. D. Multiple objective optimization with vector evaluated genetic algorithms. Proc. Int. Conj. on Genetic Algorithms and their Applications (Edited by Grefenstette J.), pp. 93-100 (1985).

12. Horn J., Nafpliotis N. and Goldberg D. E. Multiobjective optimization using the niched pareto genetic algorithm. IlliGAL Tech Report No. 93005, University of Illinois at Urbana Champaign (1993).

13. Ritzel B. J., Eheart W. and Ranjithan S. Using genetic algorithms to solve a multiple objective groundwater pollution containment problem. Water Resources Res. 30(5), 1589-1603 (1994).

14. Cieniawski S. An investigation of the ability of GAs to generate the tradeoff curve of a multi-objective groundwater monitoring problem. Unpublished M.S. thesis, University of Illinois at Urbana Champaign (1993).

15. Fonseca C. M. and Fleming E J. Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. Proc. Fifth Int. Conf. on Genetic Algorithms, San Mateo, pp. 416. 423. Morgan-Kaufmann (1993).

16. Belegundu A. D., Murthy D. V., Salagame R. R. and Constans E. W Multi-objective optimization of laminated ceramic composites using genetic algorithms. 5th AIAAINASAIUSAFIISSMO Svmp. on Multidisciplinary Analysis and Optimization, Panama City, pp. 1015 1022. AIAA Inc. (1994).

17. Osyczka A. and Kundu S. A new method to solve generalized multicriterion optimization problems using the simple genetic algorithm. Structural Optimization. 10(2), 94-99 (1995).

18. Kundu S. A multicriteria genetic algorithm to solve optimization problems in structural engineering design. Int. Conf. on Informa- tion Technology in Civil and Structural Engineering Design--Taking Stock and Future Directions, Glasgow, Scotland, 1996. In press.

19. Kundu S., Kawata S. and Watanabe A. A multicriteria approach to control system design with genetic algorithm. Proe. IFAC '96-- 13th Worm Congress, 1996. International Federation of Automatic Control; Elsevier, Oxford. To be published.

20. Dorf R. C. and Bishop R. H. Modern Control Systems. Addison- Wesley, Reading, MA (1995).

genetic algorithms for optimal feedback control design

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