Optimal Control Problem, Quasi-AssignmentProblem and Genetic Algorithm

Omid S. Fard and Akbar H. Borzabadi

Abstract— In this paper we apply one of approaches in category ofheuristic methods as Genetic Algorithms for obtaining approximatesolution of optimal control problems. The first we convert optimalcontrol problem to a quasi Assignment Problem by defining someusual characters as defined in Genetic algorithm applications. Thenwe obtain approximate optimal control function as an piecewiseconstant function. Finally the numerical examples are given.

Keywords— Optimal control, Integer programming, Genetic algo-rithm, Discrete approximation, Linear programming.

I. INTRODUCTION

In early decade Optimal Control Problem (OCP) as one ofthe most application issues has been taken into consideration.The analytical solution for OCP’s are not always available.Thus, to find an approximate solution is the most logical wayto solve the problems. Various approaches as, discretization,shooting method [3], [4], using concepts in measure theory[5], using Chebyshev polynomials [2], etc., have beenproposed to obtain approximate solutions of OCP’s. Someheuristic algorithm and their extensions are applied in topicof control problems. our aim is to apply Genetic Algorithm(GA) to construct approximate optimal control function forOCP’s. GA’s are stochastic search technique inspired by theprinciples of natural selection and natural genetics which haverevealed a number of characteristics particulary useful forapplication in optimization, engineering, computer science,among other fields. In this work, in order to apply GA toget an approximate solution for an OCP, we first convert TheOCP to a quasi Assignment Problem (QASP) by a suitablelinearizing. Then, we apply GA for the quasi assignmentproblem and we get an approximate solution as an piecewiseconstant function. Hence, the approximate solution obtainedis a precise solution for the OCP.

II. STATEMENT OF THE OPTIMAL CONTROLPROBLEM

Consider a classical OCP as follows:

minimize I(x(t), u(t)) =∫ tf

0

f◦(t,x(t), u(t)) (1)

subject to

x(t) = g(t,x(t), u(t)) (2)

O.S. Fard is with the Department of mathematics, Damghan University ofBasic Sciences, Damghan, Iran,(e-mail: osfard@dubs.ac.ir).

A.H. Borzabadi is with the Department of mathematics, Damghan Univer-sity of Basic Sciences, Damghan, Iran,(e-mail: borzabadi@dubs.ac.ir).

x(0) = x0, x(tf ) = xf . (3)

where tf is known and |u| ≤ K. Our main aim is trajectoryplanning i.e. it is to find a control function u(·) such that thecorresponding state x(·) satisfying (2)-(3) and minimize (1).In order to this work, first, we discretize the interval [0, tf ]to N subinterval [ti−1, ti], i = 1, 2, . . . , N . Since we tend todetect an approximate optimal control for problem (1)-(2),we focus on finding an optimal piecewise control functionu(·) for the problem. Corresponding each interval [ti−1, ti],i = 1, 2, . . . , N , we partition the interval [−k, k] to m equalsubinterval [uj−1, uj ], j = 1, 2, . . . , m, where u0 = −Kand um = K. Of course we need to find m + 1 constantsu0, u1, . . . , um. Thus we consider the space of time andcontrol piecewise constant segments as it is shown in Figure 1.

Figure. 1 A discrete form of control space.

One of approach for finding the best selection among of allconstants in each interval of [ti−1, ti], i = 1, 2, · · · , N , andfinally to find the best approximate control is enumerate of allcases. But we will deal with an extreme hard computationalcomplexity. Because we must check Nm piecewise constantfunctions.Our aim is to use of GA approach for detecting the bestapproximate piecewise constant control function.If u(t) =

∑Nk=1 ukξ[tk−1,tk](t) be a piecewise constant

function then by a numerical method as Euler method orRung-Kutta, we can find trajectory corresponding u(t) from(2) with initial condition x(0) = x0. Thus, if (x, u) be a pair

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of the trajectory and the control which satisfy in (2) withinitial condition x(0) = x0 and for given a small numberε > 0, ‖x(tf ) − xf‖ ≤ ε, then we can claim that, wehave been found a good approximate pair for minimizingfunctional I in (1).

III. CONVERTING OCP TO QASP

To convert the OCP to QASP, we use a similar frameworkof solving QAP (Quadratic Assignment Problem) by ACO(Ant colony Optimization). In fact, we decide to assign aconstant uk ∈ {u0, u1, . . . , um} for every interval [ti−1, ti],i = 1, 2, . . . , N . These constants are selected by GA toolboxin Matlab. The process will be continued until the objectivefunction gets its optimal value. In other word, after discretionof the OCP, the problem is converted to a QASP with anextra objective function, i.e. we add the term, ‖x(tf ) − xf‖to the original objective function and then, we apply GA forthis new criteria function.Therefore, by applying the method above, the problem (1)-(2)with conditions (3) is converted to:

minimize I(x(t), u(t)) =N∑

i=0

f◦(ti, x(ti), u(ti))

+M‖x(tf ) − xf‖ (4)

subject to

x(ti+1) = x(ti) + g(ti, x(ti), u(ti)), i = 0, 1, . . . N, (5)

x(0) = x0. (6)

where, M is a very large positive number (like as BIG-MMethod). To obtain a numerical solution for the problem(4)-(5) with initial condition(6) via GA, we focus just on thespace time and control. Because, we use of equation (5) indefinition of the fitness function for GA. Now, we are readyto state the numerical algorithm to get the optimal control.

IV. USING A GENETIC ALGORITHM

In this section, we present an algorithm based on GA, toobtain a solution for problem (4)-(5) with initial condition(6). Because of definition of the fitness function, the solutionobtained of GA, is a reasonable approximate solution fororiginal problem.Corresponding to section 3, after disceretizing the problem,including the objective function and the differential equation,to solve the problem via GA, we must add the term‖x(tf ) − xf‖ to the new objective function along with acost coefficient, M. This process causes all of the conditionsof the original problem be satisfied and also it guarantiesthe convergence of the approximated solution to the exactsolution.In this work, we use of Genetic Algorithm Toolbox of Matlab(Ver,7.04) to solve examples. The results obtained of usingthe toolbox are so satisfactory,too.

V. NUMERICAL EXAMPLESIn this section we propose our method to obtain approximate

solutions of some OCP. Before proposing of examples wedefine an error function as e(tf ) = x(tf ) − xf on [0, tf ],where x(tf ) and xf are the exact and the approximatesolution obtained from each repetition of GA for theexample,respectively.In all examples, we set tf = 1, M = 106, K = 1, and wedivide the closed interval [0, tf ](= [0, 1]) into 16 partitions,i.e. the step size �t = 0.0625. Also, we use Euler method tosolve the condition corresponding to the differential equationof the problem.

Example 1: Consider the following OCP:

minimize I =∫ 1

0

u2(t)dt

subject to

x(t) = x2(t) + u(t)

x(0) = 0, x(1) = 0.5.

After solving with the proposed method, we obtain the fol-lowing results:The final value x(1) = 0.500054, the optimal value I∗ =0.4447 and the error function e(1) = 5.43 × 10−5. Thetrajectory and control functions are shown in figure 2 and 3,respectively.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (t)

Co

ntr

ol U

(t)

Figure. 2 Control function for Example 1.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (t)

Tra

ject

ory

x(t

)

Figure. 2 Trajectory function for Example 1.

PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND TECHNOLOGY VOLUME 21 JANUARY 2007 ISSN 1307-6884

PWASET VOLUME 21 JANUARY 2007 ISSN 1307-6884 423 © 2007 WASET.ORG

Example 2: Consider the following OCP:

minimize I =∫ 1

0

u2(t)dt

subject to

x(t) =12x2(t)sin(x(t)) + u(t)

x(0) = 0, x(1) = 0.5.

After solving with the proposed method, we obtain the fol-lowing results:The final value x(1) = 0.500040, the optimal value I∗ =0.3526 and the error function e(1) = 4.095 × 10−5. Thetrajectory and control functions are shown in figure 4 and 5,respectively.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (t)

Co

ntr

ol

U(t

)

Figure. 4 Control function for Example 2.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (t)

Tra

ject

ory

x(t

)

Figure. 5 Trajectory function for Example 2.

REFERENCES

[1] L. C. Young, ”Calculus of Variation and Optimal Control Theory”,Philadelphia: W.B. Saunders, 1969.

[2] J. Vlassenbroeck, ”A chebyshev polynomial method for optimal controlwith bounded state constraints”, J. Automatica, Vol. 24, pp. 499-506,1988.

[3] H. Maurer and W. Gillessen, ”Application of multiple shooting to thenumerical solution of optimal control with bounded state variables”, J.Computing,Vol 15, pp. 105-126,1975.

[4] G. Fraser Andrews,”Shooting method for the numerical solution ofoptimal control problems with bounded state variable”, J. OptimizationTheory and Applications,Vol 89, pp. 351-372, 1996.

[5] J. E. Rubio, ”Control and Optimization, The Linear Treatment of Non-linear Problems”, Manchester University Press, England, 1986.

[6] K. L. Teo, , C. J. Goh, and K. H. Wong ,”A unified computationalapproach to optimal control problems”, Longman Scientific and Tech-nical,1991.

PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND TECHNOLOGY VOLUME 21 JANUARY 2007 ISSN 1307-6884

PWASET VOLUME 21 JANUARY 2007 ISSN 1307-6884 424 © 2007 WASET.ORG