Kyungpook Mathematical Journal 2018; 58(2): 333-346  
A New Technique for Solving Optimal Control Problems of the Time-delayed Systems
Fateme Ghomanjani
Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran, e-mail: fatemeghomanjani@gmail.com
Received: July 12, 2017; Accepted: March 9, 2018; Published online: June 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

An approximation scheme utilizing Bezier curves is considered for solving time-delayed optimal control problems with terminal inequality constraints. First, the problem is transformed, using a Páde approximation, to one without a time-delayed argument. Terminal inequality constraints, if they exist, are converted to equality constraints. A computational method based on Bezier curves in the time domain is then proposed for solving the obtained non-delay optimal control problem. Numerical examples are introduced to verify the efficiency and accuracy of the proposed technique. The findings demonstrate that the proposed method is accurate and easy to implement.

Keywords: Bezier curve, Páde approximation, optimal control problems, time-delay
1. Introduction

The control of systems with time delay has been of considerable concern. Delays occur frequently in biological, chemical, electronic and transportation systems. Wu, et al. [14] built up a computational method for solving an optimal control problem which is represented by a switched dynamical system with time delay. Kharatishidi [7] has approached this problem by extending Pontryagin’s maximum principle to time delay systems. The actual solution involves a two-point boundary-value problem in which advances and delays are exhibited. In addition, this solution does not yield a feedback controller. Time-optimal control of delay systems has been considered by Oguztoreli [11] who obtained several results concerning bang-bang controls which parallel those of LaSalle [9] for non delay systems. For a time-invariant system with an infinite upper limit in the performance measure, Krasovskii [8] has developed the forms of the controller and the performance measure. Ross [12] has acquired a set of differential equations for the unknowns in the forms of Krasovskii. However, Ross’s results are not applicable to time-varying systems with a finite limit in the performance measure. In [1], the authors presented an optimal regular for a linear system with multiple state and input delays and a quadratic criterion. The optimal regulator equations were obtained reducing the original problem to the linear-quadratic regulator design for a system without delays (see [3] and [4]). B-splines (where Bezier form is a special case of B-splines), due to numerical stability and arbitrary order of accuracy, have become popular tools for solving differential equations. The use of Bezier curves for solving time-delayed optimal control systems with Páde approximation is a novel idea. The stated technique reduces the CPU time and the computer memory comparing with existing methods such as methods in [2, 10] and at the same time keeps the solution accuracy. Although the stated technique is very easy to utilize and straightforward, the obtained results are satisfactory (see numerical results). In this paper, one may utilize the Bezier polynomials. There are many papers and books deal with the Bezier curves or surface techniques [6]. The organization of this study is arranged as follows: In Section 2, problem transformation is presented. Section 3 is referred to the Bezier curve technique. Some Numerical examples are provided in Section 4. Section 5 is devoted the conclusion.

2. Problem Transformation

Consider the time-delayed optimal control problem

minu,πJ=η(xtf,π)+0tfL(x(t),x(t-σ),u(t),π,t)   dtx˙(t)=f(x(t),x(t-σ),u(t),π,t),x(t)=ξ(t),         -σt0,η(xf,π)0

where x ∈ ℛn, u ∈ ℛm and π ∈ ℛp are the state, control and unknown parameter vectors respectively, ηRq represents the terminal inequality constraints, tf is end time, and σ is the delay time associated with the state vector x. For the sake of simplicity, our discussion will be confined to the case of a single time delay σ. However, all the results can be extended in a straightforward manner to the case of multiple time delays. The time-delayed optimal control problem (2.1) can be transformed to one without a time-delayed argument utilizing the following approximation scheme. Let X(s) be a two-sided Laplace transform of x(t) (see [10]):

x(t)X(s)-+e-stx(t)dt

Presently, X(s) is defined over the strip of convergence α < Re(s) < β, where Re(s) denotes the real part of s, and ‘⇔ denotes a two-sided Laplace transformation. The two-sided Laplace transform is utilized because x(t) = ξ(t) ≠ 0 for t ≤ 0. Given that x(t) ⇔ X(s), the two-sided Laplace transforms of x(tσ) and (t) are presented by

x(t-σ)e-σsX(S),α<Re(s)<β,x˙(t)sX(s),α1<Re(s)<β1,

where the strip of convergence after the differentiation is assumed to exist and might or might not be the same as that for X(s). To omit variables with a time-delayed argument in (2.1), let us define y(t) ≜ x(tσ) and y(t) ⇔ Y (s). The two-sided Laplace transforms Y (s) and X(s) are referred by (see (2.6))

Y(s)=e-σsX(s).

Utilizing a first-order Páde approximation, one may have

Y(s)2σ-s2σ+2X(s),(2σ+s)Y(s)(2σ-s)X(s).

If an inverse Laplace transformation was performed on the last equation, one may have (see (2.2) and (2.7))

y˙(t)2σ[x(t)-y(t)]-x˙(t)2σ[x(t)-y(t)]-f(x(t),y(t),u(t),π,t)

The time-delayed optimal control problem is approximately changed to one of minimizing J(x(t), y(t), u(t), π, t) subject to

x˙(t)=f(x(t),y(t),u(t),π,t)y˙(t)2σ[x(t)-y(t)]-f(x(t),y(t),u(t),π,t)

with the following conditions

x(0)=ξ(0),y(0)=ξ(-σ)

and the terminal condition η(xf, π) ≥ 0. An alternative approximation scheme (but equivalent) is to state Y (s) in (2.9) as follows:

Y(s)(-1+4σ2σ+s)X(s)-X(s)+Q(s)

where Q(s) is defined as

Q(s)4σ2σ+sX(s)

The inverse Laplace transforms of (2.16) and (2.17) are

x(t-σ)=y(t)=-x(t)+q(t)q˙(t)=-2σq(t)+4σx(t)

Utilizing this technique, the original optimization problem is transformed to one of minimizing J(x(t), q(t), u(t), π, t) subject to

x˙(t)=f(x(t),q(t),u(t),π,t)q˙(t)=-2σq(t)+4σx(t)

with the following conditions

x(0)=ξ(0),q(0)=ξ(0)+ξ(-σ)

and the terminal condition η(xf, π) ≥ 0.

To enhance the accuracy of the above-described approximation schemes, the time delay σ can be subdivided into smaller sections. For example, one may define

y(t)x(t-σ2)z(t)y(t-σ2)=x(t-σ)

Again, utilizing a first-order Páde approximation, one may obtain

y˙(t)4σ[x(t)-y(t)]-f(x(t),z(t),u(t),π,t)z˙(t)4σ[2y(t)-z(t)-x(t)]+f(x(t),z(t),u(t),π,t)

The time-delayed problem is changed to one of minimizing J(x(t), z(t), u(t), π, t) subject to

x˙(t)f(x(t),q(t),u(t),π,t)y˙(t)4σ[x(t)-y(t)]-f(x(t),z(t),u(t),π,t)z˙(t)4σ[2y(t)-z(t)-x(t)]+f(x(t),z(t),u(t),π,t)

with the initial conditions

x(0)=ξ(0),y(0)=ξ(-σ2),z(0)=ξ(-σ)

and the final condition (2.4). Repeated applications of this method will result progressively improved achievement. However, this is at the expense of an increase in the system order, resulting in an increase in the computation time.

3. Bezier Curve Method

Our strategy is utilizing Bezier curves to approximate the solutions xi(t) and u(t) where xi(t) are given below. Define the Bezier polynomials of degree n for t ∈ [t0, tf ] as follows:

xi(t)r=0nariBr,n(t-t0h),u(t)r=0nbrBr,n(t-t0h)

where h = tft0, and

Br,n(t-t0h)=(nr)1hn(tf-t)n-r(t-t0)r,

is the Bernstein polynomial of degree n for t ∈ [t0, tf ], ari and br are the control points. By substituting xi(t) and u(t) in (2.27)–(2.30), one may define the problem which can be solved by Maple 16.

Ghomanjani et al. [6] proved the convergence of this method where n → ∞ when the optimal control system solved by Bezier curve method (for more explanation, see [5]). For the convergence of the time-delayed optimal control problem, one may use Páde approximation, then the problem is converted to an optimal control problem (OCP), where the convergence of OCP is in [6].

Termination criterions are ||xi(t) − xi,exact(t)|| < ε, and ||u(t) − uexact(t)|| < ε

4. Numerical Application

In this section, some numerical examples are presented for illustrating the proposed technique.

Example 4.1

Consider the following time-delay system:

min J=5(x1(2))2+1202u2(t)dtx˙1(t)=x2(t),         0t2x˙2(t)=-x1(t)-x2(t-1)+u(t),0t2x1(t)=10,x0(t)=0,-1t0,

For this example the exact solution is given by [10] as follows:

u(t)={δsin(2-t)+(δ2)(1-t)sin(t-1)0t1δsin(2-t)1t2

where δ = 2.5599. Utilizing the proposed method, one may have

x3(t)x2(t-12)x4(t)x3(t-12)=x2(t-1)

The delayed differential equations (4.1) then become

x˙1(t)=x2(t)x˙2(t)=-x1(t)-x4(t)+u(t)x˙3(t)=x1(t)+4x2(t)-4x3(t)+x4(t)-u(t)x˙4(t)=-x1(t)-4x2(t)+8x3(t)-5x4(t)+u(t)x1(0)=10,x2(0)=0,x3(0)=0,x4(0)=0,

by using this method, we have

x1(t)=(1-(12)t)2a10+t(1-(12)t)a11,x2(t)=(1-(12)t)2a20+t(1-(12)t)a21,x3(t)=(1-(12)t)2a30+t(1-(12)t)a31,x4(t)=(1-(12)t)2a40+t(1-(12)t)a41,u(t)=(1-(12)t)2b0+t(1-(12)t)b1.

The optimal value of J in proposed method is 3.04246518971317. This value compares well with that given in [10] (J = 3.256613). In proposed method, one may obtain

x1(t)=10(1-(12)t)2+4.84426425991426t(1-(12)t),x2(t)=-7.57786787004287t(1-(12)t)2-1.21106606497857t2,x3(t)=8.74933030407381t(1-(12)t)-1.39164977425000t2,x4(t)=0.674964953083765t(1-(12)t)-0.683400902532152t2,u(t)=(1-(12)t)2(2.5599sin(2)-1.279950000sin(1))+3.68282830005636t(1-(12)t),

the graphs of approximated and exact solution u(t) and xi(t) for i = 1, 2, 3,4 are respectively plotted in Figs. 1, 2, 3, 4 and 5.

Example 4.2

Consider the following time-delay system (see [10]):

min J=12((x1(2))2+(x1(2))2)+1202u2(t)dtx˙1(t)=x2(t),         0t2x˙2(t)=-x2(t)-x1(t-1)+u(t),0t2x1(t)=10,x2(t)=0,-1t0,

where the exact solution is

u(t)={(μ+δ)et-2+(2μ-3δ-(μ-δ)t)et-1+δ(t+2)-μ,0t1(μ+δ)et-2+δ,1t2μ0.5226194,δ-0.5259256.

Utilizing this method, one may achieve

x3(t)x1(t-12),x4(t)x3(t-12)=x1(t-1),

presently, one may have

x˙1(t)=x2(t),x˙2(t)=-x2(t)-x4(t)+u(t),x˙3(t)=4x1(t)+x2(t)-3x3(t)-u(t),x˙4(t)=8x3(t)-5x4(t)-x2(t)-4x1(t)+u(t),x1(0)=1,x2(0)=0,x3(0)=1,x4(0)=1,

The optimal value of J in proposed method is 0.0671730978076868. This value compares well with that given in [10] (J = 0.1967). The graphs of approximated and exact solution u(t) are plotted in Fig. 6.

Example 4.3

Consider the following time-delay system (see [10]):

min J=12x1(2))2+1202x12(t)+u2(t)dtx˙1(t)=x1(t)sin(x1)+x1(t-1)+u(t),         0t2x1(t)=10,         -1t0,

now, one may have

x2(t)x1(t-12),x3(t)x2(t-12)=x1(t-1),

utilizing this method, one may achieve

x˙1(t)=x1(t)sin(x1)+x3(t)+u(t),x˙2(t)=4x1(t)-4x2(t)-x3(t)-x1sin(x1)-u(t),x˙3(t)=8x2(t)-3x3(t)-4x1(t)+x1sin(x1)+u(t),x1(0)=10,x2(0)=10,x3(0)=10,

using Bezier curve, one may have

x1,bezier=(1-12t)2p1[0]+t(1-12t)p1[1]+14t2p1[2],x2,bezier=(1-12t)2p2[0]+t(1-12t)p2[1]+14t2p2[2],x3,bezier=(1-12t)2p3[0]+t(1-12t)p3[1]+14t2p3[2],xbezier=(1-12t)2q[0]+t(1-12t)q[1]+14t2q[2],

then one may substitute (4.6) in (4.5), and solve this system by using Maple 16 software. The optimal value of J in proposed method is 161.712666656000, when the values of J in [10], Banks [2], and Wong et al. [13] are respectively J = 161.88, J = 162.019, and J = 162.104. The graphs of approximated solution u(t) and x1(t) are plotted in Figs. 7 and 8.

Example 4.4

Consider the following time-delay system (see [10]):

min J=12105×x1(2))2+1202u2(t)dtx˙1(t)=x1(t-1)+u(t),         0t2x1(t)=1,         -1t0,

now, one may have

x2(t)x1(t-12),x3(t)x2(t-12)=x1(t-1),

hence

x˙1(t)=x3(t)+u(t),x˙2(t)=4x1(t)-4x2(t)-x3(t)-u(t),x˙3(t)=8x2(t)-3x3(t)-4x1(t)+u(t),x1(0)=1,x2(0)=1,x3(0)=1,

using Bezier curve, one may have

x1,bezier=(1-12t)2p1[0]+32t(1-12t)2p1[1]+34t2(1-12t)p1[2]+18t3p1[3],x2,bezier=(1-12t)3p2[0]+32t(1-12t)2p2[1]+34t2(1-12t)p2[2]+18t3p2[3],x3,bezier=(1-12t)3p3[0]+32t(1-12t)2p3[1]+34t2(1-12t)p3[2]+18t3p3[3],xbezier=(1-12t)3q[0]+32t(1-12t)2q[1]+34t2(1-12t)q[2]+18t3q[3],

then one may substitute (4.8) in (4.7), and solve this system by using Maple 16 software. The optimal value of J in proposed method is 1.43068782747222, when the value of J in [10] is J = 1.849730. The graphs of approximated solution u(t) and x1(t) are plotted in Figs. 9 and 10.

5. Conclusions

Time-delayed optimal control problems with terminal inequality constraints can be approximately solved by a combined parameter. To this end, a Páde approximation is utilized to acquire a corresponding problem without a time-delayed argument. The results obtained by the Bezier curve are in good agreement with the given exact solutions. The study shows that the method is effective technique to solve time-delayed optimal control problems, and the technique is easy to implement and computationally very attractive without sacrificing the accuracy of the solution.

Acknowledgements

The author would like to thank the anonymous reviewer of this paper for their careful reading, constructive comments and nice suggestions which has improved the paper very much.

Figures
Fig. 1. The graphs of approximated and exact solution u(t) for Example 4.1
Fig. 2. The graphs of approximated solution x1(t) for Example 4.1
Fig. 3. The graphs of approximated solution x2(t) for Example 4.1
Fig. 4. The graphs of approximated solution x3(t) for Example 4.1
Fig. 5. The graphs of approximated solution x4(t) for Example 4.1
Fig. 6. The graphs of approximated and exact solution u(t) for Example 4.2
Fig. 7. The graph of approximated solution u(t) for Example 4.3
Fig. 8. The graph of approximated solution x1(t) for Example 4.3
Fig. 9. The graph of approximated solution u(t) for Example 4.4
Fig. 10. The graph of approximated solution x1(t) for Example 4.4
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