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.
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.
Consider the time-delayed optimal control problem
where
Presently,
where the strip of convergence after the differentiation is assumed to exist and might or might not be the same as that for
Utilizing a first-order Páde approximation, one may have
If an inverse Laplace transformation was performed on the last equation, one may have (see (
The time-delayed optimal control problem is approximately changed to one of minimizing
with the following conditions
and the terminal condition
where
The inverse Laplace transforms of (
Utilizing this technique, the original optimization problem is transformed to one of minimizing
with the following conditions
and the terminal condition
To enhance the accuracy of the above-described approximation schemes, the time delay
Again, utilizing a first-order Páde approximation, one may obtain
The time-delayed problem is changed to one of minimizing
with the initial conditions
and the final condition (
Our strategy is utilizing Bezier curves to approximate the solutions
where
is the Bernstein polynomial of degree
Ghomanjani et al. [6] proved the convergence of this method where
Termination criterions are ||
In this section, some numerical examples are presented for illustrating the proposed technique.
Consider the following time-delay system:
For this example the exact solution is given by [10] as follows:
where
The delayed differential
by using this method, we have
The optimal value of
the graphs of approximated and exact solution
Consider the following time-delay system (see [10]):
where the exact solution is
Utilizing this method, one may achieve
presently, one may have
The optimal value of
Consider the following time-delay system (see [10]):
now, one may have
utilizing this method, one may achieve
using Bezier curve, one may have
then one may substitute (
Consider the following time-delay system (see [10]):
now, one may have
hence
using Bezier curve, one may have
then one may substitute (
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.
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.