Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets

Abstract

This paper presents a novel method for finding the optimal control, state and cost of linear time-delay systems with quadratic performance indices. The basic idea here is to convert a time-delay optimal control problem into a quadratic programming one which can be easily solved using MATLAB . To implement this idea we choose a state and control parameterization method by using Chebyshev wavelets. The inverse time operational matrix of Chebyshev wavelets is introduced and applied for parameterizing state and control terms containing inverse time. The method is also applicable to linear quadratic time-delay systems with combined constraints. Illustrative examples demonstrate the validity and applicability of the approach which new expansions for initial vector functions of state and control variables are defined.Keywords time-delay system, optimal control, inverse time operational matrix of Chebyshev wavelets, multiple delays, constrained lag system.

A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems

“…A cw , E µcw , G cw , B cw , and F νcw are constant matrices which can be determined by using (5). For detailed information, see [31]. In like manner, the initial functions can be expressed as where n d µ = h µ ξ k−1 and n d ν = h ν ξ k−1 .…”

“…One can find from the results of the numerical examples that the proposed Chebyshev wavelet methods provide accurate results. Since the method for fractional linear-quadratic delay optimal control problems has the same idea to our previous method, it has some advantages (mentioned in [28] and [31]) over the existing method.…”

Legendre Wavelets with Scaling in Time-delay Systems

“…Also we have shown the advantages of Chebyshev wavelets with scaling over orthogonal functions [5] in problems arising in such systems. By doing a similar experiment as in [6,7], we can observe similar drawbacks in the use of the conventional Legendre wavelets (CLWs). The main issue of CLW method is that to model the delayed terms of many time-delay systems, we have to approximate the values of delays, for example by using greatest integer values (see [8]), which this issue causes errors in the obtained results.…”

“…where the inverse or reverse time operational matrix of ASLWs is introduced by Υ ξ . According to (6) we have…”

“…1. We can apply the method described in [6], using large k, which provides an acceptable result. This application, however, has many disadvantages.…”