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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.

AbstractIn this paper, an algorithm for solving optimal control of linear time-varying systems with quadratic performance indices is presented. By using important matrices which are derived from Chebyshev wavelets properties, the original problem is converted to a quadratic programming one. This parameter optimization method is applied on both constrained and unconstrained control systems having linear state equations of integer and fractional orders. The computing time saved by this approach is much better than with other methods in which there is no need to calculate the optimal costs of systems by substituting the approximations of the state and control vectors and their values are default outputs of the quadprog solvers.

This research presents the integration, product, delay and inverse time operational matrices of Legendre wavelets with an arbitrary scaling parameter and illustrates how to design this parameter in order to improve their accuracy and capability in handling optimal control and analysis of time-delay systems. Using the presented Legendre wavelets, the piecewise delay operational matrix is derived to develop the applicability of Legendre wavelets in systems with piecewise constant time-delays or time-varying delays. With the aid of these matrices, the new Legendre wavelets method is applied on linear time-delay systems. The reliability and efficiency of the method are demonstrated by some numerical experiments.

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann-Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods.Keywords: fractional integration operational matrix of Chebyshev wavelets; Chebyshev wavelets method; fractional time-delay optimal control; multifractional delay differential equation; quadratically constrained linear-quadratic optimal control; piecewise delay