Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials

The use of Chebyshev polynomials in solving finite horizon optimal control problems associated with general linear time-varying systems with constant delay is well known in the literature. The technique is modified in the present paper for the finite horizon control of dynamical systems with time periodic coefficients and constant delay. The governing differential equations of motion are converted into an algebraic recursive relationship in terms of the Chebyshev coefficients of the system matrices, delayed and present state vectors, and the input vector. Three different approaches are considered. The first approach computes the Chebyshev coefficients of the control vector by minimizing a quadratic cost function over a finite horizon or a finite sequence of time intervals. Then two convergence conditions are presented to improve the performance of the optimized trajectories in terms of the oscillation of controlled states within intervals. The second approach computes the Chebyshev coefficients of the control vector by maximizing a quadratic decay rate of the L₂ norm of Chebyshev coefficients of the state subject to linear matching and quadratic convergence conditions. The control vector in each interval is computed by formulating a non-linear optimization programme. The third approach computes the Chebyshev coefficients of the control vector by maximizing a linear decay rate of the L_∞ norm of Chebyshev coefficients of the state subject to linear matching and linear convergence conditions. The proposed techniques are illustrated by designing regulation controllers for a delayed Mathieu equation over a finite control horizon.