Optimal Control of Parametrically Excited Linear Delay Differential Systems via Chebyshev Polynomials

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 manuscript 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. 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 oscillations of controlled states. The second approach computes the Chebyshev coefficients of the control vector by maximizing a quadratic decay rate of L₂ norm of Chebyshev coefficients of the state subject to a linear matching and quadratic convergence condition. The control vector in each interval is computed by formulating a nonlinear optimization program. The third approach computes the Chebyshev coefficients of the control vector by maximizing a linear decay rate of L_∞ norm of Chebyshev coefficients of the state subject to a linear matching and linear convergence condition. The proposed techniques are illustrated by designing regulation controllers for a delayed Mathieu equation over a finite control horizon.