Solving Linear Quadratic Optimal Control Problems by Chebyshev-Based State Parameterization

A Chebyshev-based state representation method is developed for solving optimal control problems involving unconstrained linear time-invariant dynamic systems with quadratic performance indices. In this method, each state variable is represented by the superposition of a finite-term shifted Chebyshev series and a third order polynomial. In contrast to solving a two-point boundary-value problem, here the necessary condition of optimality is a system of linear algebraic equations which can be solved by a method such as Gaussian elimination. The results of simulation studies demonstrate that the proposed method offers computational advantages relative to a previous Chebyshev method and to a standard state transition method.