Suboptimal control of linear time-varying multi-delay systems via shifted Legendre polynomials

A direct method based on using shifted Legendre polynomials is developed to obtain suboptimal control for linear time-varying systems with multiple state and control delays and quadratic performance index. In this method, both the control and state variables are first expanded into finite shifted Legendre series. The governing delay-differential equation is then converted to a set of linear algebraic equations through use of the operational matrices of integration and delay. The problem finally becomes the simple one of finding the unknown coefficients of the control variables alone, which minimizes the quadratic form of performance index.