Abstract
The construction of an optimal state regulator for a linear system involves the solution of a matrix Riccati equation. If the system is time-varying, the closed form solution is impossible to obtain. This paper deals with an approximate construction of the optimal state regulator for a periodically varying linear system with quadratic performance index. In the present discussion, the periodic term is taken to be a perturbation to the time-invariant system equation, and a parameter ε is introduced to stand for the perturbation.
By making use of a power-series expansion in ε, when the control duration is sufficiently large, the solution of the Riccati equation is reduced to solving a nonlinear algebraic equation and obtaining steady-state solutions of a sequence of linear time-invariant differential equations. It is shown that, if the unperturbed system is completely controllable, the steady-state solutions for those linear differential equations exist and are periodic. If, in particular, the periodic coefficients in the system equation are given in finite Fourier series, the steady-state solutions are obtained merely by solving linear algebraic equations. It is further demonstrated that the l-th order approximation for the feedback law results in the (2l+1) th approximation to the optimal performance.
An example is attached to illustrate the present procedure, and the results of the approximation are compared with the optimal solution.
