An Algorithm for Determining the Weighting Matrices of the Quadratic Performance Function

Abstract

Linear optimal control theory has been recognized as a powerful tool for control system synthesis. “Optimal feedback control law” is easily obtained by solving matrix Riccati equation, but there is no way other than the trial and error method to determine proper weighting matrices in the quadratic performance function. Hence, the practical use of the theory has been restricted.
This paper proposes an algorithm of systematical determination of the weighting matrices. The algorithm concerns with the previously proposed optimal tracking systems (S_c^*)⁷⁾, and is derived by imposing the following requirements on S_c^*, using the asymptotic pole configuration and the conditional stability characteristics of S_c^* designed with very small weights on inputs^{8), 9)}.
(1) S_c^* should remain stable even if open loop gains of plants change from zero to infinity simultaneously.
(2) S_c^* should have the fastest step response under the condition of (1).
(3) S_c^* should be decoupled as much as-possible under the condition of (1) and (2).
Some examples demonstrate the efficiency of the algorithm. The algorithm can also be used for ordinaly linear optimal regulator and optimal observer syntheses with minor change.