Robust Optimal Control in Not-Completely Controllable Linear Systems

Robust control problems in linear systems are considered on the base of analysis of related linear quadratic differential game. New explicit formulae for the "best" robust optimal control input and the "worst" exogenous disturbance derived with the use of pseudo-inverse matrices ) (t D+ were originally suggested in this author's previously published paper "Reduction of Dimensionality in Choosing Robust Optimal Control in Linear Systems", WSEAS Transactions on Mathematics, Issue 4, Volume 2, October 2003, pp. 318 323. The meaning of the use of the pseudo-inverse matrices is that some uncontrollable or almost uncontrollable components are automatically eliminated in choosing optimal control. In this paper the proof of the main theorem as well as meaningful and detailed analysis are presented. We shall demonstrate that the extremal that corresponds to the saddle point of the game may be extended beyond conjugates points (under assumption that remains non-negative definite). We reveal that computational difficulties, which may arise in obtaining the solution of the game and the optimal control law, are related to the fact that the trajectories (where the state ) of the system may be exactly or approximately localized in the subspace of dimensionality less than on some time interval. We shall demonstrate how to overcome these difficulties by using "inverse" matrix differential equations of Riccati type and pseudo-inverse matrices. ) (t D ) (t x ) ,..., ( 1 n x x x = n