E.M. Khazen

Linear-quadratic diﬀerential game, linear robust control, matrix Ric-cati diﬀerential equation, pseudo-inverse matrix, conjugate points

A linear-quadratic differential game related to robust control problems in linear time-varying systems is considered. We show that difficulties, which may arise in obtaining the solution of the game and the optimal control law, are related to the fact that the trajectories x(t) (where the state x = (x1, . . . , xn)) of the controlled system may be exactly or approximately localized in the subspace of dimensionality less than n, under all possible bounded piecewise continuous control impacts acting on the inputs, provided that the initial vector x(t0) is fixed, x(t0) = x0. We demonstrate how to overcome these difficulties by using the solution D(t) of "inversem atrix differential equation of Riccati type and the pseudo-inverse matrix D+(t). We establish new sufficient conditions for the existence of the saddle point. We show that the optimal control can be extended beyond conjugate points if D(t) ≥ 0. We also derive and prove new explicit formulae for robust optimal control law using the matrices D(t) and D+(t). With the use of the pseudo-inverse matrix D+(t) uncontrollable (or almost uncontrollable) components can be automatically eliminated in choosing robust optimal control. In this paper robust optimal control problems in linear time-varying systems are investigated in the finite-horizon case.

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