Inverse LQ Regulator of Systems with Multiple Time-Varying Delays

In this paper, a linear system with multiple and mutually independent time-varying delays in the state is considered as a plant, and a method to construct a memoryless feedback law is proposed. The feedback gain is constant and is calculated via a solution of linear matrix inequalities containing the upper bounds of all derivative delays. It is shown that the resulting closed loop system is asymptotically stable and the feedback control minimizes some quadratic cost functional, so that it belongs to a class of linear quadratic regulators. It is a remarkable feature that the weighting matrix in the cost functional is time-varying. In spite of this feature, it is shown that the regulator has some robust stability against a class of static nonlinear or a class of dynamic linear perturbations in the input channel as well as the ordinary LQ regulator. The design procedure is demonstrated and the robust stability of the closed loop system is examined with a numerical example.

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