Linear time-varying system control based on the inversion transformation

This paper proposes a new state feedback control design for uniformly controllable linear time-varying systems. The new design is based on a nonlinear state transformation that converts the stabilization problem of the system state into a destabilization problem of the transformed state. Such a transformation results in a new control that has two advantages over previous controls. (1) In some of the previous control designs, forecast of future information of the time-varying system matrices over some time interval [t, t f A] is required to calculate the state feedback gain at any time instant I, while in the new control design no such forecast is needed. (2) In other previous control designs, consecutive time differentiation on the time-varying system matrices is required, while in the new control design time integration instead of differentiation is applied to the system matrices. The new control can therefore be applied to systems with time-varying parameters that are piecewise continuous in time. 0 I997 Elsevier Science Ltd. 1. Introduction The control design for linear time-varying systems has been a difficult challenge for engineers. The main difficulty arises from the fact that the eigenvalues of a time-varying system matrix cannot be linked with the system's stability property as in the time invariant case (Wu, 1974). Most state feedback control designs that are based on the eigenvalue assignment lack theoretical justification when applied to a time-varying system. Conventional approaches for the time-varying system control have relied on the robust control design, in which the time-varying system matrix is decomposed into a nominal constant matrix and a time-varying deviation. A robust time-invariant COntTOller (e. have been proposed, the resultant robust controllers can tolerate the time-varying deviation only up of a certain magnitude. The only exception occurs when the time-varying deviation satisfies a special structural condition called the matching condifion. In such a case, a stabilizing controller (Gutman 1979: Corless and Leitmann, 1981) exists for arbitrarily large but bounded time-varying deviations. However, in most cases. the time-varying deviation does not satisfy the matching condition, and stability can be assured only when the size of the time-varying deviation is small in a certain sense (Barmish and Leitmann. 1982). When information on the tim'e-varying system matrices is known exactly, there are exact control designs that can deal with arbitrarily large time-varying deviations. A particular case is when the system is periodically time-varying. For such a system, stabilizing controls have been developed …