Stabilization of Linear Systems

Abstract

This paper considers a finite-dimensional linear time-varying system and is concerned with the question: What is the relation between controllability properties of the system and various degrees of stability of the closed loop system resulting from linear feedback of the state variable? The main results are as follows: For any initial time to, and any continuous and monotonically nondecreasing function $\delta ( \cdot ,t_0 )$ such that $\delta (t_0 ,t_0 ) = 0$, the transition matrix $\hat \Phi ( \cdot , \cdot )$ of the closed loop system can be made such that $\| {\hat \Phi (t,t_0 )} \| \leqq a(t_0 )\exp [ - \delta (t,t_0 )]$ for all $t \geqq t_0 $, if and only if the system is completely controllable. Furthermore, in case of a bounded system, for any $m \geqq 0$, a bounded feedback matrix can be found such that $\| {\hat \Phi (t_2 ,t_1 )} \| \leqq a\exp [ - m(t_2 - t_1 )]$ for all $t_1 $ and $t_2 \geqq t_1 $, if and only if the system is uniformly completely controllable. Thus they can be regarded as extensions of the well-known result of Wonham [1] for a time-invariant system (i.e., the equivalence between complete controllability and the possibility of closed loop pole assignment), and also the results of Kalman [2], Johnson [3] and Anderson and Moore [4] for a time-varying system in which sufficient conditions for stabilization of the closed loop system are given.