Stabilization of Linear Systems

Authors

Masao Ikeda, Hajime Maeda, Shinzo Kodama

Year

1972

Journal Paper

Algorithm

Linear Feedback Stabilization

Source

SIAM Journal on Control

Identifiers

DOI 10.1137/0310052

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 t0, and any continuous and monotonically nondecreasing function delta(., t0) such that delta(t0, t0) = 0, the transition matrix Phi-hat(., .) of the closed loop system can be made such that ||Phi-hat(t, t0)|| <= a(t0) exp[-delta(t, t0)] for all t >= t0, if and only if the system is completely controllable. Furthermore, in case of a bounded system, for any m >= 0, a bounded feedback matrix can be found such that ||Phi-hat(t2, t1)|| <= a exp[-m(t2 - t1)] for all t1 and t2 >= t1, if and only if the system is uniformly completely controllable. Thus they can be regarded as extensions of the well-known result of Wonham 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, Johnson and Anderson and Moore for a time-varying system in which sufficient conditions for stabilization of the closed loop system are given.

Summary¶

Studies stabilization of linear time-invariant systems via state feedback, providing conditions under which a system can be stabilized and characterizing the achievable closed-loop spectra. Connects controllability theory to stabilizability and eigenvalue assignment.