On time optimal control of linear discrete-time systems by geometric approach

Abstract

This paper arose from the need for a better understanding of the minimum time state regulation problem of finite-dimensional linear time-invariant discrete systems with an incomplete state observation. For this we used a geometric approach and obtained the following interesting results. First, the necessary and sufficient condition of the existence of controllers which regulate the system from any initial state to the origin in finite steps is that the system is controllable and reconstructable. Secondly, the general minimum regulation time is obtained, which is the maximum value of the minimum regulation time. Thirdly, two typical controllers which drive the system from any initial state to the origin in the general minimum regulation time are designed. One is a time-variant system taking the system's input and output of each time as input. The other is a time-invariant system. Fourthly, it is shown that the controller does not necessarily exist which transfers the system to the origin in the minimum regulation time from any initial state.