Abstract
The controllability issues for linear discrete‐time systems with delay in state are addressed. By introducing a new concept, the minimum controllability realization index (MinCRI), the characteristic of controllability is revealed. It is proved that the MinCRI of a system with state delay exists and is finite. Based on this result, a necessary and sufficient condition for the controllability of discrete‐time linear systems with state delay is established.
Highlights
The concept of controllability of dynamical systems was first proposed by Kalman in 1960s 1
This paper discussed the controllability of linear discrete-time systems with delay in state
After introducing a new concept called MinCRI to describe the controllability feature of delay systems, we proved the existence and finiteness of MinCRI
Summary
The concept of controllability of dynamical systems was first proposed by Kalman in 1960s 1. Studying the time delay phenomena in control systems has become. X k 1 Ax k Dx k − h Bu k , 1.1 where x k ∈ Rn is the state, u k ∈ Rp is the input, A, D ∈ Rn×n, B ∈ Rn×p are constant matrices, and positive integer h is the length of the steps of time delay. System 1.1 is said to be completely controllable, if for any initial state x −h , x −h 1 , . X 0 and any final state xf , there exist a positive integer k and input u 0 , . In the case of planar systems, the exact value of the controllability realization index is obtained, and we will prove it is independent of the choices of A and D.
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