Minimal Perturbations for Zero Controllability of Discrete-Time Linear Systems: Complexity Analysis

This article deals with computational complexity of various problems related to the zero controllability of a discrete-time linear time-invariant system, assuming that purely structural conditions at the level of the connections between the system states (i.e., state-connections) and the connections from the inputs to the states (i.e., input-connections) are known. Given a generically zero controllable system, we consider the following problems: i) find a minimal set of input-connections whose removal makes the resulting system not generic zero controllability; ii) identify a minimal cost set of input-connections that must be retained from the given set of input-connections while preserving generic zero controllability property; and iii) given a not generically zero controllable system, find a smallest set of state-connections whose removal makes the resulting system generically zero controllable. Problem i) is polynomially solvable. Problems ii) and iii) are NP-hard and approximation results are provided for them. The results of i) and iii) provide clues to analyze the fragility and hardness involved in modifying a system structure. Problem ii) is useful to ensure an accurate discrete-time linear approximation of a large-scale system by maintaining generic zero controllability of the linear system.

I. Introduction

We consider a discrete-time linear system \begin{equation*} \begin{aligned} {x}(t+1)&= \bar{A} x(t) + \bar{B}u(t) \end{aligned} \tag{1} \end{equation*} where and are the states and the inputs at time , and and are the given system and input matrices, respectively.