ρ-Stability and robustness: discrete-time case

Abstract

We say that a discrete-time system is ρ-stable if, roughly speaking, ρ^k >X _k→0, where >X _k is the system state. General ρ-stability theorems are established in this paper. They concern systems governed by functional difference equations. Systems of this type are encountered in the robustness studies. These ρ-stability theorems are a generalization of the well-known Lyapunov criterion. These results are applied to the robustness quantification problem in the second part of the paper. The case of discrete-time LQ regulators is deeply investigated. Robustness properties of continuous-time LQ regulators are found as the limit when the sampling period >T tends to zero; robustness deteriorates as T increases. An upper bound is given for >T, under which the robustness remains satisfactory. The practical interest of these theoretical results is illustrated on the basis of an industrial example.