Stability and ℒ2 Gain Analysis for Discrete-Time LTI Systems with Controller Failures

Abstract

In this paper, we analyze stability and ℒ2 gain properties for discrete-time linear time-invariant (LTI) systems controlled by a pre-designed dynamical output feedback controller which fails from time to time due to physical or purposeful reason. Our aim is to find conditions concerning controller failure time, under which the system's stability and ℒ2 gain properties are preserved to a desired level. For stability, by using a piecewise Lyapunov function, we show that if the unavailability rate of the controller is smaller than a specified constant and the average time interval between controller failures (ATBCF) is large enough, then global exponential stability of the system is guaranteed. For ℒ2 gain, also by using a piecewise Lyapunov function, we show that if the unavailability rate of the controller is smaller than a specified constant, then the system with an ATBCF achieves a reasonable weighted ℒ2 gain level, and the weighted ℒ2 gain approaches normal ℒ2 gain when the ATBCF is sufficiently large.