To Filter or Not to Filter? Impact on Stability of Delay-Difference and Neutral Equations With Multiple Delays

Abstract

For control systems where the closed-loop system description is governed by linear delay-differential equations of neutral type, it is known that stability may be fragile, in the sense of sensitive to infinitesimal perturbations to parameters in the system model or arbitrarily small errors in the implementation of the controller. A natural approach to resolve this problem of ill-posedness and to break down the underlying instability mechanisms, rooted in characteristic roots moving from the left plane to the right one via the point at infinity, consists of including a low-pass filter in the control loop, provided the inclusion preserves stability. Independently of the particular control problem, the addition of a low-pass filter essentially boils down to a “regularization” of delay-difference equations and delay equations of neutral type in terms of parametrized delay equations of retarded type, where the parameter can be interpreted as the inverse of the filter's cut-off frequency. In this paper, the stability properties of these parametrized delay equations are analyzed in a general, multi-delay setting, with focus on the transition to the original delay-difference or neutral equations. It is illustrated that the spectral abscissa may not be continuous at the transition, which may impact stability. Hence, conditions for preservation of stability in terms of a robustified stability indicator called filtered spectral abscissa are presented, for which mathematical characterizations and a computationally tractable expression are provided. The application of a PD controller to a time-delay system with relative degree one is used to motivate the structure of the equations studied throughout the paper, and to explicate the implications of the presented results on control design, discussed in the last section.