Nyquist Stability Criteria for Control Systems with Stochastic Delays

Abstract

We consider a linear control loop with time-varying delays, assumed to be independent and identically distributed random variables following a known probability distribution. We provide Nyquist criteria to assert the convergence to zero of the state statistical moments. The criterion pertaining to the first order moments parallels the one for deterministic time-invariant control loops. In particular, one can determine gain and phase margins. This criterion can be used to assert almost sure stability for positive linear systems. The criterion for the second order moments can be used to assert mean square stability for general linear systems. The applicability of the results is illustrated through a numerical example.