Optimal tracking performance of MISO systems in the presence of temporally correlated multiplicative uncertainty

Abstract

This paper investigates the optimal tracking performance of a discrete-time linear time-invariant (LTI) multi-input and single-output (MISO) plant responding to a step reference signal, in the presence of temporally correlated multiplicative uncertainty. By temporally correlated, we mean the uncertainty has a practical structure of finite impulse response (FIR) and certain dynamical first and second moments, which includes multiplicative white noises and significant network-induced uncertainties. A two-degree-of-freedom (2DOF) controller is adopted and the tracking performance is measured by the expected energy of the tracking error. By a projection lemma, the criterion of achievability of asymptotic tracking is proposed in an innovative form, which explicitly characterizes how the plant properties (i.e., the potentially repeated unstable output poles, nonminimum phase output zeros, and relative degree) and the uncertainty property (captured by a rational function) may affect the performance limitation. It turns out that, when the achievability condition holds, the minimal achievable tracking performance of the closed-loop system with the uncertainty is proportional to the tracking performance limit of the system without uncertainty, made worse by a quantity related to the inverse of the largest stability margin of the closed-loop system against the uncertainty. In addition, some well-known criteria are reproduced by applying the result to the systems with random packet dropout or multiplicative white uncertainty. Several simulations are also conducted to validate the results.