Stochastic Approximation Approach to Design of Linear Controllers for Tracking Systems With Asymmetric Saturation

Abstract

Reference-tracking closed-loop systems with saturating actuators often operate in asymmetric regimes. This is because reference signals cause the operating points away from the point of saturation symmetry (even if the actuator itself is symmetric, i.e., odd, function). Stability analysis and stabilizing controller design for asymmetric systems can be carried out using the same techniques as those for the symmetric case. In contrast, currently available methods for controller design in the framework of reference tracking are not applicable to asymmetric systems. The goal of this paper is to develop such a method for single-input single-output (SISO) plants having no poles in the open right-side plane. The approach is based on a global quasi-linearization technique referred to as stochastic linearization, which approximates the saturation function by an equivalent gain and equivalent bias. The main qualitative result obtained is that the asymmetry leads to a constant disturbance acting at the input of the plant. The quantitative results are analytical expressions for this disturbance and the ensuing steady-state tracking errors. It is shown that these errors exhibit a behavior incompatible with the linear control theory. Specifically, they may be increasing or nonmonotonic functions of the controller gain. In view of this fact, the paper develops a time-domain technique for linear tracking controller design based on two loci: the saturating root locus (to account for dynamics) and the saturating tracking error locus (to accounts for statics). Methods for sketching these loci are provided and applied to controllers design.