An Investigation into Accuracy, Computation, and Robustness of Stochastic Linearization of Systems with Saturation Nonlinearities

Quasilinear Control (QLC) is a theory with a set of tools that can be used for the analysis and design of stochastic feedback control systems with nonlinear actuators and sensors. It is based on Stochastic Linearization (SL), a method of linearization that approximates a nonlinearity with an equivalent gain and a bias using statistical properties of its input. While the QLC literature has studied many problems in controls (e.g., performance analysis and controller design), important issues such as accuracy, robustness, and computation of SL have not been fully investigated. This paper fills this gap, with a special focus on feedback systems with saturating actuators, which are commonly encountered in feedback control systems and the QLC literature. Specifically, the accuracy of SL in approximating the saturated response is investigated. A mathematical relationship is established between the saturation authority and the worst-case open-loop accuracy, and the implications on closed-loop accuracy are described. To study the robustness of SL in practical situations, the sensitivities of the equivalent SL gain and bias to system parameters are investigated. Finally, several root-finding algorithms are compared in terms of their computational speed in solving the SL equations, and a novel change of coordinates is presented to improve the performance. The results provide valuable insights into and methods for addressing the computational aspects of SL that are critical for proper design and analysis of nonlinear stochastic feedback control systems.