Stochastic Linearization of Feedback Systems With Multivariate Nonlinearities and Systems With State-Multiplicative Noise

Quasilinear control (QLC) theory provides a set of methods intended for the analysis and design of stochastic feedback systems with static nonlinearities. QLC leverages the method of stochastic linearization (SL), which linearizes the nonlinear functions by utilizing the statistical properties of the inputs to the nonlinearities. In the traditional QLC literature, SL has been thoroughly applied to systems having nonlinearities with only a single input. This article investigates the case of SL applied to feedback systems with nonlinear functions of multiple inputs. More specifically, the formulas for the SL gains and bias are derived for multivariate functions and then employed to explore SL of a trivariate saturation nonlinearity and study the SL of control systems with feedback loops. The developed theory is then applied to the analysis and optimal controller design of stochastic systems having randomly varying parameters or state-multiplicative noise. Finally, a recipe for investigating the robustness of SL is provided.