Convex Optimization-based Controller Design for Stochastic Nonlinear Systems using Contraction Analysis

This paper presents an optimal feedback tracking controller for a class of Itô stochastic nonlinear systems, the design of which involves recasting a nonlinear system equation into a convex combination of multiple non-unique State-Dependent Coefficient (SDC) models. Its feedback gain and controller parameters are found by solving a convex optimization problem to minimize an upper bound of the steady-state tracking error. Multiple SDC parametrizations are utilized to provide a design flexibility to mitigate the effects of stochastic noise and to ensure that the system is controllable. Incremental stability of this controller is studied using stochastic contraction analysis and it is proven that the controlled trajectory exponentially converges to the desired trajectory with a non-vanishing error due to the linear matrix inequality state-dependent algebraic Riccati equation constraint. A discrete-time version of stochastic contraction analysis with respect to a state- and time-dependent metric is also presented in this paper. A simulation is performed to show the superiority of the proposed optimal feedback controller compared to a known exponentially-stabilizing nonlinear controller and a PID controller.