Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems

Abstract

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization of stochastic dynamical systems. Partial asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation, and hence, guaranteeing both partial stability in probability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic stochastic controller synthesis to nonlinear-nonquadratic optimal partial-state stochastic stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion.

Control Optimization of Stochastic Systems Based on Adaptive Correction CKF Algorithm

“…Nonlinear control methods generally use a differential geometry principle or feedback linearization principle to transform the system linearly, so as to achieve the purpose of control [20]. A stochastic dynamical system control method based on optimal feedback control is proposed in reference [21]. The global control is realized by real-time feedback through nonlinear local state control.…”

Partial stabilization of nonholonomic systems with application to multi-agent coordination

“…Controllers that utilize passivity-based approaches for partial stabilization have been suggested in [24], [3], [37]. The issue of achieving partial stabilization in stochastic dynamical systems is addressed, e.g., in the papers [28], [36], [44].…”