Partial‐state stabilization and optimal feedback control

Abstract

Summary 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. Partial asymptotic stability 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 Hamilton–Jacobi–Bellman equation and hence guaranteeing both partial stability and optimality. The overall framework provides the foundation for extending optimal linear‐quadratic controller synthesis to nonlinear nonquadratic optimal partial‐state stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time‐varying systems with quadratic and nonlinear nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the partial‐state stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion. Copyright © 2015 John Wiley & Sons, Ltd.

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

“…We assume that all right maximal pathwise solutions to Eqs. (1) and (2) in ðX; fF t g t!t0 ; P x0 Þ exist on [t 0 , 1), and hence, we assume that Eqs. (1) and (2) are forward complete.…”

“…Sufficient conditions for forward completeness or global solutions to Eqs. (1) and (2) are given by Corollary 6.3.5 of Ref. [20].…”

“…(i) The nonlinear stochastic dynamical system G given by Eqs. (1) and 2is Lyapunov stable in probability with respect to x 1 uniformly in x 20 if, for every e > 0 and q > 0, there exist d ¼ dðq; eÞ > 0 such that, for all x 10 2 B d ð0Þ…”

“…In Ref. [1], we extended the framework developed in Refs. [2,3] to address the problem of optimal partial-state stabilization, wherein stabilization with respect to a subset of the system state variables is desired.…”

“…In this paper, we extend the framework developed in Ref. [1] to address the problem of optimal partial-state stochastic stabilization. Specifically, we consider a notion of optimality that is directly related to a given Lyapunov function that is positive definite and decrescent with respect to part of the system state.…”