Nonlinear feedback control with global stabilization
type: Item
instance of: scholarly article
title: Nonlinear feedback control with global stabilization
zbMATH Open document ID: 0841.93067
DOI: 10.1007/BF01968501
author: Chih-Jian Wan
author: Dennis S. Bernstein
published in: Dynamics and Control
publication date: 1996-07-18T00:00:00Z
review text: Results from the second author, Int. J. Robust Nonlinear Control, No. 3, 211-229 (1993; Zbl 0795.93041) are specialized to affine control systems \(\dot x= f(x)+ g(x) u\). A cost of the form \[ J(x_0, u)= \int_0^{+\infty} (L_1 (x)+ L_2 (x) u+ u^t Ru)dt \] with \(L_2 (0)=0\), is minimized by a stabilizing feedback of the form \[ U(x)=- (1/2) R^{-1} (L^t_2 (x)+ g^t (x) \text{grad} (V) (x)) \] provided there exists a Lyapunov function \(V\) satisfying suitable conditions. This feedback is derived by finding an extremum of the Hamiltonian \(H_u (x,p)\) associated to the optimal control problem; the moment \(p\) is taken as \(p= \text{grad} (V)\). This extremum is a minimum provided \[ L_1 (x)= U^t (x) RU(x)- \text{grad} (V)^t (x) f(x) \] in which case \[ H_u (x,\text{grad} (V))= (u- U(x))^t R(u- U(x)). \] The stabilization can be made global if \(V(x)\) tends to \(+\infty\) when \(|x|\) tends to \(+\infty\). This approach allows to replace the condition \(L_f (V)\leq 0\) of Jurdjevic-Quinn type systems by the cost dependent condition \[ L_f (V)\leq (1/2) L_{g(x) R^{-1} L^t_2 (x)} (V). \] For a class of minimum phase systems with minimal relative degree, this condition is satisfied by choosing an \(L_2\) depending on a Lyapunov function \(V_0\) for the zero dynamics; \(V= V_0+ h^t Ph\) (\(h\) is the output map and \(P>0\)), will be a Lyapunov function for the closed-loop system. Such systems are shown to be feedback equivalent to Jurdjevic-Quinn type systems. The design is used for the global stabilization of the Lorenz equation and of the angular velocity equations of a rigid spacecraft with less than three actuators. It generalizes previous results.
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 49N35
Mathematics Subject Classification ID: 93D30
zbMATH DE Number: 836190
zbMATH Keywords: Lyapunov function
zbMATH Keywords: optimal control
zbMATH Keywords: stabilization
zbMATH Keywords: global
zbMATH Keywords: Jurdjevic-Quinn type systems
zbMATH Keywords: minimum phase
zbMATH Keywords: Lorenz equation
zbMATH Keywords: angular velocity equations
reviewed by: A. Akutowicz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 836190