On stabilization of nonlinear systems with enlarged domain of attraction

doi:10.1016/0005-1098(92)90188-L

Automatica

Volume 28, Issue 3, May 1992, Pages 623-626

https://doi.org/10.1016/0005-1098(92)90188-L Get rights and content

Abstract

This paper presents a systematic procedure to design a control law that guarantees asymptotic stability of an equilibrium with an enlarged domain of attraction for a class of nonlinear systems. To this end, we use a natural extension of well-known linear time-invariant transformation methods to write the system in controller canonical form with a forcing term. The control law is implemented in two steps, first we calculate a nonlinear state feedback that cancels the nonlinear terms of the unforced system, so that for the resulting system a suitable Lyapunov function candidate is available. Then, we add a linear state feedback that stabilizes the equilibrium and maximizes its domain of attraction. Interestingly enough, the latter maximization problem reduces to a classical, linear time invariant H_∞ minimization problem. An example is given to illustrate the procedure.

References (11)

A. van der Schaft
On a state space approach to nonlinear H_∞ control
Syst. Control Lett.
(1991)
K. Glover
All optimal Hankel norm approximation of linear multivariable systems and the $L_{∞}$ error bounds
Int. J. Contr.
(1984)
K. Glover et al.
A state space approach to H_∞ optimal control
A. Isidori
P. Khargonekar et al.
Robust stabilization of uncertain linear systems: Quadratic stabilization and H_∞ control theory
IEEE Trans. Aut. Control
(1990)

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Citation Excerpt :
It is worth noting that the proof of Theorem 1 is constructive and gives explicit control design procedures. We note that Theorem 1 gives a local result and the methods proposed in Ichikawa and Ortega (1992) and Saeki, Araki, and Kondo (1980) may be applied to estimate the stability region. We have presented a reduced-order control scheme to stabilize a class of nonlinear interconnected systems with mismatched uncertainty.
In this paper, a class of nonlinear interconnected systems with similar structure is considered. The interconnected system consists of matched and mismatched uncertainties. Based on constrained Lyapunov equations and by exploiting the structure of the interconnected system, a continuous output feedback reduced-order control scheme is presented to stabilize the system robustly. Our approach allows more general forms of known and uncertain interconnections than existing work. The effectiveness of the proposed reduced-order control scheme is illustrated through a numerical example.
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^☆: The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Isidori under the direction of Editor H. Kwakernaak.

^☆☆: The second author's work is sponsored by the Japan Society for Promotion of Science and Sophia University.

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Brief paperOn stabilization of nonlinear systems with enlarged domain of attraction☆,☆☆

Abstract

Syst. Control Lett.

All optimal Hankel norm approximation of linear multivariable systems and the L∞ error bounds

Int. J. Contr.

A state space approach to H∞ optimal control

Robust stabilization of uncertain linear systems: Quadratic stabilization and H∞ control theory

IEEE Trans. Aut. Control

Brief paper
On stabilization of nonlinear systems with enlarged domain of attraction☆,☆☆

All optimal Hankel norm approximation of linear multivariable systems and the $L_{∞}$ error bounds

A state space approach to H_∞ optimal control

Robust stabilization of uncertain linear systems: Quadratic stabilization and H_∞ control theory