Optimal singular control for nonlinear semistabilisation

The singular optimal control problem for asymptotic stabilisation has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensu...

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Bibliographic Details
Published in:International journal of control 2016-06, Vol.89 (6), p.1222-1239
Main Authors: L'Afflitto, Andrea, Haddad, Wassim M.
Format: Article
Language:English
Subjects:
Asymptotic properties
Control systems
Controllers
Dynamical systems
Hamilton-Jacobi-Bellman theory
least squares Riccati solutions
Mathematical analysis
Mathematical models
Minimum cost
nonlinear control
Nonlinear programming
Nonlinearity
Optimal control
Production planning
semicontrollability
semiobservability
semistabilisation
singular control
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Summary:The singular optimal control problem for asymptotic stabilisation has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Three approaches are presented to address the nonlinear semistable singular control problem. Namely, a singular perturbation method is presented to construct a state-feedback singular controller that guarantees closed-loop semistability for nonlinear systems. In this approach, we show that for a non-negative cost-to-go function the minimum cost of a nonlinear semistabilising singular controller is lower than the minimum cost of a singular controller that guarantees asymptotic stability of the closed-loop system. In the second approach, we solve the nonlinear semistable singular control problem by using the cost-to-go function to cancel the singularities in the corresponding Hamilton-Jacobi-Bellman equation. For this case, we show that the minimum value of the singular performance measure is zero. Finally, we provide a framework based on the concepts of state-feedback linearisation and feedback equivalence to solve the singular control problem for semistabilisation of nonlinear dynamical systems. For this approach, we also show that the minimum value of the singular performance measure is zero. Three numerical examples are presented to demonstrate the efficacy of the proposed singular semistabilisation frameworks.
ISSN:0020-7179
1366-5820
DOI:10.1080/00207179.2015.1126356