On optimal predefined‐time stabilization

• Published in: International journal of robust and nonlinear control Vol. 27; no. 17; pp. 3620 - 3642
• Main Authors: Jiménez‐Rodríguez, E., Sánchez‐Torres, J. D., Loukianov, A. G.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 25.11.2017; Wiley Subscription Services, Inc
• Subjects: Computer simulation; Controllers; Dynamic stability; Hamilton–Jacobi–Bellman equation; Liapunov functions; Lyapunov functions; Manifolds; Nonlinear systems; Numerical methods; optimal control; Optimization; Performance indices; predefined‐time stability; Robustness (mathematics); Sliding mode control; Time functions
• ISSN: 1049-8923; 1099-1239
• DOI: 10.1002/rnc.3757

Summary This paper addresses the problem of optimal predefined‐time stability. Predefined‐time stable systems are a class of fixed‐time stable dynamical systems for which the minimum bound of the settling‐time function can be defined a priori as an explicit parameter of the system. Sufficient conditions for a controller to solve the optimal predefined‐time stabilization problem for a given nonlinear system are provided. These conditions involve a Lyapunov function that satisfies a certain differential inequality for guaranteeing predefined‐time stability. It also satisfies the steady‐state Hamilton–Jacobi–Bellman equation for ensuring optimality. Furthermore, for nonlinear affine systems and a certain class of performance index, a family of optimal predefined‐time stabilizing controllers is derived. This class of controllers is applied to optimize the sliding manifold reaching phase in predefined time, considering both the unperturbed and perturbed cases. For the perturbed case, the idea of integral sliding mode control is jointly used to ensure robustness. Finally, as a study case, the predefined‐time optimization of the sliding manifold reaching phase in a pendulum system is performed using the developed methods, and numerical simulations are carried out to show their behavior. Copyright © 2017 John Wiley & Sons, Ltd.