Solution for optimal control of general nonlinear systems using generating functions. Issue 3 (11th April 2022)

• Record Type: Journal Article
• Authors: Wang, Shi; Wang, Yaonan; Liang, Qiaokang; Chen, Yijing
• Is Part Of: Asian journal of control. Volume 24:Issue 3(2022)
• Page Start: 1183
• Page End: 1196
• Publication Date: 2022-04-11
• Subjects: generating functions -- Hamilton–Jacobi equation -- optimal control -- semi‐tensor product of matrices; Automatic control -- Periodicals; Control theory -- Periodicals; 629.805
• DOI: 10.1002/asjc.2818
• Languages: English
• ISSNs: 1561-8625

Abstract

Abstract: In this paper, we present an analytical local solution for optimal control problems applicable to a wide class of general nonlinear systems. We first introduce the optimal control problem for a general nonlinear system and formulate the associated Hamilton–Jacobi–Bellman equation. Starting from the necessary conditions for optimality represented by the Hamiltonian system of the optimal control problem, we solve the Hamilton–Jacobi equation for the associated generating functions. The generating functions are obtained in the form of series expansions by using canonical transformation theory and the semi‐tensor product of matrices. The coefficients of the generating functions satisfy simply the algebraic Riccati equation and linear algebraic equations for the second‐degree homogeneous component and higher degree components, respectively, from which an algorithm and data structure can be derived. This enables us to obtain analytical optimal feedback controllers and the corresponding Lyapunov function. Through theoretical analysis, it is established that the proposed methodology guarantees stability of the closed‐loop system. This approach is not affected by the complexity of the performance index and state equations, and most importantly, it does not require any initial admissible control law. A simulation example is presented to illustrate the effectiveness and applicability of the proposed approach.