Optimal Control Computation for Nonlinear Fractional Time-Delay Systems with State Inequality Constraints

• Published in: Journal of optimization theory and applications Vol. 191; no. 1; pp. 83 - 117
• Main Authors: Liu, Chongyang, Gong, Zhaohua, Yu, Changjun, Wang, Song, Teo, Kok Lay
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computation; Control systems; Engineering; Inequality; Integral equations; Interpolation; Mathematics; Mathematics and Statistics; Methods; Nonlinear control; Nonlinear systems; Numerical analysis; Numerical integration; Numerical methods; Operations Research/Decision Theory; Optimization; Optimization techniques; Taylor series; Theory of Computation; Time delay systems; Time optimal control; Fractional optimal control; Numerical integration; Numerical optimization; 90C55; Inequality constraint; 34K37; 49M37; Fractional time-delay system
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-021-01926-8

In this paper, a numerical method is developed for solving a class of delay fractional optimal control problems involving nonlinear time-delay systems and subject to state inequality constraints. The fractional derivatives in this class of problems are described in the sense of Caputo, and they can be of different orders. First, we propose a numerical integration scheme for the fractional time-delay system and prove that the convergence rate of the numerical solution to the exact one is of second order based on Taylor expansion and linear interpolation. This gives rise to a discrete-time optimal control problem. Then, we derive the gradient formulas of the cost and constraint functions with respect to the decision variables and present a gradient computation procedure. On this basis, a gradient-based optimization algorithm is developed to solve the resulting discrete-time optimal control problem. Finally, several example problems are solved to demonstrate the effectiveness of the developed solution approach.