Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints

Abstract

Consider the class of nonlinear time-delayed optimal control problems with continuous linear constraints. This class of problems is difficult to solve numerically. In this article, a computational method based on a semi-infinite programming approach is given. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the control, an approximated semi-infinite nonlinear problem is obtained. To solve this approximate problem, we use the dual parametrization method. The dual problem is a max-min problem, which can be solved by optimizing over two levels. The upper level can be solved by simulated annealing and the lower level can be solved by using any unconstrained optimisation technique, such as the quasi-Newton method.

The control parameterization method for nonlinear optimal control: A survey

“…Contrary to what is widely claimed in the literature, these partial derivatives are continuous functions: since the state trajectory and the solutions of the variational and costate systems depend continuously on ξ and τ , the derivative formulae in (32) and ( 35) also depend continuously on ξ and τ . In principle, (32) and ( 35) can be used in conjunction with a gradient-based optimization method to optimize the switching times. However, there are several difficulties with this approach:…”

A symplectic local pseudospectral method for solving nonlinear state‐delayed optimal control problems with inequality constraints

“…By setting Q = 40 and N = 20, we can obtain a referenced value of cost functional as J * = 162.2217195092138. This case is also studied by semi‐infinite programming approach . In Table , the computational results of the cost functional, the number of iterations, and the CPU time using different Q and N are reported.…”

“…In Table , the computational results of the cost functional, the number of iterations, and the CPU time using different Q and N are reported. The symbol N p in the work of Lee and Wong denotes the number of subinterval used to implement the dynamic programming. From Table 1, it is seen that, as Q and/or N increases, ε J decreases dramatically while the CPU time consumes does not grow rapidly and converged solutions can be obtained after approximately 12 iterations.…”

Solving optimal control problems for delayed control-affine systems with quadratic cost by numerical continuation

“…The literature is abundant of numerical methods to solve (OCP) τ , for example [2], [3], [4], [5], [6], [7], [8]. Nevertheless, some applications, like atmospheric reentry and satellite launching [2], require great accuracy like indirect methods can provide.…”

“…7 and C T (u 4 (•)) = 8.68821 • 10 −7 while the optimal values of the initial adjoint vectors obtained are τ (s) p x (0) • 10 8 p y (0) • 10 8 p θ (0) • 10 6 p δ (0)…”