Direct optimal control for time-delay systems via a lifted multiple shooting algorithm

<p style='text-indent:20px;'>Aiming at efficient solution of optimal control problems for continuous nonlinear time-delay systems, a multiple shooting algorithm with a lifted continuous Runge-Kutta integrator is proposed. This integrator is in implicit form to remove the restriction of smaller integration step sizes compared with delays. A tangential predictor is applied in the integrator such that Newton iterations required can be reduced considerably. If one Newton iteration is applied, the algorithm has the same structure as direct collocation algorithms whereas derives a condensed nonlinear programming problem. Then, the solution of variational sensitivity equation is decoupled from forward simulation by utilizing the implicit function theorem. Under certain conditions, this function evaluation and derivative computation procedure is proved to be convergent with a global order. Complexity analysis shows that the computational cost can be largely reduced by this lifted multiple shooting algorithm. Then, parallelizable optimal control solver can be constructed by embedding this algorithm in a general-purpose nonlinear programming solver. Simulations on a numerical example demonstrate that the computational speed of multi-threading implementation of this algorithm is increased by <inline-formula><tex-math id="M1">\begin{document}$ 36\% $\end{document}</tex-math></inline-formula> compared with non-lifted one, and increased by a factor of <inline-formula><tex-math id="M2">\begin{document}$ 6.64 $\end{document}</tex-math></inline-formula> compared with traditional sequential algorithm; meanwhile, the accuracy loss is negligible.</p>