Solving optimal control problems using a gegenbauer transcription method

In this paper we describe a novel direct optimization method using Gegenbauer-Gauss (GG) collocation for solving continuous-time optimal control (OC) problems (CTOCPs) with nonlinear dynamics, state and control constraints. The time domain is mapped onto the interval [0; 1], and the dynamical system formulated as a system of ordinary differential equations is transformed into its integral formulation through direct integration. The state and the control variables are fully parameterized using Gegenbauer expansion series with some unknown Gegenbauer spectral coefficients. The proposed Gegenbauer transcription method (GTM) then recasts the performance index, the reduced dynamical system, and the constraints into systems of algebraic equations using optimal Gegenbauer quadratures. Finally, the GTM transcribes the infinite-dimensional OC problem into a parameter nonlinear programming (NLP) problem which can be solved in the spectral space; thus approximating the state and the control variables along the entire time horizon. The high precision and the spectral convergence of the discrete solutions are verified through two OC test problems with nonlinear dynamics and some inequality constraints. The present GTM offers many useful properties and a viable alternative over the available direct optimization methods.