Generation of higher order pseudospectral integration matrices.
ABSTRACT A new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of Chebyshev polynomials. Applications to initial value problems, boundary value problems, linear integral and integro-differential equations are presented. The present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods.
- SourceAvailable from: Saeid GholamiResearch Journal of Applied Sciences, Engineering and Technology. 01/2014; 7(4):801-806.
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ABSTRACT: 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.Proceedings of the 2012 Australian Control Conference. 01/2012;
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ABSTRACT: The theory of Gegenbauer (ultraspherical) polynomial approximation has received considerable attention in recent decades. In particular, the Gegenbauer polynomials have been applied extensively in the resolution of the Gibbs phenomenon, construction of numerical quadratures, solution of ordinary and partial differential equations, integral and integro-differential equations, optimal control problems, etc. To achieve better solution approximations, some methods presented in the literature apply the Gegenbauer operational matrix of integration for approximating the integral operations, and recast many of the aforementioned problems into unconstrained/constrained optimization problems. The Gegenbauer parameter α associated with the Gegenbauer polynomials is then added as an extra unknown variable to be optimized in the resulting optimization problem as an attempt to optimize its value rather than choosing a random value. This issue is addressed in this article as we prove theoretically that it is invalid. In particular, we provide a solid mathematical proof demonstrating that optimizing the Gegenbauer operational matrix of integration for the solution of various mathematical problems by recasting them into equivalent optimization problems with α added as an extra optimization variable violates the discrete Gegenbauer orthonormality relation, and may in turn produce false solution approximations.Advances in Computational Mathematics 01/2012; · 1.47 Impact Factor