A Chebyshev polynomial method for optimal control with state constraints
Automatica (Journal of IFAC)
A modified tau spectral method that eliminated spurious eigenvalues
Journal of Computational Physics
An improved pseudospectral method for fluid dynamics boundary value problems
Journal of Computational Physics
Spectral integration and two-point boundary value problems
SIAM Journal on Numerical Analysis
Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems
SIAM Journal on Scientific Computing
Consistent Approximations for Optimal Control Problems Based on Runge--Kutta Integration
SIAM Journal on Control and Optimization
Journal of Computational and Applied Mathematics
Integral equation method for the continuous spectrum radial Schrödinger equation
Journal of Computational Physics
State analysis of linear time delayed systems via Haar wavelets
Mathematics and Computers in Simulation
Runge-Kutta based procedure for the optimal control of differential-algebraic equations
Journal of Optimization Theory and Applications
Spectral methods in MatLab
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
A MATLAB differentiation matrix suite
ACM Transactions on Mathematical Software (TOMS)
The Euler approximation in state constrained optimal control
Mathematics of Computation
Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control
SIAM Journal on Numerical Analysis
Spectral method for constrained linear-quadratic optimal control
Mathematics and Computers in Simulation
Journal of Global Optimization
Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators
SIAM Journal on Scientific Computing
Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)
Pseudospectral integration matrix and boundary value problems
International Journal of Computer Mathematics
Computational Optimization and Applications
Computational Optimization and Applications
Error estimates of mixed finite element methods for quadratic optimal control problems
Journal of Computational and Applied Mathematics
ACM Transactions on Mathematical Software (TOMS)
Journal of Computational Physics
Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods
Finite Elements in Analysis and Design
Automatica (Journal of IFAC)
Differentiation by integration with Jacobi polynomials
Journal of Computational and Applied Mathematics
Inexact Restoration for Runge-Kutta Discretization of Optimal Control Problems
SIAM Journal on Numerical Analysis
Brief paper: Pseudospectral methods for solving infinite-horizon optimal control problems
Automatica (Journal of IFAC)
Computational Optimization and Applications
A Legendre-Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
Optimal Gegenbauer quadrature over arbitrary integration nodes
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
This paper reports a novel direct Gegenbauer (ultraspherical) transcription method (GTM) for solving continuous-time optimal control (OC) problems (CTOCPs) with linear/nonlinear dynamics and path constraints. In (Elgindy et al. 2012) [1], we presented a GTM for solving nonlinear CTOCPs directly for the state and the control variables, and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment. This article extends the GTM to deal further with problems including higher-order time derivatives of the states by solving the CTOCP directly for the control u(t) and the highest-order time derivative x^(^N^)(t),N@?Z^+. The state vector and its derivatives up to the (N-1)th-order derivative can then be stably recovered by successive integration. Moreover, we present our solution method for solving linear-quadratic regulator (LQR) problems as we aim to cover a wider collection of CTOCPs with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature. The proposed numerical scheme fully parameterizes the state and the control variables using Gegenbauer expansion series. For problems with various order time derivatives of the state variables arising in the cost function, dynamical system, or path/terminal constraints, the GTM seeks to fully parameterize the control variables and the highest-order time derivatives of the state variables. The time horizon is mapped onto the closed interval [0,1]. The dynamical system characterized by differential equations is transformed into its integral formulation through direct integration. The resulting problem on the finite interval is then transcribed into a nonlinear programming (NLP) problem through collocation at the Gegenbauer-Gauss (GG) points. The integral operations are approximated by optimal Gegenbauer quadratures in a certain optimality sense. The reduced NLP problem is solved in the Gegenbauer spectral space, and the state and the control variables are approximated on the entire finite horizon. The proposed method achieves discrete solutions exhibiting exponential convergence using relatively small-scale number of collocation points. The advantages of the proposed direct GTM over other traditional discretization methods are shown through four well-studied OC test examples. The present work is a major breakthrough in the area of computational OC theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points, states and controls expansion terms. Moreover, the GTM produces very small-scale NLP problems, which can be solved very quickly using the modern NLP software.