Solution of linear two-point boundary value problems via a collocation method and application to optimal control (original) (raw)
Related papers
Analytical Solution for Optimal Control by the Second Kind Chebyshev Polynomials Expansion
Iranian Journal of Science and Technology:Transaction A, 2017
Second kind Chebyshev polynomials are modified set of defined Chebyshev polynomials by a slightly different generating function. This paper presents new and efficient algorithm for achieving an analytical approximate solution to optimal control problems. The proposed solution is based on state parameterization, such that the state variable is approximated by the second kind Chebyshev polynomials with unknown coefficients. At first, the equation of motion, boundary conditions and performance index are changed into some algebraic equations. This task converts the optimal control problem into an optimization problem, which can then be solved easily. The presented technique approximates the control and state variables as a function of time. After optimizing, the system is converted into a feedback mode for having the closed loop profits. The results proved the algorithm convergence. Finally by analyzing two numerical examples, the reliability and effectiveness of the proposed method by comparing two different methods is demonstrated.
Chebyshev Polynomials and Spectral Method for Optimal Control Problem
This paper presents efficient algorithms which are based on applying the idea of spectral method using the Chebyshev polynomials: including Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind and shifted Chebyshev polynomials of the first kind. New properties of Chebyshev polynomials are derived to facilitate the computations throughout this work. In addition the convergence criteria for the proposed algorithms are derived. The use of the three algorithms has been demonstrated with example.
New operational matrices approach for optimal control based on modified Chebyshev polynomials
Samarra Journal of Pure and Applied Science
The purpose of this paper is to introduce interesting modified Chebyshev orthogonal polynomial. Then, their new operational matrices of derivative and integration or modified Chebyshev polynomials of the first kind are introduced with explicit formulas. A direct computational method for solving a special class of optimal control problem, named, the quadratic optimal control problem is proposed using the obtained operational matrices. More precisely, this method is based on a state parameterization scheme, which gives an accurate approximation of the exact solution by utilizing a small number of unknown coefficients with the aid of modified Chebyshev polynomials. In addition, the constraint is reduced to some algebraic equations and the original optimal control problem reduces to optimization technique, which can be solved easily, and the approximate value of the performance index is calculated. Moreover, special attention is presented to discuss the convergence analysis and an upper...
A Chebyshev approximation for solving optimal control problems
Computers & Mathematics with Applications, 1995
This paper presents a numerical solution for solving optimal control problems, and the controlled Duffing oscillator. A new Chebyshev spectral procedure is introduced. Control variables and state variables are approximated by Chebyshev series. Then the system dynamics is transformed into systems of algebraic equations. The optimal control problem is reduced to a constrained optimization problem. Results and comparisons are given at the end of the paper. Keywords-Chebyshev approximation; Optimal control problem; Ordinary and partial differential equations.
A Chebyshev spectral method for the solution of nonlinear optimal control problems
1997
This paper presents a spectral method of soluing the controlled Duflng oscillator. The method is based upon constructing the Mth degree interpolation polynomials, using Chebyshevs nodes, to approximate the state and the control Llectors. The differential and integral expressions that arise from the system dynamics and the performance index are converted into some algebraic equations. The optimum condition is obtained by applying the method of constrained extremum.
International Journal of Computer Applications, 2014
In this paper the quasilinearization technique along with the Chebyshev polynomials of the first type are used to solve the nonlinear-quadratic optimal control problem with terminal state constraints. The quasilinearization is used to convert the nonlinear quadratic optimal control problem into sequence of linear quadratic optimal control problems. Then by approximating the state and control variables using Chebyshev polynomials, the optimal control problem can be approximated by a sequence of quadratic programming problems. The paper presents a closed form solution that relates the parameters of each of the quadratic programming problems to the original problem parameters. To illustrate the numerical behavior of the proposed method, the solution of the Van der Pol oscillator problem with and without terminal state constraints is presented.
Chebyshev finite difference approximation for the boundary value problems
Applied Mathematics and Computation, 2003
This paper presents a numerical technique for solving linear and non-linear boundary value problems for ordinary differential equations. This technique is based on using matrix operator expressions which applies to the differential terms. It can be regarded as a non-uniform finite difference scheme. The values of the dependent variable at the Gauss-Lobatto points are the unknown one solves for. The application of the method to boundary value problems leads to algebraic systems. The method permits the application of iterative method in order to solve the algebraic systems. The effective application of the method is demonstrated by four examples.