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 bound of the error for the presented approximate solution is derived. Finally, some important illustrative examples of obtained results are shown and proved that powerful method in a simple way to get an optimal control of the considered.
Axial dynamic load identification of hydraulic turbine based on Chebyshev orthogonal polynomial approximation
