New operational matrices approach for optimal control based on modified Chebyshev polynomials

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.

Probability Function Estimation of Rock Random Variables Using Chebyshev Orthogonal Polynomial

The Chebyshev orthogonal polynomial with sample moments (the origin moments) were used to approximate the probability density function (PDF) or cumulative distribution function (CDF) of variable (CPA method). Three examples from observed datas of uniaxial compressive strength of a kind of hard rock were presented for illustrative purposes. The results show the PDF or CDF of rock variables can be accurately derived from CPA method with sample moments. The relative errors of estimation by CPA method is much smaller than that of TDF method (PDF fitted by some standard theoretical distributions). It is suggested that the presented method can be used in stochastic reliability analysis of rock engineering.