Abstract
This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.
Highlights
Consider linear system described by x Ax t Bu t,t t0, (1) x t0 x0.where x n,u m are the state and control vector, respectively
In general the problem can be transformed to the Riccati differential equation [1], solving the Riccati equation arised from Optimal Control Problem (OCP) is not very simple
Parametric Iteration Method (PIM) is an approximation method for solving linear and nonlinear problems and at beginning it was proposed for solving nonlinear fractional differential equations [6], by modifying He’s variational iteration method [7]
Summary
In general the problem can be transformed to the Riccati differential equation [1], solving the Riccati equation arised from OCP is not very simple. The differential equations of the OCP are approximated by algebraic equations [2]. These methods are flexible and for programming with computer are compatible, but they have their weaknesses for instance they react quite sensitively on the selection of time-step size [3]. Analytic solutions can rarely be found for such TPBV problem and authors often solve it approximately for example Yousefi, Dehghan and Tatari [5] applied He’s. Variational Iteration Method (VIM) to find the optimal solutions. We are going to solve (3) by use of the Parametric Iteration Method (PIM) with emphasis on preference of PIM over VIM
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