A Numerical Approach for Solving Optimal Control Problems Using the Boubaker Polynomials Expansion Scheme

Abstract

In this paper, we present a computational method for solving optimal control problems and the controlled Duffing oscillator. This method is based on state parametrization. In fact, the state variable is approximated by Boubaker polynomials with unknown coefficients. The equation of motion, performance index and boundary conditions are converted into some algebraic equations. Thus, an optimal control problem converts to a optimization problem, which can then be solved easily. By this method, the numerical value of the performance index is obtained. Also, the control and state variables can be approximated as functions of time. Convergence of the algorithms is proved. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.

Restarted State Parameterization Method For Optimal Control Problems

“…First, from Equation (1), the expression for u(t) as a function of t, x(t) and x(t) is determined, i.e. [32]:…”

“…In [15], an algorithm for solving optimal control problems and the controlled Duffing oscillator is presented; in the algorithm the solution is based on state parameterization such that the state variable can be considered as a linear combination of Chebyshev polynomials with unknown coefficients and later, extended state parameterization to solve nonlinear optimal control problems and the controlled Duffing oscillator [31]. The authors in [32], presented a computational method based on state parametrization of state variable by using Boubaker polynomials for solving optimal control problems and the controlled Duffing oscillator. This paper is organized into following sections of which this introduction is the first.…”

Optimal trajectory planning for an omni-directional mobile robot with static obstacles: a polynomial based approach

“…Control efforts constraint (13) and maximum limitation velocity and acceleration of the robot 2 2 max (t) (t) , …”

“…This method transforms the optimal control problem into a parametric optimization problem which from computational point of view is too easier than the main optimal control problem. The parameterization method related to parameterization variables divided to parameterization of control variables which is applied in [11], parameterization of state variables which is presented in [12], and parameterization of both the state and control variables which proposed in [13].…”

Numerical treatment of nonlinear optimal control problems

“…Optimal control theory has received a great deal of attention and has found applications in many fields of science and engineering [20,22,23,15,16,17,18,21,24]. Because of the complexity of most applications, optimal control problems are most often solved numerically.…”