Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

Abstract

In this paper, the optimal control problems (OCPs) are converted into a constrained optimization problem based on state parameterization via the Bspline functions (BSFs). In fact, the state variable can be considered as a series of the BSFs with unknown coefficients, and the OCPs are transformed into a constrained optimization problem. With the proposed method, the control and state variables also the performance index can be obtained approximately. Also, the convergence theorem of the presented approach is proved in details, some illustrative examples are reported. Also, an example, which has analytic noncontinuous state and control variable, is presented to show the efficiency and reliability of the purposed method, compared with other existing methods.

Optimal control design for linear time-varying systems by interpolated variational iteration method

“…This problem has no analytical solution, thus the numerical procedures must be applied (Kafash and Alavizadeh, 2020).…”

“…However, the optimization needed high CPU time, which was the drawback of the algorithm. As a direct method, Kafash and Alavizadeh (2020) proposed a procedure using the state parametrization via B-spline functions with unknown coefficients. The method reduced the problem to a quadratic programming problem, which could be solved by a system of linear equations.…”

“…• The proposed method obtains a feedback controller, unlike parametrization methods, such as (Kafash and Alavizadeh, 2020;Kafash et al, 2012Kafash et al, , 2014Malmir and Sadati, 2020), where the controllers are obtained open loop.…”

“…We also optimize the method to eliminate the redundant calculations. The highlights of our proposed method are as follows:• The speed of the proposed method has been multiplied several times compared to the original IVIM and its convergence has been proven.• The proposed method obtains a feedback controller, unlike parametrization methods, such as (Kafash and Alavizadeh, 2020; Kafash et al, 2012, 2014; Malmir and Sadati, 2020), where the controllers are obtained open loop.• The solutions are iteratively achieved in the proposed method, and there is no need to solve a system of equations or an optimization problem at each iteration.• In parametrization methods, there is no relationship between the coefficients of the basis functions from one iteration to the next one, and at each step the coefficients must be calculated independently, but in this method, the coefficients are calculated iteratively.• The acceleration proposed in this paper can be widely applied to other interpolated iterative methods.…”

“…Comparison between optimized IVIM and B-spline parameterization method(Kafash and Alavizadeh, 2020), Example 5.1.…”