Numerical Solution of a Class of Nonlinear Optimal Control Problems Using Linearization and Discretization

Abstract

In this paper, a new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional, in three phases. In the first phase, using linear combination property of intervals, changes nonlinear system to an equivalent linear system, in the second phase, using discretization method, the attained problem is converted to a linear programming problem, and in the third phase, the latter problem will be solved by linear programming methods. In addition, efficiency of our approach is confirmed by some numerical examples

On the stability of a class of nonlinear control systems

“…The NC equation (14) can be converted into the equivalent AC equation. For this, by the relations (3) and (4), 1] x 2 tan π 8 u 3 + π 4 ,…”

“…, n. Also we attend that the control in ith equation of the system (1) does not appear in other equations of it. The aim of this paper is to survey and analyze the stability of the nonlinear control system (1). For this purpose, we first convert the NC system (1) into the equivalent affine control (AC) system by linear combination property of intervals (LCPIs) which is proposed and used by Noori Skandari et al [1][2][3] to convert some nonlinear problems in optimization into the equivalent linear problems.…”

“…The aim of this paper is to survey and analyze the stability of the nonlinear control system (1). For this purpose, we first convert the NC system (1) into the equivalent affine control (AC) system by linear combination property of intervals (LCPIs) which is proposed and used by Noori Skandari et al [1][2][3] to convert some nonlinear problems in optimization into the equivalent linear problems. Then, the CLF [5,[10][11][12] and control VL function (CVLF) [15][16][17][18][19] are utilize to analyze the stability of the obtained AC system, and thus, we get several analytical results for stability of equivalent NC system (1) and a control stabilizer is constructed.…”

Optimal control approach for discontinuous dynamical systems

“…Direct methods are based on the discretization and parameterization, which can lead to nonlinear programming (NLP) problems. Methods such as pseudo-spectral method, measure theoretical approaches, linearization methods, control parameterization methods, finite difference methods, and time-scaling transformation methods belong to this class (see previous studies [37][38][39][40][41][42][43][44][45][46][47] ). However, indirect methods are based on the Pontryagin minimum principle and Hamiltonian-Jacobi-Bellman equations, which can lead to a problem with initial and boundary conditions (see previous studies [48][49][50] ).…”

Solution of optimal control problems with Payoff term and fixed state endpoint by using Bezier polynomials

“…EdrisiTabriz et al used B-spline functions to solve constrained quadratic optimal control problems [9]. A new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional [16,18]. Bernstein polynomials have been utilized for solving different equations by using various approximate methods [13].…”