Control of a class of nonsmooth dynamical systems

Abstract

In this paper, the authors propose an approach to control a class of nonsmooth continuous and discontinuous dynamical systems which is based on a generalization of derivative and Chebyshev pseudospectral (PS) methods. First a generalized derivative for nonsmooth functions, which is proposed by Kamyad et al. is considered. Therefore the obtained generalized derivative of a nonsmooth function is approximated with the Fourier series. So, the nonsmooth control problem is approximated by a smooth optimal control problem and in what follows, it is solved by Chebyshev PS method and the approximate solution for the state and control of the main problem is achieved. Finally, some numerical examples of the control of nonsmooth dynamical systems are solved to illustrate the efficiency of the authors’ approach.

An Efficient Pseudo-Spectral Method for Nonsmooth Dynamical Systems

“…The modification of the PS methods permits to prove sufficient conditions for feasibility and convergence of the approximate solution. These methods were largely developed for solving nonlinear problems, for example, in solving partial differential equation (PDE) problems, aerospace central problems and orbit transfers [see Canuto et al (1988); Elnagar et al (1995); Erfanian et al (2015); Ross (2001, (2002); Gong et al (2006); Noori Skandari et al (2014); Trefethen (2000); Vlassenbroeck and Van Doreen (1988)]. Among the PS methods, CPS method is more attractive for a number of reasons.…”

“…Some authors utilized the generalized derivatives of nonsmooth functions in dealing with the solutions of nonsmooth systems (Erfanian et al 2013(Erfanian et al , 2015Noori Skandari et al 2014). Auer et al (2011) combined a generalized derivative with the algorithm of a verified special solver to obtain inclusion of the solutions for nonsmooth systems.…”

Fractional Chebyshev pseudospectral method for fractional optimal control problems

“…For the case M = 0, we have l = 0 and l ′ = 1. Moreover, according to the final time T and terminal point x(T), we must consider one of Equations (10)- (14) and add its discrete form to Equations (26). After solving the obtained algebraic equations, we get the pointwise approximate optimal solutions (18) and continuous approximate optimal solutions (17).…”

“…After solving the obtained algebraic equations, we get the pointwise approximate optimal solutions (18) and continuous approximate optimal solutions (17). Notice that, before solving algebraic system (26) ̄1, … ,̄N), we need to calculate the fractional differentiation matrixes D = (D k ) N×(N+1) ,D = (D k ) N×(N+1) and…”

“…This method, in comparison to other methods, has higher accuracy and better results. Up to now, the pseudospectral methods have been utilized by many researchers to solve different types of continuous-time problems (see other works 9,[24][25][26][27][28][29]. In this paper, we consider the shifted Chebyshev-Gauss-Lobatto points as interpolating points and apply the fractional Chebyshev pseudospectral (FCP) method based on a new fractional Lagrange polynomial for indirectly solving FOCPs.…”