Fractional Optimal Control of Continuum Systems

Abstract

This paper presents a formulation and a numerical scheme for fractional optimal control (FOC) of a class of continuum systems. The fractional derivative is defined in the Caputo sense. The performance index of a fractional optimal control problem is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The scheme presented relies on reducing the equations of a continuum system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. The numerical scheme involves discretizing the space domain into several segments, and expressing the spatial derivatives in terms of variables at spatial node points. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain the Euler–Lagrange equations for the problem. The numerical technique presented in the work of Agrawal (2006, “A Formulation and a Numerical Scheme for Fractional Optimal Control Problems,” Proceedings of the Second IFAC Conference on Fractional Differentiations and Its Applications, FDA ‘06, Porto, Portugal) for the scalar case is extended for the vector case. In this method, the FOC equations are reduced to the Volterra type integral equations. The time domain is also discretized into a number of subintervals. For the linear case, the numerical technique results in a set of algebraic equations that can be solved using a direct or an iterative scheme. An example problem is solved for various orders of fractional derivatives and different spatial and temporal discretizations. For the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other continuum systems.

Numerical Method for Solving Fractional Optimal Control Problems

“…A recent paper by Agrawal [3] used Volterra integral equation to solve the same problem where the derivatives are defined in the sense of Caputo. Similar attempts have been made by several researchers for solving the fractional order optimal control problem of distributed systems [5,6].…”

Optimal control based on neural observer with known final time for fractional order uncertain non-linear continuous-time systems

“…In Frederico and Torres (2008), Neother theorem has been described by Frederico and Torres for optimal control of fractional order systems regarding Caputo concept and some laws have been presented for optimal fractional order control problem. In Tangpong and Agrawal (2009), fractional optimal control problem has been presented for continuum systems regarding Caputo concept and a numerical iterative method has been used to solve the problem. In Agrawal (2008b), optimal fractional order control problem with respect to Caputo concept has been presented and Voltra integral equations have replaced the resulting differential equations and linear solver has been used to obtain solution of algebraic equa-tions.…”

Solving Fractional Variational Problem Via an Orthonormal Function

“…A fractional optimal control problem (FOCP) is an optimal control problem in which the performance index or the differential equations in the dynamics of the system or both contain at least one fractional order derivative 448 SOLVING FRACTIONAL VARIATIONAL PROBLEM term [39]. Tricaud and Chen have solved a large class of FOCPs (linear, nonlinear, time-invariant, time-variant, SISO, MIMO, state or input constrained, etc.)…”