Application of optimization algorithm to nonlinear fractional optimal control problems

Subject Areas : Journal of Computer & Robotics

Asma Moradikashkooli ¹ , Hamid Haj Seyyed Javadi ² , Sam Jabbehdari ³

1 - Department of Computer Engineering, North Tehran Branch, Islamic Azad University, Tehran, Iran
2 - Department of Computer Engineering, Shahed University, Tehran, Iran
3 - Department of Computer Engineering, North Tehran Branch, Islamic Azad University, Tehran, Iran

Received: 2023-09-18 Accepted : 2023-10-27 Published : 2023-11-28

Keywords: generalized Laguerre polynomials, Operational matrix, Optimization algorithm, Nonlinear fractional optimal control problems, Coefficients and parameters,

Abstract :

In this study, an optimization algorithm based on the generalized Laguerre polynomials (GLPs) as the basis functions and the Lagrange multipliers is presented to obtain approximate solution of nonlinear fractional optimal control problems. The Caputo fractional derivatives of GLPs is constructed. The operational matrices of the Caputo and ordinary derivatives are introduced. The established scheme transforms obtaining the solution of such problems into finding the solution of algebraic systems of equations by approximating the state and control variables using the mentioned basis functions. The method is very accurate and is computationally very attractive. Examples are included to provide the capacity of the proposal method.

Full-Text:



Journal of Computer & Robotics 17 (2), Summer and Autumn 2024, 13-22

 

 

Application of Optimization Algorithm to Nonlinear Fractional Optimal Control Problems

Asma Moradikashkooli a, Hamid Haj Seyyed Javadi b,*, Sam Jabbehdari a

a Department of Computer Engineering, North Tehran Branch, Islamic Azad University, Tehran, Iran

b Department of Computer Engineering, Shahed University, Tehran, Iran

Received 18 September 2023; Accepted 29 October  2023

 

Abstract

 

In this study, an optimization algorithm based on the generalized Laguerre polynomials (GLPs) as the basis functions
and the Lagrange multipliers is presented to obtain approximate solution of nonlinear fractional optimal control problems.
The Caputo fractional derivatives of GLPs is constructed. The operational matrices of the Caputo and ordinary derivatives
are introduced. The established scheme transforms obtaining the solution of such problems into finding the solution of
algebraic systems of equations by approximating the state and control variables using the mentioned basis functions. The
method is very accurate and is computationally very attractive. Examples are included to provide the capacity of the
proposal method.

 

Keywords: Generalized Laguerre polynomials; Nonlinear fractional optimal control problems; Optimization algorithm;
Operational matrix; Coefficients and parameters.

1.Introduction

 

 

Optimal control problems (OCPs) have recently been investigated in few studies. Postavaru and Toma  presented a computational method based on the fractional-order hybrid of block-pulse functions and Bernoulli polynomials for solving Fractional optimal control problems (FOCPs). Heydari and Razzaghi  considered the piecewise Chebyshev cardinal functions as an appropriate family of basis functions to construct a numerical method for solving a category of FOCPs.

Tricaud and Chen  introduced rational  approximation for solving a wide class of FOCPs. Li et al.  investigated a spectral Petrov-Galerkin method for an OCPs governed by a two-sided space-fractional diffusion-advection-reaction equation. Wang et al.  used linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time for a control constrained distributed OCPs subject to a time fractional diffusion equation with non-smooth initial data. Kheyrinataj and Nazemi  described an artificial intelligence approach using neural networks
to solve a class of delay OCPs of fractional order with equality and inequality constraints. Hoseini et al.  applied an approximate technique based on fractional shifted Vieta-Fibonacci functions for solving a type of FOCPs. Mohammadi and Hassani  used generalized polynomials for solving two-dimensional variable-order FOCPs. Zaky  applied
a Legendre collocation method for distributed-order FOCPs. Lima  investigated the solution of  FOCPs by using the orthogonal collocation method and the multi-objective optimization stochastic fractal search algorithm. Fakharian and Hamidi Beheshti  used Adomian decomposition method for solving linear and nonlinear OCPs. Hadizadeh and Amiraslani  constructed a numerical algorithm based on Adomian decomposition method for the nonlinear feedback operators for the time-variant optimal control with nonquadratic criteria. Fakharian et al.  applied Adomian decomposition method to solve the Hamilton-Jacobi-Bellman equation arising in nonlinear optimal problem. Phuong Dong et al.  presented a general formulation for the OCPs to a class of fuzzy fractional differential systems relating to SIR and SEIR epidemic models. Also, they investigated these epidemic models in the uncertain environment of fuzzy numbers with the rate of change expressed by granular Caputo fuzzy fractional derivatives of order . Li et al.  investigated a sensitivity analysis of OCPs for a class of systems described by nonlinear fractional evolution inclusions on Banach spaces. Nemati et al.   applied the Ritz spectral method to solve a class of FOCPs. The developed numerical procedure is based on the function approximation by the Bernstein polynomials along with fractional operational matrix usage. Ghanbari and Razzaghi  introduced an alternative numerical method based on fractional-order Chebyshev wavelets for solving variable-order FOCPs. Marzban  provided a new framework based on a hybrid of block-pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear FOCPs involving delay. Rezazadeh and Avazzadeh  formulated a numerical method based on using shifted discrete Legendre polynomials and collocations method to approximate the solution of two-dimensional OCPs with a fractional parabolic partial differential equation constraint in the Caputo type. Hassani et al. proposed hybrid method based on the transcendental Bernstein series and the generalized shifted Chebyshev polynomials for two dimensional nonlinear variable order FOCPs.

The optimization method plays a significant role in signal and image processing, control theory, physics, engineering, chemistry and mathematics. Heydari and Atangana  proposed an optimization scheme based on the Lagrange multipliers scheme for solving variable-order space-time mobile-immobile advection-dispersion equation involving derivatives
with non-singular kernels. Pakdaman et al.  approximated the solution of fractional differential equations by using the fundamental properties of artificial neural networks for function approximation. Soradi-Zeid  introduced an optimization algorithm, called King, for solving variable order FOCPs. Heydari and Avazzadeh  applied an optimization method through the Legendre wavelets for solving variable-order fractional Poisson equation. Dehestani et al.  used fractional-Lucas optimization method for evaluating the approximate solution of the multi-dimensional fractional differential equations. S M et al.  introduced an optimization-based physics-informed neural network scheme for solving fractional differential equations. Hassani et al.  solved the nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials. Dahaghin and Hassani  proposed an optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Hassani et al.  proposed an optimization method standing on a basis formed by the transcendental Bernstein series for solving nonlinear variable-order fractional functional boundary value problems. Alam Khan et al.  used bat optimization algorithm for computing the approximate solution of fractional order Helmholtz equation, with Dirichlet boundary conditions. Idiri et al.  used the parametric optimization method to find optimal control laws for fractional systems. Kheyrinataj and Nazemi  applied fractional Chebyshev functional link neural network-optimization method for solving delay FOCPs with Atangana-Baleanu derivative.

In the current paper we focus on a class of FOCPs with the Caputo fractional derivative in a dynamical system and propose a new direct computational method based on the new families of basis functions namely Generalized Laguerre polynomials (GLPs) to obtain an approximate solution for them. The problem formulation is as follows:

 

 

with the fractional dynamical system:

 

J F

 

and the initial conditions:

 

G
,

 

where  is a positive integer,  for  are real constants, F and G are continuous functions, assuming that the function F  is considered bounded below and  denotes the fractional derivative of order  in the Caputo sense.

The present paper is devoted to proposing an optimization algorithm based on new basis functions, generalized Laguerre polynomials (GLPs), for solving a family of FOCPs. First, the functions  and  are approximated based on the GLPs in terms of the unknown coefficients and parameters. We generate a new operational matrices with the help of GLPs and the Caputo derivative. Then, by substituting the estimated values of  and  into the cost functional J, an algebraic equation in terms of the unknown coefficients and parameters is achieved to be optimized. Imposing the necessary condition for optimality on the mentioned equation, a system of algebraic equations is obtained. By solving this system, the unknown coefficients are calculated. The method is illustrated by means of some examples and the numerical approximations compared with the analytical solutions. The method is simple to implement and accurate.

The rest of the paper is organized as follows. In the next section, a brief overview of the fractional calculus, some basic definitions of Caputo fractional derivative, Laguerre polynomials (LPs), GLPs, function approximation and convergence analysis are given. In Section 3, an optimization algorithm is offered for solving FOCPs. Section 4 is devoted to
several examples to display the applicability and the efficiency the proposed method. Section 5 provides the conclusion.

 

 

2.Fundamental Definition

 

Here, we provide a brief review of fractional
calculus, LPs and GLPs which shall be used in the proposed scheme.

2.1.Caputo Fractional Derivative

 

Definition 2.1. (see  and references there in) Let  be differentiable function, and let
 be the order of derivative. Then Caputo fractional derivative is defined as follows:

 

 

where  denotes the gamma function.

Collary 2.2. From the definition above for , we have

Where .

 

2.2. Laguerre Polynomials

 

Definition 2.3. (see  and references there in) The LPs, , are the solutions to second order linear differential equation of  .

Definition 2.4. (see  and references there in) The representation of power series for LPs, , is provided by

The first LPs are given by:

In general, the considered function  with the first  LPs terms is approximated such that

 

 

where

and

2.3. Generalized Laguerre Polynomials

 

Definition 2.5. The GLPs, (t), are formed with a change of variable. Accordingly,  is changed to
, (), on the LPs and defined as

 

 

where  indicate control parameters. If , then GLPs fully coincide with classical LPs.

The expansions of the states and controls,  and , in the terms of GLPs are respectively shown in the form of matrices

where

R   , S   ,

and

R  =
S   =
,

 

 

 

with  and  denoting the control parameters.­­­­

 

 

2.4. Oprational Matrices

 

In the literature there are lot of articles in which we can see the derivation of OMs of differentiation in the Caputo sense. In this section we will derive the OMs of fractional derivatives and ordinary derivatives based on the GLPs.

The fractional derivatives of orders , of  can be written as

 

,

 

where Dis the  ⨉  OMs of fractional derivatives of order  defined by­

D

The first-order derivative of  can be written as:

 

D

 

where the +1) ⨉   matrix D is called the operational matrix of ordinary derivative and its elements can be computed as follows:

 

D

Generally, the r-order derivative operational matrix of D can be expressed as follows:

 

D

 

2.5. Function Approximation

 

Let  and assume that , be the vector defined in Eq. ,

 and  be an arbitrary element in . Since  is a finite dimensional vector subspace of ,  has a unique best approximation out of  such as , that is

,  .

Since , there exist the unique coefficients , such that

  R  .

 

 

2.6. Convergence Analysis

 

The following theorem will be useful in subsequent results.

Theorem 2.6. Let →ℝ be a function,
 and . If there exist  ℝ and  such that for the function

 R  

we have

 

Then the error bound is presented as follows:

∥ R  ,

where

∥ R  {∥ }.

 Let   in this case:

∥ R   ∥.

We notice that   which implies that
  Now, by the Taylor theorem for all , we have:

 

Therefore

 

and

 

D

 

REMARK . Note that the same results for  can be investigated.

 

3.Algorithm of the Solution for FOCPs

 

Now, we use the above obtained results to solve the FOCPs  with the fractional dynamical system  subject to the initial conditions . To this end, we approximate the state variable, i.e.  and the control variable, i.e.  by  and , respectively as follows:

 

∥

∥

∥

where 

R   ,
S    ,

 

are unknown vectors that called vectors of the free coefficients which should be computed and  and ,  are the vectors which defined in Eqs. .

At this stage, by Eq. (20), we approximate  in terms of the operational matrix of  fractional derivative of GLPs as follows:

 

,

By substituting Eqs. (20) into Eq. (1), the performance index S  is approximated as:

R  D .

 

where K  and L are unknown control parameters vectors with elements ’s and ’s, respectively.

Also, by substituting Eqs.  and  into the fractional dynamical system , we have:

S   S R , S , K , L    =   F   R   S ≜F R , S , K , L

Furthermore, by taking the collocation points  for  where  into Eq. , we obtain the following system of algebric equations:

R , D G  R   S   G̃ R , S , K , L

*Corresponding Author. Email: [e-mail]

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