Indirect Method for Optimal Control Problem Using Boubaker Polynomial

Abstract: In this paper, a computational method for solving optimal problem is presented, using indirect method (spectral methodtechnique) which is based on Boubaker polynomial. By this method the state and the adjoint variables are approximated by Boubaker polynomial with unknown coefficients, thus an optimal control problem is transformed to algebraic equations which can be solved easily, and then the numerical value of the performance index is obtained. Also the operational matrices of differentiation and integration have been deduced for the same polynomial to help solving the problems easier. A numerical example was given to show the applicability and efficiency of the method. Some characteristics of this polynomial which can be used for solving optimal control problems have been deduced and studied for any future work.


Introduction:
Control theory is a branch of optimization theory concerned with minimizing or maximizing a given performance index which satisfying the system state equations and constraints [1].The main goal is to find an optimal open loop control u * (t) or an optimal feedback control u * (t, x) that satisfies the dynamical system and optimizes in some sense performance index.Analytical solutions of optimal control problems are not always available, so a numerical solution for solving optimal control problems is the most logical way to treat them.[2] The linear quadratic control (LQP) is a special case of the general nonlinear optimal control problem (OCP), the (LQP) is stated as follows; Minimize the quadratic continuous time ∫ subject to the linear system state equations; ̇ A particular form of the (LQP) that arises in many control system problems, where A represents an B is

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, and u(t) is [3] The solution is known to be λ(t)x(t) .Where x(t) satisfies the following equation Indirect methods are generally based on a reduction of the control problem to a problem involving a differential equation such as the HJB(Hamilton-Jacobi-Bellman) or TPBV(Two Point Boundary Value )problem that is based on the principle of optimality which in most casesare very difficult to solve, So the idea is using the solution of the first order necessary conditions for optimality that are obtained from Pontryagin's minimum principle for problems without inequality constraints, then the optimality conditions can be formulated as a set of differential algebraic equations [4], and to reduce them to an algebraic equations in terms of the orthogonal functions and the operational matrix of differentiation ( or integration) matrix associated with this function.In [3,[5][6][7][8][9] the same method has been used with different kinds of polynomials (e.g.,Chebyshev, Laguerre, Bernstein).
The spectral method was used in this paper to find the solution for these equations by the aid of Boubaker polynomials as the basis function, presenting it as an efficient tool with spectral method technique for solving a linear quadratic problem.We have derived the powers in terms of Boubaker polynomials which will help us in solving our problems.Then we have in matrix form the powers of t as follows;

1-Boubaker Polynomials
A r e c u r s i v e r e l a t i o n i s g i v e n b y ; Using the recursive relation for then we have ̇ ̇ t h e r e c u r s i v e r e l a t i o n i s , The differentiation operational matrix for Boubaker polynomials, which is orthogonal polynomials, was deduced as follows; . 0 r e = 1, 2 1 .a n d a r e i n t e g e r s . ...

3-Operational Matrix of Integration
T h e f o llo w i n g g 's r e p r e s e n t o n ly t h e n u m e r a t o r .
( 5 ) , i f > 4 & i s a p o s i t i v e i n t e g e r . ( e g e r s .

4-Spectral method Technique
Spectral method is used to solve finite Linear quadratic optimal control problems with the aid of classical polynomials usually like, Hermite, Laguerre polynomials… etc, as the basis functions [4].In this work, Boubaker polynomials have been used with the following procedure, -Writing the necessary conditions to determine the optimal solution of the problem equations ( 1) and ( 2), which are the followings, ̇ ̇ with the initial conditions x(0)=x 0 , and the final conditions λ(t f )=0.
-Choosing a set of state and adjoint variables and approximating them using a basis function to approximate The remaining 2(n-q) state and adjoint variables are obtained from the system state and adjoint equations.
-Form the 2q (NN) system of algebraic equations as follows; Differentiate the basis functions B i (t), i=1,2,…,N then introducing Boubaker polynomials differentiation operational matrix D B to yield ̇ where the matrix D B is given by equation (5).
-Solve the above resulting square of equations using Gauss elimination procedure with pivoting, to find the entries of  and  .
-Find the approximate value of the performance index J in equation (1).
To illustrate the procedure, the following numerical example is given.

With boundary conditions
The state variable and adjoint variable are approximated by third order Boubaker polynomials, then can be found from equation (13) while is from equation (16). 3 Substituting in equations (13-14), using equation (11) we get from equation( 16) The following state and control approximations are: with the approximate value of J * =12, which is a good approximation to the exact value.

Conclusions:
Boubaker polynomials can be used for transforming an optimal control problem to algebraic equation by the aid of indirect method (spectral method technique).The operational matrix of differentiation was derived and applied for solving the optimal conrol problem by reducing it into algebraic problem, also the operational matrix of integration was derived to be used in a future work, and then an example has been presented which showed the applicability of this method.
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