Solution of Optimal Control Problems with Payoff Term and Fixed State Endpoint by Using Bezier Polynomials

abstract: In this paper, a new numerical method for solving the optimal control problems with payoff term or fixed state endpiont by quadratic performance index is presented. The method is based on Bezier polynomial. The properties of Bezier polynomials in any intervel as [a, b] are presented. The operational matrices of integration and derivative are utilized to reduce the solution of the optimal control problems to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.


Introduction
One of the widely used methods to solve optimal control problems is the direct method.There is a large number of research papers that employ this method to solve optimal control problems (see for example [2,3,5,7,8,9,10,11,12,13,14,17] and the references therein).Razzaghi, et. al. used direct method for variational problems by using hybrid of block-pulse and Bernoulli polynomials [14].Optimal control of switched systems based on Bezier control points presented in [6].Edrisi-Tabriz 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].An accurate method is proposed to solve problems such as identification, analysis and optimal control using the Bernstein orthonormal polynomials operational matrix of integration [17].
In this paper, we present a computational method to solve optimal control problems with payoff term and fixed state endpoint by using Bezier polynomial.The method is based on approximating the state variables and the control variables with Bezier polynomials [5,13,14].Our method consists of reducing the optimal control problem to a NLP one by first expanding the state rate ẋ(t) the control u(t) as a Bezier polynomial with unknown coefficients.These linear cardinal Bezier polynomials are introduced.In order to approximate the integral and differential parts of the problem and the performance index, the operational matrix of integration P Φ and differentiation D Φ are given.
The paper is organized as follows: In Section 2 we describe the basic formulation of the Bezier functions required for our subsequent development.Section 3 is devoted to the formulation of optimal control problems.Section 4 summarizes the application of this method to the optimal control problems, and in Section 5, we report our numerical finding and demonstrate the accuracy of the proposed method.

Some Properties Of Bernstein And Bezier Polynomials On [a,b]
The Bernstein basis polynomial of degree n on [a,b] are defined as [17] where i is integer nummber and the binomial coefficients are given by Some properties of these polynomials are , where δ is the Kronecker delta function.
(ii) B i,n (t) has two roots, each of multiplicity i and n − i , at t = a and t = b respectively.
(iv) The Bernstein polynomials form a partition of unity i.e.
(v) It has a degree dowing property in the sense that any of the upper-degree polynomials (degree > n − 1 ) can be expressed as a linear combinations of polynomials of degree n − 1.We have, Bernstein Polynomials on [a,b] satisfy in the following relations: .

Definition Of Bezier Polynomials On [a,b]
We will express Bezier (polynomials) curves in terms of Bernstein polynomials, defined explicitly by where the c i , i = 1, 2, . . ., n are given by c i = c[a <n−i> , b <i> ] and they are control points or Bezier pionts and a <n−i> means that a appears n − i times.For example, (i) Symmetry:

The Operational Matrices Of Derivative And Integration For The Bezier Polynomials
Suppose Φ n (t) on [a,b] is given by where T denotes transposition.
The differentiation of vector Φ n (t) can be expressed as where D Φ is the (n + 1)(n + 1) operational matrix of derivative for the Bezier polynomials given as follows: The integral of the vector Φ n (t) defined in Eq. (2.3) is given as where P Φ is the (n + 1) × (n + 1) operational matrix of integration for the Bezier polynomials given as: where P φ is the (n + 1) × (n + 1) operational matrix of integration for the Bezier polynomials on [0,1] and W = (w i,j ) and V = (v j,k ) are (n + 1) × (n + 1) matrices as: where and (2.10)

Function Approximation
Any function f ∈ L 2 [a, b] can be approximated using Bezier polynomials as where C = [c 0 . . .c n ] T can be obtained as (2.12) Let R =< Φ n , Φ n > which is a (n + 1) × (n + 1) matrix and is called the dual matrix of Φ n (t), and it can obtain as: where Proof: We know that Set{1, x, x 2 , ..., x n } is a basis for polynomials space of degree n .Therefore we define . Using Taylor expansion we have: where ξ x ∈ (a, b).Since C T B is the best approximation of f out of S n and y 1 ∈ S n using (2.15) we obtain

✷
We can rewrite Eq. (2.14) as: which shows that the error vanishes as n → ∞

Problem Statement
Consider the following class of nonlinear systems with inequality constraints, where A(t) = (a i,j (t)) n×n and B(t) = (b i,j (t)) m×m are matrices functions and x(t) and u(t) are n × 1 and m × 1 state and control vectors respectively.The problem is finding the optimal control u(t) and the corresponding state trajectory x(t), a ≤ t ≤ b satisfying Eqs.(3.1) and (3.2) while minimize (or maximize) the quadratic performance index where G(t) = (g i,j (t)) n×n , Q(t) = (q i,j (t)) n×n are symmetric positive semi-definite matrices and and R(t) = (r i,j (t)) m×m is a symmetric positive definite matrix.

Variational Problems
Consider the following variational problem: with the boundary conditions x where x(t) = [x 1 (t), x 2 (t), . . ., x n (t)] T .The problem is to find the extremum of Eq. (3.4), subject to boundary conditions (3.5) and (3.6).The method consists of reducing the variational problem into a set of algebraic equations by first expanding x(t) in terms of Bezier polynomials with unknown coefficients [14].

Illustrative Examples
This section is devoted to numerical examples.We implemented the proposed method in last section with MALAB (2012) in personal computer.To illustrate our technique, we present four numerical examples, and make a comparison with some of the results in the literatures.with the conditions x(0) = 1, ẋ(π/4) = 0. (5.27) The exact solution is x(t) = sin(t) + cos(t).5, show the results for Z. Figure 3 shows the plots of errors for x(t) for n = 3, 5.

Conclusion
In this paper we presented a numerical scheme for solving linear constrained quadratic optimal control problems.The Bezier polynomials was employed.Several test problems were used to show the applicability and efficiency of the presented method.The obtained results show that the new approach can solve the problem effectively.

Figure 1 :Solution
Figure 1: Plots of errors for state (left) and control (right) functions for n=3

Figure 2 :
Figure 2: Plots of errors for state (left) and control (right) functions for n=5 figure(a): graph of state error for n=3

Figure 3 :
Figure 3: Plots of errors for x(t) for n = 3 (left) and for n = 5 (right)

Table 1 :
The approximate values of Z, and absolute value of error for Example 5.2.

Table 2 :
The approximate values of Z for n=3, for Example 5.3

Table 3 :
The approximate values of Z for n =5 for Example 5.3 Tables2 and 3show the approximate values of Z toghether with absolute values of errors for different values of T and n = 3, 5.

Table 4 :
The approximate values of x(t), for Example 5.4

Table 4 ,
shows the approximate values of x(t) toghether for different values of t and n.Table