A New Computational Method for Optimal Control Problem with B-spline Polynomials

The main purpose of this w ork is to propose direct method which is employed by using state vector parameterization (SVP) to convert the quadratic optimal control problems into qu adratic programming problem. The state vector par ameterization is based on the s pline polynomial which inc ludes: B-spline as a basis functions to approximate the system state variables by a fi nite length of the basis functions series of unknown parameters. An example as application of this method is given .


1-Introduction:[7]
Optimal control is a special type of the optimization problem and has tremendous application.It deals with the study of systems.The early developments of the theories were given by engineers whose systems were machines and their interactions were controls.In order to understand a system, mathematical equation representing it exactly or to reasonable approximation is written.Usually a system is represented by one of the equations: differential, partialdifferential, integral, Integrodifferential, difference, stochasticdifferential and stochastic-integral equations, and such equations are known as models of the system.Till around 1950 mathematicians and physicists were by and large engaged in finding the mathematical models of various systems.The absence of physical laws in certain case created problems.However, these were overcome by knowledge of inputs and outputs of system or just statistical data.The side-by side existence and uniqueness of solution was studied which helped to understand the system properly.Since 1948, the scientists have started identifying the factors affecting the behavior of the system.Year's later serious thinking started as

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to how these factors could be controlled to give the best possible result.This concept has been pursued in a systematic manner in the last 30 years and nowadays is know as optimal control.
The system objective is a given state or set of state which may vary with time .Restrictions or constraints, as they are normally called ,are placed on the set of controls (inputs) to the system ; controls satisfying these constraints belong to the set of admissible control

1-Formal Problem Statement
The optimal control problems considered in this work are defined in terms of the system dynamics, the boundary conditions, and the cost criterium • System dynamics: the system dynamics will be defined in terms of state space equations ) ,..., , , ,..., , , ( where The function f is called the vector field.

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B in B-spline stands for basis.Higher degree basis functions are given by convolution of multiple basis functions of one degree lower.In the mathematical subfield of numerical analysis a B-spline is a spline function which has minimal support with respect to a given degree, smoothness, and domain partition.A fundamental theorem states that every spline function of a given degree, smoothness and domain partition, can be represented as a linear combination of B-spline of that same degree and smoothness, and over that same partition.The term B-spline was coined by Isaac Jacob Schoenberg short for basis spline.In the computer science subfields of computer-aided design and computer graphics the term B-spline frequently refers to a spline curve parameterized by spline functions that are expressed as linear combinations of B-spline (in the mathematical sense above).

Definition (1):[8]
Given m+1 knots t i in [0,1] with t0 < t1< t2 <…< tm a B-spline of degree n is a parametric curve The Pi i=0,1,…,m+1 are called control points or anchor points or de Boor points.A polygon can be constructed by connecting the de Boor points with lines starting with P0 and finishing with Pn this polygon is called the de Boor polygon.
The m-n basis B-spline of degree n can be defined using the Cox-de Boor recursion formula When the knots are equidistant we say the B-spline is uniform otherwise we call it non-uniform.
The B-spline can be defined in another way like: Express B-spline Polynomials of Degree n-1 In terms of n Linear Combination of B-spline Polynomials of Degree n.

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where the bi,j are the coefficients of the power basis that are used to determine the respective B-spline polynomials.
The matrix in this case is lower triangular .In the quadratic case (n=2), the matrix representation is The mth derivative of B-spline polynomials Bk,n(t) is given by The product of two B-splines polynomials Bi,n(t) and Bj,m(t) of degrees n and m respectively can be written as B-spline polynomial of degree (n+m)and given by the following For n,m=0,1,…, i=0,1,…n and j=0,1,…,m; we have

3-State Vector Parameterization (SVP) for Solving Linear Quadratic Optimal Control LQOC Problem [1]
Consider the LQOC problem First, when n=1 , we have () The control points i a i=0,1 can be evaluated as follows: Put t=0 and t=1 into (9) to get   The following optimal parameters a can be found a = (1 0.7482852 0.65006441 0.64805174)T Substitute these optimal values into (18) to evaluate the approximate optimal value J* .Here the optimal value is found to be J*=0.76159967.Moreover ,the optimal trajectory and the optimal control are given by: x(t)=( 1 0.7482852 0.65006441 0.64805174) B(t) u(t)=(-0.75515243-0.44815371 -0.19844891 -0.006038) B(t) where B(t)= (B0,3(t) B1,3(t) B2,3(t) B3,3(t) )T Table (1) lists the values of the optimal trajectory x(t) and the optimal control u(t) obtained by the algorithm SVPB with N=3 as well as the absolute error .This problem is also solved by expanding x(t) into different orders Bspline series with N=4,5 and 6, the optimal values of J* for each case are listed in Table (2),from which ,it can be concluded that as N increases ,we get more accurate value for J. From table (.2) we can conclude that the sequence Jk converges to J*=0.761594156 super linearly i.e.
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Initial and Final Values[7]The initial and final values of B-spline polynomials at t=0 and t=1 respectively are given The initial values of Bk,n(t), 0 ≤ t≤ 1 Rewrite eqn.(11)and (13) in the matrix form as 3 or ….. the same step follow 4-Examples of OC Problems:[2] In this section the performance of the proposed methods discussed in the previous section will be compared using example.• x(t) is approximated by 3 rd order Bspline series of unknown parameters i.e.,

& Tech. Journal, Vol. 28, No.18, 2010 A New Computational Method for Optimal Control Problem with B-spline Polynomials .
PDF created with pdfFactory Pro trial version www.pdffactory.comEng.

& Tech. Journal, Vol. 28, No.18, 2010 A New Computational Method for Optimal Control Problem with B-spline Polynomials .
PDF created with pdfFactory Pro trial version www.pdffactory.comEng.