A direct Approximation Method to solve OCP Using Laguerre Functions

This paper presents an approximate method to solve unconstrained optimal control problem (OCP).This method is classified as a direct method in which an OCP is converted into a mathematical programming problem.The proposed direct method is employed by using the state parameterization technique with the aid of Laguerre polynomials and Laguerre functions to approximate the system state variables. To facilitate the computations within this method, new properties their proofs of Laguerre polynomials and Laguerre functions are given with proof.Furthermore, we will derive the condition under which the proposed method with Laguerre functions converges to the solution of the OCP equation. We will also show that for N (the number of basis functions) sufficiently large, the approximate states stabilize the system.The proposed method has been applied on several numerical examples and we find that it gives better or comparable results compared with some other methods.


Introduction
Optimal control is a special type of the optimization problem and has tremendous applications. 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. However, nowadays the system covers a much wider range, such as the human body or a particular system of the human body, a particular industry or even the whole economy of a country. ) Alonzo & Bryan ,2002, Beeler & Tran,1999, Herdman & Morin, 2002, Jankowski,2002 In order to understand a system, a mathematical equation representing it exactly or to a reasonable approximation is written. Usually a system is represented by one of the equation: differential, partial-differential, integral, integro-differential, difference, stochastic-differential, stochasticintegral, and such equations are known as models of the system. The objective of optimal or an optimal ) (t u  control is to determine an optimal open loop control that forces the system to satisfy physical constraints ) , ( t x u  feedback control and at the same time minimizes or maximizes a performance index. ) Auzinger & Kneisl ,2002,Findeisen & Diehl, 2001(.Systems governed by ordinary differential equations arise in many applications as, e.g. , in astronautics, aeronautics, robotics, and economics . The task of optimizing these systems leads to the OCP investigated in this paper.The general OCP formulation is to find a control vector ) (t u that minimize the functional ) Hussein, 1998 subject to a system of n differential equations ) ), Here, the l vector of control variables is denoted by , are assumed to be continuously differentiable. The controls are assumed to be bounded and measurable and f t may be finite or infinite. The work throughout this work is concerned with the QOC problems and is associated with both finite and infinite time of minimizing a running cost or performance index subject to linear control dynamics. (Binder & Blank, 2001). The mathematical formulation of the problems addressed in this paper is given by: P1: Finite time horizon linear quadratic optimal control FLQOC problem, where the optimization index (performance index) , eqn. (1) , is over a finite time interval subject to the linear system state equations P2: Infinite time horizon linear quadratic optimal control ILQOC problem, where the optimization index , eqn.
(3), is over an infinite time interval.

The Basic Idea of the Method
The direct methods can be applied by using either the discretization or the parameterization techniques for : control variables, both state and control variables or state variables. Throughout this work, state parameterization technique is used. In this technique, Laguerre polynomials and Laguerre functions will be used to approximate the solution to the finite and infinite OCP respectively. (Marta & Werner 1995, Oskar & Bulirsch,1992.
To apply state vector parameterization, it is first necessary to place the solution to the differential equation in a Hilbert space. To do so we restrict attention to a compact subset  of the stability region of a known stabilizing control. When the solutions to the OCP equation are restricted to this set, they exist in the Hilbert space is called a Laguerre series approximation of The following three properties of Laguerre series will be especially useful to solve OCP by means of state parameterization,  The recurrence relation )

Journal of Kirkuk University -Scientific Studies , vol.1, No.2 ,2006
Moreover; an important new formula of Laguerre polynomials is derived in this paper.

Lemma (1):
The first derivative of Laguerre polynomials ) (t L n , is formulated as: Proof: ) Hussein ,1998( The mathematical induction principle is used to prove this lemma. eqn. (11) is true for Let us assume that eqn. (11) is true for a particular positive integer k n  , i.e., Now, we wish to see that eqn. (12) is true for 1   k n . Using the recurrence relation (8) with

Remark
This lemma is useful for our computational work because it relates Laguerre series approximations of time functions to Laguerre series approximations of the time derivatives of these time functions. The idea of the state vector parameterization, using the Laguerre polynomials i L , as a basis functions, is to approximate the state variables as follows: Approximate the state variables by a finite length Laguerre series, i.e.
are the unknown parameters and the , are determined from the system state equations as a function of the unknown parameters of the state variables. Note that, two cases can be distinguished when applying SVP technique.

Case (1):
If the numbers of the states and the control variables are equal m n  , then each state variable will be approximated by a finite length Laguerre series and the control vector is obtained as a function of the state variables (13). Therefore, in this case, the matrix B in eqn. (4) is assumed to be nonsingular, then the control variables can be obtained as: Then by substituting eqn. (13) into eqn. (14), yields: , then reordering eqn. (15) by collecting the coefficients of the same

Case (2):
If the number of the state variables is greater than the number of control variables, i.e., m n  , then there is no need to approximate all the state variables. In this case, a set of the state variables are approximated which will enable us to find the remaining state variables and control variables as a functions of this set, so that the quadratic optimal control problem (3)-(5) is reduced to a quadratic programming problem with fewer unknown parameters. Then the initial conditions (5) are replaced by equating constraints as follows: Eqn. (17) represents the equality constraints. By substituting the approximations (13) and (16) of the state and control variables respectively into the performance index (3), we can get the approximate performance index value  J , so that the LQOC problem (3)-(5) can be converted into a quadratic function of the unknown parameters ij a as follows: Rewrite eqns. (13) and (16) The first term of the integrand in (18) can be written as: Also the second term of the integrand in (18) can be written in the same way. Now, the QOC problem (3)-(5) is converted into parameters optimization problem which is quadratic in the unknown parameters and the new problem can be stated as: The constraints (21) are due to the initial conditions (17).

Remark
The state equations which are not satisfied yet are added to the initial conditions to represent equality constraints. Finally, the optimal value of the vector  a can be obtained from the standard quadratic programming method discussed Now to solve the following infinite LQOC Problems using Laguerre functions, An important new formula of Laguerre polynomials is derived in this paper.

Lemma (2):
The first derivatives of Laguerre functions is given by:

Proof:
Since the nth Laguerre functions Then differentiating Laguerre functions with respect to t , yields The algorithm represents the state vector parameterization with the use of Laguerre functions which can be summarized as follows:  Approximate the state variables by using a finite number of basis Laguerre functions, such that: with the aid of the differentiation formula for Laguerre functions which was given through lemma (2), we obtain  Compute the Hessian for  J using (22).  Use the initial values of Laguerre functions to find the equality constraints

The Convergence Test
In the proposed method, the state vector will be approximated globally, over the entire domain of the problem. To do so, we assumes a global functional form for the solution, typically an expansion in terms of a set of orthogonal functions (basis set) or at least linearly independent set, It is not possible to perform computations on an infinite number of terms, therefore; we must truncate the series in eqn. (29). In place of (29), we take we must select coefficients in eqn. (30) such that the norm of the residual function ) (t r is less than some convergence criterion  , where Now we will return to the question of how large N must be later. An important of SVP method is that the residual function decreases very rapidly. In fact, SVP in general are of infinite order, that is, the norm of the residual ) (t r approaches zero faster than any finite power of   N 1 . There is a convergence test that must be used with SVP method. It is to do with the number of terms kept in the basis set N . The most useful test of convergence in terms of N comes from examining the 2 L norm of i x , n i , , 2 , 1   (the state variables that is approximated using Laguerre functions), i.e., for all N greater than some value 0 N . Since we do not know ) (t x , we replace the presumably better approximation In other words, when the sum of the squares of the remaining coefficients becomes negligible, then we have a satisfactory approximation to the solution.

Numerical Examples
Example (1): The following finite linear quadratic problem is considered , Minimize The optimal value of the performance in this problem is 06936094    (Hsieh,1965) using a modified steepest method and by (Neuman &Sen,1973) using collocation and approximation by cubic splines, while (Hussian ,1998) treated this example by using state parameterization with Chebyshev polynomials. These results besides the present results are listed in Table (2).

Example (2):
The algorithm using Laguerre functions is tested on the following infinite nonlinear QOC problem. Minimize The linearized system about for this problem is treated using Laguerre functions. The parameters i a 1 , i a 2 and i b 1 ; (3). Table (3) The Parameters of Laguerre Functions for N=5 In Table (4) a comparison between the computed optimal value obtained by using the algorithm with Laguerre functions for different orders.

Discussion
In this paper, approximate approaches are proposed to solve QOC problems for both finite and infinite time performance indices depending on Laguerre polynomials and Laguerre functions. The following points can be stated :  From the numerical results, it is clear that using Laguerre basis function to approximate the states will produce accurate solution as N increases and the numerical solution converges to the correct optimal trajectories as the length of series increases.  Since the proposed algorithm depending on Laguerre polynomials doesn ' t deal with the infinite time problems, the finite time version of the infinite time problems can be considered, but the infinite time optimal performance index can be approximated by a finite time optimal performance index if the states is the optimal state of the finite time problem at the end time, therefore; the algorithm depending on Laguerre functions avoids the problem associated with the algorithm using Laguerre polynomials to solve all infinite time problems with satisfactory results.
The main advantages of the proposed algorithms are: + The OC problem is converted into quadratic programming problem with a few linear constraints, which can be solved using the standard quadratic programming . + The number of unknown parameters is kept as small as possible. + The coefficients ij a and ij b ; decrease rapidly as N increases.