Parametric Iteration Method for Solving Linear Optimal Control Problems

This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.


Introduction
x Ax t Bu t t t x t x where are the state and control vector, respectively.
are constant matrix and 0 , x is the initial state.The Optimal Control Problem (OCP) is to find a control law which minimizes the quadratic cost functional where are symmetric positive semi-definite matrices and is symmetric positive definite matrix. , In general the problem can be transformed to the Riccati differential equation [1], although solving the Riccati equation arised from OCP is not very simple.Another proposal for directly solving the OCP is discretizing the original problem and solving it numerically.Herein, the spectral collocation methods differ from other computational methods in their special discretization at carefully selected nodes for example, the so-called Legendre-Gauss-Lobatto nodes.Then the differential equations of the OCP are approximated by algebraic equations [2].Although these methods are flexible and for programming with computer are compatible, but they have their weaknesses for instance they react quite sensitively on the selection of time-step size [3].
According to the classic optimal control theory, as pointed out in [4], by using Pontryagin's maximum principle, we can obtain the following Two-Point Boundary Value (TPBV) problem and the optimal control law for OCP can be written as where is known as the costate variable.

 
Analytic solutions can rarely be found for such TPBV problem and authors often solve it approximately for example Yousefi, Dehghan and Tatari [5] applied He's Variational Iteration Method (VIM) to find the optimal solutions.In this paper, we are going to solve (3) by use of the Parametric Iteration Method (PIM) with emphasis on preference of PIM over VIM.

Parametric Iteration Method
PIM is an approximation method for solving linear and nonlinear problems and at beginning it was proposed for solving nonlinear fractional differential equations [6], by modifying He's variational iteration method [7].The idea of PIM is very simple and straightforward.Consider the following differential equation: where A is a nonlinear operator (5) where L denotes a linear differential , t denotes the time, and   u t is an unknown variable.To explain the basic idea of M, we first consider Equation (4) as below: Lu t Nu t g t t   

PI
operator with respect to u, N is a nonlinear operator with respect to u and   g t is the source term.We then construct a family of iterative formulas as: where and denote the so-ca ter and  Accordingly, the successive approximations , th 1 will be readily obtained by choosing the zero mponent   0 u t satisfying the general property One logical guess for   0 u t inear can so w

Solution of Optimal Control Problem via
In solving the OCP described by ( 1) and ( 2), the PIM constructs the following sequences to directly ap-be stablished by lving its corresponding l homogeneous equation Otherwise it ca freely chosen with possible unknown constants.Note that choosing   0 u t can affect on the form of the solutions.
T auxiliary parameter h is an accelerating factor nitial c he n be hich can be identified optimally by the technique proposed in this paper.We show that a suitable value of h, directly improves the rate of convergence.The auxiliary function

 
H t prepare us to have various basis functions to change the solution terms to a desired form.Relation (6) shows that the sequence constructed by PIM is dependent on h and   H t , and this directly ables us to identify and control the main and rate of convergence and this is the main preference of PIM over VIM.
It should be emphasized that though we have the great fre L do edom to choose the linear operator , the auxiliary parameter h, the auxiliary function   H t , and the initial approximation   0 u t , which is funda ntal to the validity and flexibili f PIM, we can also assume that all of them are properly chosen so that solution of (6) exists, as will be shown in this paper later.
Finally, the exact solution may be obtained by using

PIM
order to proximate the solutions of the TPBV problem (3),  

  x t , then  
x t is the optimal trajectory of system (1), and if   t  is the limit of (10), then the optimal con nction Analytically, as mentioned in [4,5], by having the answers of the system (3), i.e.
Now by substituting (11) and (12) we have: and satisfy in conditions of system (3), beca

Res h
l examples by the PIM to ness of the method indi- In this section, we solve severa show the efficiency and useful cating on the influence of parameter h on decreasing the iterations and increasing the convergence rate and accuracy of approximations.Whenever the form of approximations has no importance, we take   1 H t  .As pointed out in section 3, we solve OCPs by solving the corresponding TPBV problems (3).
Example 1.Consider the following optimal control system [4]: The PIM constructs the following sequences to approximate the solutions: The exact solutions are:       . : 2 , 0 0.9.
for   k t and its e According to [4,5], .In Figure 3 According to Equations ( 9) and (10), the iteration f mulas are: Copyright © 2012 SciRes.AM     The exact solutions are:  and 5 show the exact and approximate solutions.This problem was solved by VIM in [5] and their presented solutions are only in a small region [1.4,1.7].

Conclusion
There are various methods for solving linear OCPs, but in practice, the preferred method is that which be executable by computers and the PIM is one of them, because, moreover it's simple structure, it has an accelerator parameter h which directly increases the convergence rate t Copyright © 2012 SciRes.AM and decreases the number of iteration e interesting for using in the softwars.One idea to estimate optimal h mentioned in the paper.In general finding optimal auxiliary parameter h and auxiliary functio s and this ability will b n  

H t used fo
, are open problems.This easy to use method can be r nonlinear systems too.

Figure 1 .
Figure 1.Plot of exact and approximation solutions.

Figure 3 .
Figure 3. Exact and approximate solution and affection of h.

Figure 4 .
Figure 4. Plot of first coordinate for various h.

Figure 5 .
Figure 5. Plot of second coordinate for various h.
This shows the flexibility and excellence of the PIM.Figure2is plot of the error for various iterations.It is clear that accuracy of PIM is higher than VIM. ,