ABSTRACT

In this paper, Homotopy perturbation method is used to find the approximate solution of the optimal control of linear systems. In this method the initial approximations are freely chosen, and a Homotopy is constructed with an embedding parameter, which is considered as a “small parameter”. Some examples are given in order to find the approximate solution and verify the efficiency of the proposed method.

1. Introduction

Optimal control problems arise in a wide variety of disciplines. optimal control theory has also been used with great success in areas as diverse as economics to biomedicine [1]. Apart from traditional areas such as aerospace engineering [2], robotics [3] and chemical engineering. We know that generally optimal control problems are difficult to solve. particularly, their analytical solutions are in many cases are not questionable. Thus, the key to solve many of these real world problems are numerical methods. There is a new method proposed by some authors new for solving optimal control problem based on Pontryagin’s maximum principle or Hamilton-JacobiBellman equation, such as the relaxed descent method, variation of extermal, quasilinearization, gradiant projection method [4-8]. An easy way that some author used for solving problem is to transform the problem to new problem. In [9] the problem is solved by converting the problem to differential inclusion form. In [10] the problem is converted to measure space and then solved and in [11] the problem is solved by genetic algorithm, Others deal with the optimal control problem directly. For example see [12-17].

In this paper we solve the optimal control problem by combine perturbation method. To this end, there are quite a few fundamentally diverse approaches, some of which can be found in [18,19]. The homotopy method is a powerful numerical method for solving nonlinear algebraic and functional equations. The main advantage over classical methods is that the method enjoys global convergence. However, it is not used as widely as these, mainly owing to being poorly covered in the Russian literature.

The Belgian mathematician Lahaye was the first to use the homotopy method for the numerical solution of equations. He considered the case of a single equation. He used discrete continuation by the Newton method. Later, Lahaye [20] also considered systems of equations. Davidenko [21,22] stated the method in the most effcient differential form and applied it to a wide class of problems such as the inversion of matrices, the computation of determinants, the computation of matrix eigenvalues, and the solution of integral equations. Subsequently, in [23, 24] the homotopy method was applied to boundary value problems and simplest variational problems. An essential contribution to the development of the method was made by Shalashilin, Grigolyuk, and Kuznetsov; their papers [25,26] are the most comprehensive publications on the homotopy method in Russian. The homotopy method has been developed for optimal control problems by Avvakumov [27], Since the 1980s. Allgower and Georg made an essential contribution to the popularization of the method. Their review [28] stimulated the development of the method. Of the recent publications, we note the monograph [28], where the homotopy method was combined with the Newton method or the gradient method in infinite-dimensional spaces.

Consider the following optimal control problem

where, , and, are the time invariant given matrices. The control function is an admissible control if it is piecewise continues in t for each t in the given interval. It is assumed the control is bounded, that is, a closed, bounded, subset of exists, such that the control function takes its values form. The input can be derived by minimizing the quadratic performance index, where and are symmetric positive semi-definite and is symmetric positive definite. By using Pontryaging’s maximum principle, the optimal control law, can be achieved for system (1.1) (see [34]). In this paper, we try to find an approximate value for by means of the perturbation homotopy method. Other numerical methods for approximating based on orthogonal functions are available in [29].

2. Homotopy Perturbation

Non-linear techniques for solving linear and non-linear problems have been dominated by the perturbation methods, which have found wide applications in engineering. But, like other non-linear analytical techniques, perturbation methods have their own limitations, Firstly, almost all perturbation methods are based on small parameters so that the approximate solutions can be expressed in a series of small parameters. This so called small parameter assumption greatly restricts applications of perturbation techniques, as is well known, an hefty gigantic of linear and non-linear problems have no small parameters at all. Secondly, the determination of small parameters seems to be a special art requiring special techniques. An appropriate choice of small parameters leads to ideal results, however, an unsuitable choice of small parameters results in bad effects. In 1997, Liu [30] proposed a new perturbation technique which is not based upon small parameters but upon artificial parameters, which are built in the equations.

One may consider the following nonlinear differential equation (see [31-36])

(2.1)

with natural boundary conditions or tangentiality conditions as:

(2.2)

where A is a general differential operator, B is a boundary operator, is a known analytic function and is the boundary of the domain.

The operator can, generally, be divided into two parts L and N, where L is Linear, while N is nonlinear, so that (2.1) may written as:

(2.3)

By homotopy perturbation technique, we construct a homotopy which satisfies

(2.4)

or

(2.5)

where is an embedding parameter, and is an initial approximate solution of Equation (2.1).

Obviously from Equation (2.5)

(2.6)

(2.7)

By changing continuously from zero to unity the Equations (2.6) and (2.7) show that will change from to. In topology, this changing is called deformation, and are called homotopy functions.

In this method, using the homotopy parameter, we assume that the solution of Equation (2.5) is a power series of:

(2.8)

Letting results in the approximate solution of Equation (2.1) as:

(2.9)

Series (2.9) is convergent for most cases, the convergent rate depends upon the nonlinear operator A(v).

3. Solution of the Optimal Control System

In this section, we apply the homotopy perturbation method to solve the optimal control system (1.1).

Consider Hamiltonian of the control system (1.1) as:

(3.1)

where is known as the costate variable. By Pontryagin’s maximum principle, the optimal control must satisfy the following equation:

(3.2)

where is a solution of the adjoint equation

(3.3)

with the terminal condition

(3.4)

Thus, from Equation (3.2), the optimal control law is

(3.5)

From control system (1.1) and adjoint Equation (3.3) one have:

(3.6)

Implementing the optimal control as a closed loop if the solution to the adjoint Equation (3.3) is assumed like Equation (3.4) as a linear function of the states in the form( see [29]),

(3.7)

By using Equations (3.3), (3.6) and (3.7), we have

where the first equality follows from Equation (3.7) and the second one from Equation (3.6). Hence

(3.8)

Since the above equation must hold for all nonzero, must satisfy the following matrix Riccati equation

(3.9)

Considering Equations (3.5) and (3.7), we can see that the optimal control law is given as

(3.10)

and can be computed using the following relation

(3.11)

where, and

with conditions, and

4. Numerical Examples

In this section, we present some examples to show the reliability and efficiency of the method described in the previous section. In the following examples, we assume.

Example 4.1. Consider a single-input scalar system as follows (see [29]):

According to system (1.1), we have and by using (3.12), we have

thus

So

by using (2.8), let and so,

from equating the terms with identical power of p,

(4.1)

(4.2)

(4.3)

where are considered as initial approximations, and imposing boundary condition, so

by using (4.3), we have

From (2.9), we have

Figures 1 and 2 show the approximated value of and, respectively.

Example 4.2. Consider a single-input scalar system as follows (see [29]):

According to system (1.1), we have and, by using (3.12), we have

thus

So

by using (2.8), we have

from equating the terms with identical power of p,

where are considered as initial approximations, Setting and imposing boundary condition, so

by using (4.3), we have

From (2.9), we have

.

Figures 3 and 4 show the approximated value of x(t) and u(t), respectively.

Example 4.3. Consider a single-input scalar system as follows:

According to system (1.1), we have

and and by using (3.12), we have

thus

So

by using (2.8), we have

from equating the terms with identical power of p,

where, are considered as initial approximations, Setting and imposing boundary condition, the approximate and exact value for is the

exact value for is and the approximate value for and obtained from this algorithm in this following form

Figure 5 compares the exact and approximate solution of x(t), and Figure 6 shows the residual function.

5. Conclusions

In this paper, we solve the optimal control problems us-

ing Homotopy perturbation method. Embedding parameter can be taken into account as a perturbation parameter. Full advantage of the traditional perturbation techniques can be taken by the novel method. The initial approximation can be freely chosen with unknown constants, which can be identified via various methods [35].

At last, Homotopy perturbation method is applicable method which calculates the approximate solution of linear and nonlinear problems, particularly optimal control problems.

REFERENCES