Solving a Class of Nonlinear Optimal Control Problems via He's ...

International Journal of Control, Automation, and Systems (2012) 10(2):249-256 DOI 10.1007/s12555-012-0205-z

ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

Solving a Class of Nonlinear Optimal Control Problems via He’s Variational Iteration Method Mohammad Shirazian and Sohrab Effati Abstract: This paper presents an analytical approximate solution for a class of nonlinear quadratic optimal control problems. The proposed method consists of a Variational Iteration Method (VIM) together with a shooting method like procedure, for solving the extreme conditions obtained from the Pontryagin’s Maximum Principle (PMP). This method is applicable for a large class of nonlinear quadratic optimal control problems. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Two illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method. Keywords: Optimal control problem, Pontryagin’s maximum principle, suboptimal control, variational iteration method.

1. INTRODUCTION The optimal control of nonlinear systems is one of the most challenging and difficult subjects in control theory. For the general optimal control problem (OCP) for nonlinear systems, however, an analytical solution does not exist. This has inspired researchers to propose approaches to obtain an approximate solution for it. There exists two major approaches. One of them is to directly face the problem and attempt to find a minimum of the objective (or Lagrangian) functional. The other one is using the optimality necessary conditions and attempt to solve them [1]. There are a number of direct methods. One of them is to cast the nonlinear OCP in the form of a nonlinear programming problem [1,2]. Another one is to treat the problem with a Measure theory approach [3]. This changes the nonlinear OCP to a linear programming and gives a piecewise constant control law. In the past two decades, the indirect methods have been extensively developed. It is well-known that the nonlinear OCP leads to a nonlinear Two-Point Boundary Value Problem (TPBVP) or a Hamilton-Jacobi-Bellman (HJB) partial differential equation. Many recent researches have been devoted to solve these two problems. The HJB equations are solved by different approaches, such as power series approximation [4], State Dependent Riccati Equation (SDRE) [5], Successive Galerkin __________ Manuscript received November 2, 2010; revised June 4, 2011; accepted November 23, 2011. Recommended by Editorial Board member Guang-Hong Yang under the direction of Editor Young Il Lee. Mohammad Shirazian and Sohrab Effati are with the Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad, Iran (e-mails: [email protected], [email protected]). © ICROS, KIEE and Springer 2012

Approximation (SGA) [6], and etc. The other indirect way of treating the nonlinear OCP is to solve a nonlinear TPBVP, obtained by PMP. One of these approaches is Successive Approximation Approach (SAA) which designs an suboptimal controller for a class of nonlinear systems with a quadratic performance index. In this approach, a sequence of nonhomogeneous linear time-varying TPBVP’s is solved to produce a finite-step iteration of the nonlinear compensation sequence obtaining the suboptimal control law [7]. However, SAA needs to solve a linear time-varying TPBVP’s which cannot be solved easily and thus, reduces the efficiency of this method. Recently, in [8], a novel method which implements Modal series to solve a class of nonlinear OCP’s with quadratic performance index, has been proposed. This method which requires solving a sequence of linear timeinvariant TPBVP’s, has less efficiency for large scale problems. Other various methods for solving the TPBVP obtained from PMP, are available in [9]. On the other hand, in the context of numerical analysis, the VIM which was proposed originally by He [10,11], has been proved by many authors to be a powerful mathematical tool for various kinds of linear and nonlinear ODE’s or PDE’s. Unlike the traditional numerical methods, VIM needs no discretization, linearization, transformation or perturbation. The method, has been widely applied to solve nonlinear problems, and different modifications are suggested to overcome the demerits arising in the solution procedure (see for example [12,13]). Among a various number of VIM applications, the use of this method in solving Riccati equations (see for example [14,15]), made it a powerful tool in the context of control theory. In [16], the authors used the original or basic VIM for linear quadratic OCP’s. They transfer the linear TPBVP obtained from PMP to an initial value problem and then

Mohammad Shirazian and Sohrab Effati

250

implement the basic VIM to get a feedback controller. This paper concerns with a class of nonlinear quadratic OCP’s. As discussed earlier, using PMP, one can obtain a nonlinear TPBVP. Applying the main ideas of the shooting method to the basic and also a modified version of VIM, we solve this TPBVP. Two illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method. This paper is organized as follows. Section 2 describes the quadratic nonlinear OCP and its necessary and sufficient extreme conditions. Solving procedure based on VIM is given in Section 3. Section 4 introduces an algorithm to produce the desirable suboptimal control and state. Two illustrative examples are given in Section 5.

x(t0 ) = x 0 , x(t f ) = x f ,

1 tf T ( x (t )Qx(t ) + uT (t ) Ru (t ))dt , 2 ∫ t0

u* = − R −1 g T (t , x)λ.

the

(5)

For the sake of simplicity, let us define the right hand sides of (4) as follows: Φ(t , x, λ ) := f (t , x) + g (t , x)[− R −1 g T (t , x)λ ], ∂f (t , x) T  Ψ (t , x, λ ) := −  Qx + ( ) λ ∂x  n

+ ∑ λi [− R −1 g T (t , x)λ ]T

x = Φ(t , x, λ ), λ = Ψ (t , x, λ ),

(1)

with x(t ) ∈ » n denoting the state variable, u (t ) ∈ » m the control variable and x0 and xf the given initial and final states at t0 and tf , respectively. Moreover, f (t , x(t )) ∈ » n and g (t , x(t )) ∈ » n×m are two continuously differentiable functions in all arguments. Our aim is to minimize the quadratic objective functional J [ x, u ] =

Also

(6) ∂gi (t , x)  . ∂x 

Thus the TPBVP in (4) changes to:

Consider the following nonlinear dynamical system t ∈ [t0 , t f ],

g n (t , x)]T with gi (t , x) ∈ » m , i = 1,..., n. optimal control law is obtained by

i =1

2. NONLINEAR QUADRATIC OCP AND OPTIMALITY CONDITIONS

x (t ) = f (t , x(t )) + g (t , x(t ))u (t ),

where λ (t ) ∈ » n is the co-state vector with the ith component λi (t ), i = 1,..., n and g (t , x) = [ g1 (t , x),...,

(2)

t ∈ [t0 , t f ],

(7)

x(t0 ) = x 0 , x(t f ) = x f .

Unfortunately, there is no analytical solution to this nonlinear TPBVP, in general. So, it is of high importance to calculate analytic approximate or numerical solutions for it. In recent decades, some new numerical and analytic approximate methods have been proposed for solving such a difficult problem in the context of ordinary differential equations (see for example [7,8]). In the next section, we propose another analytic approximate method based on VIM, for this purpose.

subject to the nonlinear system (1), for Q ∈ » n×n , R ∈

3. SOLVING TPBVP BASED ON VIM

» m×m positive semi-definite and positive definite matrices, respectively. Since the performance index (2) is convex, the following extreme necessary conditions are also sufficient for optimality:

In this section, we use a shooting method like procedure combined with the VIM for solving TPBVP in (7). So, we first solve the following initial value problem, using a (modified) version of VIM,

x = f (t , x) + g (t , x)u * , λ = − H ( x, u* , λ ), x

u* = arg min u H ( x, u , λ ),

x = Φ(t , x, λ ), λ = Ψ (t , x, λ ),

(3)

x(t0 ) = x 0 , x (t f ) = x f , 1 where H ( x, u, λ ) = [ xT Qx + uT Ru] + λT [ f (t , x) + g (t , x)u] 2 is referred to as the Hamiltonian. Equivalently, (3) can be written in the form of:

x = f (t , x) + g (t , x)[− R −1 g T (t , x)λ ], ∂f (t , x ) T  λ = −  Qx + ( ) λ ∂x  n

∂g (t , x)  + ∑ λi [− R −1 g T (t , x)λ ]T i , ∂x  i =1 x(t0 ) = x0 , x(t f ) = x f ,

(4)

t ∈ [t0 , t f ],

(8)

x(t0 ) = x 0 , λ (t0 ) = α ,

where α ∈ » n is an unknown parameter. This parameter will be identified after sufficient iterations of VIM, as discussed hereinafter. It is well known that VIM gives rapidly convergent successive approximations of the exact solution, if such a solution exists or otherwise, approximations can be used for numerical purposes. According to the VIM [10,11], for solving an initial value problem such as (8), we can construct a correction functional as follows: t

xn +1 (t ) = xn (t ) + ∫ Λ1 ( s ){xn ( s) − Φ( s, xn ( s), λn ( s))}ds, t0

t λn +1 (t ) = λn (t ) + ∫ Λ 2 ( s ){λn ( s ) − Ψ ( s, xn ( s ), λn ( s ))}ds t0

(9)

Solving a Class of Nonlinear Optimal Control Problems via He’s Variational Iteration Method

with n ≥ 0, and the initial values x0 (t ) = x(t0 ) = x 0 and λ0 (t ) = λ (t0 ) = α . By taking variation with respect to xn and λn , and considering the restricted variations δΦ( s, x , λ ) = δΨ ( s, x , λ ), = 0 we get n

n

n

n

δ xn +1 (t ) = δ xn (t ) + Λ1 ( s )δ xn ( s ) |s =t −∫

t

t0

which yields the following stationary conditions: ( s ) = 0, 1 + Λ1 ( s ) |s =t = 0, Λ 1 ( s ) = 0. 1 + Λ ( s ) | = 0, Λ

It can be readily identified that Λ1 ( s ) = Λ 2 ( s ) = −1. Therefore, the VIM formula becomes: t

λn +1 (t ) = λn (t ) − ∫

{ xn ( s) − Φ( s, xn (s), λn ( s))} ds,

t t0

{

}

λn ( s ) − Ψ ( s, xn ( s ), λn ( s )) ds

(10)

0

with n ≥ 0, x0 (t ) = x , and λ0 (t ) = α . Theorem 1: Assume that {xn (t )} and {λn (t )} are the solution sequences produced by VIM formula (10), which converge respectively to xˆ(t , α ) and λˆ (t , α ), as n → ∞. Then xˆ (t , α ), λˆ (t , α ) are the exact solutions of (8). Accordingly, xˆ(t , αˆ ), λˆ (t , αˆ ) are the exact solutions of (7) when αˆ is the real root of xˆ(t f , α ) − x f = 0. Proof: Taking limits of both sides of (10) as n → ∞, since lim xn (t ) = lim xn +1 (t ) = xˆ (t , α ) and lim λn (t ) = n →∞

n →∞

n →∞

lim λn +1 (t ) = λˆ (t , α ), we have

n →∞

∫ t { xˆ (s, α ) − Φ(s, xˆ(s,α ), λˆ(s, α ))} ds = 0, t

0

∫ t {λˆ(s,α ) − Ψ (s, xˆ (s,α ), λˆ(s, α ))} ds = 0. t

0

Differentiating from both sides with respect to t, yields

xˆ (t , α ) = Φ(t , xˆ (t , α ), λˆ (t , α )), λˆ (t , α ) = Ψ (t , xˆ (t , α ), λˆ (t , α )). Moreover, if t = t0, then from (10), xn +1 (t0 ) = xn (t0 ), λn +1 (t0 ) = λn (t0 ) for every n ≥ 0. Thus, xn (t0 ) = 0

and λn (t0 ) = λ0 (t0 ) = α or equivalently, xˆ(t0 , α ) = x and λˆ (t0 , α ) = α . Hence, xˆ(t , α ) and λˆ (t , α ) are the exact solutions of (8). In addition, they x0 (t0 ) = x

1 tf T ( xn (t )Qxn (t ) + unT (t ) Run (t ))dt , 2 ∫ t0

(12)

uˆ(t ) := lim un (t ) = − R −1 g T (t , lim xn (t )) lim λn (t )

2

t0

(11)

converge to the optimal control law and optimal objective value, respectively. Proof: Since g is a continuous function, taking limit from (11) and putting α = αˆ , results in

t0

xn +1 (t ) = xn (t ) − ∫

=0 by αˆ completes the proof. Theorem 2: Under the Theorem 1 assumptions, the sequences {un (t )} and {Jn} defined by

Jn =

t ( s )δλ ( s)ds = 0, −∫ Λ 2 n

s =t

choose the unknown parameter α ∈ » n such that xˆ(t f , α ) = x f . Denoting this real root of xˆ(t f , α ) − x f

un (t ) = − R −1 g T (t , xn (t ))λn (t ),

( s )δ x ( s )ds = 0, Λ n 1

δλn +1 (t ) = δλn (t ) + Λ 2 ( s )δλn ( s ) |s =t

2

251

0

are the exact solutions of (7), only if the final state condition x(tf ) = xf is satisfied. So, it is straightforward to

n →∞ −1 T

n →∞

n →∞

= − R g (t , xˆ (t , αˆ ))λˆ (t , αˆ ), where uˆ(t ) is the optimal control law, since xˆ(t , αˆ ) and λˆ (t , αˆ ) are the exact solutions of extreme conditions in (7). Similarly, taking limit from (12), yields tf 1 Jˆ := lim J n = lim ∫ ( xnT (t )Qxn (t ) + unT (t ) Run (t ))dt n →∞ 2 n→∞ t0 1 tf = ∫ ( lim xnT (t )Q lim xn (t ) n →∞ 2 t0 n→∞ T + lim un (t ) R lim un (t ))dt n →∞

n →∞

1 tf = ∫ ( xˆ (t , αˆ )T Qxˆ (t , αˆ ) + uˆ (t )T Ruˆ (t )) dt. 2 t0

It follows that Jˆ is the optimal objective value and the proof is completed. The VIM formula (10) is directly dependent on the integration. In our optimal control problem, Φ and Ψ might be two nonlinear functions which cause the righthand-side integration of (10) to be very time consuming and complicated, even at the first few iterations of VIM. To overcome this undesirable case, if x, λ, Φ and Ψ are analytic functions, the following correction, which omits the time consuming and repeated calculations from VIM, can be applied [15]: xn +1 (t ) = xn (t ) − ∫ λn +1 (t ) = λn (t ) − ∫

t t0 t t0

{Tn ( s) − Tn −1 ( s)}ds,

{

}

Tˆn ( s) − Tˆn −1 ( s) ds,

(13)

where n ≥ 0, and Tn ( s ) and Tˆn ( s) are the n-th order Taylor expansions of

xn ( s ) − Φ ( s, xn ( s ), λn ( s ) )

and

λn ( s ) − Ψ ( s, xn ( s ), λn ( s ) ) , respectively, around t0. Also x0 (t ) = x 0 , λ0 (t ) = α , T−1 ( s ) = Tˆ−1 ( s) = 0. As can be seen in (13), the integrands are subtractions of two consecutive Taylor expansions, that is the n-th term of Taylor expansion. Therefore, the complicated


252

integrations in (10) reduces to integration of n-th term of Taylor expansion. We should point out that the foregoing procedure is similar to the shooting method. In fact, for solving a TPBVP, shooting method begins with an unknown initial condition. Propagating the differential equation from initial to final time, the unknown initial condition is calculated such that the final condition is satisfied (for more details, see [1]). Remark 1: If x(tf) is free, then the conditions in (7) become x = Φ(t , x, λ ), λ = Ψ (t , x, λ ),

t ∈ [t0 , t f ],

x(t0 ) = x 0 , λ (t f ) = 0.

In this case, we will solve the initial value problem (8). Then α can be calculated imposing final condition λ(tf) = 0, in the same manner discussed above. Remark 2: It should be mentioned that it is complicated to discuss about the real roots of xN (tf , α) – xf = 0, in general, since VIM is directly dependent on the structure of the problem. But it is natural to obtain more than one α ∈ » n by implementing some numerical methods. In this case, we propose to choose an α, which its associated x(t) and u(t) give the minimum value of the objective functional J[x, u]. It also ensures that x(t) and u(t) are the nearly optimal state and control. 4. SUBOPTIMAL CONTROL DESIGN AND PROPOSED ALGORITHM The solution guidelines for TPBVP (7) has been discussed in previous section. In this section, we give a more reliable way for finding the desired suboptimal control and the suboptimal state and then we present an algorithm for this end. From Theorem 2, we conclude that for large number of iterations, N, suboptimal control law is derived by u* ≅ u N (t ) = − R −1 g T (t , x)λN (t , αˆ ),

1 tf T ( x (t )Qx(t ) + uTN (t ) Ru N (t )) dt. 2 ∫ t0

(15)

For the accuracy analysis, we consider the following criterion. The suboptimal control (14) has the desirable accuracy, if for given ε1 , ε2 > 0, the following two conditions hold jointly, J N − J N −1 < ε1 , ‖x(t f ) − x f ‖< ε2 , JN

the familiar Euclidean norm. This comes from the fact that any two arbitrary norm in » n are equivalent. If the tolerance limits ε1 and ε2 , are sufficiently small, the suboptimal value is very close to the optimal value J *, according to Theorem 2, and the boundary state conditions will be satisfied tightly. Now, we present an algorithm of the proposed method with low computational complexity, in order to maintain the accuracy of solutions. Algorithm: Step 1: Let N = 1 and ε1 , ε2 > 0 small tolerances. Step 2: Solve (8) implementing modified VIM in (10) and (13) to λN (t , α ). Step 3: Solve xN (t , α ) – xf = 0

be two sufficiently the basic or the find xN (t , α ) and to

calculate

an

n

approximation value for α ∈ » , say αˆ . Step 4: Substitute αˆ in (14) to determine a suboptimal control uN (t) and calculate the suboptimal state by solving the nonlinear system (1). Step 5: If criterion (16) holds, go to Step 6, otherwise let N = N + 1 and go to Step 2. Step 6: Stop the algorithm. uN (t) is the desirable suboptimal control and the state calculated in Step 4, is the desirable suboptimal state. Remark 3: If x(t f ) is free, then the stop criterion in (16) changes to J N − J N −1 < ε, JN

(17)

where ε > 0 is any given error tolerance. We illustrate this procedure through two examples.

(14)

and the approximate suboptimal state is x(t ) ≅ xN (t , αˆ ). To get a more accurate suboptimal state, one can put the suboptimal control derived by (14), in original nonlinear system (1), and solve it wihout using the final state condition. Then applying this pair of suboptimal control and state to the objective functional (2), yields in the suboptimal value of the problem, i.e., JN =

where x(.) is the suboptimal state obtained by applying the suboptimal control (14) to the nonlinear system (1) and ⋅ denotes any arbitrary norm in » n , for example

(16)

5. ILLUSTRATIVE EXAMPLES The following two examples are given to illustrate the simplicity and efficiency of the proposed method. The codes are developed using symbolic computation software MAPLE 13 and the calculations are implemented on a machine with Intel Core 2 Due Processor 2.53 Ghz and 4 GB RAM. Example 1: Consider the following nonlinear optimal control problem 1

minimize J = ∫ u 2 (t ) dt 0

1 2 x (t ) sin x(t ) + u (t ), t ∈ [0,1] 2 x(0) = 0, x(1) = 0.5.

subject to : x =

Using the extreme necessary and sufficient conditions,


we have x (t ) = 1 x 2 (t ) sin x(t ) − 1 λ (t ), t ∈ [0,1], 2 2 λ (t ) = −λ (t ) x (t ) sin x (t ) − 1 λ (t ) x 2 (t ) cos x(t ), 2 x(0) = 0, x(1) = 0.5,

and the optimal control law is u (t ) = − 1 λ (t ). For 2 solving this TPBVP, first, we substitute the final state condition x(1) = 0.5, by λ (0) = α , i.e., x (t ) = 1 x 2 (t ) sin x(t ) − 1 λ (t ), t ∈ [0,1], 2 2 λ (t ) = −λ (t ) x(t ) sin x(t ) − 1 λ (t ) x 2 (t ) cos x(t ), 2 x(0) = 0, λ (0) = α ,

where α ∈ » is an unknown parameter. To solve this initial value problem, the successive formula (10) becomes t

xn +1 (t ) = xn (t ) − ∫ {xn (s) − 1 xn2 (s)sin xn ( s) + 1 λn (s)}ds, 0 2 2 t

λn +1 (t ) = λn (t ) − ∫ {λn ( s) + λn ( s) xn ( s ) sin xn ( s) 0

+ 1 λn ( s) xn2 ( s) cos xn ( s)}ds, 2 x0 (t ) = 0, λ0 (t ) = α , n ≥ 0.

(18)

This is the basic VIM which gives the following solutions: x1 (t ) = − 1 α t , λ1 (t ) = α , 2 x2 (t ) = 1 α t 2 cos( 1 α t ) + (− 1 α − sin( 1 α t ))t 4 2 2 2 2 1 + (1 − cos( α t )), α 2 2 2 1 λ2 (t ) = α − α t sin( 1 α t ). 4 2

The integration can not be calculated for the next iteration, i.e., x3 (t ), λ3 (t ). The relative error of objective value and the final state error are: J 2 − J1 = 1.20706 ×10−1 , J2 x(t f ) − x f = 1.39776 ×10−2.

Obviously, given ε1 = 2 × 10−3 and ε2 = 5 × 10−5 , the VIM formula (18) is not suitable for this problem. Thus it is better to apply the modified VIM introduced in Section 3. Denote the n-th order Taylor expansions of {xn ( s) − 1 xn2 ( s) sin xn ( s) + 1 λn ( s)} and {λn ( s ) + λn ( s ) 2 2 2 1 xn ( s) sin xn ( s) + λn ( s ) xn ( s) cos xn ( s)}, by Tn(s) and 2 Tn ( s). Using the modified VIM formula (13), the following results are obtained:

x1 (t ) = − 1 α t , 2 x2 (t ) = − 1 α t , 2 1 x3 (t ) = − α t , 2 x4 (t ) = − 1 α t , 2 1 x5 (t ) = − α t , 2

253

λ1 (t ) = α , λ2 (t ) = α ,

λ3 (t ) = α − 1 α 3t 3 8 λ4 (t ) = α − 1 α 3t 3 8 1 λ5 (t ) = α − α 3t 3 + 1 α 5t 5 8 192

By considering the final state condition, we should have xi (1) = 0.5, for i = 1, …,5. That is, − 1 α = 0.5 or 2 equivalently, α = −1. For ε1 = 2 × 10−3 and ε2 = 5 × 10-5, as in [8], the accuracy analysis is summarized in Table 1. According to Table 1, after N=5 iterations, the desired suboptimal control is as follows: 1 1 1 1 5 u* (t ) ≅ u5 (t ) = − λ5 (t ) = − (−1 + t 3 − t ). 2 2 8 192

In analogy with the modal series approach, Table 1 of [8], the suboptimal control are very close together in the sense that, ‖u5VIM (t ) − u5Modal ‖∞ = 3.23 × 10−4 , but in our modified VIM approach, after five iterations, the final state condition is more tightly satisfied. This problem has also been solved by Rubio in [3] via measure theory in which to find an acceptable solution, a linear programming problem with 1000 variables and 20 constraints should be solved. It is noticeable that the error of measure theory method can not be controlled but in our method it is controlled by criterion (16) or (17). The comparisons of these three methods are presented in Table 2, for the same ε1 and ε2 . Also the suboptimal Table 1. Simulation results for different iterations, Example 1. N

JN

J N − J N −1 JN

1 2 3 4 5

0.25000 0.25000 0.23493 0.23493 0.23533

0 6.41330×10-2 0 1.69325×10-3

‖x(t f ) − x f ‖

1.60200×10-2 1.60200×10-2 4.47430×10-4 4.47430×10-4 4.18320×10-6

Table 2. Comparing results of proposed modified VIM, modal series and measure theory approaches, Example 1. Method Proposed modified VIM (N=5) Modal series method (N=5) Measure theory method

Objective Objective value value relative error

Final state error

0.2353

1.7 × 10−3

4.2 × 10−6

0.2353

1.7 × 10−3

2.8 × 10−5

0.2425

-

4.3 × 10−3


254

Fig. 1. The suboptimal control, Example 1.

Fig. 3. The suboptimal control, Example 2.

Fig. 2. The suboptimal state, Example 1.

Fig. 4. The suboptimal state x1(t), Example 2.

control and state of the proposed modified VIM and modal series method are demonstrated in Figs. 1 and 2. Example 2: Consider the nonlinear system described by x1 = x2 + x1 x2 , x2 = − x1 + x2 + x22 + u, x1 (0) = −0.8, x2 (0) = 0,

and the cost functional J=

1 1 2 ( x1 + x22 + u 2 ) dt. 2 ∫0

The extreme conditions are λ1 = −( x1 + λ1 x2 − λ2 ), λ2 = −( x2 + λ1 (1 + x1 ) + λ2 (1 + 2 x2 )), x1 = x2 + x1 x2 , x2 = − x1 + x2 + x22 − λ2 , x1 (0) = −0.8, x2 (0) = 0, λ1 (1) = λ2 (1) = 0,

and the optimal control is u = −λ2 . Implementing the algorithm described in Section 4 and using Remark 3 with ε = 10−3 , the desired suboptimal control is obtained for N = 10 iterations, as:

Fig. 5. The suboptimal state x2(t), Example 2. u* (t ) ≅ u10 (t ) = −0.59860 + 0.35934 t + 0.18145 t 2 + 0.084636 t 3 − 0.25200 ×10−2 t 4 − 0.012546 t 5 − 0.96525 ×10−2 t 6 − 0.26026 ×10−2 t 7 − 0.17137 ×10−3 t 8 + 0.42385 ×10−3 t 9 + 0.24035 ×10−3 t10 .

Figs. 3, 4, and 5 show the solutions after N=10 algorithm iterations, compared to the collocation method [17]. We should mention that for some iterations, N, solving λ1, N (1, α ) = 0, produces different α’s. In these cases, we use Remark 2. For example, if N = 2, the roots of λ1,2 (1, α ) = 0 and λ2,2 (1, α ) = 0 where α = (α1 , α 2 ),


Table 3. Calculation of α for N = 2, Example 2. Objective value α2 α1 -2.331429725 5.922045520 -1.390615796

-0.5493566136 -1.086679580 0.5360361938

0.680946053 1.760396681 0.410273933

Table 4. Comparison of the basic and modified VIM, based on CPU time and objective value relative error, Example 2. VIM N

J N − J N −1 JN

CPU time (sec.)

1 2 3 4 5 6 7 10 20 30

1.07743 5.03437×10-2 1.58334×10-2 6.14665×10-3 1.83968×10-3 -

0.031 0.031 0.046 0.125 1.544 45.693 ≥1800 -

[3]

[4]

[5]

Modified VIM J N − J N −1 CPU time (sec.) JN 1.07991 4.67492×10-2 1.57128×10-2 5.60038×10-3 3.14261×10-3 3.33818×10-3 6.96060×10-4 2.14354×10-6 4.51176×10-9

0.031 0.031 0.032 0.047 0.047 0.063 0.093 0.296 7.051 67.049

[6]

[7]

[8]

are as in Table 3. From Table 3, it is clear to choose α1 = −1.39061579 and α 2 = 0.5360361938. . For N=3 one can use these α1 and α2 as initial guesses to solve λ1,3 (1, α ) = 0 and λ2,3 (1, α ) = 0.

[10]

To analysis the accuracy and effectiveness of the basic and modified VIM, the CPU time and the relative error of objective value are summarized in Table 4, for several iterations.

[11]

[9]

6. CONCLUSIONS [12] In this paper, a novel method for solving a class of nonlinear optimal control problems has been proposed. This method can solve the TPBVP obtained from PMP by the well known VIM. The recursive solution for suboptimal control does not need complex on-line computations compared to the SDRE, SGA and SAA techniques. In addition, two illustrative example demonstrated the good results and comparisons with modal series and measure theory methods. This technique can be applied to a broad class of nonlinear optimal control problems. An algorithm of low computational complexity has been presented to produce the desirable suboptimal control law. Future works are focused on VIM modification in order to solve the described TPBVP.

[13]

[14]

[15]

[16] [1]

[2]

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Mohammad Shirazian received his B.S. and M.S. degrees in applied mathematics from Ferdowsi University of Mashhad, in 2007 and 2009 respectively. He is now pursuing a Ph.D. degree in applied mathematics (control and optimization) at Ferdowsi University of Mashhad, Mashhad, Iran. His research interests include optimal control theory and its applications, numerical methods for solving ODEs and PDEs, fuzzy theory, and neural networks.

Sohrab Effati received his B.S. degree in Applied Mathematics from Birjand University, Birjand, Iran, his M.S. degree in Applied Mathematics from Institute of Mathematics at Tarbiat Moallem Tehran University, Tehran, Iran, in 1992 and 1995, respectively. He received his Ph.D. degree in Control Systems from Ferdowsi University of Mashhad, Mashhad, Iran, in April 2000. He is an Associate Professor with the Department of Applied Mathematics at Ferdowsi University of Mashhad in Iran. His research interests include control systems, optimization, fuzzy theory, and neural network models and its applications in optimization problems, ODE and PDE.