Research Journal of Applied Sciences, Engineering and Technology 4(12): 1822-1827, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 31, 2012 Accepted: April 11, 2012 Published: June 15, 2012
Abstract: This study presents a numerical method for solving nonlinear optimal control problems including terminal state constraints, state and control inequality constraints. The method is based on Triangular
Orthogonal Functions (TFs). By approximating the dynamic systems, performance index and boundary conditions into triangular orthogonal series, the optimal control problem is converted into algebraic equations with unknown coefficients. Then the problem can be easily solved by some iterative method. Illustrative examples are included to demonstrate the accuracy and applicability of the presented method.
Key words: Triangular orthogonal functions, Optimal control, Direct method, Inequality constraints, Nonlinear
INTRODUCTION
In large number of problems arising in analysis, mechanics, geometry and so forth, it is necessary to determine the maximum or minimum of a certain functional. Because of the important role of this subject in science and engineering, considerable attention has been received by this kind of problem. Such problems are called optimal control problems. A widely used method to solve this kind problem is the direct method.
The main idea of direct method for solving the optimal control problem is to convert the problem of finding the extreme of a functional into a problem of finding a solution in a finite set of parameters. In other words, the direct method converts nonlinear differential equations with boundary-initial value problem into a parametric problem with algebraic equations.
In the past, several orthogonal functions and wavelets are applied in the direct solution of the optimal control problem. The problem is solved by using the Haar wavelets (Hsiao, 2004). Walsh wavelets (Daubechies,
1992) and Legendre (Yousefin and Razzaghi, 2005) wavelets were are also applied for analyzing these problems and approximating the solution of integral equations .Being different from above orthogonal functions, we will use a new set of orthogonal functions derived from block pulse functions to implement the direct method. in (Deb et al.
, 2006), presented a new set of orthogonal functions which they called it triangular orthogonal functions and then, the calculated the operational matrix for integration.
In this study, we present a computational method for solving nonlinear optimal control problems with state and control inequality by using triangular orthogonal functions. We apply this set of orthogonal functions in parameterization of both state and control. Then the dynamic systems and functional can be approximated by triangular orthogonal series. Also, the inequality constraints are adjoined into the optimal control problem by using the auxiliary functions. After that, the optimal control problem is deduced to solve a system of algebraic equations. These equations can be solved by some iterative method.
REVIEW OF TRIANGULAR
ORTHOGONAL FUNCTIONS
Triangular orthogonal function: many authors (Deb interval [0,T) et al is given by:
 i
 

1 ,
0 , i
, t
 i
 m m otherwise
1
In the past decades,
., 1994; Jiang and Schaufelberger,
1992) proposed a set of Block Pulse Functions (BPF) containing m component functions in the semi-open where i = 0, 1, m-1and m is a positive integer.
(1)
With the help of BPFs, various control systems were analyzed effectively, but a staircase solution provided by the BPF domain analysis may introduce relatively more error than an equivalent piecewise linear solution.
Based on BP
Fs, Anish Deb et al . (1994) proposed Triangular
Functions (TFs). They dissected a BPF into two TFs according to:
N
(m)
(t) = T1
(m)
(t)+T2
(m)
(t) (2)
Corresponding Author: Zhenyu Han, College of Information and Control Engineering, China University of Petroleum, Qingdao
266580, PR China
1822

Res. J. Appl. Sci. Eng. Technol., 4(12): 1822-1827, 2012 whereT1
(m)
(t) and T2
(m)
(t) are called the Left-Handed
Triangular Function (LHTF) (Deb et al.
, 2006) and the
Right-Handed Triangular (RHTF) vector, respectively.
The i-th component of the LHTF vector T1
(m)
(t) is defined as:
T 1 i
( )
 

1
 t
 ih
,
0 , h ih
   i

1
 h otherwise
(3) where the matrices P1
(m) and P2
(m) are the operational matrices for integration in TF domain and the P1
(m) and
P2
(m) are as follows:
P 1
 h
2








0
0
0
1
0
0
1
0

 
0 0 0  1
     


1
1
0





 and P 2
 h
2






1
0
0
1
1
0
1
0

 


0 0 1  1
    

1
1
1








(12) and the i-th component of the RHTF vector T2
(m)
(t) is defined as:
T 2 i
( )
 
 t
 ih
, h
0 , ih
 


1
 h otherwise
(4) where P1 and P2 are m operational matrixes for integration. The integration matrixes of T2(t) are the same as (11) and (12).
Using Eqs. (11) and (12), the integration of f(t)
0
(0,1) can be written as: where, i = 0,1,..., m-1.
The orthogonality property for the LHTF set and
RHTF set are given by:

0
1 f d

C
T
 t
0
T 1 d
 
D
T

0 t
T 2 d


C
T
( 1 1

)

D
T
( 1 1

( C

D ) ( 1 1

)

)
(13) and
 1
0
T 1 i
( ) 1 j
( )
 
 h
3
0 ,
, i
 j i
 j
 1
0
T 2 i
( ) 2 j
( )
 
 h
3
0 ,
, i
 j i
 j
(5)
(6)
Knowing C and D, which are in fact (m+1) number samples of f(t) from 0 to 1, we can determine approximate integration of f(t).
Problem statment: Find the control vector U(
J
) and the corresponding state vector X(
J
),
J 0 (0, t f
), which minimize
(or maximize) the functional: where h = 1/m.
It should be noted that the set of TFs is a complete orthogonal set in the Hilbert space L 2 [0,1]. Thus any function in this space can be expanded in terms of Tfs.
Function approximation: A function f(t) 0 L 2 [0,1], may be expanded into an m-term TF series as:
T 2

 c c c i c m

1

T 1
( )

C T 1 ( )

D T 2

 d d d i d m

1

(7)
J = H(X(t f
),tf)+

0 t f
G X

U
   (14) where the constant coefficients c i and d i
are given by: c i
= f(ih) (8) d i
= f[(i+1)h] (9) c i+1
= d i
(10)
Operational matrix of integration: The integration of the T1is given by:

0
1
T 1
( m ) d
 
P 1
( m )
T 1
( m )

P 2
( m )
T 2
( m )
(11) subject to,

(

U
   
0 t f
], X 0
 x
0
(15)
S i
(X(
J
),U(
J
),
J
)
#
0,
J 0
[0, t f
], i = 1, …, v (16) where, X(
J
) and U(
J
) are l×1and q×1 vectors, respectively. The vector function F and scalar functions
H,G and S i
are generally nonlinear and assumed to be smooth with respect to X and U. Also, t f
denotes the final time which may be free. It is assumed that the problem
(14)-(16) has a unique solution. The time transformation
J
= t f t is introduced in order to use TFs defined on t
,
[0,
1]. Using this transformation, Eqs. (14)-(16) are replaced by:
J = h (x (1), t f
)+

0
1
( ( ), ( ), ( ), , f
) dt
 ( )

( ( ), ( ), , f
), t

0 t f
], x 0
 x
0
(17)
(18) s i
(x (t), µ(t), t, tf) # 0, t 0 [0, t f
], i = 1, ...
<
(19)
1823

METHODOLOGY
The proposed method : Expand each of x  f
( ) and each of u j
(t), i =1, 2,..., l, j =1, 2,..., q in terms of TFs as:
 i
( )

Res. J. Appl. Sci. Eng. Technol., 4(12): 1822-1827, 2012 m
 i


0
1
C T i
( )
 m i


1 
0
C T 2 i
( )

C 1
T
1 ( )

C 2
T
T 2 t
(20) f x t u t t t f
) i

, ,

 ,
F l
1 i
T
T 1 t

F where
F 1 i
T 






 
1 , 2 , 1 , 2 0



 tf

,
1 ,
 i


2 ,
1 ,
1 ,
2 , D D
2 , m
 m
2
1
,
,
1 m t f
,

 t f


,






( )
 m

 i

1
0

1 1 i
( )
 m i


1

0
2 2 i
( )
1 1 ( )

D 2
T
T 2 ( )
(21)
(28)
(29)
F 2 i
T 









1 , 2 , D 1 , D 2 ,
1 m

,
 t
 f


,
1 ,
 i 

2 ,
1 ,
D 1 ,
2 ,
D
D 1 ,
2 1
D t f
2

,
2 m
, t f


,






(30)
Similarly we have: x
0
 ~ (
T 1

T 2 ) (22) where i = 0, 1, ... 1..
By substituting Eqs. (20) and (28) in Eq. (18) we get:
C 1
T
1 ( )

C 2
T
T 2 ( )

F 1
T
1 ( )

F 2
T
T 2 ( ) (31) where
0 is a l×m matrix, each column of which is x
0
By integrating Eq. (20) from 0 to t we get:

 
0 t x     x
0

 t

C 1
T
T 1

0

C 1
T 
C 2
T


C 2
T
T 2

 x T 1

T 2 )
1 1

2 2 ( )

 ~ (
1

T 2 )
(23)
The independent variable t can be expanded in terms of TFs as follows: t
 r 1
T
1 ( )
 r 2
T
T 2 ( ) (24)
Here, a theorem needs to be derived.
Theorem: (Babolian et al ., 2007) Suppose that s
1
T1(t)+s
2
T2(t) = z
1
T1(t) + z
2
T2(t), which s
1,
s
2
, z
1
and z
2 are m- vectors of coefficients. Then we have s
1
= z
1 and s
2
= z
2
.
Proof.
By distinct properties of T1 i and T2 i for i=0,1, m-
1, it is sufficient to show the: s
1
T1(t)+s
2
T2(t) = z
1
T1(t)+z
2
T2(t)
Y
(s
1
= z
1
, s
2
= z
2
)
For this, in contrary assume that s
1 definition of T1 i
and T2 i
= z
1
. From
, for each ih # t<(i+1)h we have: where r 1



0 '
1 2
, m m
,  , m
 m
1 

T
(25) s
1 

1

 t
 ih
  h

 s
2
 t
 ih
 h

 s
1
 z
1

 s
1
 z
1
 s
2
 z
2
 z
1


1

 t
 ih

 t
 ih
  h
 h
 z
2
 t
 ih
 h
(32) r 2



1 2 3
, , m m m
,  , 1


T
(26)
The system dynamics approximation: Using Eq. (21) and (23), the system dynamics can be approximated as follows:
( ( ), ( ), , f
) i

 
(
, ,  , l
1 , 2 , D D t t f
),
Using Eq. (7) we get:
(27)
But in Eq. (32) the equality cannot hold. Indeed, in any possible cases for s
1
, s
2
, z
1
and z
2 contradiction. Hence, the proof is complete.
we have a
From the above theorem and Eq. (31) we obtain:
C1 !
F1 = 0
C2 !
F2 = 0
(33)
(34)
The performance index approximation: Using Eq. (21) and (23), we get: h x 1 t f
)

( 1 , 2 , 1 , 2 , t f
) (35)
1824

g(x(t), u(t), t, t f
) =
Res. J. Appl. Sci. Eng. Technol., 4(12): 1822-1827, 2012
(C1, C2, D1, D2, t, t f
) (36) s j
(x(t), u(t), t, t f
)+z 2 j
= (K1 T j
T2(t)+K2 T j
T2(t)) = 0, j = 1, 2,..., v (46)
Similarly to Eq.(28) we obtain: g(x(t), u(t), t, t f
) = G1 T T1(t)+G2 T T2(t) (37) where G1and G2 can be obtained similarly to Eq. (29) and
(30) as:
G 1
T 





 g C C 2 , D D 2 0 t f
),



1 , 2 , 1 , 2 ,
1 m


, tf  


, 2 , , 2 , m
 m
1
, t f 







(38)
G 2
T 







2 , D D 2 ,
1 m
 
, tf
(


,
,



2 ,
,
,
2 , D D
, , t f
2
)
,
2 m
, t f


,





(39)
By substituting Eq. (37) in Eq. (17) we get:
1 2 , 1 2 1 2 )

 h C C 2 , D D tf )
 
0
1
( G 1
T
1 ( )

G 2
T
T 2 ( ))
( 1 , 2 , 1 , 2 , f
)

( G 1
T 
G 2
T
)( ( )

( ))
(40) where K1 T j
(30) as:

, K2 T j
K 1
T j
K 2
T j

can be obtained similarly to Eq. (29) and

 k C C 2 , D D Z Z 2 0 t f
) k j 

C C D D Z Z
1
1 , 2 , 1 , 2 , 1 , 2 , m
, t f 

 , k j 

1 , 2 , 1 , 2 , 1 , 2 , m
 m
1
, t f 




 k j


C C D D Z Z
1
1 , 2 , 1 , 2 , 1 , 2 , m
, t f 
 k j


C C D D Z Z
2
1 , 2 , 1 , 2 , 1 , 2 , m
, t f
 , ( 1 , 2 , 1 , 2 , 1 , 2 , t f
)



J C C 2 , D D Z 1
1
,  , Z 1 v
, Z 2
1
,  , Z 2 v
)

0
1 v  j

1
 j

K 1
T j
1

K 2
T j
T 2
 
1
T j
1 ( )

K 2
T j
T 2

(47)
(48)
Now the inequality constraints in Eq. (19) into another minimization problem by finding K1 j
, K2 j
, j = 1,
2, v, to minimize the integral:
(49)
Replacing the inequality constraints: The inequality constraints in Eq. (19) can be replaced by the equality constraints as (Marzban and Razzaghi, 2003): s j
(x(t), u(t), t, t f
)+z 2 j
(t) = 0, j = 1, 2,..., v (41) where z j
(t), j = 1, 2, ..., v are auxiliary functions.
Expanding each z j obtain:
(t), j = 1, 2, ..., v in terms of TFs, we z j
(t)
.
Z1 T j
T1(t)+Z2 T j
T2(t) (42) and z 2 j
is as follows: z 2 j
.
(Z1 T j
T1(t)+Z2 T j
T2(t)) T (Z1 T j
T1(t)+Z2 T j
T2(t)) (43)
The s j
can be rewritten as: s j
(x(t),u(t), t, t j = 1, 2,.., v f
) = s  j
(C1,C2, D1, D2, t, t f
),
(44)
We define: where
T j
> 0, j=1, 2, …v are weighting constants chosen as follows: errors generated by constraint violation for the weights
T and
Let J
T n-1 en
and J en-1 denote two consecutive corresponding
respectively. Then we choose
T n n
such that the following criterion is satisfied:
|J en
!
J en-1
|<
,
(50) where, g
is a sufficiently small required number.
Therefore, the minimization problem of Eq. (17) subject to Eq. (18) and (19) has been reduced to a parameter optimization problem which can be stated as follows:
Find C1,C2, D1, D2, Z1 j minimize: and Z2 j
, j = 1,2, v to



, 2 , , 2

, 2 , , 2



 
1 , 2 , 1 , 2 , 1
1
, Z 1 v
, Z 2
1
,...
Z 2 v
1 , 2 , 1 , 2 , 1
1
,...
Z 1 v
, Z 2
1
,...

Z 2 v

(51) where Eq. (51) is subject to Eq. (20), (21), (22), (33), (34) and (46)

( 1 , 2 , 1 , 2 , , f
)

( 1 , 2 , 1 , 2 , , f
)

( Z 1
T j
1 ( )

Z 2
T j
T 2 t Z 1
T j
1 ( )

Z 2
T j
T 2 t j
 v
Expanding Eq. (41) in TFs, we obtain:
(45)
ILLUSTRATIVE EXAMPLES
Example 1: This example is adapted from (Kleinman et al ., 1968) and also studied by using classical
Chebyshev method (Vlassenbroeck, 1988). Find the control vector u(t) which minimize:
1825

Res. J. Appl. Sci. Eng. Technol., 4(12): 1822-1827, 2012
Table 1: Simulation results of example 1
Methods
Classic Chebyshev (Vlassenbroeck, 1988) m = 5, K = 12 m = 10, K = 22 m = 12, K = 26
Present m = 16 m = 32 m = 64
1.5
1.0
J
 k

1
0.196000
0.178000
0.173580
0.182687
0.173564
0.170835
0.5
0
X
1
-0.5
X
2
-1.0
0 0.2
0.4
0.6
0.8
Fig. 1: x
1
(t), x
2
(t) obtained in example 1 for k = 64
14
12
10
8
2
0
6
4
-2
-4
0 0.2
0.4
0.6
0.8
Fig. 2: u(t) obtained in example 1 for k = 64
J
  
1
0 x
2
1
( )
 x
2
2
( )

.
( )

1.0
1.0
(52)
Subject to:

1
( )
 x t
    x t
  
(53)
(54) x
2
(t) # 8(t-0.5) 2 -0.5
x
1
(0) = 0 x
2
(0) =
!
1
(55)
(56)
(57)
In Table 1, we compared the solution obtained using the proposed method with other solutions in the literature
(Vlassenbroeck, 1988).
The computational result for x
1
(t), x
2
(t) and u(t) using the present method for k = 64 are given in Fig. 1 and 2 respectively.
Table 2: Simulation results of example 2
J
 k

1
Methods
Quasilineartion and Chebyshev polynomials (Jaddu, 2002)
2.26330
N = 6, R
1
= 1000
N = 8, R
1
= 1000
N = 10, R
1
= 1000
Present m = 16 m = 32 m = 64
2.16015
2.14208
2.18834
2.15142
2.14056
Example 2: The Van der Pol problem (Jaddu, 2002) is considered here. Find the control vector u(t) which minimize:
J


0
5
 x
1
2
   x
2
2
   u
2
   subject to:

1
  x
2

2
   x
1
 
( 1
 x
2
1
 x
2
  u
 u
  x
1
(0) = 1, x
2
(0) = 0 x
1
(t f
) = 1, x
2
(t f
) = 0
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
0 1 2 3 4
Fig. 3: x
1
(t), x
2
(t) obtained in example 2 for k = 64
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0 1 2 3 4
Fig. 4: u(t) obtained in example 2 for k = 64
5
5
(58)
(59)
(60)
(61)
(62)
(63)
1826

In Table 2 we compared the solution obtained using the proposed method with other solutions in the literature.
The computational result for x
1
(t), x
2
(t) and u(t) using the present method for k = 64 are given in Fig. 3 and 4 respectively.
CONCLUSION
In this study, an efficient and accurate method is presented to solve nonlinear optimal control problems subject to inequality constraints and terminal state constraints. The algorithm is based on expanding the dynamic systems and performance index into a sequence of triangular orthogonal functions. Using the auxiliary functions, the inequality constraints are added into the optimal problem. Then the presented method deduces the optimal control problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the accuracy and applicability of the presented Method.
ACKNOWLEDGMENT
The financial support provided by National Nature
Science Foundation of China (60974039) and Key
Laboratory of Mathematics Mechanization.
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