An Efficient Fractional Programming Approach to Linear System Model Reduction Problem
SANJAY JAIN
Department of Mathematics
Government College, Ajmer-305 001, India [email protected]
ABSTRACT
This paper presents a new and efficient way of implementing fractional programming to use in the linear system model reduction problem. Linear system model having higher order fractional objective function and it is reduced to lower order fractional objective function by using our proposed method. This proposed method is simple, computationally straightforward and takes least time. The reduced order fractional programming problem will always be stable if the higher-order fractional programming problem is stable. The proposed method has been illustrated by an example.
Key words : Model Reduction, Fractional Programming Problem, Objective Function.
Introduction
Many real-life problems may not be adequately described in the frames of linear programming problems. In the early 60’s, Hungarian Mathematician Martos B formulated and considered a so-called hyperbolic programming problem, which in the English language special literature is referred as a linear fractional programming problem. After introduction of linear fractional progra- mming problem s, this branch has attracted the attention of more and more researchers and specialists because
there is a broad field of real-world problems, where the use of linear fractional programming problem is more suitable.
The stability in a system implies that small changes to the system stimulus do not result in large effects on system response. Almost every working system is designed to be stable.
Within the boundaries of parameter va ria t ions pe rm it t e d by s ta bilit y considerations, we can then seek to im prove the syste m perform anc e.
Generally the transfer function of
stable linear systems is described by
n
i i i n
i
i i
z b
z a z
D z z N
G
0 1
0
) (
) (
,
which is a form of fractional function and having some restrictions. This shows that there is existence of fractional programming problem. The exact analysis of most linear systems of having higher order fractional programming problem (FPP) is both tedious and lengthy calculations. It is always desirable to reduce such a linear system of higher order FPP by a lower-order FPP. Complex control systems are generally represen- ted by mathematical models derived from well-known theoretical considera- tions.
The analysis and design of such systems (Controllers, Compensators, SISO, MIMO) are often carried out by using lower-order models, which retain the dominant characteristics of the highe r-orde r original s ys t e m a nd approximates its response as closely as possible for the same types of inputs.
Stability of a linear time-invariant system can be determined by finding the roots of its characteristic polynomial. For first and second-order polynomials, this is trivial. Algebraic methods are available for third-order polynomials, but are somewhat difficult to apply.
For polynomials of order higher than three, numerical procedure or computer program are usually required.
Several researchers have studied variety of model reduction methods1-14 in time domain, frequency domain, or a combination of both for continuous system. All these methods involve special procedures leading to large computa- tional complexity. The main aim of this paper is to present a simple method to get a reduced lower-order model from the given higher-order model involving less number of computations.
The Problem formulation
Higher order fractional objective function is occurs mainly in controller design system, linear time invariant s ingle -input-s ingle -out put (S I S O ) system and many more systems. Here we consider transfer function and restrictions of a system in the form of nth order stable fractional programming problem (whose solution exists) as is:
0 ) (
) (
0 1
0
z and
c z A to
subject
z b
z a z
D z z N
F
Optimize n
i i i n
i i i n
(1)
where ai (0 i n-1), bi (0 i n) > 0, and are scalar constants.
T he c orre s ponding t ra ns fe r function and restrictions of reduced system is in the form of stable kth (k<n) order reduced fractional programming problem:
0 ) (
) (
0 1
0
z and
c z A to
subject
z e
z d z
D z z N
R
Optimize k
i i i k
i i i
k k k
(2)
where di(0 i k-1), ei (0 i k) > 0, and are scalar constants.
In this paper, assuming the original higher order fractional progra- mming problem described by problem (1), The problem is to find a reduced order model in the form of problem (2) such that the reduced order fractional programming problem retains the important characteristics of the original fractional programming problem12,13. The Method of Model Reduction
The stepwise step procedure for determining the reduced order model is as follows:
Step-1 Consider the nth order stable fractional programming problem is as:
0 ) (
) (
0 1
0
z and
c z A to
subject
z b
z a z
D z z N
F
Optimize n
i i i n
i
i i n
Step-2 Transformed above FPP into Fn(s) by applying linear transformation z = s +1, where s is positive to retain stability. The transformed FPP in s is as:
( ) )
( s D
s s N F Optimize n
...
...
2 2 1 0
1 1 2
2 1 0
s b s
b s b b
s a s
a s a a
n n
n n
0
s and
c s A to
subject
(3 )
Step-3 T he propos e d m e thod t o obtain the kth order reduced FPP with unknown parameters represented as:
0
...
...
2 2 1 0
1 1 2
2 1 0
s and
c s A to
subject
s e s
e s e e
s d s
d s d s d
R
Optimize k
k k k k
(4)
Step-4 By assumption that reduces order system retains the characteristics of the original system. Hence Gn(s)=
Rk(s)
n n
n n
s b s
b s b b
s a s
a s a a
...
...
2 2 1 0
1 1 2
2 1 0
k
k k k
s e s
e s e e
s d s
d s d d
...
...
2 2 1 0
1 1 2
2 1 0
(5)
Step-5 O n c ros s -m ult iplying a nd rearranging the equation (5), we have
Step-7 The unknown parameters are determined by taking any positive values for d0 or e0, for simplification, choosing d0 =1 or e0=1, and using the above relations, the other unknown parameters are evaluated.
Step-8A pplying the invers e linea r transformation (s=z–1) to the numerator and denominator of Gk(s), we get reduced
n k n k
k n k n
s d b s
d b d b d b s d b d b d b
s e a s
e a e a e a s e a e a e a
1 1 2
0 2 1 1 2 0 0
1 1 0 0 0
1 1 2
0 2 1 1 2 0 0
1 1 0 0 0
...
...
(6 )
Step-6 Equating the coefficients of the corresponding terms in the equation (6), the following relations are obtained:
1 1
3 4 2 3 1 2 2
3 1 2 1
3 3 2 2 1 1 2
2 1 1 0
3 2 2 1 1 0 3
2 2 1 1 0
0 2 1 1 2 0 0 2 1 1 2 0
0 1 1 0 0 1 1 0
0 0 0 0
. . . ...
...
...
. . .
k n k n
k k
k k
k k
k k
k k
k k
k k
k k
k k
d b e a
d b d b d b e
a e a e a
d b d b d b e
a e a e a
d b d b d b e
a e a e a
d b d b d b e a e a e a
d b d b e a e a
d b e a
order model Gk(z).
To explain the whole procedure we consider an example of higher order fractional programming problem and it reduces to lower order fractional programming problem.
5. Example
Consider the eighth order fractional programming problem as
Max. Fn(z),
018 . 0 132 . 0 786 . 0 548 . 1 456 . 0 63 . 0 348 . 3 046 . 5 8
018 . 0 44 . 0 262 . 0 516 . 0 152 . 0 21 . 0 116 . 1 682 . ) 1
( 8 7 6 5 4 3 2
2 3
4 5
6 7
z z
z z
z z
z z
z z
z z
z z
z z Fn
0
z and
c z A to
subject
(7 )
The above eighth order FPP is transformed into F
n(s) by applying linear transformation z = s +1, we have
Max. Fn(s),
2 16 808 . 76 459 . 210 864
. 335 576
. 322 33
. 185 954
. 58 8
988 . 1 16 064 . 49 182 . 79 712 . 74 808 . 41 886 . 12 682 . ) 1
( 8 7 6 5 4 3 2
2 3
4 5
6 7
s s s
s s
s s
s
s s s
s s
s s s
Fn
0
s and
c s A to
subject
(8)
Let we want to reduce eighth order FPP into second order FPP, so consider a second order model as:
Max. Rk(s),
2
2 1 0
1 0
s e s e e
s d s d
Rk
0
s and
c s A to
subject
(9)
where d0, d1, e0, e1 and e2 are unknown parameters.
According to method, Fn(s) = Rk(s)
2 16 808 . 76 459 . 210 864
. 335 576
. 322 33
. 185 954
. 58 8
988 . 1 16 064 . 49 182 . 79 712
. 74 808 . 41 886
. 12 682
. 1
2 3
4 5
6 7
8
2 3
4 5
6 7
s s s
s s
s s
s
s s s
s s
s s
= 2
2 1 0
1 0
s e s e e
s d d
Comparing the like terms in (10), we have 1.988 e0 = 2 d0
1.988 e1 + 16 e0 = 1.992d1 + 16d0
1.988 e2 +16 e1+49.064 e0 = 16 d1+76.808 d0 16 e2 + 49.064 e1+ 79.182 e0 = 76.808 d1 + 210.454 d0 49.064 e2 +79.182 e1+ 74.712 e0 = 210.454 d1+ 335.864 d0 79.182 e2 + 74.712 e1+ 41.808 e0 = 335.864 d1+ 322.576 d0 74.712 e2 + 41.808 e1+ 12.886 e0 = 322.576 d1+ 185.33 d0
41.808 e2 + 12.886 e1+1.682 e0 = 185.33 d1+ 58.954 d0 12.886 e2 +1.682 e1 = 58.954 d1+ 8 d0
1.682 e2 = 8 d1
For solving above equations choosing either e0=1 or d0=1, then other unknown parameters can be evaluated easily.
Now the second order reduced FPP is obtained as:
2
2 1.00503 2.77166 8.57768 1
80346 . 1
s s
s s
R
0
s and
c s A to
subject
(11 )
Applying inverse linear transformation (s = z – 1) to (11), we have
8.57768 14.38270 6.81105 80346
. 0 80346 . 1 .
2 2 2
z z
z z R
z R Max
0
z and
c z A to
subject
(1 2 )
So finally, from equation (7) eighth order fractional programming problem and now by cross multiplying, we have
1.988 e0+(1.988 e1+16 e0)s+(1.988 e2+16 e1+49.064 e0)s2+(16 e2+49.064 e1 +7 9.1 82 e0) s3+(49.064 e2+7 9.1 82 e1+7 4.712 e0)s4+(79 .1 82 e2+74.712 e1+41.808 e0) s5+(74.712 e2+41.808 e1+12.886 e0)s6+(41.808 e2 + 12.886 e1+1.682 e0) s7 + (12.886 e2 +1.682 e1) s8 +1.682 e2 s9
= 2d0+(1.992d1 + 16d0) s + (16 d1+76.808 d0) s2+(76.808 d1+210.454 d0) s3 + (210.454 d1+ 335.864 d0) s4+(335.864 d1+322.576 d0) s5+(322.576d1
+185.33 d0) s6+ (185.33 d1+58.954 d0) s7+(58.954 d1+ 8 d0) s8 +8 d1 s9 (10)
is reduces to second order fractional programming problem
8.57768 14.38270 6.81105 80346
. 0 80346 . 1 .
2 2 2
z z
z z R
z R Max
0
z and
c z A to
subject
6. CONCLUSION
The proposed method is com- put at iona lly simple a nd doe s not involve much procedure. The given example of eighth order fractional programming problem shows that the second order reduced fractional progra- mming problem may be obtained in a straightforward manner. This proposed method used/ extended to design of controllers, compensators as well as multi-input-multi-output (MIMO) system reduction for continuous and discrete systems.
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z z
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