J. Comp. & Math. Sci. Vol. 1(4), 403-410 (2010).
An Efficient Fractional Programming Approach to Linear System Model Reduction Problem SANJAY JAIN Department of Mathematics Government College, Ajmer-305 001, India drjainsanjay@gmail.com 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 programming problems, 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 variations permitted by stability considerations, we can then seek to improve the system performance. Generally the transfer function of
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
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Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
stable linear systems is described by n 1
G z
N ( z) D( z)
a z
i
b z
i
i
i 0 n
, i
i0
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 represented by mathematical models derived from well-known theoretical considerations. 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 higher-order original system and 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 computational 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 single-input-single-output (SISO) 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: n 1
Optimize Fn z
N ( z) D( z )
a z
i
b z
i
i
i 0 n
i
i 0
subject to and
(1)
Az c z0
where ai (0 i n-1), bi (0 i n) > 0, and are scalar constants. The corresponding transfer function and restrictions of reduced system is in the form of stable kth (k<n) order reduced fractional programming problem:
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
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:
k 1
di z
i
e z
i
N k ( z) Optimize Rk z i k0 Dk ( z )
i
i0
subject to
(2)
Az c
Optimize Fn s
z0
and
where di(0 i k-1) , ei (0 i k) > 0, and are scalar constants. In this paper, assuming the original higher order fractional programming 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: n 1
Optimize Fn z
N ( z) D( z )
a z
i
b z
i
i
i 0 n
i
i0
subject to and
Az c z0
405
N ( s) D( s)
a0 a1 s a 2 s 2 ... a n 1 s n 1 b0 b1 s b2 s 2 ... bn s n
subject to
A s c
(3)
s0
and
Step-3 The proposed method to obtain the kth order reduced FPP with unknown parameters represented as:
Optimize Rk s subject to and
d 0 d 1 s d 2 s 2 ... d k 1 s k 1 e 0 e1 s e 2 s 2 ... e k s k A s c s0
(4)
Step-4 By assumption that reduces order system retains the characteristics of the original system. Hence Gn(s)= Rk(s)
a 0 a1 s a 2 s 2 ... a n 1 s n 1 b0 b1 s b2 s 2 ... bn s n
d 0 d1 s d 2 s 2 ... d k 1 s k 1 (5) e0 e1 s e2 s 2 ... ek s k Step-5 On cross-multiplying and rearranging the equation (5), we have
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Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
a 0 e0 a 0 e1 a1e0 s a 0 e2 a1e1 a 2 e0 s 2 ... a n1ek s n1 k b0 d 0 b0 d 1 b1 d 0 s b0 d 2 b1d1 b2 d 0 s 2 ... bn d k 1 s n1 k
(6)
Step-6 Equating the coefficients of the corresponding terms in the equation (6), the following relations are obtained:
a0 e0 b0 d 0 a 0 e1 a1e0 b0 d1 b1 d 0 a0 e2 a1 e1 a 2 e0 b0 d 2 b1 d 1 b2 d 0 . . a 0 ek 1 a1ek 2 a 2 ek 3
. ... b0 d k 1 b1 d k 2 b2 d k 3
a0 ek a1 ek 1 a 2 ek 2 ... b1 d k 1 b2 d k 2 b3 d k 3 a1ek a 2 ek 1 a 3 ek 2 ... b2 d k 1 b3 d k 2 b4 d k 3 . . . a n 1ek bn d k 1 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-8Applying the inverse linear transformation (s=z–1) to the numerator and denominator of Gk(s), we get reduced
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
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
407
Max. Fn(z),
Fn ( z )
1.682 z 7 1.116 z 6 0.21 z 5 0.152 z 4 0.516 z 3 0.262 z 2 0.44 z 0.018 8 z 8 5.046 z 7 3.348 z 6 0.63 z 5 0.456 z 4 1.548 z 3 0.786 z 2 0.132 z 0.018
subject to
Az c (7)
z0
and
The above eighth order FPP is transformed into Fn(s) by applying linear transformation z = s +1, we have Max. Fn(s), Fn ( s )
1.682 s 7 12.886 s 6 41.808s 5 74.712 s 4 79.182 s 3 49.064 s 2 16 s 1.988 8s 8 58.954 s 7 185.33s 6 322.576 s 5 335.864 s 4 210.459 s 3 76.808 s 2 16 s 2
subject to and
A s c s0
(8)
Let we want to reduce eighth order FPP into second order FPP, so consider a second order model as: Max. Rk(s),
Rk s
d 0 d1 s e0 e1 s e2 s 2
subject to and
A s c s0
(9)
where d0, d1, e0, e1 and e2 are unknown parameters. According to method,
Fn(s) = Rk(s)
1.682 s 7 12.886 s 6 41.808 s 5 74.712 s 4 79.182 s 3 49.064 s 2 16 s 1.988 8 s 8 58.954 s 7 185.33s 6 322.576 s 5 335.864 s 4 210.459 s 3 76.808 s 2 16 s 2 =
d 0 d1 s e0 e1 s e 2 s 2 Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
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Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
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 e 1 +79.182 e 0 ) s 3 +(49.064 e 2 +79.182 e 1 +74.712 e 0 )s 4 +(79.182 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) 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.
Applying inverse linear transformation (s = z – 1) to (11), we have
Now the second order reduced FPP is obtained as:
R2 z
1.80346 s 1 R2 s 1.00503 2.77166s 8.57768s 2
subject to and
A s c s0
(11)
Max. R2 z 1.80346 z 0.80346 8.57768 z 2 14.38270 z 6.81105
subject to and
Az c z0
(1 2)
So finally, from equation (7) eighth order fractional programming problem
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
409
Max. Fn(z), Fn ( z )
1.682 z 7 1.116 z 6 0.21 z 5 0.152 z 4 0.516 z 3 0.262 z 2 0.44 z 0.018 8 z 8 5.046 z 7 3.348 z 6 0.63 z 5 0.456 z 4 1.548 z 3 0.786 z 2 0.132 z 0.018
Az c
subject to
z0
and
is reduces to second order fractional programming problem
Max. R2 z
2.
1.80346 z 0.80346 R2 z 8.57768 z 2 14.38270 z 6.81105
subject to
Az c z0
and
3.
6. CONCLUSION The proposed method is computationally simple and does not involve much procedure. The given example of eighth order fractional programming problem shows that the second order reduced fractional programming 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.
4.
5.
6.
REFERENCES 7. 1. Devarajan, N. and Sivanandam, S. N., Simple and direct methods for model reduction of linear time invariant discrete system, National
8.
system conference proceedings, Vikram Sarabhai Space Centre, Trivandrum, India, 124-129 (1996). Devarajan, N and Sivanandam, S. N., Model reduction of linear discrete system using routh array, National system conference proceedings, REC, Calicut, India, 123-128 (1998). Gutman, P.O., Mannerfelt, C.F. and Molander, P., Contributions to the model reduction problem, IEEE Transaction on Automatic Control, Vol. AC-27, 454-455 (1982). Jain, S. and Mangal, A., Algorithm for fractional programming problem with homogeneous constraints, Ganita Sandesh, Vol. 17, 55-60 (2003). Lucas, T.N., Factor Division-A useful algorithm in model reduction, IEE Proceedings, Vol. 130, 362 (1983). Mitra, M. and Kumar, V., Order reduction of linear discrete system using eigan spectrum analysis, International conference on computer application in electrical engineering, Roorkee, India, 497502 (1997). Mukherjee, S. and Mishra, R. N., Review of some model reduction techniques, Journal of Franklin Institute, Vol. 323, 23( 1987). Pal, J., An algorithmic method for
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
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Sanjay Jain, J.Comp.&Math.Sci. Vol.1(4), 403-410 (2010)
the simplification of linear dynamical scalar systems, International journal of control, Vol. 323, 257 (1986). 9. Palaniswami, S. and Sivanandam, S. N., A system scheme for model reduction of linear discrete systems, International conference on computer modeling, simulation and communication, Jaipur, India, 290296 (1999). 10. Prasad, R., Sharma, S.P. and Mittal, A.K., Linear model reduction using the advantages of Mihailov criterion and factor division, IE(I) JournalEL, Vol. 84, 7-10 (2003). 11. Sastry, G.V.K.R and Reddy, G. S., New Routh approximations for order reduction of discrete time
large scale systems, National system conference proceedings, Coimbatore, India, 210-215 (1995). 12. Shamash, Y., Continued-fraction method for the reduction of Discrete â&#x20AC;&#x201C;time Dynamic Systems, International Journal of Control, Vol. 20, 267-275 (1974). 13. Shamash, Y., Linear system reduction using Pade approximation to allow retention of dominant models, International Journal of Control, Vol. 21, 257-272 (1975). 14. Volkwein, S., Model reduction for nonlinear control systems, IFIP TC 7 Conference on system modeling and optimization, CRACOW, POLAND (2007).
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