A Review on Order Reduction of System using Routh Approximation Method

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International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264)

Vol. 3, No. 2, April 2015Available at www.ijrmst.org

2321-3264/Copyright2015, IJRMST, April 2015 114

A Review on Order Reduction of System using

Routh Approximation Method

Santosh Kumari#1

, Ruchi Taneja#2

#ECE Deptt. Prannath Parnami Institute of Management & Technology, Hisar,Haryana,India

[email protected]@gmail.com

AbstractIt is usually possible to describe the dynamics of

physical systems by a number of simultaneous linear differential

equations with constant coefficients. But for many processes (like

chemical plants and nuclear reactors etc) the order of the matrix

may be quite large. So it would be difficult to work with these

complex systems in their original form. The analysis and

synthesis of such higher order systems are difficult and generally

not desirable on economic and computational considerations. Insuch cases, it is common to study the process by approximating it

to a simpler model. . For instance, the response of an airplane is

quite commonly approximated by a second order transfer

function. Through model order reduction, a complex higher

order system can be converted to a simple lower order system.

The reduced models are constructed such that all the parameters

can be preserved with acceptable accuracy.

Keywords accuracy, Integral Square Error, Model OrderReduction, order of system, RAM, stability, transfer function

I. INTRODUCTION

In the design of control systems, it is often required to work

with mathematical models of high order. The analysis andsynthesis of higher order systems are difficult and generally

not desirable on economic and computational considerations.

Therefore, for computation, control, and other purposes, it is

often in demand adequately to comprise a high-order system

by a low-order model. The processes or devices can bemodelled by partial differential equations. To simulate such

models, spatial discretization via e.g. finite element

discretization is necessary, which results in a system of

ordinary differential equations, or differential algebraic

equations. After spatial discretization, the number of degrees

of freedom is usually very high. It is therefore very timeconsuming to simulate such large-scale systems of ordinary

differential equations or differential algebraic equations.A great number of problems are brought about by the

present day technology and societal and environmental

processes which are highly complex and 'large in dimension

and stochastic by nature'. The notion of 'large scale' is highly

subjective one in that one may ask: 'How large is large?'.

There has been no accepted definition for what constitutes a

large scale system. Many viewpoints have been presented on

this issue. One viewpoint has been that a system is considered

large scale if it can be decoupled or partitioned into a numberof interconnected subsystems or small scale systems for either

computational or practical reasons. Another view point is that

a system is large scale when its dimensions are so large that

conventional techniques of modeling, analysis, control, design

and computation fail to give reasonable solutions with

reasonable computational efforts. In other words a system is

large when it requires more than one controller. So, it is

highly desirable to obtain a reduced order model from thecomplex higher order original system.

Through model order reduction, a small system with

reduced number of equations is derived. The reduced model is

simulated instead, and the solution of the original differential

equation can then be recovered from the solution of thereduced model. As a result, the simulation time of the original

large-scale system can be shortened by several orders of

magnitude.

Model order reduction is a technique for reducing

thecomputational complexity of higher ordermathematical

models innumerical simulations.Many modernmathematicalmodels of real-life processes pose challenges when used

innumerical simulations,due to complexity and large size.Model Order Reduction aims to lower the computational

complexity of such problems, for example, in simulations of

large-scaledynamical systems andcontrol systems. By a

reduction of the model's associatedstate space dimension

ordegrees of freedom,an approximation to the original model

is computed. This reduced order model can then be evaluated

with lower accuracy but in significantly less time.

Goals of model order reduction are:

Simplifying the understanding of system fordesign purpose.

To reduce the simulation time with the model of

system. Reducing the controller size.

To economics in terms of hardware when

realizing the system.

Minimizing the computational complexity of the

model for system analysis.

Through model order reduction, a complex higher order

system is converted into a simplified lower order system. Letthe transfer function H(s) of higher order system is given by:
http://en.wikipedia.org/wiki/Computational_complexityhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Numerical_simulationhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Numerical_simulationhttp://en.wikipedia.org/wiki/Dynamical_systemhttp://en.wikipedia.org/wiki/Control_systemhttp://en.wikipedia.org/wiki/State_spacehttp://en.wikipedia.org/wiki/Degrees_of_freedomhttp://en.wikipedia.org/wiki/Degrees_of_freedomhttp://en.wikipedia.org/wiki/State_spacehttp://en.wikipedia.org/wiki/Control_systemhttp://en.wikipedia.org/wiki/Dynamical_systemhttp://en.wikipedia.org/wiki/Numerical_simulationhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Numerical_simulationhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Computational_complexity


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b1Sn-1

+......................+bn

H(s) = (1)a0S

n+.......................+an

Let us assume that a reduced order model R(s) of order 'r'


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Step 4: Reverse the coefficients of Pk(s) and Qk(s) again to

find reciprocal Rk(s). This reciprocal Rk(s) is the desired

reduced order transfer function.

In step 3, Pk(s) and Qk(s) can be found by using the followingformula

Pk(s)= kPk-1(s) + Pk-2(s) + k (5)Qk(s)= kQk-1(s) + Qk-2(s) (6)

k = 1, 2, 3, 4...

With, Q-1(s)= 0 and P-1(s)= 0

Q0(s)= 1 and P0(s)= 0

TABLE IIBETA ROUTH TABLE

Advantages of Routh approximation method:

Stability of system is preserved, when the original

system is stable.

Poles and zeros of the reduced system approaches

poles and zeros of the original system as the order of

the approximation is improved. Steady state of system is preserved.

Disadvantages of Routh approximation method:

Gives no guarantee that transient state of system ispreserved.

IV.PERFORMANCE MEASURES

A. Integral Square Error

Integral square error is formed by integrating the square of

the system error over a fixed interval of time; it is frequently

used in linear optimal control and estimation theory.

ISE= (7)

B. Impulse Response Energy

Impulse response describes the reaction of the system asafunction of time. If h(t) is the impulse response of the

transfer function H(s) then the impulse response energy bedefined by the integral of impulse response and is given by

h2= (t) dt (8)

It is assumed that H(s)is the transfer function of a stable

system.

V. CONCLUSION

A study of model order reduction to reduce the order of a

system using suitable reduction method, preserving stability

with acceptable accuracy of the reduced system is done. A

survey of some pre-existing model order reduction methods

concludes that any Routh approximant of a stable system is

stable. It not only preserves stability but also has otherinteresting and useful properties. Since Routh approximation

method retains some of the initial time moments only so the

reduced model obtained by Routh Approximation Method

tends to approximate the steady state response of the original

system. This work can be extended to find some highly useful

algorithms for model order reduction which can be used toimprove the existing methods of model order reduction.

REFERENCES

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1b =b

0 1

1

b =b2 3

2

b =b0 2

2

b =b2 4

1

b0

1 =1

a

0

3 1 1

b =b - a0 2 1 2

3 1 1

b = b - a2 2 1 4

2

b0

2 =2

a0

4 1 2

b = b - a0 2 2 2

...........
http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)


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[10] T.H.S. Abdelaziz and M. Vala sek,"Pole-placement for SISOlinear systems by state-derivative feedback", IEE Proc.-ControlTheory Appl., Vol. 151,No.4,July,2004.

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Feedback Control Analysis and design with MATLAB, 2007.

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[14] M Gopal Digital Control and State Variable Methods ThirdEdition, New Delhi, 2009.

[15]

Singh, V. P., and Chandra, D. Routh Approximation Based ModelReduction Using Series Expansion of Interval Systems, IEEE

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(ICPCES),1,1-4, 2010.

A Review on Order Reduction of System using Routh Approximation Method

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