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System order reduction by differentiation technique and implementation using mat lab.

INTRODUCTION

In a paper, Lucus suggested a model reduction method based on the differentiation technique to overcome the limitations of the method given Lepschy and Viaro [3], Gutman et al [2], proposed a reduction method which preserves stability and involves the differentiation of the numerator and denominator polynomials of the original system to reduce the order of the system. Lepschy and Viaro [3] presented a mixed method using the Gutman method and Pad'e technique, avoiding the use of gain factor unlike Gutman method. The denominator is to be obtained by differentiation method and the numerator of the model using Pad'e approximation to retain the first (k-1) time moments. But the major limitations of the Lepschy method are that it involves the application of reciprocal transformation twice and needs the time-moments of the original system before-hand. The Lucas method is based on using the differentiation technique for both the numerator and denominator and claims that it overcomes the limitations of Lepschy method, avoiding the use of gain factor and reciprocal transformations. Even the Lucas method has serious limitations like the necessity for formulating two Routh-type arrays for the numerator and denominator of the reduced model and always retaining only one time-moment of the original system which severely affects the time-response approximation. Another significant drawback of this method is that it can not be implemented for systems having a difference of more than one in the orders of the denominator and numerator polynomials. It is established in literature [7,8] that a better time-response approximation can be obtained by retaining initial Markov parameters and time-moments. To facilitate this and to overcome the limitations of the existing methods in the literature based on the differentiation technique; a new procedure is presented in this paper. The proposed method avoids the necessity of reciprocal transformations unlike Gutman method and Lepschy method. New algorithms are suggested here avoiding the formulation of Routh-type arrays unlike Lucas methods. The Lucas method is capable of any order k of the model. Another advantageous and useful feature of the proposed method is that it generates stable reduced order models of order k, retaining t time-moment and m Markov parameters of the original system. Therefore, this method gives a better time-response approximation than the other methods which can retain only time-moments in the reduced models.

DESCRIPTION OF THE METHOD

Let the full nth order system be represented by its transfer function

[G.sub.n](s) = [B.sub.n-1][s.sup.n] - 1 + ...... + [B.sub.1]S + [B.sub.0]/[A.sub.n][s.sup.n] + ...... + [A.sub.1]S + A0 (1)

Then the reduced-order model of order k which retain initial m Markov parameter and t time-moments of [G.sub.n] (s) using the proposed method is defined as,

[R.sub.k](s) = [b.sub.k -1][s.sup.k-1] + ...... + [b.sub.1s] + [b.sub.0]/[a.sub.k][s.sup.k] + ...... + [a.sub.1]s + ao = [N.sub.k](s)/[D.sub.k](s) (2)

DENOMINATOR [D.sub.K] (S)

The denominator polynomials [D.sub.k] (s) (k = 1, 2, ..., n) of the reduced order models are obtained by the following new algorithms based on the Differentiation Method avoiding the formulation of Routh-type arrays unlike method.

For k =1, D1(s) = A0 + [n - 1[C.sub.n - 1]/n[C.sub.n - 1]] A1s

For k = 2 [D.sub.2] (S) = [A.sub.0] + [n - 1 [C.sub.n - 2]/n[C.sub.n - 2]] [A.sub.1]S+ [n - 2] [C.sub.n-2]/ n[C.sub.n - 2] [A.sub.2][S.sup.2]

In general, the kth order denominator is given by,

Dk(s) = [k+1.summation over (i=1)] Ai - 1 [n - i + 1 [C.sub.n-k]/n[C.sub.n-k]] [S.sup.i-1]; k = 1,2,3....,n (3)

Where p[C.sub.q] [p!/q!(P - q)!]

The reduced order denominators obtained by the proposed new algorithm for different values of k are same as the denominator polynomials obtained by the other differentiation method. Hence the stability aspect is well retained in the proposed algorithm and always stable reduced order models are generated for stable original system.

NUMERATOR NK (S)

The numerator Nk (s) of the model, which will retain the first t time-moments and m Markov parameters of the original system Gn (s) is defined as, Nk (s) = Nkt (s) + Nkm (s) with k = t + m

= [T.sub.1] + [T.sub.2] s + ...... + Tt [s.sup.k - m + 1] +

Mm [s.sup.k - m] + ...... [M.sub.2] [s.sup.k - 2] + [M.sub.1][s.sup.k -1]. (4)

In general, Tt = a0/A0 ([B.sub.t-1]) (5)

And Mm = 1/An { [m.summation over (i=1) B n - 1 ak - (m - i) - [m-1.summation over (j = 0)] MjA n - (m - j)} (6)

with [M.sub.0] = 0

APPLICATIONS AND RESULTS :

Consider the 4th order system given by,

G(S) = [S.sup.2] + 15S + 50/[S.sup.4] + 5 [S.sup.3] + 33 [S.sup.2] 79S + 50

A 2nd order model is to be obtained using the proposed method, retaining one initial Makov parameter and one time moment of the original system ie t = 1, m =1.

The denominator of the 2nd order model using equation (3) is obtained as,

D2 (s) = [4C2/4C2] A0 + [3C2/4C2][A.sub.1]S + [2 [C.sub.2]/4 [C.sub.2]] [A.sub.2][S.sup.2]

= [4 * 3/4 * 3] A0 + [3 * 2/4 * 3]A1s + [2 * 1/4 * 3] [A.sub.2][s.sup.2]

= 5.5 s2 + 39.5 s + 50

The numerator [N.sub.2](s) to retain one initial Markov parameter and time-moment obtained using equations (4), (5) and (6), with [T.sub.1]+50, [M.sub.1]+5.5 is, [N.sub.2] (s) =5.5s +50..

Therefore the 2nd order obtained by the proposed method is,

[R.sub.2](S) = 5.5S + 50/5.5 [S.sup.2] + 39.5S + 50; (m = 1, t = 1)

The 2nd order model obtained using the Lepschy method (2) is,

[V.sub.2](S) = -24.5S + 50/5.5 [S.sup.2] + 39.5S + 50; (m = 0, t = 2)

The method of Gutman et al [2] and Lucas method cannot be implemented for this system as per their limitation that they are not applicable whenever the order of the denominator n of the original system differs from its numerator order m by more than one for m < n. The step-Reponses of the 2nd order models viz, [R.sub.2](s) and [V.sub.2](s) are compared with the of the original system G(s). It can be inferred from the figure that the model obtained by using the proposed method gives a better time-response approximation than that obtained by the Lepschy method [3]. This is because of the fact that the proposed method retain both the time moments and Markov parameters whereas Lepschy method retains only time-moments.

EXAMPLE--2

Consider the 8th order system given by,

G(S) = 18[S.sup.7] + 514 [S.sup.6] + 5982 [S.sup.5] + 36382 [S.sup.4] + 122664 [S.sup.3] + 222088 [S.sup.2] + 185760 s + 40320/[S.sup.8] + 36 [S.sup.7] + 546 [S.sup.6] + 4536 [S.sup.5] + 22449 [S.sup.4] + 67284 [S.sup.3] + 118124 [S.sup.2] + 109584 S + 40320

It is proposed to obtain 3rd order models using the proposed technique to achieve a better step-response approximation. The 3rd order model obtained by the proposed algorithm using equation (4), (5) and (6) for m = 2, t = 1 is,

[R.sub.3](S) = 21627[S.sup.2] + 6809.574S + 40320/1201.5 [S.sup.3] + 12656.143 [S.sup.2] + 41094S + 40320; (m = 2, t =1)

The 3rd order model obtained using the method of Gutman et al [2] and Lucas method [6] is,

[L.sub.3](S) = 10575 .619 [S.sup.2] + 53074 .2857 S + 40320/1201 .5 [S.sup.3] + 12656 .143 [S.sup.2] + 41094 S + 40320; (m = 0, t = 1)

The 3rd order model obtained using the Lepschy method [3] is,

[V.sub.3](S) = 12776.80481 [S.sup.2] + 117269.9823 S + 40320/1201.5 [S.sup.3] + 12656.143 [S.sup.2] + 41094 S + 40320

Compares the step responses of the 3rd order reduced models [R.sub.3](s), [L.sub.3](s) and [V.sub.3](s) with that of the of the original system G(s). It can be observed from the figure that being a biased model, [R.sub.3](s) obtained by the new method gives a better time response approximation than those of the methods.

EXAMPLE--3

Consider the system given by,

G(S) = 24 [S.sup.3] + 60 [S.sup.2] + 72 S + 54/[S.sup.4] + 10 [S.sup.3] + 35 [S.sup.2] + 50 S + 24

It is proposed to obtain a 2nd order reduced model using the proposed method to improve time response approximation of the existing methods based on differentiation technique [2,3,4,6].

The 2nd order model obtained using the proposed method is,

[R.sub.2](S) = 72 S + 24/5.833 [S.sup.2] + 25 S + 24; (m = 0, t = 2)

The 2nd order model obtained using the method of Gutman et al [2] and the Lucas method is,

[L.sub.2](S) = 24 S + 24/5.833 [S.sup.2] + 25 S + 24; (m = 0, t = 1)

The 2nd order model obtained using the Lepschy method is,

[V.sub.2](S) = 24 S + 24/5.833 [S.sup.2] + 25 S + 24; (m = 0, t = 2)

The step-responses of the reduced models [R.sub.2](s), [L.sub.2](s) and [V.sub.2](s) are compared with that of the original system G(s). It can be observed that the proposed model gives a better time-response approximation than the available methods in literature based on the differentiation technique [2,3,4,6].

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CONCLUSION

A new procedure for obtaining biased models for high order linear time invariant system is suggested. The method is based on differentiation technique and generates stable reduced order models with flexibility or retaining both initial time moments as well as Markov parameters of the original high order system. Therefore it gives a better approximation in time response than the differentiation methods available in literature. The proposed method is simple, direct and avoids the formulation of Routh type arrays, necessity of calculating the time-moments of the original system before-hand, use of gain factor and reciprocal transformations etc, unlike other methods and hence is computationally superior and useful.

REFERENCES

[1.] Chen, C.F. and L S Shieh, L.S.1968. 'A Novel Approach to Linear Model Significant', International Journal of Control, Vol 8, December 1968 : 561.

[2.] Gutman, P.O., Mannerfelt, C.F. and Molander, P.1982.. 'Contributions to the Model Reduction Problem'. IEEE Transaction of Automatic Control, Vol AC-27, April 1982: 454.

[3.] Lepschy, A. and Viaro, U. 1983. 'A Note on the Model Reduction Problem', IEEE Transaction on Automatic Control, Vol AC-28, 1983 : 525.

[4.] Lucas, T.N.1988. 'Differentiation Reduction Method as a Multipoint Pad'e Approximant'. Electron Letters IEE (U K), Vol 24, 1988: 60.

[5.] Lucas, T.N. 1988. 'Scaled Impulse Energy Approximation for Model Reduction', IEEE Transaction on Automatic Control, Vol AC-33, August, 1988 : 791.

[6.] Lucas, T.N. 1992.'Some Further Observations on the Differentiation Method of Model Reduction', IEEE Transaction on Automatic Control, Vol AC-37, September, 1992: 1389.

[7.] Pal, J. 1983. 'Improved Pad'e Approximants using Stability Equation Method', Electron Letters IEE (U K), 1983 : 426.

[8.] Sastry, G V K R and Murthy, V Krishna .1987. 'State Space Model Reduction using Simplified Routh Approximation Method (SRAM)', Electron Letters, IEE (U K), Vol 22 : 1300.

P. Muna Swamy * and R.V.S. Satyanarayana **

ECE Department, Narayana Engineering College, Nellore.(Andhra Pradesh.), India Email : swamymuna@yahoo.co.in

* Associate Professor, ** Associate Professor, EEE Department, S.V.U. College of Engineering, Tirupathi. (Andhra Pradesh.), India. Email : rvssatyanarayana@hotmail.com
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Author:Swamy, P. Muna; Satyanarayana, R.V.S.
Publication:Bulletin of Pure & Applied Sciences-Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Jan 1, 2008
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