bharath bhikk aji and t orsten s oderstr omsystems using singular p erturbations bharath bhikk aji...

Reduced order models for di�usion systems using

singular perturbations

Bharath Bhikkaji and Torsten S�oderstr�om

Department of Systems and Control, Information Technology,Uppsala University, P O Box 27, SE 75121 Uppsala, Sweden, email

[email protected], [email protected]

October 16, 2000

Abstract

In this paper, we consider a special case of the one dimensional heat

di�usion across a homogeneous wall. This physical system is modeled by

a linear partial di�erential equation, which can be thought of as an in�-

nite dimensional dynamic system. To simulate this physical system, one

has to approximate the underlying in�nite order system by a �nite order

approximation. In this paper we �rst construct a simple and straightfor-

ward approximate �nite order model for the true system. The proposed

approximate models may require large model order to approximate the

true system dynamics in the high frequency regions. To avoid the usage of

higher order models, we use a scheme similar to singular perturbations to

further reduce the model order.

Keywords: Di�usion, State-space model, Singular perturbations.

1

1 Introduction

A lot of physical process, like ow of heat across a temperature gradient, ow ofchemicals across a concentration gradient are examples of di�usion phenomena.Systems which involve di�usion phenomena are called di�usion systems, see [3],[9]. Di�usion systems are normally modeled by a linear partial di�erential equa-tions (LPDE). LPDE's can be thought of as in�nite dimensional linear dynamicsystems. To simulate a di�usion system, one has to approximate the underlyingin�nite dimensional dynamical system by a �nite order model. In some cases, inorder to achieve certain realistic error bounds in the simulation of the system,one would be forced to use an approximate model of large model order.

Singular perturbation is a technique to reduce the model order of a largescale dynamic systems (dynamic systems with large model order), which involvesinteracting dynamic phenomena of varying time constants. The theme behindsingular perturbation is that, one assumes the given large scale system can becharacterized by two modes, a fast mode and a slow mode. Model reductionis achieved by neglecting the fast phenomena or the fast mode, and later, thereduced model is improved by adding certain correction factors. For more detailsrefer to [8], [5] and [7].

In this paper, we consider the simple case of a heat di�usion in a homogeneouswall. We �rst show that the LPDE, that results from this problem, is actuallyan in�nite order dynamical system and hence needs model reduction. We pro-pose certain �nite order approximations for the given in�nite order dynamicalsystem, and these �nite order models occur as natural choice for the true system.The proposed approximations may require large model orders to match the truedynamics in the high frequency regions. To overcome this problem of using ahigher order model, we use a scheme, which is similar to singular perturbationsapproach, to further reduce the model order without compromising' much on theerror in the high frequency regions.

This paper is organized into �ve sections. In Section 2 we de�ne and exten-sively discuss the problem to be solved and also discuss our approach to solve theproblem. In Section 3 we derive the true in�nite order dynamics of the system,while in Section 4 we construct an approximate �nite order model for the in�niteorder system and discuss about them. In Section 5 we reduce higher order modelsto obtain lower order models that have less error in high frequency regions. InSection 6 we numerically analyze the convergence of the proposed approximatemodels.

2

-�

-

Ti T Te

qi d

x

Figure 1: Heat di�usion through a homogeneous wall.

2 A case study: the heat di�usion equation

2.1 Statement of the problem

Consider a homogeneous wall of thickness d units. Let x be the coordinate acrossthe wall, Te the temperature on the exterior side, Ti the temperature on theinterior side and qi the supplied heat on the interior side, cf Figure 1.

Let T = T (x; t) denote the temperature at position x and at time t. Thenthe heat ow and the temperature are described by

@T (x; t)

@t= �

@2T (x; t)

@x2(1)

with boundary constraints

qi(t) = ��@T (x; t)@x

jx=0; (2)

Te(t) = T (d; t) (3)

For simplicity we consider the intitial condition T (x; 0) = 0. In the model (1)-(3) the di�usion coeÆcient � and and the thermal conductivity � are parametersthat depend on the material of the wall.

Needless to say, both the case selected for study and its geometry are simple.While the geometry is chosen for technical ease, the simple case is aimed atgetting insight into more diÆcult systems for future work.

The problem to be solved is, given the parameters � and � and the boundarydata fqi(kh); Te(kh); gNk=1, for a particular given sampling interval of h units,estimate the interior temperature fTi(kh)gNk=1.

Note that, it has been inherently assumed that the thermal properties ofthe wall are not a�ected by the variations of external temperature or the heatsuppled. This can also be seen from the model, (1)-(3), as the parameters � and� are assumed to be constants.

Since the wall is independent of the boundary data fqi(kh); Te(kh); gNk=1, onecan think of the wall as a continuous time dynamic system, see [12], with u(t) =

3

Wall

-

-

-

Te

qi

Ti

Figure 2: Wall viewed as a dynamic system

(qi(t); Te(t))T as the input and y(t) = Ti(t) as the output, see Figure 2. The

impulse response h(t) of the wall or its Laplace transform, the transfer functionH(s) is determined from the governing equations (1)-(3). Once the transferfunction H(s) or the impulse response h(t) is determined, Ti(t) can be computedeither as a convolution of the input u(t) with the impulse response h(t) or as ainverse Laplace transform of the product of U(s) and H(s), where U(s) is theLaplace transform of the input u(t).

In the next section we determine the transfer function H(s) of the wall andshow that it is a transcendental function, or in system's terminology, it is ofin�nite order. Since the transfer function H(s) of the wall is of in�nite order, onehas to approximate H(s) by a �nite order model, which is referred to as modelreduction in system's terminology. Hence, we propose certain �nite order transferfunctions, which arise as a natural choice of approximation for the in�nite ordertransfer function H(s). It turns out that one needs to use a large model orderto approximate H(s) more closely in the high frequency regions. To overcomethis problem of using a higher order approximation, we use a model reductiontechnique, which is similar singular perturbations to reduce these higher ordermodels into lower order ones. These models approximate H(s) better in the highfrequency regions than those corresponding lower order models directly used.

We strike a note of caution here when we say that, "the lower order mod-els obtained by reducing the higher order models, approximate H(s) better inthe high frequency regions than those corresponding lower order models directlyused". It would be more appropriate to say that the lower order models, obtainedby using singular perturbations, approximate H(s) better in certain desired highfrequency regions and not in general throughout the high frequency regions.

It is well known that there are a lot of well established numerical schemes, likeGalerkin, Finite di�erence and Collocation, see [4], that can be used to computeTi(t) from (1)-(3). These numerical schemes, when applied to (1)-(3), can alsobe interpreted as schemes which construct approximate transfer functions forthe true transfer function H(s), see [11] and [1]. Even though Galerkin, Finite

4

di�erence and Collocation have a well established theory for their convergence,they are very general in their approach and do not exploit the structure of thegiven problem. In this paper we take a systems approach rather than a numericalanalysis approach and propose certain models, which exploits the structure of thegiven problem. These models turn out to approximate the true system better thanthe Galerkin, Finite di�erence and Collocation, for a given model order.

3 The system dynamics

The de�ned system is linear and has two inputs, qi and Te, and one output, Ti.Interestingly enough, one can rather easily derive the transfer function of thesystem, which will give an input-output description of the dynamics, [11]. Forconvenience we repeat the derivation here.

First take the Laplace transform of (1) which gives

sT (x; s) = �d2

dx2T (x; s) (4)

where T (x; s) is the Laplace transform of the temperature T (x; t). Regarding sas a �xed parameter, (4) is a standard ordinary di�erential equation. It is foundthat the general solution to (4) can be written as

T (x; s) = A(s) cosh(x

rs

�) +B(s) sinh(x

rs

�) (5)

Further, taking Laplace transforms on (2),(3) and evaluating the constants A(s)and B(s) gives the following expression for temperature at the interior side.

Ti(s) =tanh(d

ps

�)

�p

s

�| {z }qi(s) +

1

cosh(dp

s

�)| {z }Te(s): (6)

G1(s) G2(s)

Note that both G1(s) and G2(s) are bounded transcendental functions, with thesame set of poles

Pn = ��2

d2(n� 1

2)2�; n = 1; 2; : : : ; (7)

and only G1(s) has zeros, at

Zn = ��2

d2n2� n = 1; 2; : : : : (8)

The static gains of G1(s) and G2(s) are

G1(0) =d

�; G2(0) = 1: (9)

5

Since all the poles of G1(s) and G2(s), (7), are real and negative, the systemis stable and the transfer functions G1(s) and G2(s) are bounded. Boundednessof G1(s) and G2(s) implies that, the output Ti(t) depends continuously on theinput data qi(t) and Te(t). Continuous dependence along with the uniquenessof the solution renders the problem, (1)-(3), well posed in numerical analysisterminology.

It is worth noting that both G1(s) and G2(s) are of low pass character, seeFigures 3, 4 and 5. Hence the approximate models should also be of low passcharacter with the same static gain. Otherwise the approximate models would notreach the steady state as t!1 in the time domain. The low pass character ofG1(s) and G2(s) would also enable us to choose the approximate model dependingon the frequency contents of the input signals.

If one takes a closer look at the Figures 3, 4 and 5, one can see that j G2(i!) j!0 at a rate much faster than j G1(i!) j, as ! ! 1. The faster convergence ofj G2(i!) j! 0 is apparently due to the fact it has no zeros and it is an all polesystem. Due to its fast convergence j G2(i!) j may not need a higher orderapproximation, as j G2(i!) j is almost close to 0 in the high frequency regions.In the case of j G1(i!) j, it behaves more like �

d

ps

�

in the high frequency regions,

as tanh(dp

s

�) � 1 for a large s = i!. Hence one may have to use a higher order

models to approximate G2(s) well in the high frequency regions.Finally, we look at the e�ect of the parameters � and � on the transfer func-

tionsG1(s) andG2(s). From (6) one can see that the parameter �merely re-scalesthe frequency axis, as G1(s) and G2(s) can also be considered as functions of s

�.

The parameter � just scales the magnitude of G1(s) by1

�and has no e�ect on

G2(s). To make our analysis more general, we scale the frequency axis by usingthe transformation s 7! �s, to discount the e�ect of the parameter � in (6). Fora given �, one can always revert back to the original frequency domain by usingthe inverse transformation s 7! s

�on the scaled frequency domain.

4 Approximate models

Without much ado, we directly introduce the approximate models. Since G1(s)is characterized by the poles (7) and the zeros (8), a natural candidate for ap-proximating G1(s) is

Gn

1 (s) = K1

(s� z1)(s� z2) : : : (s� zn�1)

(s� p1)(s� p2) : : : (s� pn); (10)

where

K1 = �d p1p2 : : : pnz1z2 : : : zn�1

(11)

6

is the normalizing constant which renders

G1(0) = Gn

1 (0): (12)

Further, pi and zi, i = 1; 2; : : : ; n are the poles and zeros, respectively, of the truetransfer functions with their frequencies scaled by �, that is

pi =Pi�

(13)

and

zi =Zi�

(14)

where Pi; i = 1; 2; : : : ; n are as de�ned in (7) and Zi; i = 1; 2; : : : ; n � 1 are asde�ned in (8).

Following a similar pattern as above and noting that G2(s) has no zeros, wede�ne

Gn

2 (s) = K2

1

(s� p1)(s� p2) : : : (s� pn); (15)

where

K2 = (�1)np1p2 : : : pn (16)

is the normalizing constant which renders

G2(0) = Gn

2 (0) (17)

and pi, i = 1; 2; : : : ; n are as de�ned in (13).One can see from the Figures 3, 4 and 5, that the approximate transfer

functionsGn

1 (s) andGn

2 (s) track the true true transfer functions, G1(s) and G2(s),upto a certain frequency but deviate after that. In fact one can say thatG1(s) andG2(s) are well approximated by Gn

1 (s) and Gn

2(s) at the low frequency regions,but the approximation deteriorates as the frequency increases. Note that byconstruction both j Gn

1 (i!) j and j Gn

2 (i!) j! 0 as ! ! 1. Hence one wouldexpect the error j Gi(i!)�Gn

i(i!) j to increase from zero at ! = 0 and attain a

maximum at some �nite ! and then decrease to zero as ! ! 1. This can alsobe seen in the Figures 6 and 7.

To check the consistency of the proposed approximate models Gn

1 (s) andGn

2 (s), we plot, in Figure 8, max! j Gi(i!) � Gn

i(i!) j for di�erent model or-

ders. Note from Figure 8 that max! j Gi(i!)�Gn

i(i!) j decreases as n increases.

From Figure 8 we further conclude that max! j Gi(i!)�Gn

i(i!) j! 0 as n!1.

Hence we also conclude that Gn

i(s)! Gi(s), i = 1; 2.

7

Table 1: � in m2=hr for typical building materials.

Material � in m2=hr

Aluminium 0.3383

Concrete 0.0037

Wood 3.1579 x 10�4 - 6.3158 x 10�4

Mineral wool 4.2857 x 10�5 - 2.5714 x 10�4

One has to be very careful, when one speaks about about high and low fre-quency regions, as both G1(s) and G2(s), in the Figures 3, 4 and 5, are plottedfor normalised frequencies. Table 1 the values of �, in m2=hr, for certain typicalbuilding materials. Hence a normalised frequency of 104 rads=hr would corre-spond to 37 rads=hr for concrete and 3383 rads=hr for aluminum. What is ahigh frequency for a particular material may not be high for some other materi-als. Nevertheless, when we speak about frequencies with respect to the transferfunctions, we mean only normalised frequencies.

Since j G1(s) j! 0 at a very slow rate, at 1

dpsas s !1, it has considerable

magnitude even at the high frequencies and hence needs to be well approximatedin the high frequency regions. Note that, form the Figures 3, 4 and 5, j Gn

1 (s) j!0 at a faster rate than j G1(s) j. This is due to fact that Gn

1(s) has a pole excess,(10), so Gn

1 (s) � K1

sfor a large s. This faster convergence deteriorates the

approximation as the frequency increases. If possible one would like to have animproved approximation in the high frequency regions with a lower order model.

In the case of G2(s), things are much simpler, since j G2(s) j! 0 at anexponential rate, see Figures 3, 4 and 5, which is due to the fact that G2(s)has no zeros to nullify the e�ect of the poles. Note that, from the Figures 3, 4and 5, both j G2(s) j and j Gn

2 (s) j are very close to zero for frequencies above1000 rads=hr. Hence improvement of the approximate transfer function Gn

2 (s),with a low model order is necessary only if the input Te(t) has high frequencycomponents of very high magnitude.

As we have mentioned before, the choice of the model order depends verymuch on the spectral contents of the input signals. If one restricts oneself to ascenario of using the proposed model for the materials used for building houses,then the input Te(t) or the external temperature, is normally constructed from theTest Reference Year Data (TRY data), see [2], available at that place where theapproximate model is going to be used. TRY Data, at any given place, consistsof hourly samples of the temperature over a period of one year with a minimumdi�erence of 0:1ÆC between two non-identical samples of the temperature. TRYdata normally do not have high frequency components of considerable magnitude.This is apparently due to the fact that, temperature at any given place does not

8

show marked di�erences in short intervals of time. One would have to viewthem in large windows of time like months or seasons to see large variations intemperature. Hence even for a frequency of 10 rads=hr, is too high a frequencyvalue for the external temperature Te(t), to have considerable magnitude. Thusa lower order model, say n = 7 or 8 or 9 may suÆce for G2(s), due to the lowfrequency contents of the input, in the case of materials used for building houses.However, if one wants to use the proposed model, in situations where the inputTe(t) has high frequency components, then one has no choice, but to use a higherorder model depending on the frequency contents of the input.

The input qi(t) may or may not contain high frequency components, depend-ing on the application, for which the approximate model, Gn

2 (s), is going to beused. In most practical applications involving building materials, qi(t) is used asa control signal to keep the internal temperature, Ti(t), constant or close to adesired value. In such cases qi(t) would mostly bootstrap Te(t) and would havefrequency contents almost similar to that of Te(t). In such cases again one maynot need very high model order, possibly a model of order n = 7 or 8 or 9 wouldserve the purpose. But if the application involves usage of a qi(t), which has highfrequency components with considerable magnitude then one has no other optionbut to use a higher order model.

As we seen before, from Figures 3, 4 and 5, the approximations Gn

1 (s) andGn

2 (s) deteriorate as the frequency increases. The primary reason for the poorperformance of Gn

1 (s) and Gn

2 (s) at the high frequencies is due the non inclusionof the higher order poles and zeros in the lower order models. For any generaltransfer function, the lower order poles and zeros (poles and zeros close to theorigin) play a larger role in determining the magnitude of transfer function in thelow frequency regions, whereas the higher order poles and zeros (poles and zerosthat are quite far away from the origin) have negligible e�ect on the magnitudeof transfer functions at the low frequency regions. But as the frequency increasesthe higher order poles and zeros start playing a greater role in determining themagnitude of the transfer functions, as ! !1 almost all poles have equal e�ect.Hence to get better approximations for G1(s) and G2(s) in the high frequencyregions, one would have to incorporate the higher order poles and zeros in (10)and (15). In e�ect one has to increase the model order for better approximationsin the high frequency regions.

Since the inputs qi(t) and Te(t) are band limited signals there may arise a lotof situations, where the spectral densities of qi(t) and Te(t), may decay rapidlyto zero after a particular frequency. A typical example would be, the spectraldensities of qi(t) and Te(t) having the shape of a low pass �lter or a band pass�lter. In such situations, the following model reduction technique may come inhandy.

9

10−2

100

102

104

10−3

10−2

10−1

100

101

frequency [rads/hr]

ma

gn

itu

de

[dB

]

10−2

100

102

104

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

frequency [rads/hr]

ma

gn

itu

de

[dB

]

n=3 n=3

Figure 3: The approximate transfer functions (10) and (15) along with theirrespective true transfer functions G1(s) and G2(s) with the model order n = 3.The true transfer function (of in�nite order) is plotted with a dotted line and theapproximate transfer function (of �nite order) is plotted with a solid line. Thefrequency axis is scaled by the parameter � for both G1(s) and G2(s) and themagnitude of G1(s) is scaled by �.

10

10−2

100

102

104

10−3

10−2

10−1

100

101

frequency [rads/hr]

ma

gn

itu

de

[dB

]

10−2

100

102

104

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

frequency [rads/hr]

ma

gn

itu

de

[dB

]

n=5 n=5

Figure 4: The approximate transfer functions (10) and (15) along with theirrespective true transfer functions G1(s) and G2(s), with the model order (n = 5).The true transfer function (of in�nite order) is plotted with a dotted line and theapproximate transfer function (of �nite order) is plotted with a solid line. Thefrequency axis is scaled by the parameter � for both G1(s) and G2(s) and themagnitude of G1(s) is scaled by �.

11

10−2

100

102

104

10−2

10−1

100

101

frequency [rads/hr]

ma

gn

itu

de

[dB

]

10−2

100

102

104

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

frequency [rads/hr]

ma

gn

itu

de

[dB

]

n=7 n=7

Figure 5: The approximate transfer functions (10) and (15) along with theirrespective true transfer functions G1(s) and G2(s), with the model order (n = 7).The true transfer function (of in�nite order) is plotted with a dotted line andthe approximate transfer function (of �nite order) is plotted with a solid line.Thefrequency axis is scaled by the parameter � for both G1(s) and G2(s) and themagnitude of G1(s) is scaled by �.

12

101

102

103

104

105

10−4

10−3

10−2

10−1

frequency in rads/hr

Err

or

|G1(i

ω)

− G

n 1(i

ω)|

in

dB

n=3

n=5

n=7

Figure 6: The error, j G1(i!) � Gn

1 (i!) j, in the approximate transfer function,Gn

1 (i!), for model orders n = 3; 5; 7. The frequency axis is scaled by the param-eter � and the magnitude of G1(s) and Gn

1 (s) is scaled by �.

13

100

101

102

103

104

10−22

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2


Err

or

|G2(i

ω)

− G

n 2(i

ω)|

in

dB

n=7

n=5

n=3

Figure 7: The error, j G2(i!) � Gn

2 (i!) j, in the approximate transfer function,Gn

2 (i!), for model orders n = 3; 5; 7. The frequency axis is scaled by the param-eter �.

14

100

101

102

10−3

10−2

10−1

model order

Err

or

ma

xω|G

1(i

ω)

− G

n 1(i

ω)|

in

dB

100

101

102

10−6

10−5

10−4

10−3

model order

Err

or

ma

xω|G

2(i

ω)

− G

n 2(i

ω)|

in

dB

Figure 8: The maximum error, max! j G1(i!)� Gn

1(i!) j and max! j G2(i!) �Gn

2 (i!) j, in the approximate transfer functions Gn

1 (i!) and Gn

2 (i!) respectively.The frequency axis is scaled by the parameter � and the magnitude of G1(s) andGn

1 (s) is scaled by �.

15

5 Model reduction using singular perturbation

As we have mentioned in the previous section, the poor performance of the lowerorder approximations is due to the non-inclusion of the higher order poles andzeros. By using a lower order model and thereby not including the higher orderpoles and zeros, one cannot achieve a good approximation in the high frequencyregions. To have better approximations there one must include the higher orderpoles and zeros and thereby use a higher order model. As a trade o�, we device amodel reduction scheme, which is similar to the singular perturbation techniqueused for reducing the model order of sti� linear systems, see [8],[5] and [7]. Inthis scheme we take a model of large order and then reduce it to lower ordermodel without completely neglecting the higher order poles and zeros. The lowerorder model thus obtained by reducing the higher order model, tracks the truetransfer function for a longer frequency window than the corresponding lowerorder transfer functions obtained directly by using (10) and (15). This is notwithout a cost, as these approximate models do not converge to zero as thefrequency tends to in�nity. First, we rewrite (10) and (15) using partial fractiondecompositions as

Gn

1(s) = K1[A1

s� p1+

A2

s� p2+ : : :+

An

s� pn] (18)

Gn

2(s) = K2[B1

s� p1+

B2

s� p2+ : : :+

Bn

s� pn] (19)

where

Ai =(z1 � pi)(z2 � pi)(z3 � pi) : : : (zn�1 � pi)

(p1 � pi)(p2 � pi) : : : (pi�1 � pi)(pi+1 � pi) : : : (pn � pi); (20)

and

Bi =1

(p1 � pi)(p2 � pi) : : : (pi�1 � pi)(pi+1 � pi) : : : (pn � pi): (21)

A justi�cation for the expressions (20) and (21) is provided in the appendix.Let us �rst consider Gn

1 (s). As a �rst step towards model reduction we sepa-rate the given transfer function Gn

1 (s) into fast and slow modes.

Gn

1 (s) = Gm

1 (s) +H1(s) (22)

where

Gm

1 (s) = K1[A1

s� p1+

A2

s� p2+ : : :+

Am

s� pm]; (23)

H1(s) = K1[Am+1

s� pm+1

+ : : :+An

s� pn] (24)

16

Note that Gm

1 (s) corresponds to the slow component or slow phenomena, as allits poles are dominating the poles of H1(s). Vice versa, H1(s) corresponds tothe fast component or fast phenomena, as all its poles are relatively fast whencompared to the poles of H1(s). Looking at it in the time domain by takinginverse Laplace transform of (23) and (24), one can see that the inverse Laplacetransform of both (23) and (24) are linear combinations of exponentials with theinverse Laplace transform of (24) converging to zero at a rate faster than theinverse Laplace transform of (23).

As in singular perturbations, we �rst obtain an intermediate lower order modelby neglecting the fast component, that is, we obtain an intermediate reduced or-der model Gm

1 (s), by dropping H1(s) in (22), and then add some correction fac-tors. These correction factors are obtained by approximating the fast component,H1(s), by its static gain H1(0), and thus we have the reduced order model

Gm

R1 = Gm

1 (s) +H1(0) (25)

for (22).Similarly by rewriting (19), as a sum of fast modes and slow modes and

approximating the fast mode by its static gain, we get a reduced order model

Gm

R2 = Gm

2 (s) +H2(0); (26)

for Gn

2 (s).If one looks at the approximate models (25) and (26) in the time domain, one

can see that the fast components H1(s) and H2(s), which correspond to

K1

nXi=m+1

Ai exp (�pit) (27)

and

K2

nXi=m+1

Bi exp (�pit); (28)

respectively, in the time domain, have been approximated by

K1

nXi=m+1

Ai

�pi Æ(t) (29)

and

K2

nXi=m+1

Bi

�pi Æ(t); (30)

17

respectively. To put it in words, the quickly decaying exponentials have beenapproximated by delta functions or impulse functions. This kind of an approx-imation has been extensively used in uid dynamics for a long time, see pages481� 519 in [6].

In Figures 9, 10 and 11, we have plotted Gm

Ri(s) along with Gm

i(s) and true

transfer functions Gi(s), for i = 1; 2. One can see that Gm

Ri(s) tracks Gi(s), for a

longer period ( for a larger frequency window) than Gm

i(s), but Gm

Ri(s) does not

converge to zero as s!1. This is due to the fact that

lims!1

Gm

Ri(s) = Hi(0): (31)

As we have mentioned before, this model reduction scheme comes handy onlywhen the spectral density of, qi(t) and Te(t), decays quickly after a particularfrequency. Since one can use the extra region of approximation to capture thetail of the spectrum of qi(t) and Te(t) and avoid a few extra computations. Tomake it clear, consider a case where the normalised frequency contents of qi(t)are concentrated within the frequency window of [0 ; 100] and decay very rapidlyafter 100 rads=hr, then instead of using a higher order model of n = 7 or 8 or 9,once can use a reduced order model with m = 5 and n = 11, that is 5 slowcomponents and 6 fast components, see Figures 10, and thereby save a few extracomputations.

6 Concluding discussion

We conclude this paper by discussing a few questions, which we think are im-portant. In section 1, we claimed that the models proposed, (10) and (15), inthis paper are better than the models obtained by using conventional schemeslike Finite di�erence, Chebyshev Collocation, Chebyshev Galerkin and Cheby-shev Tau. In [1], we have shown by simulations, that for a given model ordern, Chebyshev Collocation gives a better approximation for this problem, whencompared to Chebyshev Galerkin and Chebyshev Tau. From [11] one can alsoconclude that for a given model order n Chebyshev Collocation gives a betterapproximation, for this problem, than Finite di�erence approximation. Henceit is enough if we compare the models proposed in this paper with ChebyshevCollocation. In Figures 12, 13, 14, 15, 16 and 17 we have plotted the true transferfunctions along with their respective Chebyshev Collocation approximation andthe proposed models (10) and (15). In the case of G2(s), from the Figures 15,16 and 17, we can see that the proposed model, Gn

2 (s), approximates G2(s) muchbetter than a Chebyshev Collocation model of same order. In the case of G1(s),one cannot infer from the Figures 12, 13, 14, which one is better. In the lowfrequency regions both, Gn

1 (s) and the Chebyshev Collocation approximation,seems to approximate G1(s) quite well. Since G

n

1 (s)! 0, one can say that Gn

1 (s)approximates G1(s) better, in the high frequency regions, than the Chebyshev

18

10−1

100

101

102

103

10−3

10−2

10−1

100

101

n=3


ma

gn

itu

de

in

[d

B]

Approximate True Singularly Perturbed

Figure 9: The approximate transfer function (10), plotted in solid line, along withthe true transfer function in the dotted line and the singularly perturbed model(25) in dashed dots. The approximate transfer function, (10), in the �gure, isplotted with n = 3 and singularly perturbed model (25) is plotted with m = 3and n = 10.

19

10−1

100

101

102

103

10−3

10−2

10−1

100

101


ma

gn

itu

de

in

[d

B]

n=5


Figure 10: The approximate transfer function (25), plotted in solid line , alongwith the true transfer function in the dotted line and the singularly perturbedrurbed model (25) in dashed dots. The approximate transfer function,( 10), inthe �gure, is plotted with n = 5 and singularly perturbed model (25) is plottedwith m = 5 and n = 10.

20

10−1

100

101

102

103

10−2

10−1

100

101


ma

gn

itu

de

in

[d

B]

n=7


Figure 11: The approximate transfer function (10), plotted in solid line, alongwith the true transfer function in the dotted line and the singularly perturbedmodel (25) in dashed dots. The approximate transfer function, (10), in the �gure,is plotted with n = 7 and singularly perturbed model (25) is plotted with m = 7and n = 15.

21

Collocation approximation, as the Chebyshev Collocation approximation doesnot tend to zero as the frequency tends to in�nity. But in the grey areas betweenthe high and the low frequency region (the interval [100 ; 1000] rads=hr in theFigures 12, 13 and 14) it is not clear, which is better.

If one compares the singularly perturbed reduced order models Gn

R1, (25),with a Chebyshev Collocation approximation, see Figures 18, 19, 20, it is fairlyclear that the singularly perturbed reduced order models perform better thanChebyshev Collocation approximation. At this point, it must be mentioned thata singular perturbation of the type (25) is also possible for Chebyshev Collocationapproximation, but this involves considerable computational costs, as one doesnot have the analytical expressions for the approximate transfer functions ob-tained by using Chebyshev Collocation. Hence our claim that the approximatemodels proposed in this paper are better than the models obtained by usingconventional schemes like Finite di�erence, Chebyshev Collocation, ChebyshevGalerkin and Chebyshev Tau is justi�ed to a certain extent.

Having said the above, we list the advantages of using the models (10), (15),(25) and (26) proposed in this paper. First and the foremost, one can select twodi�erent model orders for approximating G1(s) and G2(s), which is not the casewith any standard PDE solvers in numerical analysis. This is a big plus in terms ofmodel reduction. The analytic expressions of the proposed models (10), (15), (25)and (26) are simple and easy to implement. This is not the case with ChebyshevCollocation, Chebyshev Galerkin and Chebyshev Tau, where it is diÆcult, ifnot impossible, to obtain the analytical expressions for the approximate transferfunctions and setting up the di�erential equations for a numerical simulationin Chebyshev Collocation, Chebyshev Galerkin and Chebyshev Tau, could be atedious task.

7 Appendix

We prove the expressions (20) and (21) in this appendix. The proof goes bymathematical induction. We use a new notation here. For a given n in (20) and(21), we refer to their partial fraction coeÆcients Ai asA

n

i, Bi as B

n

iand K1, K2

as Kn

1 and Kn

2 respectively.We �rst prove (20). Let n = 2, then we have

G21(s) = K2

1

(s� z1)

(s� p1)(s� p2)(32)

where

K1

1 = �p2p1z1

(33)

22

10−1

100

101

102

103

10−3

10−2

10−1

100

101

frequency in [rads/hr]

ma

gn

itu

de

in

[d

B]

n=3

Approximate True Approximate Collocation

Figure 12: The approximate transfer function (10), plotted in solid line, alongwith the true transfer function in the dotted line and Collocation approximation,refer to [10], in dashed dots.

23

10−1

100

101

102

103

10−3

10−2

10−1

100

101


ma

gn

itu

de

in

[d

B]

n=5



24

10−1

100

101

102

103

10−2

10−1

100

101


ma

gn

itu

de

in

[d

B]

n=7



25

10−1

100

101

102

103

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n=3


ma

gn

itu

de

in

[d

B]

ApproximateTrue Collocation


26

10−1

100

101

102

103

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100


ma

gn

itu

de

in

[d

B]

n=5



27

10−1

100

101

102

103

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100


ma

gn

itu

de

in

[d

B]

n=7



28

10−2

10−1

100

101

102

10−1

100

101


Ma

gn

itu

de

in

dB

n=3

Singularly perturbed model True transfer function Chebyshev Collocation

Figure 18: The Singularly perturbed model (25), plotted dashed dots, along withthe true transfer function in the dotted line and Collocation approximation, referto [10], in dashed lines for a model of order n = 3.

29

10−1

100

101

102

103

10−2

10−1

100

101


Ma

gn

itu

de

in

dB

n=5



30

10−1

100

101

102

103

10−2

10−1

100

101


Ma

gn

itu

de

in

dB

n=7



31

By using partial fractions, (32) can be rewritten as

G1

1(s) = K1

1 [A11

s� p1+

A22

s� p2] (34)

where

A1

1 =z1 � p1p2 � p1

(35)

A22 =

z1 � p2p1 � p2

(36)

which is in agreement with (20) and (21).Assume (20) to be true for n = 2; : : : ; N , we claim it is true for n = N + 1,

and thus for any n by induction.Let n = N + 1. Then

GN+1

1 (s) = KN+1

1

(s� z1)(s� z2) : : : (s� zN )

(s� p1)(s� p2) : : : (s� pN+1)(37)

= KN

1

(s� z1)(s� z2) : : : (s� zN�1)

(s� p1)(s� p2) : : : (s� pN)

pN+1

zN

s� zNs� pN+1

(38)

= GN

1 (s)pN+1

zN

s� zNs� pN+1

: (39)

Since we have assumed (18) and (20) to be true for n = 2; : : : ; N , using (20) in(39) we have

GN+11 (s) = KN

1

pN+1

zN[NXi=1

AN

i

(s� pi)]s� zNs� pN+1

(40)

= KN

1

pN+1

zN[NXi=1

AN

i(s� zN )

(s� pi)(s� pN+1)]: (41)

As we did in (32) and (33), we rewrite (41) using partial fractions, which gives

GN+1

1 (s) = KN

1

pN+1

zN

NXi=1

AN

i[(zN � pi)

(pN+1 � pi)

1

(s� pi)

+(zN � pN+1)

(pi � pN+1)

1

(s� pN+1)] (42)

which from (20) is

GN+1

1 (s) = KN+1

1

NXi=1

AN+1

i

(s� pi)+KN+1

1

NXi=1

AN

i(zN � pN+1)

(pi � pN+1)

1

(s� pN+1)(43)

32

Thus we are done, if we prove that

NXi=1

AN

i(zN � pN+1)

(pi � pN+1)= AN+1

N+1(44)

To prove (44), �rst note that, by equating (10) and (18), for any given n, we have

(s� z1) : : : (s� zm�1) =nXi=1

An

i(s� p1) : : : (s� pi�1)(s� pi+1) : : : (s� pn): (45)

Hence by equating the coeÆcients of sn�1 in (45) we have

nXi=1

An

i= 1; (46)

for any given n. Note that

(zN � pN+1)

(pi � pN+1)= 1� zN � pi

pN+1 � pi: (47)

Hence using (47) in (44) we have

NXi=1

AN

i(zN � pN+1)

(pi � pN+1)=

NXi=1

AN

i�

NXi=1

AN

i

zN � pipN+1 � pi

: (48)

Note that

AN

i

zN � pipN+1 � pi

= AN+1

i; (49)

therefore using (46) and (49) in (48), we get

NXi=1

AN

i(zN � pN+1)

(pi � pN+1)= 1�

NXi=1

AN+1

i; (50)

which by (46)

NXi=1

AN

i(zN � pN+1)

(pi � pN+1)= 1� (1� AN+1

N+1) (51)

= AN+1

N+1(52)

Thus

GN+1

1 (s) = KN+1

1

N+1Xi=1

AN+1

i

(s� pi): (53)

Hence (20) is proved.A similar induction proof can be made for (21).

33

References

[1] B. Bhikkaji and T. S�oderstr�om. Reduced order models for di�usion systems.August 2000. Report 2000-019. Department of Information Technology, Up-psala University.

[2] J.J. Bloem. editor, System Identi�cation Competition. Joint Research Cen-tre, European Commission, Luxembourg, 1996.

[3] H. S. Carlslaw and J. C. Jaeger. Operational Methods In Applied Mathemat-

ics. Oxford University Press, London, 1947.

[4] D. Gottlieb and A.Orzag. Numerical Analysis of Spectral Methods: Theory

and Applications. SIAM, Philadelphia, 1977.

[5] R. E. O'Malley Jr, P. V. Kokotovic, and P. Sannuti. Singular perturbationsand order reduction in control theory - an overview. Automatica, Vol 12,

Number 2, pp. 123-132, 1976.

[6] J. Kevorkian and J. D. Cole. Perturbation Methods in Applied Mathematics.Springer-Verlag, New York, 1981.

[7] H. K. Khalil. Nonlinear Systems. Macmillan Publishing Company, NewYork, 1992.

[8] P. Kokotovic, H. K. Khalil, and J. O' Reilly. Singular Perturbation methods

in Control: Analysis and Design. Academic Press, New York, 1986.

[9] K. Kunisch. Numerical Methods for Parameter Estimation Problems. InInverse Problems in Di�usion Process. SIAM, Philadelphia, 1995.

[10] T. S�oderstr�om and B. Bhikkaji. Reduced order models for di�usion systemsvia collocation methods. August 2000. Report 2000-018. Department ofInformation Technology, Uppsala University.

[11] T. S�oderstr�om and S. Remle. Parameter estimation for di�usion models.Proc of 12th IFAC Symposium on System Identi�cation. Santa Barbara,CA, 21-23 June 2000.

[12] T. S�oderstr�om and P. Stoica. System Identi�cation. Prentice Hall Interna-tional, Hemel Hempstead, UK, 1989.

34

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