Model Reduction Using Rational Approximation Techniques

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1<)<)1

MODEL REDUCTION USING RATIONAL APPROXIMATION TECHNIQUES M.-P.C. Cai and E.B. Lee f)cpllrtrtlCnr of Electrical En}iincerin}i , University of Minnesota, Minnellpolis, MN 55455, USA

Most of model reduction for finite dimensional sLahle systems is covcred hy the ahove methods related to balancing. However, many of the systems we deal with in terms of designing stahilizing and optimal rohust controllers are not finite dimensional or stahle. Frequently there are time delays involved and the controlled system involves spatial variables (distrihuted parameters) . Thus before using onc of the standard model reduction algorithms it is necessary to find a sLahle finite dimensional approximation.

ABSTRAcr An effectivc way to get a redueed order model for use in control system analysis and design is to apply frequency domain analysis and associawd computational techniques of model reduction. This can he accomplished hy direct experimentation in which spectral data (amplitude ratio and phase shift) is ohtaincd hy input/output measurements or hy creating a file of frequency response data from a given analytical transkr function. It is then sufficient to ohtain a rational transfer function fit to the data hy Fourier series methods,aftcr which one of the finite dimensional algorithms ean he exploited to extract a reduced order model. Here we descrihe computationally ellcctive user friendly software, hascd on MATLAB. for doing this frequency domain analysis and model reduction with associated ll ~- norm (unifOlm nOlm on j9\) error estimates. The H ~- norm error can often he decreased hy increasing the order of the approximant so as to get models which are suflicient for rohust controllcr designs.

Approximation of possibly unstahle infinite-dimensional systems in the frequency domain is a typical application of model reduction. This approximation task has heen studied hcfore using the Hankel operator approximation theory and Fourier transform techniques. The major advantage of the later is that the approximate model can he directly and easily calculated hascd on the Fourier transform algorithm. In the next section wc shall descrihe hrieOy how to take frequency response data into a (high order) model hy FourierLaguerre series approximation. This is a task of fitting a rational function to fn:ljuency response data generated from an irrational transfer function or to frequency response data experimentally obtained at a lar&e numher of freque~cies, After the high order model IS ohtamed ha lanced truncaUon IS applied to get a low order model, as discussed in the next section.

I. INTRODUCTION The construction of adeljuate models for control system analysis and synthesis continues to he a major theme of control systcms rcscarch. On the onc hand wc 'would like to model all aspect of thc cxternal (or cvcn intcrnai) hehavior of the systcm; hut at the same time we necd simple dfcctive models for realistic analysis and synthesis. Tuning the models to the task at hand is an ellective way to proceed.

In model reduction algorithm the main idea is to eliminate any weak suhsystem which contributes little to the system input/output relationship. It involves a tradeoff hetween model order and the degree to which the characteristics of the system are reflected by the model. Because the relative importance of various characteristics is highly dependent on the applications, there is no universal algorithm for model reduction. The application of concern here is to have a good model for fcedhack controller synthesis and analysis. 2. FITTING FREQUENCY RESPONSE DATA AND MODEL REDUCTION

We shall dcscrihe here a rcccntly developed algorithm and associated software to find adeljuate models for possihly infinite dimensional unstahle systems with possihly multiinputs and multi-outpuLs which can he used in the construction of stahili/.ing controllers or other controller synthesis tasks. The algorithm was developed to find finite dimensional approximants for infinite dimensional systems, hut can he applied in many situations where frequency response data is availahle; for example, to replace the normal curve filting procedures to find transfer function models with least squares error function or other norms on j9\ (for examples, see Sponos and MingOli 1I 99~ I). Also, the freljuency response data can he weighted to highlight certain frequcncy rangcs and free up other frequency rangcs (sec Zhou 119(21).

One way to ohLain a finite dimensional approximation is to use a partial fraction expansion to decompose T(s) into its stable and unstahle parts so that T(s)=Ts(s)+ T u(s), in which Ts(s) and Tu(-s) are hoth analytic in H (opcn right half plane). It is a fact that for any Laplace transformahle function H(s), we can express it in the form

We shall now hrielly dcscrihe some of the finite dimcnsional model reduction techniljues which are used as part of our algorithm for efkctivc model reduction. Previously we ha vc descrihed some of the tradeolT ljuestions involved in the software hased on MA TLAB and given a numher of examples of its use (Cai and Lee 1I 99~1). Here we will descrihe in the next section the use of curve lilting procedures on j9\ and give further examples in section III of use of the software.

H(s)

= LH, ~,(s) l =O

where

~'(S)=..fIT(A-s)' A +s A +s

In the past, many methods of model redm:tion have heen developed. Early methods were mainly hased on retaining the dominant poles in the reduced-order model, as in aggregation methods, or on matching some moments of the original system, as in the Pade approximation techniques. In I 9!\ I, the principle component analysis hased on the concept of ha lanced realization was proposed hy Moore for the model reduction of linear continuous time systems. This was extended to the discrete-time case hy Perneho and Silverman in 1982. After that, all optimal Hankel-norm approximation and their error hounds was proposed hy Glover for linear multi-variahle systems. In 19!\9, a Schur method for model reduction was proposed hy Safonov and Chiang. And in 1989, Gu, Khargonekar and Lce comhined frequency domain measurements and model reduction in one algorithm providing primarily a way to ohtain rational approximanLs for infinite dimensional systems. We shall he mainly interested in descrihing a computer implementation of this rational approximation algorithm.

and

H,ER"'xP.

This is a Fourier-Laguerre series representation for H(s). Letting H(s)

..fIT =- TJs) A +s

we have the expression for the sLahle part

T, (s)

- (A--=..:: )' . = LH, '=0

,1+ S

Because T u( os) is also analytic in H, we also have an expression of the ahove form for the unsLahle part as

335

Thus,

3. EXAMPLES OF MODEL REDUCTION

T(s)

- (A.A. - s )' . 'LH,

=T,(s) + Tu(s) = ,___

Implemcntation of the approximation scheme discussed aboY\! can be easily done hy using a program written in MATLAB.

+s

This looks more like a formal power series if we use a change of variable

l-z s:= 1..--, l+z

(or

We now descrihe some of the aids availahle in the program manage.m for doing this model reduction/approximation.

A. -s A. +s

First there is the question of what Laguerre parameter "- to choose in the hilinear transformation. It is suggested that "should he one to ten times the handwidth of T(s)ls=jw' Thus in the program manage. m the frequency response IT(jw)1 is calculated at a sufficient numher of points Wk so that the handwidth can he approximately selected.

z:=-)

which is a con formal mapping from H to D, the unit disk; getting thereby

l-z l+z

F(z):=T(I..-)

Next the numher of points M in the FFT algorithm is selected; usually a 1024, or 2D4X point FFT is used in MATLAB software. Then the numher of terms N for truncation of the Fourier-Laguerre series is to he selected, hoth for unstable as well as stahle parL~. The program manage.m helps one do this hy displaying the discrete time signal for the stahle as well as unstahle parts. See figure 2 for a typical plot of this. Note periodic property means for the stahle part we look at the first part of the 1024 point spectrum (stahle part corresponds to phase lag, while unstahle part to phase lead). In figure 2 after ahout 75 terms the data is approximately zero so we select N=75 to do an initial approximation.

= 'Lf,z' = F,(z) + Fu(z) k=--

where Fu(z)

='Lf-,z-'

and F,(z)

1:1

='Lf.z' . k=O

Obviously, Fu(z) corresponds to the unstable part Tu(s) and Fs(z) corresponds to the stable part Ts(s). For the stahle part N-I

F,(z) = 'Lf,z' and

-I

Fu(z) = 'Lf.z'.

k=O

To select the reduced order n of the unstahle part (we want same numher of poles in right half plane for our model as the system has) we look for gaps in the Hankel singular values. Manage.m program plots these as an aid to find n .

k=-N

Using 2M-point IFFT, we have

To show more details of the algorithm and the aids availahle in the program manage.m we now consider a couple of examples. In each case the frequency response data was generated hy creating a file from a given (analytical) transfer function. Many other examples have heen tried with very promising results.

where

j w =ex p(- 2rr AI ) .' 2M

Example I Consider a system with frequency response data (at 200 sample points) in the range from D.l to 100 radians per second as shown as amplitude and phase data in figure I. This was ohtained from a sixth order system givcn in transfer function form as example I in the paper of Aquirrc 1992.

and fM(k) is fk approximately. Also, using the bilinear transform on the approximation of F(z) again, we get the approximation ofT(s) dircctly.

From this frequency response data the software manage. m determined the hand width to he ahout 0.4; thus we selected the Laguerre parameter to he ;\=2 (for the hilinear transformation). The discrete-time signal (Fourier-Laguerre coefficients) hased on a 1024 point FFT algorithm are shown in figure 2, this is stahle part of spectrum. While figure 3 is a plot of the unstahle discrete-time signal showing that there is no unstahle part. Based on figure 2 we select a 75th order model for the stahle part. Next the Hankel singular values of the 75th order model are calculated and displayed to aid in selecting the size of the reduced order model, sec figure 4. Note hig drop is from a sixth order model to a second order one . Using one of the halanced truncation techniques manage.m gives fifth, fourth, and third order reduced models with frequency response comparisons as shown in figures 5, 6. and 7 . Figure X is the error for the fifth order approximant, while figure 9 lists the finite plane zeros and poles of the fifth order approximant.

For the convergence of the unstable part, approximation theorems are established hy Gu, Khargonekar, Lee and Misra 1992: Let F(z) be defined as hefore and Fu(z) have McMillan degrc:; n. Define N

S~(z);;: 'LfM(-k)z-' .1:=1

with fM(k) as before, N>n. Suppose that dF(eiW)/d(ci W) is L2[O,27tJ. Then,

where F:~N is an n-th order approximate function of S~ (z) obtained using the balanced truncation scheme (or the optimal Hankel approximation method).

Example 2 Consider a linear system containing one delay type tnm as given hy the transfer function T(s) = J(K)() . sJ_ 23s 2 + Ins + 29Xle· s - 463 Note this has real axis poles in the right half plane at s=3.972 and s=5.X6I, so we are dealing with an unstahle infinite dlmen~IOnal system. The discrete-time signal for the unstahle pan, .flgure 10, shows many non-zero Fourier-Laguerre coeffICIents. And a calculation of the Hankel singular values for th~. unstahle part shows a large gap hetween six and seven, (see fl~ure 1.1), suggesting a IOtal of six unstahle poles . ExtractIOn 01. the unstahle poles is easily done using the manage.m software; a plot of approximate location of the ten poles to the right of the line Re s~-2 is shown in figure 12.

However, the McMillan degree n of Fu(z) and the rest of the Hankel singular values of S~ (z) converge to the true Hankel singular values of Fu(z) and the rest of the Hankel singular values of S~ (z) converge to zero as (M,N) approaches (00,00) with M>N. Otherwise S~ (z) can't he an approximate function of Fu(z). Thus, as M, N are hoth large, a gap between singular values crn(S~) and crn+l(S~) would be significant if crn(Fu) is not too small. In this case, the McMillan degree of Fu (z) can bc identified in the approximation process. Thus, we are able to turn frequency response data into a rational function approximation of a possibly unstahle infinite dimensional system.

foi~ure n gives a frequency response comparison for a tenth orL_'r approxImate. model ohtained using the manage. m software. ~IS IS a SIxth order approximation for the unstahlc part and a fourth order approximation for the stahle part. FIgure 14 IS an elTor comparison for the tenth order model.

Ncxt we extract the most significant part of the model hy one of the standard finite dimensional model reduction techniques.

336

.10·3

4. REFERENCES

....

2

Aquirre, L. A. (1992). The least squares Pade method for model reduction. Int. J. Sysfem.v Sci , 23. 1559-1570. Cai, M.-P. and E. B. Lee (\ 99]). Identification of linear systems using rational approximation techniques. In: Stochastic Theory and Adaptive Control [T. Duncan and B. Pasik-Duncan, ed.l, Springer-Verlag, New York. Curtain, R. F. and K. Glover (19116). Rohust stahilization of infinite-dimensional systems hy finite dimensional controllers. Syst. Contr. Left.. 7,41 -47 . Glover. K. (1984). All optimal Hankel norm approximations of linear multivariahle systems and their L~-e rror hounds. Int. J. Con tr. , 39, 11 15-119]. Gu. G . , P.P. Khargonekar and E.B. Lee (19119). Approximation of Infinite-Dimensional Systems. IEEE Trans. on AutorrUlt. Contr., 34, 610-6111. Gu, G., P.P. Khargonekar. E.B. Lee. and P. Misra (1992). Finite-Dimensional Approximation of Unstahle InfiniteDimensional Systems. SIAM J. Contr. and Opt., 30 , 704-716. Lauh. A. J. (1986). Computation of System Balancing Transformations. Pmc. 25th Con! Decision IInd Control. 548-55~. . . Moore, B. C. (19l! I). Principal Component AnalYSIS m Linear Systems: Controllahility, Ohservahility. and Model Reduction. IEEE Trans. Automat. Contr.. 26. 17-]\. Safonov, M.G .. and R. Y. Chiang (1989). A Schur method for halanced-truncation model reduction. IEEE Trans. . Automat. Contr., 34.729-7]]. Spanos, J. T . and D. L. Mingori (199]). Newton algonthm for fitting transfer functions to frequency respo~se measuremenL~. Jour. Guidance, Control and DynamICs. 16. ~4-~9. . f Sreeram, V. and P. Agathoklis (1991). Model ReduCllon 0 Linear Diserete System via Weighted Impulse Response Gramians. Int. J. Contr., 53, 129-144.

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The research reported on here was supported hy NSF Grant DMS 9002919.

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