The 2-D σ-reciprocal system for model reduction

1056 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 11, NOVEMBER 1997

The 2-D -Reciprocal System for Model ReductionBaohui Xie and Vassilis L. Syrmos,Senior Member, IEEE

Abstract—In this paper we define the�-reciprocal system of atwo-dimensional (2-D) linear time-invariant system representedby the Roesser model. We introduce the 2-D�-reciprocal sys-tem and show that the 2-D generalized singular perturbationalapproximation (2-D GSPA) is related to direct truncation of its�-reciprocal system. The stability analysis and theH1-norm ofthe error for the GSPA reduction of 2-D denominator separablesystems are presented. A model reduction algorithm based onthe analysis of the�-reciprocal system is given for 2-D systems.Simulation results show that theH1-norm of the error of the2-D GSPA reduction is less or equal to theH1-norm of the errorif the 2-D singular perturbational approximation (2-D SPA) orthe direct truncation method are used.

Index Terms—�-reciprocal system, generalized singular pertur-bational approximation, model reduction, two-dimensional sys-tems.

I. INTRODUCTION

I N THE PAST decade there has been an increasing interestin two-dimensional (2-D) systems. Two-dimensional filters

are extensively used in processing 2-D sampled data such asseismic data, digitized photographic data, and gravitational andmagnetic maps [1]–[3]. The theory of 2-D systems has maturedto the extent that it now forms an important part of digitalimage processing [2].

The concepts of balanced realization, balanced truncationand singular perturbational approximation have been extendedto 2-D systems [4]–[6]. These model reduction methods areeffective and numerically economical for both 1- and 2-Dsystems. If the requirements imposed on the approximationerror need to be small at some frequency, or to be small over allfrequency domain, the reduced-order systems can be obtainedby using the generalized singular perturbational approximation(GSPA) model reduction method [7].

For a better comparison of the 2-D model reduction tech-niques, we summarize some basic 1-D results. The GSPAmodel reduction method is one of the most popular methodsfor obtaining the reduced-order representations for the 1-Dlinear time-invariant systems. One of the main advantages ofthis technique is its conceptional and computational simplicity.Another advantage is due to the application of the balancedgeneralized singular perturbational approximation (BGSPA)method, which has the same -norm of the error as thebalanced truncation. Nevertheless, the frequency characteristic

Manuscript received December 18, 1995: revised November 1, 1996. Thiswork was supported by the National Science Foundation under Grant NCR-9210408 and by the Defense Advanced Research Projects Agency underContract MDA-972-93-1-0032 and Contract MDA-972-95-3-0016. This paperwas recommended by Associate Editor R. Ober.

The authors are with the Department of Electrical Engineering, Universityof Hawaii, Manoa, HI 96822 USA.

Publisher Item Identifier S 1057-7122(97)04402-4.

of the error of the BGSPA truncation can vary according tothe requirements of the designer [7]. Two extreme cases ofthe GSPA truncation are obtained, namely, whenequals toone and tends to zero. When equals to one, the GSPAmodel reduction method becomes the classical singular pertur-bational approximation (SPA) [8], which is usually associatedwith a fast-slow decomposition of the state space, with theapproximation achieved by setting the “fast” states to theirsteady-state values.

The -reciprocal system [7], introduced to emphasize theproperties of the GSPA model reduction method, allows us toshow that the GSPA truncation can be converted to a directtruncation of the -reciprocal system. Consequently, manyproperties of the direct truncated system can be extended tothe GSPA truncation of systems. For example, in the 1-D case,the BGSPA truncated system retains the stability and has thesame error-bound as the balanced truncated system.

Balanced SPA (BSPA) [9] is another well-known modelreduction method. This method is a combination of the bal-anced realization and the SPA truncation. Although the error-bounds for the balanced truncation and the BSPA truncationare identical in the 1-D case, the resulting models havedifferent high- and low-frequency characteristics. The BGSPAtruncation is based on searching through all possiblefordifferent GSPA over a balanced realization so that the-norm of the error is minimum. Therefore, one can achieveless or at least the same -norm of the error by using theBGSPA model reduction method than the balanced truncationor the BSPA truncation.

Balanced realizations are of significant interest since theyhave good absolute-error truncation properties and preservethe stability of the reduced-order model in the 1-D case. Whenstudying the 2-D balanced realizations and model reductions,“pseudobalanced” [5], “structurally balanced” [6] or “quasi-balanced (QB)” [4] systems were introduced due to differentdefinitions of Lyapunov equations (or inequalities). Thesebalanced systems are not equivalent in the general sense,but are closely related to each other. The pseudobalancedand quasi-balanced realizations are the same for the 2-Ddenominator separable system. In this paper we use the quasi-balanced system as our balanced realization because it iscomputationally simple and efficient. In general, balancedtruncations of 2-D systems do not guarantee the stability ofthe reduced systems.

Our first contribution is the development of the GSPAmodel reduction technique for 2-D systems. This result is anextension of the 1-D case. It is shown that the GSPA reductionof a quasi-balanced (QB) realization has the same error-boundas the quasi-balanced truncation.

1057–7122/97$10.00 1997 IEEE

XIE AND SYRMOS: THE 2-D -RECIPROCAL SYSTEM 1057

Our second contribution is the definition of the 2-D-reciprocal system. We show that the 2-D GSPA is related to adirect truncation of the 2-D -reciprocal system. An importantapplication of such a result is the model reduction method ofthe 2-D quasi-balanced GSPA (2-D QB-GSPA). With the aidof the 2-D -reciprocal system, we are able to show that theproperties of the quasi-balanced truncation can be extended tothe 2-D QB-GSPA truncation.

In our paper, we show that stability is preserved for the2-D QB-GSPA truncation of a stable denominator separablesystem. We also prove that the 2-D QB-GSPA truncation andthe 2-D quasi-balanced truncation have the same-normof the error.

The organization of this paper is as follows. We developthe background in Section II. In Section III, we define the 2-D GSPA model reduction method and the 2-D-reciprocalsystem. We show that the 2-D GSPA truncation can beconverted to a direct truncation of the 2-D-reciprocal system.Stability is shown to be preserved for denominator separablesystems and the -norm of the error of the 2-D quasi-balanced GSPA truncation is proven to be the same as the 2-Dquasi-balanced truncation. In Section IV, simulation results aregiven for the 2-D QB-GSPA model reduction method and themethod is compared with the 2-D quasi-balanced truncationand 2-D SPA truncation. Finally, in Section V we conclude.

II. PRELIMINARIES AND BACKGROUND

Consider a 2-D discrete-time system in state-space formgiven by the Roesser model [10]:

(1)

(2)

where

is the horizontal state vector,is the vertical state vector, the input vector, and

the output vector.Then the transfer function corresponding to this state-space

realization is given by

(3)

where

(4)

Let denote the largest singular value of a complexmatrix and let be a real rational transfermatrix. Then the -norm of is defined as

Note that the -norm of 1-D rational matrix is definedas

Definition 1: A matrix is said to be2-D stable if

for allDefinition 2: [11] A 2-D system is said to be denominator

separable if its transfer function has the following form:

where and are coefficients of the polynominal. Adenominator separable system has a state-space realizationwith and/or

Next, we introduce some notation used throughout thispaper. For convenience, we partition the system defined in(1) and (2) as follows:

We define the subsystem in the following format:

(5)

(6)

(7)

(8)

(9)

(10)

where we assume that all submatrices conform to the order oftheir subsystems.

Next, we briefly review the balanced realization and thetruncation of the 2-D systems. The following definition hasbeen introduced in [4] for quasi-balanced 2-D systems.


Definition 3: A 2-D state-space realization is called quasi-balanced if there existsand such that

(11)

(12)

(13)

(14)

This definition of a balanced realization was proposed byZhou [4] in the sense of balancing the diagonal blocks of theLyapunov equations. It is shown in the Appendix that thereexist unique solutions and for (11)–(14) ifthe matrix

where denote the Kronecker product [12], is nonsingular.The unique solutions and can be obtainedthrough an iterative procedure introduced in [4].

Suppose that the realization is quasi-balanced with balanced gramians

and If

and whereand are in decreasing order and and

then a quasi-balanced truncation to the high-order 2-D system is given by truncating the less dominantsubsystem governed by and withthe singular values of the balanced gramiansand Thus the reduced model is given by

In the above quasi-balanced truncation, we assume that thepartition of and or and does not split statesassociated with a multiple or We also assume that

, and are positive definite. In a denominatorseparable system, , and are positive definite ifthe system is locally reachable and locally observable [5].

In general, the quasi-balanced truncation of a stable 2-Dsystem does not guarantee the stability and the minimalityof the reduced system A good approximation isobtained only if the reduced system is stable. Therefore, inorder to obtain a quasi-balanced truncation of a high orderdenominator nonseparable 2-D system, a stability test is alwaysnecessary. If the system is denominator separable and stable,the stability is preserved and a frequency error-bound can bederived [4].

Theorem 1: [4] Suppose that the 2-D system is denomi-nator separable with and is quasi-balanced with

andThen the reduced model

is denominator separable and stable. Moreover, thefrequency response error can be bounded as

(15)

or alternatively

(16)

Recall that in the 1-D case, the balanced generalized sin-gular perturbational approximation is a good model reductioncandidate on the grounds of the provable properties of thefrequency-domain error. We will extend the GSPA to the 2-D system. Later, we will show that in the 2-D case, thereduced system by the QB-GSPA has the same frequency-domain error-bound as the quasi-balanced truncated system.For a denominator separable system, the-norm of the errorof the 2-D QB-GSPA will be shown to be the same as the onedefined in Theorem 1.

III. GENERALIZED SINGULAR PERTURBATIONAL

APPROXIMATION FOR 2-D SYSTEMS

AND 2-D -RECIPROCAL SYSTEM

Define the 2-D generalized singular perturbational approx-imation system as follows:

(17)

(18)

(19)

(20)

with transfer function

(21)

Then, it can be shown after some algebra that

(22)

Observe that when (21) is the classical SPAsystem.

We notice that the 2-D quasi-balanced truncation and the2-D QB-SPA truncation are two special cases of the 2-DGSPA model reduction methods. The reduced order modelresulting from the 2-D quasi-balanced truncation is a fastsingular perturbational approximation of the balanced systemsince the slow singular values are truncated. The reducedorder model resulting from 2-D QB-SPA truncation is a slowsingular perturbational approximation of the balanced systemsince the eigenvalues of in (17)–(20) have slow singularvalues. These properties are similar to the properties in the1-D case. To understand the scheme of the 2-D GSPA, weintroduce the 2-D -reciprocal system.


Definition 4: Assume that exists.Define as the -reciprocal system of where

(23)

(24)

(25)

(26)

A frequency domain characterization of the -reciprocal mapping is given by the followingproposition. From this proposition, we can see why

is called the-reciprocal system ofProposition 1: Consider a realization of a 2-D system

with transfer function

and let be its -reciprocal system with transfer function

Then

where

and

The proof of Proposition 1 is given in the Appendix.Proposition 2 reveals the connection between the-

reciprocal and the generalized singular perturbational approx-imation. This property provides an efficient way to explainthe characteristics of the GSPA model reduction method, suchas stability and error performance.

Proposition 2: The GSPA truncation of a 2-D system isthe -reciprocal of the system obtained by performing a directtruncation of the 2-D -reciprocal of the full-order system.

The proof of Proposition 2 can be found in the Appendix.Proposition 2 is useful in all cases in which we want toextend the properties of the reduced-order systems obtained bythe direct truncation to systems obtained by using the GSPAtruncation. We will see later in Theorem 2 and Theorem 3,that Proposition 2 plays a key-role in proving the stability anderror-bound of the GSPA truncation of the 2-D denominatorseparable systems.

In general, the GSPA truncation of a stable system does notguarantee the stability of its reduced system. However, thestability of a denominator separable system will be preservedfor its reduced system.

Theorem 2: Suppose that a 2-D system described in (1) and(2) is stable and denominator separable, i.e., and/or

Then the QB-GSPA truncation of such system isalso stable and denominator separable.

Proof: From Theorem 1, the quasi-balanced realizationof a stable and denominator separable system is also stableand denominator separable. From Proposition 2, it is clear thatthe reduced system is denominator separable. Preservation ofstability can also be shown directly from Theorem 1.

Error-bound analysis is always important in model reductiontechniques. Although we can not derive a common error-bound for the GSPA truncation of an arbitrary system, wecan show that the GSPA truncation has the same-norm ofthe error as the direct truncation of the system. Therefore, theQB-GSPA truncation has the same error-bound as the quasi-balanced truncation. Later, we will get to the error-bound ofthe QB-GSPA truncation for a denominator separable systemin Corollary 1.

Theorem 3: GSPA truncation of a 2-D system has the same-norm of the error as the direct truncation of the system.

The proof of Theorem 3 is given in the Appendix.It can be easily seen from Theorem 3, that the QB-GSPA

truncation of a denominator separable system has the same fre-quency error-bound as the direct truncation of quasi-balanceddenominator separable system. The error-bound for such sys-tem is given in (15) or (16).

Corollary 1: Let the assumption in Theorem 2 hold true.Then the frequency response error of the QB-GSPA reducedmodel can be bounded as in (15) or (16).

Although the GSPA truncation of a quasi-balanced systemhas the same frequency error-bound as the direct truncationof a quasi-balanced system, their frequency characteristicsare quite different when we vary and For example,when and it becomes the classical SPAmodel reduction method, the error is small at low frequencies.However, when and the error is small athigh frequencies. Based on these observations we constructour GSPA model reduction method.

Algorithm 1: The QB-GSPA model reduction techniqueconsists of the following steps:

Step 1) Compute a quasi-balanced realization of the orig-inal system and choose the order of the reducedmodel.

Step 2) Varying the parameters perform a searchprocess which minimizes the -norm of the errorbetween the original system and the approximatedone obtained via the GSPA truncation.

Step 3) Perform a minimality test on the reduced-ordersystem. If minimal, go to next step. Otherwise, goback to Step 2.

Step 4) If the original system is denominator separable andstable, done. Otherwise, go to next step.

Step 5) Perform a stability test on the reduced-order sys-tem. If stable, done. Otherwise, go back to Step2.

The key point of the proposed approach lies in varying theparameter to search for the minimum of the error


Fig. 1. The frequency response of the truncated systems.

Fig. 2. The frequency response of the error systems.

between the original and the approximated systems. Therefore,the -norm of the error from the approximation obtained inthis way is less than or at least equal to those obtained when thesingular perturbational approximation or the direct truncationmethods are used.

The denominator separability is necessary to guarantee thestability and boundness of the -norm of the error. Noticethat our algorithm is not constrained only to denominatorseparable systems. If a system is denominator nonseparable,our algorithm can still be applied but we need to pay attentionto the stability of the reduced system. In such cases, the sta-bility test is always necessary. Therefore, in order to make thereduced-order system stable, we need to adjust the parameter

and not only to get the smallest possible -norm ofthe error but also guarantee the stability of the reduced system.

The full-order system in the example presented inSection IV is a denominator nonseparable system. As wecan see, the reduced-order model is also stable and has anerror less than the one from the quasi-balanced truncation orthe SPA truncation.

IV. SIMULATION RESULTS

In this section we provide one example to obtain an eighth-order model approximation of a twelfth-order 2-D discrete-time system using Algorithm 1.

Consider the state-space realization given in the examplein [4]. We can find the quasi-balanced realization of thesystem using the algorithm given in [4]. Wefind that for the quasi-balanced system

andwhere

......

At first, we apply a direct truncation to the quasi-balancedrealization After removing the last four states,we get the quasi-balanced truncation Us-ing the SPA to the quasi-balanced realization, we obtain thethe QB-SPA realization :

Now, we apply Algorithm 1 to the quasi-balanced systemFor different values of we search

for the minimum -norm of the errors. It was found thatthe -norm of the error is minimum, which is 0.1193,


when 1.5. The resulting system fromthe QB-GSPA truncation is then given by

Additionally, the reduced models using QB, QB-SPA, orQB-GSPA model reduction methods are still stable and min-imal. The frequency response of the reduced-order systemsusing the three model reduction methods discussed above areshown in Fig. 1. The -norm of the errors is also shown inFig. 2. The maximum amplitude of the frequency response ofthe full-order system is about 1.13.

Looking back at the results shown in Fig. 2, the error of theQB-SPA truncation is small at low frequencies, while the errorof the quasi-balanced reduction is high at low frequencies. TheQB-GSPA truncation has a smoother error response than theQB-SPA truncation or the quasi-balanced reduction. We alsonotice that the -norm of the errors from the quasi-balancedtruncation and the QB-SPA truncated system are 0.2743 and0.1457, respectively. The -norm of the error from theQB-GSPA truncation is 0.1193, which is smaller than that ofthe QB-SPA or quasi-balanced truncation as expected. Thecovariance of the frequency response errors are 0.0015, 5.5

and 4.7 for the QB, QB-SPA, and QB-GSPAtruncations, respectively, which indicates that the frequencyresponse error from the QB-GSPA truncation is not only lessbut also smoother than the errors from the quasi-balancedtruncation or the QB-SPA truncation methods.

The approximation errors are not too small compared tothe full-order system amplitude response. The reason is thatthe truncated singular values are not small compared to othersingular values. It should also be pointed out that our example

is a low-pass filter. The QB-SPA model reduction has goodperformance at very low frequencies. To achieve the bestoverall error performance throughout the whole frequencydomain, the QB-GSPA model reduction at 1.5 is a bettercandidate than the QB or QB-SPA model reduction methods.

V. CONCLUSIONS

In this paper, the definition of the 2-D-reciprocal systemhas been introduced. Some preliminary results concerningthe properties of this system, with special emphasis on theconnections to the 2-D GSPA model reduction method, havebeen presented. On the basis of these results, the 2-D GSPAmodel reduction algorithm has been derived. A numericalexample has been studied to compare the error-bounds ofthe quasi-balanced truncation, QB-SPA and QB-GSPA modelreduction methods.

The 2-D GSPA model reduction algorithm can be appliednot only to a denominator separable system but also to adenominator nonseparable system. When this method is ap-plied to a denominator nonseparable system, special attentionmust be paid to the stability of the reduced-order system. Thesimulation example given in Section IV is not a denomina-tor separable system. If the stability and minimality of thereduced-order system are preserved, the resulting reduced-order system with minimum -norm of the error is a goodapproximation for the full-order system.

The results and the techniques presented in this paper canalso be easily generalized to multidimensional systems.

APPENDIX

In this Appendix, we first show that the equations

(A.1)

(A.2)

(A.3)

(A.4)

have unique solutions and if the matrix

(A.5)

where denote the Kronecker product [12], is nonsingular.To solve (A.1) and (A.2), we stack the equations and get

(A.6)

(A.7)

where and is the stackingoperator and defined as

...if


where are column vectors. Rewriting (A.6) and(A.7), we get

(A.8)

(A.9)

Therefore, there exist unique solutions for (A.8) and(A.9) if the matrix (A.5) is nonsingular.

Similarly, there exist unique solutions for (A.3)and (A.4) if the matrix (A.5) is nonsingular.

Proposition 1:Proof:

Proposition 2:Proof: Partition as

where

From the block-matrix inversion lemma and the partionedsystem in (5)–(10)

Partition conformly as

From the -reciprocal representation (24), we obtain

which is

(A.10)

(A.11)

Therefore, from (18), (A.10), and (A.11), we have

Similarly, we can get and


Theorem 3:Proof: Define the one-to-one mapping

and

where and are defined in (23), (24)and (25). Now, from Proposition 2 and using the mappingdefined above, we have

Therefore,

(A.12)

Using Proposition 2, (A.12) can be written as

(A.13)

Furthermore, from (22), we have

(A.14)

Substituting (A.12), (A.13) and (A.14) into the -norm ofthe error, we have

ACKNOWLEDGMENT

The authors would like to thank the Editor and the Review-ers for their incisive and constructive comments which havehelped significantly in improving the presentation of the paper.

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[1] T. Kaczorek,Two-Dimensional Linear Systems. New York: Springer-Verlag, 1985.

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Baohui Xie received the B.S. degree from theDepartment of Radio Engineering and the M.S.degree from the Department of TelecommunicationEngineering, both at Bejing University of Postsand Telecommunications, in 1987 and 1990,respectively. She is currently working towards thePh.D. degree with the Department of ElectricalEngineering, University of Hawaii, Manoa.

From 1990 to 1992, she was with GuangzhouPeace Computer Integrated TelecommunicationCompany. Her current research interests include

robust/optimal feedback design in control systems, 2-D systems, and flexiblestructures, model/controller approximation, digital signal processing, andindustrial applications of control theory.

Vassilis L. Syrmos (M’91) received the diplomafrom Democritus University, Xanthi, Greece, in1987, and the Ph.D. degree from the Georgia In-stitute of Technology, Atlanta, in 1991, both inelectrical engineering.

Since 1991, he has been with the Departmentof Electrical Engineering, University of Hawaii,Manoa, where he is an Associate Professor. Hisinterests include geometric and algebraic approachesin linear system theory, numerical methods forrobust/optimal feedback design in control systems,

sampled-data systems, and flexible structures. He is the author and/or coauthorof 30 journal papers and the bookOptimal Control(New York: Wiley, 1995,2nd ed.)

Dr. Syrmos is an Associate Editor ofCircuits, Systems, and Signal Process-ing. He has been a Summer Boeing Faculty Fellow.

The 2-D σ-reciprocal system for model reduction

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