Realization algorithms for expansions in generalized orthonormal basis functions

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REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZ...

H-3a-11-6

Copyright © 1999 IFAC

14th Triennial World Congress,

Beijing~

P.R. China

REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZED ORTHONORMAL BASIS FUNCTIONS P.S.C. Heuberger*, Z. Szab6*"', T.J. de Hoog*, P.M.J _ Van den Hof,""l and J .. Bokor **

* .lvlechanical Engineering Systems and Control Group Delft University of Technology A-fekelweg 2, 2628 CD Delft. The Netherland.'J ** COTnputer and A utomation Research Institute

Hungarian Academy of Sciences Kende u. 1/'1-17, H 1502 Bndapest: H1tngary

Abstract: In this paper a realization theory and associated algorithms are presented for the construction of minimal realizations 011 the basis of a sequence of expansion coefficients in a generalized orthonormal basis. Both the exact and the partial realization problem are addressed and solved, leading to extended versions of the classical Ho-Kalman algorithm.In the construction of the realization algorithms, fruitful use is Inade of a system analysis in the transform domain, being induced by the choice of basis functions. The resulting algorithms can also be applied in approximate realization and in systenl approxirnation. Copyright © 1999 IFAC Key\vords: Orthogonal basis functions; discrete-time systems; system approximation; realization; identification.

1. IKTRODUCTION In recent years rene\ved attention has been paid

to the development and use of orthonormal basis functions in systern theory, and particularly in system identification. Considering linear system descriptions in terms of orthogonal basis functions expansions~ linearly parameterized models can result by restricting the models to finite expansions. In this Vv~ork the choice of basis functions is kno,vll to be rather crucial for deterlnining the length of the expansion that is needed for arriving at accurate system descriptions. The flexibility that is av~ilable in the basis construction tnCC11anism can essentially contribute to a fast convergence of Author to Vl,rhom correspondence should be addressed. E-mail: P.!\1.J. [email protected]. The research reported here has been supported by the Dutch Technology Foundation {ST\V) under contract D\VT55.3618. 1

the series expansion for a specific system. For Laguerre functions (Wahlberg, 1991) ~ this flexibility rests in the choice of a single pole location, while tVv''O pole locations can be fixed in the two-parameter Kautz; functions (Kautz, 1954; "\Vahlberg, 1994). Generalized versions of these approaches have been developed by lleuberger et al. (1995) using repeated blocks of all-pass sections of user-chosen order, and have been analyzed for system identification purposes in "~an den Hof et al. (1995). Ninness and Gustafsson (1997) have presented and analyzed an alternative structure \vhere the need for repetition of all-pass sections has been removed.

Here we address the following problem: Consider a linear tinle-invariant systeul in the form of a. series expansion P(z) ::::::

L

CkFk(Z)

k=l

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REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZ...

where {Fk (z)}k=1,2, ... is a sequence of (orthonormal) basis functions in 1{2 ~ The problem is to construct a minirnal state space realization (A) B, C, D) of P on the basis of the sequence of coefficients {Ck} k=l~··~ ,1'1· For N ::::= ()Q and Fk (z) == Z~k the problerll reduces to a minimal (exact) realization problem, as solved by the celebrated Ho-Kalman algorithm (Ho and Kalman, }966). Using the same basis functions~ and considering finite N, the corresponding minimal partial realization algorithm is analyzcd and solved in Tether (1970), including t.he forlllulation of eonditions under \vhich the problem has a unique solution4 In this latter ~it­ uation, the required algorithm for obtaining a solution is basically the sarne Ho-Kahnan algorithrn as applied in the infinite data case. Szab6 and Bokor (1997) have extended the exact realization theory to the situation of Hambo ba..sis functions, for the case of infinite data (J.'l == 0::::,). In this paper it will be shown how these results connect to system descriptions in the related transform domain, and additionally the partial realization problem (1\r < (0) \vill be addressed and solved, and cOIlsequences will be indicated also for the approximative case. The paper is structured as follows: in section 2 t.he theory on generalized orthonormal basis functions is summarized. In sections 3 and 4 some results are forIllulated tha.t are instrumental in handling the realization problems. The exact realization problem is addressed in section 5 and the

partial problem in section 6. Some comments on the approximate realization probleln conclude the paper. Unless otherwise stated, all systems in this paper are scalar discrete time systems~ I\1ultivariable extensions are straightforward. In the discrete tirne domain the space 11. 2 can either be defined to include only strictly proper systems or to include proper systems as well. In this paper \Ve will use the latter definition. For brevity reasons all proofs are oInitted, see (Heuberger et al. ~ 1998).

2.

ORTHONORI\1~t\L

BASIS F'UNCTIONS

In this sect.ion the main ingredients of the theory on orthonormal basis functions will be briefly revie,ved. For details see Heuberger et al. (1995), Van den Raf et aL (1995)1 Heubergcr and \-ran den Hof (1996), NiuIless and Gustafsson (1997). The basis functions are constructed from state trajectories related to balanced realizations of square inner functions (Le. stable all-pass systeuls). A transfer function Gb E 1{'2 is inner if it satisfies 1G b (eiJ.,l)I = 1 for all W ,E (-1r,1rJ. Given a balanced realization {A, B) C, D} of an inner function Gb (z), it is straightforvvard to show that the

14th World Congress of IFAC

elelnents q)1 (2), ... , 1'nb (z) of the nb dimensional vector function

are nlutually orthonorrnal in the standard 11 2 inner product sense. Furthermore consecutive multiplica.tion of these functions with Gb (z) results in an orthonorrnal set, given by the components {$(k-l)nb+ 1 , ... , epknb} of the nb dimensional vec... tor functions

kEN and it can be sho"\~rn that the set of all these functions {tPi(Z) }~l constitutes a basis for the strictly proper part of 1i2. Directly resulting from the basis for strictly properst.able systerIlsin 1i2, a basis {Vlr-Ct)} for the related Hilbert space of £2 [1, 00 )-signals follo\vs, by considering the inverse z-transforms of the 1-£2signals. These functions are the ilupulse responses of the functions Vk (z) "vith the property: VI

(t)

::=

At - 1B

(2)

'Uk+l

(t)

~

(Gb(q) . I)Vk (t).

(3)

To avoid confusion these functions will also be denoted by Vk (t){A,B,C!D}

(4)

Considering this general clas~ of basis functions, for any strictly proper system lI(z) E 1-l 2 there exists a unique series expansion: ,:;,0

H(z) ==

L

Lk 17k(z)

k=l

Specific choices of Gb(z) lead to well known classical basis functions, such as the standard pulse basis T-Tk (z) = z-k, the Laguerre basis and the t~vo-parameter Kautz functions (Kautz, 1954; \\t'"ahlberg, 1994), see Heuberger et aL (1995). These basis functions give rise to a general theory on system transformations induced by these 50called Hambo basis functions, such that a system P(z) admits an alternative description, denoted by peA) in the transform domain. For details see (Heuberger and \ran den Hof, 1996). This function P(..\) can be expressed through variable transforrnations, using the nb x nb inner function N(z) ;==""4

+ B(z

~ D)-le

(6)

with l\1cIvlillan degree 1 and balanced realization (D, C, B~ A)~ If we write P(z) ::=; L~l Pk Z - k , then

peA)

=

L

Pk l\rk CA)

k=l

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REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZ...

14th World Congress ofIFAC

the matrices {Pk , Q k} are the solution of the following set of Sylvester equations:

1.L' here

or differently denoted: P(:.\.) == P(z)}z-l=lV(A). The rnapping T; f{2 ---t 1/}{b Xnb defined by T(P) := peA) is referred to as the Ha1nbo sy,c;tc'mtransform. From a signal processing point of view this alternative representation of systems fits into the Hmatrix representation framework as discussed by Audley and Rugh (1973).

..4PkAT ~4QkAT

and where

Given this background material the problem ad-

Problem 1. Given a system G(z) and an 01'thonormal ba.'Jis {Vk(Z)}~l' .~uch that G(z) == L~l LrV'k (z) construct an algorithm to derive a nlinimal state space realization of G(z)} based

Q:=

ii

H

H

H

xn~

(7)

Q;:b]T

E Ilinb

xnt

(8)

k.

In this section it will be shown how the Inaterial of the previous sections allovls the calculation of the Harnbo systeIIl transforIIl, given an expansion in terms of orthonormal basis functions. The system transform "\vill be expressed in terms of l'viarkov paraJneters, connecting directly to the Hankel matrix of the systeul transforrn. Hence, given an expansion G(z) == 2:'%:1 Lfl,lk(Z) \ve will show how~ this can be translated into an expansion G()') == I:~o .J."'\1kA -k. The result is given in terms of the notation (7,8)~ and <8> denoting Kronecker product.

In

Proposition 3. G(A) = E~o MkA. -k, where 1\;10 == [Lf @ [JP l0k = [L'k+l ® IJP

(10)

PTopo,~ition 4. G(A) ~ L~o Mk J\-k J where Mo (LTB)I+[(LT ..4 )®I]P (11)

Mk

The key property of the basis functions, that will be used in this paper ~ is the fact that the Hambo transform of the elements of V; (z) can be calculated explicitly, as stated in the following proposition:

Proposition 2. Let the vector valued function V 1 (z) [
+ [L.[ 0IJQ

(9)

In the course of the pa.per we ~rill need a similar expression for the Hambo transform of the shifted function G (z) := zG(z):

THE H.A.:.\rlBO TRANSFORl\1 OF B.A.SIS FUNCTIONS


Qr

E !Rn"

TRANSFORivl DOMAIN

The solution of the problem is based on an extension of the Ho-Kalman algorithm and the solution of the classical rnininlal partial realization problem, given by Tether (1970).

3~

[Qr

p~]T

4. HANKEL l\,'lATRICES IN THE

This problem ~rill be approached through the Hambo sy~tenl transforrrl, revealing the relation betv-leen the expansion coefficients {L'k} and the Markov parameters {Mk } of the transformed systern GCA-) and thus creating the Hankel matrix of the transforlned systeln~ denoted by if. Furthermore the relation betv..reen ii and the Hankel matrix of G(z), denoted by H~ \vill be established. Equivalent relations are given for the shifted sys~m G (z) := zG(z) and its Harnbo tran~ornl

respectively

is the k th Euclidean unit vector.

P:= [PIT P2T

on the expansion sequence {L 1 , · · · , £1"}.

fI

ek

::;;;; Qk

To facilitate the notation in the following sections \ve introduce a compact form for the set of all matrices {Pk ~ Q k,} :

dressed in this paper can be formulated as follows:

G(z), with Hankel matrices scherne:

+ .4e k BT = P k + BCekBT + Bcpl..4 T

(L'[;+lB)[

+ [(L,f A)

=:

Q9

+ [(Lk~1~4) 0I]P I] Q

(12)

T'he latter two Propositions show hov.-· the set of expansion coefficients can be efficiently transformed into Markov parameters of the Hambo systern transform of the (shifted) transfer function, which immediately results in the Hankel matrices

jj and H. This section will be concluded with establishing the connection between .the Hankel matrices in the transform domain (if, H) and the corresponding Hankel rnatrices in the standard domain (H~ H)-

4155

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REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZ...

14th World Congress of IFAC

This connection turns out to be deterrnined by a. set of transformation matrices, that are derived

(4) Furthermore B g and C g are created from

By

from the all-pass function Gb (z).

Proposition 5. Given a system G(z) and its Hambo transform G(>") with Hankel matrices H respectively ii, then there exist unitary Tnatric.e~~ T], T 2 such that H = T 1- 1 iiT2 == TiT iiT2 , with (T1)'lj == Vi(j){A,B1C)D}

(T2 )ij == Vi (j) {AT lG, whe1'e

'Vi

T

,B

T

(13) ,D

T }

(14)

(J){F~G~H,J} is defined by eq. (2-4).

This property shows that the Hankel matrices in both domains have eq~l rank and that any full rank decomposition of H immediately results in a full rank decomposition of H. This property will be of use in the genera.lization of the Ho-Kalman realization algorithm, discussed next.

= Ll

0 ] and C g = [Jp 0] r.

[1

The extended mininlal realization problem addressed here is given in Problem 1, with 1'.7 = 00. For the solution of this problelll explicit use is TIlade of the tr~nsfornlationproperty H = TIT HT2

and H= TIT HT2 , given by Proposition 5. Substituting these relations in Algorithm 1 results in a realization algori!.hm based on the ~1aI'kov pararneters in if and H. The missing link is then the relation bet\veen the expansion coefficients {L k } and these Hankel matrices. T'hat relation is established by Propositions 3 and 4. This approach leads to the follo\ving algorithm:

_41gorithrn 2. (Generalized Ho-Kalma.n). Given a series expansion G(z) =: L~=o LfVk(Z), the following steps result in a minimal state space realization {Ai}, B g , C y } of G(z):

(1) Calculate 'U'ith Propositions 3 and 4- the Ma'rkov paTa'f11ete1~S {Mk}:..{Mk} of the Hambo

5. EXACT REALIZATION

system transforms G(A), G(~).

In this section the generalization of the HoKalrnan algorithm \\I~ill be discussed, generalized in terms of the GOBF basis. It is assumed that the Hankel Inatrices involved are fully known, Le. the rnatrices are of infinit.e di mension, or eqllivalently all I\1arkov parameters {Gi} ~1 are known. This problem has a solution if and only if there exist integers ]\/", .l\{ such tha.t rank(HN' ~N) ::::::: rank(HpP+i,iV+i) ~ no for all i, j ::::= 0,1,2,·",

(2) Create Hankel matrices jj, H, with elements

Hij (3) Let

H,

= NJi +j - 1 and Hij ~Mi+j-l. == LS. be a full rank decomposition of

ii

(4) Then

__

r·

r+

B-g-- LlT2

1

(15)

GrL-t and in this case no is the rninirnal state space dimension. lrnder the assumption that the unknov.."u system is finite dimensiona1 1 a minimal state space realization can be deterulined using the Ho-Kalrnan algorithm:

Algorithm 1. (Ho-Kalman). G
.~

[1] 0

and C-g--

[1 0] Trf 1

An important feature here is the fact that only one row c.q. colulun of the transformation matrices T 2 is required to calculate the state space realization. In general the calculation of B g and Og involves infinite dimensional matrix calculations. In the special case that G(A) is a finite impulse response model~ this will be reduced to finite operations. This occurS v.,rhen the underlying system G(z) ha~ only a finite lluITlber of non-zero expansion coefficients: G(z) 2:~~1 Lf1/k" (z). For numerical exa.mples of this application see (Szab6 and Bokor, 1997).

TT:

=

parameters of a finite dirnen8ional system G(z), the following steps result in a minimal state space

6. PARTI_l\L REALIZ_A-'TION

realization {Ag;Bg:Cg} ofG(z): (1) Let H == r .6. be a full rank decornposition , i.e. rank(T)~Tan.k(D..)~rank(H). (2) Then fI obeys the relation H= r . A g • ~. (:9) Hence ~4g = r+ ..if. ~ +, 'UJith (.)...L. indicating the Moore-Penrose pseudo-inverse.

== r ...4 g . H . E+

relation!!

(5) Hence A g obeys Ay === (6) Furthermore B.g and C g are created fro'Tn

where

GGc C+ 1

~

H obeys the

In this section the generalization of the so-called partial realization problem is described and a solution to this problem is presented. The partial realization problem (Tether, 1970) deals ","ith the case of lirnited informatjon~ i.e~ only a finite num-

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REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZ...

14th World Congress of IFAC

ber of IvIarkov-pararneters is known~ The problem is the constrlletion of a finite dimensional minimal state space realization that fits these lVIarkov paranleters~ The rninilual partial realization probleln airns at finding such a realization with a minimal 1:1crv1illan degree over all possible realizations. The following lemma gives conditions for the existence of a unique IniniInal partial realization.

(1) lIse the Ho-Kalman algorithm to create minimal state space realizations {.<4~ B, C} and

{A, B cl;, 1

-~k~l-

J.~k= CA

+ lV ==

B

(2) Observe that the infinite Hankel matrices

Ht H,

constituted by these T'ealizations have full rank decomposition.s

Lemma 6~ (Tether" 1970). Let {Y1 , · · · , YNo } be an arbitrary sequence of p X 'nt rnatrice,l; and let H i :j , i + j ~ .1VoJ be a corresponding block Hankel matrix. Then a rninimal partial realizatZ:on given by the Ho-Kalman algorithrrt i8 unique (rnodulo similarity transformations) if and only if there exist positive integers N'1 IV such that (1) .I'l'

such that for 1 ::; k :::; No

rB AB ... ]

1'10

(2) rank (HIV' ;N )=rank(Hjv'+l._N )=rank(HNJ ,J\r+l)· (J) Apply the generalized Ho-Kalman algorithnl 2 (step 5 and 6) to derive

Generalizing this property to the GOBF case is lnore involved than in the exact realization case, because proposition 5 is only valid for infinite Hankel matrices and not for finite matrices. The key idea to overcome this problem is to extend these finite matrices to infinite dimensions, using

A g ~f+fELi+

=

:;:;: R I l R2 S2S1 1

the Ho-Kalman algorithm in the transform dornain. This will result in (minimal) !calizations

[~] = R 3

By = ZiT2

of the underlying systems G(z) and G(z), which are guaranteed to be unique under rank conditions given ~Y lemma 6. Since a minimal realization for

G(z), G(z)

(rTr)-l (r7~f)(3.LiT) (LS.Li T)-l

Cg =

[1 0 J T!r == R.1

'Using the Lyapunov/Sylvester equations:

au tonlatically gives a full rank decom-

RI =

position for the associated infinite Hankel matrices in terms of the product of the observability and eontrollability lnatrices

ATRIA + ere

--

== AT R 2 A + eTe T T SI == AS1A + BB -52 == AS2AT + BET R 3 == .:IT R 3 D + BeT R4 == D T R 4 A + ETC . ,..-..

R2

.-

we can apply Algorithm 1 to obtain .-

~-

H = f . L\ = r . A g . L\. -+

Ag =

r+r LS..~ +,

which expression can be calculated using Sylvester and Lyapunov equations . .:'\.long the same line of reasoning the Inatrices B g and C g are derived. This leads to the following algorithm:

Algorithm 3. (Generalized Afini'mal Partial Realization). Let {L k }::r~1 be the first No expansion coefficients of a scalar system G(z) and let {l\.lk' lV[k }t'~Ol be the A-farkov parameters of the Hambo system transjor'm of G(z) respectively its shift G (z), as defined by Propositions 3 and 4. ~4SS1l7ne that both {Mk}f~ll and {Mk}:~;-l satisfy the conditions of Lemma 6. Then a unique minimal state space realization {.<4 g ,Bg ,Cg } ojG(z) is ob-

tained as follows:

Given this algorithm, the question will arise \vhy

and when it would be of use. This ,vill typically be the case in an approxinlation or identification setting, using an ortbonormal basiS" function approach. One of these situations is where a finite number of expansion coefficients is estimated, i.e. estimation of the parameters {L k } in a rnodel structure N

yet) ~

E

L["{lk(q)u(i)

+

e(t)~

k=l

and the number of significant coefficients Lk is relatively high, such that a direct state space realization would result in a high order rnodel. One approach in this case \vould be to apply model reduction techniques to obtain a Io~"er order nlodel. However, such a procedure will not

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REALIZATION ALGORITHMS FOR EXPANSIONS IN GENERALIZ...

9. REFERENCES

make use of the intrinsic information contained in the expansion coefficients or the directly related Markov parameters of the transformed system. The merit of Algorithm 3 is that it 111akes full use of this information and hence will improve the quality of the resulting approximation. It ,vi]] furthermore give much nlore insight in t.he M cT\·1illan degree of the underlying system. It is important to note that the presented partial realization algorithrn intrinsically requires t\VO applications of the standard Ho-Kalman mechanism. In the classical case this can be reduced to one application, because there a realization of {Alk}~~l immediately results in a. realization of {J.\1 k } f~ l' Also in t he generalized case these sets with ~vIa.rkov parameters are obviously closely related and it might be possible to further sirnplify the algorithul.

7. APPROXIlVIATE

REALIZATIO~

The generalized algorithms can be applied in an approximate fashion as is the case with the standard Ho-Kalrnan algorithm (Kung, 1978). rvfost commonly the full rank decomposition of if \\"ill be computed by rneans of a singular value decomposition. One can then simply truncate the S\lD by setting the smaller singular values t.o zero. and proceed as in the non-approxirnative case. v\?'hen algorithm 2 is applied in an approximate sense in the situation ,vhere all expansion co efficien ts are kno"vn, the resulting rea.lization vlil1 be exactly the saUle as the one obtained by application of the Ho-Kalman algorithm in the original domain. This is caused by the unit.ary transforulations in Proposition 5. The situation becomes different in the case \vhere only a finite number of ~arkov paraJIleters is given. Note that in that partial realization case we need onlx to truncate the SVD of ii and t

not that of iI. The consequences of this method when compared to the standard partial realization algoritllm still have to be further explored.

8. CONCL GSIONS The problem has been addressed of constructing a minimal state space reali~ation on the basis of a sequence of expansion coefficients in a generalized orthonormal basis function expansion. The classical Ho-KalIIlan algorithru, designed for minimal realization on the basis of Markov parameters~ has been extended for expansions in a general class of basis functions. '''''hereas in the classical situation one algorithm essentially solves both the exact (infinite data) and the p~rtial (finite data) probIem, in the generalized case different algorithms result. The presented algorithnls can also be used for constructing reduced order state space models on the basis of estimated. expansion coefficients.

i~ud[ey,

D.R. and \V.J. Rugh (1973). On the H-matrix system representation. IEEE Tr-ans. AUt01U. Control, 18, 235-242. Heuberger: P.S.C~, P.I\.I.J. \lan den Hof and O.H. Bosgra (1995). A generalized orthonormal basis for linear dynamical systenls. IEEE Trans~ Autom. Control~ AC-40, 451---465. Hcuberger, P.S.C. and P.I\1.J. \Tan den Hof (1996). The Haulbo transform: a signal and systems tra.nsform induced by generalized orthonormal basis functions. In: Prepr. 13th Triennial IF..4 C World Congress, San Francisco~ 357-362.

Heuberger: P.S.C.~ Szabo, Z., de Hoog, T.J., 'lan den Hof, PJvl.J. and J. Bokor (1998). Realization algorithms for expansions in generalized orthonormal basis functions. Internal report N225, I\'1echanical Engineering Systerns and Control Group, Delft lJniversity of Technology, Ho, B.L.~ and R.E. Kalulan (1966). Effective construction of linear state-variable models fronl input-output functions. Regelungstechnik, vol 14, 545-548. Kautz, W.H. (1954). Transient synthesis in the tinlC dOlnain. IRE Trans. Cire. Th., CT-l~ 29-39.

Kung,

S~

(1978). A ne\v identification and model

reduction algorithm via singular value decornpositions~ Pr-oc 12th Asilomar Conf. Circuits Byst. and Computers~ Pacific Grove~ CA, Nov. 6-8, 1978, 705-714. Ninness, B.M. and F. Gustafsson (1997) ..A. unifying construction of orthonormal bases for systelu identification. IEEE Tra.ns. Av.tom. Con-

trol, A C-42, 515-521. Szab6, Z. and J. Bakor (1997). Minimal state space realization for transf~r functions represented by coefficients using generalized orthonormal basis. In: Proceedings 36th IEEE Confel'ence on Deci8ion and Control, San Diego~ Ca, 169-174. Tether, A.J., (1970). Construction of minimal linear state-variable models from finite inputoutput data. IEEE Trans. Au.tom. Control, AC-i5, 427-436. \lan den Hof, P.l\·1.J., P.S.C. Heuberger and J. Bokor (1995). System identification with generalized orthonormal basis functions. Automatica, 31~ 1821-1834. \Vahlberg, B. (1991). Systerrl identification using Laguerre models. IEEE Trans. Autom. Control, AC-36, 551-562. V\Tahlberg, B. (1994)~ System identification using Kautz models. IEEE Tran8. Autom. Control, AC-39, 1276-1282.

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