Proper and Stable, Minimal MacMillan Degree Bases of ... - IEEE Xplore

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IEEE AUTOMATIC TRANSACTIONS ON VI. DISCUSSION

In the solution of the optimal control problem, the operator

G(T)=QLTQ,-L:LQ, corresponds to the gradient vector of the cost function. In this case, our iterative procedure closely resembles a gradient minimization technique in which we set

ST”= - M[G(Tn))I Tn+l=Tn+8Tn. Furthermore, if we set the onedimensional problem

JmQ=J(Tn-Xd4G(TdI) the minimum of J, is achieved when X , = tr{GnG:},/ trI-VGn)C:l where G,, = d[G(T,)],X ( - ) =

QL(*)QY.

Clearly,

X.= llGnllz/(Gn, Since (G,, X(GJ) 5 llXll llGnll*,we see thatA,, > llxll - I . Therefore, it X in such a way that the sequenceof cost is possibleto choose the value of functions is monotonically decreasing.

CONTROL, VOL. NO. AC-29,

12, DECEMBER 19S4

among the proper bases of 36) there is a subfamily of proper bases which are 1) stable, 2) have no zeros in GlJ {a}and therefore are column (row) reduced at infinity, and 3) their MacMillan degreeis minimum amongthe MacMillan degreesof all other proper bases of 3(s) and it is given by the sum of the MacMillan degrees of their columns taken separately. It is shown that this notion is the counterpart of Forney’sconcept of a minimal polynomial basis for the family of proper and stable bases of 36). I. INTRODUCTION

In this paper, the structure of proper and stable bases of rational vector spaces is investigated. The algebraic structure of the set of all proper rational vectors, which have no poles inside a “forbidden” region Q of the finite complex plane, and which are also containedin a given rational vector space 3(s), is known to be that of a Noetherian &.(s)-module M* [1]-[3] [&(s): the ring of rational function with no poles in 6 : = Q U { w }]. The structure of the various bases M* of is examined via the notion of a “simple” basis TIof M* [3]. A simple basisof M* has the property that its MacMillandegree is givenby the sumof the MacMillan degreesof its columns taken separately. Based on this concept, the existence and constructionof “simple, proper, andQ-stable” bases of 3(s) having minimal MacMillan degree among all other proper bases of 3(s) is established. These proper bases we call “simple, minimal, MacMillan degree, proper, and Q-stable” bases of 3 . 9 , andit is shown that this notion is the counterpartto Forney’s [4] concept of a minimal polynomial basis for 3(s) for the case of the &(s)-module M*.

REFERENCES

II.NOTATION AND PRELU~INARIES T. B. Sberidan, “Three models of previewcontrol,” IEEE Trans. Human Fact. Electron., vol. HFE-7, June 1966. M. Tomizuka, “Optimal continuous finite preview problems,” IEEE Trans. Automat. Contr., vol. AC-20, June 1975. M. Tomizuka and D. H. Fung, “Designofdigital feedforwardlpreviewcontrollers for processes with predetermined feedback controllers,” ASME, J. Dynam. Syst. Meas., Contr., vol. 102, Dec. 1980. D. A. Pierre, “Steady-state error conditions for use in the design of look-ahead digital control systems,” IEEE Trans. Automat.Contr., vol. AC-27, Aug. 1982. W. A. Porter, “A basic optimization problem in linear systems,’’ Math. Syst. Theory, vol. 5 , no. 5, 1971. R. M. DeSantis, R. Saeks, and L. Tung, “Basic optimal estimation and control problems in Hilbert space,” Math. Syst. Theory, vol. 12, 1978. A. Feintuch and R. Saeks, Systems Theory: A Hilberi Space Approach. New York Academic, 1983. R. M. De-Santis and W. A. Porter, “Optimization problems in partially ordered Hilbert resolution spaces,” Int. J. Contr., 1983. -, ”Operator factorization inpartiallyordered Hilbert resolution spaces,” Math. Syst. Theory, 1983. A. Schumitzki, “State feedback control for general linear systems,’’ in Proc. Int. Symp. Math. Theory Networks Syst., Delft Univ., The Netherlands, July 3-6, 1979, pp. 194-200. A.Sageand C. Whtie, Optimal Systems Control. Englewood Cliffs. NJ: hentice-Hall, 1977.

Let W be the field of reals, a[s]the ring of polynomials, R(s) the field of rational functions and RpAs) the ring of proper rational functions. Let 0 be a region of the finite complex plane symmetrically located with respect to the real axis and which excludes at least one point (Y E 2 , and letQcbethecomplementofQwithrespectto’~,i.e.,~ = QUQC.LettE R(s), and factorizeit as: t = (no*fi)/(do-{) where nn,do E 3[s] coprime with all their zeros not outside Q and 3, d E 3[s] coprime with all their zeros outside Q. Let 6 := Q U{ c- } , and denote by &(s) the subring of R(s) consisting ofall t E W(s)with no poles in 6,i.e., of “proper and Qstable” rationalfunctions. It is known [ 5 ] , [6] thatwith :‘degree” function a@(-): l&) -t BU { + a}defined by &(t) : = deg-d - degas 2 0, &(O) := + w, R&) is a Euclidean domain. Two matrices( T I ,Td E ,Pxm(s) x W x m ( s )are called “equivalent in 6”if there exist E&&)unimodularmatrices TL, TR suchthat TLTITR = T2. Consequently, every T E W x m ( s )is equivalent in 6 to its “Smith-MacMillan form in 6”[8]:SF whose invariantfactors E)/$+, ei E &(s), $i E R&) give the pole-zero smcture of Tin 6 (see [7],[8],[l] for details). The following is a direct consequence of the above. Proposition I It?]: Let A E Rgm(s),B E g ” ( s ) withp : = I + t 2 m. Then the following statements are equivalent: 1) A and B are right coprime in 6 ;2) T : = E @“(s) has no zeros in 6 , 3 ) there exists an I&)-unimodular matrix TL E 2 g X p ( s )such that TLT = [Arn] = $; 4) there exist X F @“l(s), Y E G X ‘ ( s ) such thatX A + YB = I,; and 5 ) r a n k E [:$$ = m vso E Q undlim,,, [:3= E E PX, with rankx E = m. A matrix T E Bffx”(s) satisfyingtheequivalentconditions of Proposition 1 is called Ra(s)-left unimodular. Notice that a &&)-left unimodular matrix might have only finite poles andzeros in Qc. If T E Y/“(s), TI E @““(s), TR E 3 : X m ( s ) are related via ($3,

61

Proper and Stable, Minimal MacMillan Degree Bases of Rational Vector Spaces ANTONIS I. G.VARDUWUS AND NICOS KARCANIAS

T,

T=

TR

(1)

then T, is called a right divisor in 6 of (the rows of) T. I f p 2 m and Abstract-The structure of proper and stable bases of rational vector ranka(,) T = m, then any TR E R:””(s) that satisfies(1) for some R&spaces is investigated. We prove that if 3(s) is a rational vector space, then left unimodular matrixT~E @“(s) is called a greutest (common)right divisor in 6 of (the rows of) T . Notice,thatinsuch a case, T, “contains” all the zeros of Tin 6 [ 8 ] . Finally, for a T E with Manuscript received February 22, 1984. This paper is based on prior submissions of August 5 , 1982 and July 13, 1983. r&(,) T = r we define: &(T) = min {A,(*), among the &(.)‘s of all A. I. G. Vardulakis is with the Department of Mathematics,Faculty of Sciences, rth-order minors of T ) if r > 0 and & ( T ) := + m if r = 0. It can be Aristotle University of Thessaloniki,Thessalonki, Greece. easily verified that if p = m then 6 @ ( T )represents the total number of N. Karcanias is with the Department of Systems Science, City University, Northampzeros of T i n 6 (see [8] for details). ton Square, London EClV OHB,England.

ern@

IEEE TRANSACTIONS ON AUTOMATIC CONTROL,

1119

VOL. AC-29, NO. 12, DECEMBER 1984

rn. PROPER AND O-STABLE, MINIMAL MACMILLAN DEGREE BASES OF

&(s)-module M*. First, and unlike theproper submodulesMiof M*, all

RATIONAL VECTOR SPACES

bases of M* are column reducedat s = 03, since (by the definition of M*)

they are all &.(s)-left unimodular (see [3, Proposition 91). Second, and due to the abovefact, it also follows from Remark1 that if T E @/"(s) We examine now the algebraic structure of the set of all t E E$@" '(s) which are contained in therationalvectorspace 3(s) spannedby the is a basis of M*, then we can always determine an R&)-unimodular such that T := TUR E #"(s) is a simple basis columns ti E Rpxl(s), j E m of a general rational matrixT E Rpxm(s). matrix UR E agXm(s) First, the existenceof proper and &stable basesfor 3(s) follows fromthe of M* which satisfies: 6 d T ) = 6,,,(T). The following theorem,which is our main result, proves the fact that given a basis T of M* we can always results in [9], [lo], i.e., if T E Rpxm(s)is a basisfor 3(s) and T = BA-I is a "fractional representation" of Twhere B E A E 2gxm(s) determine an &(s)-unimodular matrix UR € R~""(s) such that T := is a simpre basis of M* which hasdesired poles (in P)and whose then clearly B is a proper and Q-stable basis for 3(s). Thus, let TI E &""(s) be abasis for 3(s) andconsidertheR&)-module M I , MacMillan degree 6dn is minimum among the MacMillan degreesof generated by (the columnsof) T I .If TI is not &(s)-left unimodular, then all other proper or proper and &stable bases of 3(s), i.e., that 6 d f )I basis of 3(s). We thus prove that if 3(s) is a by extracting from it non-R&)-unimodular right divisors in 6:T I R , Tm, 6 d T ) V T E g:"(s) ...T x . . . E lF$""(s) suchthat 0 < 6p(Tl~)< 6@(Tm)< < rational vector space, then among the proper and Q-stable bases of 3(s) &'(TiR) and TI = T i,I T i R for some (not necessarily R&-left unimodu- which have no zeros in 6 (Le., they are R,&)-left unimodular) thereis a subfamily of simple Re(s)-left unimodularbases which have any desired ~ar)Ti+ E Ppxm(s), i = 1 , 2 , . * ,then the W&)-modules Mi+ I , i = 1, are minimumamongthe 2, generated by (thecolumns of) T i + I , i = 1, 2, . form an setofpolesandwhoseMacMillandegrees MacMillan degrees of all other proper bases of 3(s). T h i s result is ascending sequence of submodules[1]-[3]: M I C Mz C * . * t Mi, 1. If now for some i = 1,2, * TiR = : TRis ag(c.)R*D- in 6 of TI, so that formally stated in the following. T, = TT, for some R@(s)-leftunimodular T E F$'"(s), then the ~ ~ ( s ) - Theorem I : Let T E W " ( s ) , p 2 m, be W&)-left unimodular and modulegenerated by T , andwhichwedenote by M*, satisfies an let M* be the &(s)-module generated by its columnst,(s). Then T(s) can always be factorized (in a nonunique way) as ascending chain condition on submodules 11 11, i.e., M I C M2 C .. . C Mi+ 3 M* for some i = 1, 2, ... and coincides with thesetof all T = TUR proper and Q-stable rational vectors t E # '(s) which are contained in where t = [i,,.* * , im] E @;"(s) is &+)-left unimodular and simple [11, 121. basis of M* which has no finite zeros and UR(S)E Rgxm(s)is a&)In the sequel, we examinethe structure of thevarious(R&)-left unimodular)basesof M*. As we show,these bases can be further unimodular and the set of its finite zeros2 contains as a subset the setof classified according to properties of their MacMillan degree SA.). In finite zeros (which, if any, are in Q c ) of T. Furthermore, if uj 2 0, j E m , are the (Fomey) invariant dynamical indexes of fie rational vector order to proceed, we need a few more known results. We start with the space 3(s) spanned by the columns of T, and also of T, ordF (3(s))-: = following. = Proposition 2 [3]: Let T E grxm(s), p 5 m , rank, T = m be Cy=I uj is the (Forney invariant dynamical order of 3(s) then 1) u j , j E m, and 2) 6 d T ) = 6, & ) = uj = OrdF ( 3 ( s ) ) and column reduced at s = QJ [ 121, [13]. Let dj E 2[s]be the monic least common multiple ofthep denominators which appear in thejth column ti 6 d T ) is minimum among the MacMillan degrees of all other proper bases of 3(s). ofTandwrite:tj=nj/dj,njEIpx1[s],jEm.LetN:=[nl,---,nJ Proof: Let T = ND-', N E apxm[s],D € Rmxm[s] be. a right E RPxm[s]and D : = diag (dl, dm) E Rmxm[s]. Then coprime MFD of T, where, due to the assumption that T is b&.(s)-left unimodular, thefiNte zeros of N and D are confined in W , and factorize 6.dtj126dT) (2) Nas N = NLNRWhereNR E Rmxm[s] is a g.c-r.de of (the rowsof) N jsI and NL is a minimal polynomial basisof the rational vector space3(s) [4] (i.e., NL is 1) column proper and 2) has relatively right prime rows [4], with equality holding iff N,D are right coprime.

em@),

-

Ti&'

-.

e . . ,

2

[16]).Ifnj(s)=[n1j,~~~,n~j]TE~~X1[~],jEm,arethecolumns

Proof: See [3, Proposition 151. Definition 1 [3j: A column reduced at s = QJ matrix T = [ t , , . . ., tm] @:m(s)with p 2 m and T = m whichsatisfies (2) with equality is called a "simple" basis of the R,Xs)-module M generated by tj

NL then, by definition [4],deg.nj := m a i E p {deg-nU}= uj, j E m . Let now D: = diag ( d * . dm)with deg.dj = uj and d j arbitrary monic polynomials with zeros in Q'. Then, T can be wriaen as e ,

.

Remark 1: It canbe easily seen1 that givena column reduced at s = 03 basis T E g:m(s) of M , then wecanalwaysdeterminean E$&)unimodular(biproper) UR E arrX"'(s) suchthat T = TU,is column reduced at s = 00 and simple basis of M with 6dT) = 6 d T ) . Lemma I: Let 1) T E %:"(s), p z m, rank2, T = m and 2) T = E E R p x m with rankaE = m. Let uj 2 0 , j E rn the (Forney) invariant dynamical indexes of the rational vector space3(s) spanned by the columns of T, ordF(a(@): = C uj the (Forney) invariant dynamical order of T and ZXT) the number of finite zeros of T. Then h

~

+

7-

6 d T ) = OrdF (3(S))

+ Z,(T).

(3)

Proofi 1) and 2) imply, respectively, that T has no poles and no zeros at s = 03 [3]. Hence, 6dT)is the number of finite polesof T. The lemma then followsas a particular case of a more general result (see [12], and [14, Theorem 2.11, or [15, Theorem 31 or [3, Corollary 81. Finally, we will need the following. Lemma 2: Let Ti E with lims+m Ti =: E; E R p x m and & Ej = m, i = 1, 2. If there exist a Q E RMxm(s)such that T, = T2Q,then Q E @:"(s) and Iims+- Q = Qo E R m x mwith ra& Qo =

m. Having introduced the above concepts and results, we return now to the examination of the various R&)-left unimodular bases of the maximal

' see

)J&l(L)&l)-l

p m f and notation of Theorem 1 below and write: T = ND-1 = = TUil,

-

T = [NLD-'][DNRD-']:= TUR.

(4)

Now, sincep 5 m and AT&) is a minimal basis: rank2 [NL]:= m3 and rank, D q = r e NL = m for every s E 3 , i.e., NL, Dareright coprime. By construction, T := NL&l is proper andsimple and b-, = [NL]:. Therefore, tis proper and has no zeros in U (QJ}.Also, by construction, Thas no poles in a, therefore, T E @.""(s) is R&)-left unimodular with no finite zeros at all. From Lemma 2, it follows that U, := DNRD-' has lims-.m UR =: Qo E R m x mwith Q o = m. However, URhas all its finite zeros and poles in' Q and SO UR E is &&)-unimodular and some of its zeros, Le., the zeros of NRare zeros of T. Now, 6-&) := deg-dj = uj and X a,&) = X uj = : ordF (3(s)) = deg*detD = : 6 d T ) . Finally, to see that 6&) is a mn im i um, notice that from Lemma 1we have that for every proper basis T of 3(s) with lirns+- T(s) = E and r a n k R E = m, 6 d T ) = ordF (3(s)) + Z f ( T ) where nowZf(T)2 0 denotes the numberof finite zeros (if any) of T(s) in Qc. Hence, from 2) it follows that 6dp) 5 6 d T ) for every proper basis T(s)of 3(s). Definition 2: An R&-leftunimodularand simple basis T E G m ( s of ) M* which has no finite zeros (and thus satisfies 1) and 2) of Theorem 1) is defined as a simple, minimal MacMillan degree,proper and Q-stablebask ofthe rationalvectorspace 3(s) spanned by its columns [SMMD-2&) basis].

[NI,

* Notice that an I&)-unimcdular matrix has (pssibly) only f ~ t poles e and zeros not outside 0'. By [N}: we denote the "highest column degree coefficient matrix" of N(s)E E13PXm[s][16].

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. NO. AC-29,

IV. CONCLUSIONS

In this paper, we have investigated the algebraicstructure of the set of all rational vectors which are contained in a given rational vector space 3(s) and which also have no poles in aset 6 := Q U { -1, Le., they are “proper and Q-stable.” Relying on the fact thatthe above setis an El&)module M*,and by investigating thestructure of the various basesof M*, we have established the notionof “simple, proper, Q-stable and minimal MacMiUan degree” bases of 3(s) as the counterpart to Forney’s concept of minimal polynomial bases of 3(s), for the case of the &($-module M* . These concepts andresults can be used for the resolution of algebraic control problems which involvequestions of properness, stability, andlor minimality of solutions of rational matrix equations [17]. For example, the “stable exact model matching problem” (SEMMF’) can be tackled and thedifficultiesinvolved in theconstruction of solutions to thestable minimal design problem (SMDP) can be elucidated [18]. REFERENCES J. Hammer and M. Heymann, ‘‘Factorization of h e a r systems: A generalized framework,” Linear Algebra Appl., vol. 50, pp. 321-352, 1983. -, “Causal factorization and hear feedback,” SIAM J. Contr. Optimiz., vol. 19, no. 4, p p . 445-468, 1981. A. I. G . Vardulakis and N. Karcanias, “Classiiication of Proper bases of rational vector spaces: Minimal MacMillan degree bases.” Int. J. Contr., vol. 38, no. 4, pp. 779-809, 1983. G . D. Forney, “Minimal bases of rational vector spaces with applications to multivariable linear systems,” SIAM J. Contr., vol. 13, pp. 493-520, 1975. A. S. Morse, “System invariants under feedback and cascade control,” in h o c . Lecture Notes Econ. Math. Syst., vol.131; also in Int. Symp. Udine,Italy 1975, pp. 61-74. N. T. Hung and B. D. 0. Anderson. “Triangularization technique for the design of multivariablecontrol system,”IEEE Trans. Automaf. Contr.,vol. AC-24, pp. 455-460, 1979. C.C. MacDuffee, The Theory of Matrices. New York Springer-Verlag, reprinted byNew York Chelsea, 1946. A. I. G . Vardulakis and N. Karcanias, “Smructure,Smith-MacMillan form and Int. J. coprime MFD’s of a rational matrix inside a region P = 0 U {a},” Contr., vol. 38, no. 5, pp. 927-957, 1983. M. Vidyasagar, “On the use of rightcoprime factorizations in distributed feedback systems mntaining unstable subsystems.” IEEE Trans. Circuits Syst., V O ~ .CAS-25, pp. 916921, 1978. C. A. Desoer, R. W. Lui, 1. Murray, and R. Saeks, “Feedback system design: The fractional representation approach to analysis and synthesis,” IEEE Trans. Automat. Contr., vol. AC-25, no. 3, pp. 339412, 1980. S. MacLaneand G . Birkhoff, Algebra. London,England:McMillan, 1979. GI Verghese, “Infinite frenuency behaviour in generalized dynamid systems,” Ph.D. dissertation, Dep. Elec. Eng., Stanford Univ., Stanford, CA, 1978. G . Verghese and T. Kailath, “Rational maaix structure,” IEEE Truns. Automat. Contr., vol.AC-26, pp. 434-439, 1981. S. Kung and T. Kailath, “Some notes on valuation theory in linear systems,” presented at me IEFE Conf. Decision Con&., San Diego, CA, 1979. G . Verghese, P. Van Dooren, and T. “hoperties of the system matrix of a generalized state-space system,” Inr. J. Contr., vol. 30, ppI 235-243, 1974. W. A. Wolovich, Linear Multivariable Systems. New York: Springer-Verlag, 1974. L. Perneh, “An algebraic theory for the design of controllers for h e a r multivariable systems, Part I,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 171-182,1981. A. I. G . Vardulakis and N. Karcanias, “On the stable exact model matching and stable minimal design problems.” in MultivariableControl:Concepts and Took. Amsterdam, The Netherlands: Reidel, 1984, p p . 233-262.

12, DECEMBER 1984

the direct method of Lyapunov. A new Lyapunov function for stability study of the powersystem is constructed by the Lagrangecharpit method. The Lyapunov fonction is thenused to estimatethecritical reclosing times for power-system transient stability. The critical redosing times given by the proposed Lyaponov function are compared to those obtained by numerical integration. I. INTRODUCTION

In recent years, many authors haveappliedthe direct method of Lyapunov to studyingpower-system-stabilityproblems,takinginto account the effectof control apparatus suchas velocity governor (GOV) or automatic voltage regulator (AVR). In [l], Pai and Fbi constructed a Lyapunov function of the single-machine system considering a voltage regulatorapproximated by asimpleexponentialrepresentation,while much of the literature has de& with the velocity governor. Afterwards, another Lypunov function was presented in [2], taking into account the automaticvoltageregulatorwithfeedback loop. However,these two papers could not deal well with the product-type nonlinearity of generator output, whichappears in considerationof the effect offlux decay. Hence, the resulting Lyapunov function given in [11 and [2] is the energy function in essence. This difficulty is due to violation of the sector condition of such nonlinearity. In this note, the Lagrange-Charpit method [3] is applied to the singlemachine systemtaking the automatic voltage regulator into consideration. The aforementioned difficulty is overcome through both the LagrangeCharpit approach and some arrangement of the product-type nonlinearity. Thus, an expanded Lyapunov function can be derived.

II.CONSTRUCTION OF EXPANDED LYAPUNOV FUNCTION According to [2], the dynamic equation of the system shown in Fig.1, taking into account flux decay and AVR action, is written as

Kailath.

Transient Stability of the Power Systemwith the Effect of the Automatic Voltage Regulator HAYAO W A G 1

where Pi = aV/axi and 4 is an arbitrary nonnegative function whose opposite sign gives the time derivative of the Lyapunov function obtained. The auxiliary equation for (3) is given by

dx,

_=

Abstract-In this note, thetransientstability of thesingle-machine system with the effect of the automatic voltage regulatoris studied, using Manuscript received April 22, 1983; revised May 23, 1984 and March 26, 1984. This pper is based on a prior submission of July 13, 1982. The author is with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.

dx4

dV

f4

-4

...=_=-

fl

- dP1

ac ac ah j-1

0018-9286/84/1200-1120$01.00 01984 IEEE

Pi--+--+PI ax, ax,

-

av