計測自動制御学会東北支部 40 周年記念学術講演会
2004 年 12 月 22 日，23 日 日本大学工学部

講演番号：C13

多次元システムに関する同時安定化問題について
森 和好 (会津大学)

On Simultaneous
Stabilizability of Multidimensional SystemKazuyoshi MORI (The University of AIZU)

Abstract— The relationship between the simultaneous stabilizability and the strong stabilizability of the multidimensional system is investigated. We present that the simultaneous stabilizability over a commutative ringcannot be given by the simultaneous stabilizability over its local rings in general. We, however, show that if
we have one-side coprime factorization, solving the simultaneous stabilizability of the multidimensional system
can be recast as solving the strong stabilizability of the multidimensional system.
Keywords: Multidimensional Systems, Simultaneous Stabilizability
I. I NTRODUCTION
In this paper, we consider the relationship between the simultaneous stabilizability and the strong stabilizability ofthe multidimensional system.
Simultaneous stabilization problem is a problem to stabilize two or more plants by a controller. When any
stabilizable plant admits a doubly coprime factorization,
it is known that solving the simultaneous stabilizability
can be recast as solving the strong stabilizability[1, Theorem 5.4.3], [2, Theorem 3.1]. However, in the case of themultidimensional system, it seems that we do not know yet
whether or not any stabilizable plant admits a doubly coprime factorization [3] (Lin gave an affirmative conjecture
in [4]). Also the strong stabilizability of the multidimensional system has studied by Ying et al. in [5] and [6].
The objective of this paper is to address the relationshipbetween the simultaneous stabilizability and the strong stabilizability of the multidimensional system.
In this paper, we have two approaches: one is to use localglobal principle and the other to assume one-side coprime
factorization.
It is known that using the local-global principle[7]gives the stabilizability of plants and stabilizing controllers of plants without coprime factorizability[8]–[10].
Parametrization of stabilizing controllers without coprime
factorizability[11], [12] is also given by using the localglobal principle. Even though the local-global principle isa powerful tool, in this paper, we will show that the simultaneous stabilizability over an original ring, which isthe set of stable causal transfer functions, cannot be givenby the simultaneous stabilizabilities over the local rings in
general.
We will also consider the case where we assume there exists one of right- or left-coprime factorization. In this case,we will obtain the same result to the case where any stabilizable plant admits a doubly coprime factorization. Thus,
we will see, under this assumption, that solving the simultaneous stabilizability can be recast as solving the strong
stabilizability.
This paper is organized as follows. After this introduction, we begin on the introduction of the coordinate-free
approach, we used in this paper, in Section I, including definitions. In Section II, we consider the simultaneous stabilizability with the local-global principle. Then, in Section III, we investigate the case where plant admits one-side
coprime factorization.
II. C OORDINATE -F REE A PPROACH

in this paper. Then we give the formulation of the feedback
stabilization problem.
A. Notations
Commutative Rings In this paper, we consider that any
commutative ring has the identity different from zero.
Let denote a (unspecified) commutative ring. The totalring of fractions of is denoted by.
We will consider that the set of stable causal transfer
functions is a commutative ring denoted by . From thesense of the transfer functions we consider that the commutative ring satisfies the invariant basis property (cf. [13]).
In addition to , we will use the following three kinds ofring of fractions. The first one appears as the total ring of
fractions of , which is denoted byor simply by ;that is,is a nonzerodivisor . This
will be considered as the set of all possible transfer functions. The second one is associated with the set of powers
of a nonzero element of . Let denote a nonzero elementof . Given a set, which is a multiplicative subset of , we denote by
the ring of fractions
with respect to the multiplicative subset ; that is,of. The last one is the total ringof fractions of, which is denoted by; that is,is a nonzerodivisor of.
coincides with the toIf is a nonzerodivisor of ,tal ring of fractions of . Otherwise, they do not coincide.

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Matrices The set of matrices over of size
is de. Further, the set of square matrices overnoted byof size is denoted by. The identity and the zero matrices are denoted byand, respectively, if the sizes
are required, otherwise they are denoted by and .
Matrixoveris said to be nonsingular singularover if the determinant of the matrix is a nonzerodivisor a zerodivisor of . Matrices and over are rightcoprime over if there exist matrices and over suchthat
holds. Further, an ordered pairofmatrices and is said to be a right-coprime factorization
over of if (i) is nonsingular over , (ii)over, and (iii)andare right-coprime over .
As the parallel notion, the left-coprime over and the leftcoprime factorization over of are defined analogously.
If a plant has both a right- and a left-coprime factorizations
over , then the plant is said to admit a doubly coprime factorization over . For short, we may omit “over ” when.

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B. Feedback Stabilization Problem

We start by giving the preliminary of the coordinate-free
approach. In the following we introduce the notations used

The stabilization problem follows that of Desoer et al. of[14], Sule in [8], and Mori and Abe in [9], who consider

165

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[

_ ` ^

-f 6
6
6

a _

_
^

b

\

?
?
?
-f
-

a

b

t -

c

Fig. 1. Feedback system .

dthe feedback system [1, Ch.5, Figure 5.1] as in Figure 1.
For further details the reader is referred to [1]. The plant
we consider has inputs and outputs, and its transfer
matrix, which is also called a plant itself simply, is denoted. We can always represent
by and belongs toin the form of a fraction, whereandwith.
nonsingulareT1f'EgT
ThUQWRWVYXijTk;lRmV:X0Qnl 
Qo&f'EgpSQql f'Egr RsP g SRtl u f 
Rs Rvl 
Definition
x1f'Eg and yzx1g{E.f , a matrix| jTry}}B11 For isTwdefinedasg~f| jT.y$  jG f NTMyMVYX  TjG g N1yMT}VYX (1)y jG f NOTMyM VYX jG g N6yMT$ V:Xx
providedjG f N6TyM is a nonzerodivisor of . This| jTry} isthat
the transfer
matrix from :Y   to  0 X ,*,r is a nonzeroof the feedback system| d . If (i) 80jG X N6TyM
f , thenand (ii) jT.y$n 
we say thatdivisor ofg{~.fthe plant T is stabilizable, T is stabilized by y , and y is astabilizing controller of T .
|Since the transfer matrix jT.y$ of the stable causal
feedback system has all entries in , we call the above notion -stabilizability. One can further introduce the notion-stabilizability as follows.
of

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| jTrSy55TDefinition 2 Let
be a nonzero element of . If (i)is a nonzerodivisor ofand (ii), then we say that the plant is-stabilizable,is-stabilized by , and is an-stabilizing controllerof .

jG f N6TyM
 #  g{~.f
#
T

y

y

T

#

9#

#

The causality of transfer functions is an important physical constraint.

Definition 3 (Definition 3.1 of [15]) Let
be a prime
ideal of , with, including all zerodivisors. Define the subsets and s of as follows:

U

 

WW4$2m4 W%&$%4 )/ sThen every transfer function in q s  is called causal
 strictly causal . Analogously, if every entry of a transfer matrix  is in  s  , the transfer matrix  is calledcausal  strictly causal . A matrix over is said to be  nonsingular if the determinant is in 4 , and  -singularotherwise.

To apply the coordinate-free approach to the multidimensional system with structural stability, and are given
as





{. § jSn R ¢¡ X 0/././'+¡<£  $B¤ in ¥ £ ¦§¨£
¡ q&nmhj{S% R ¢¡ X 0/././'+¡<£  Xthe constant term of  is zero. 

¥ £

where
denotes the closed unit polydisc.
Finally, we introduce the notion of simultaneous stabilization.

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 f'E.g y 6g{E.f
y
T© TX
y
T}© T X

Definition 4 Let and be plants in. If inis a stabilizing controller ofand , then is said to bea simultaneously stabilizing controller ofand , andandare simultaneously stabilized by . If therethat
exists a simultaneously stabilizing controller ofand ,then they are said to be simultaneously stabilizable.

T©

TX

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-stabilizability instead of For the case we consider
stabilizability, we will use “simultaneously-stabilizing
controller,” “simultaneously-stabilized,” and “simultaneously-stabilizable” analogously.

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III. L OCAL -G LOBAL P RINCIPLE AND S IMULTANEOUSS TABILIZABILITY
We consider in this section the simultaneous stabilization
problem and the local-global principle. The following is alocal-global principle with a fixed stabilizing controller.

T©

T
2gEf X

 f'Eg

Theorem 1 Let
and be plants in. Let
a transfer matrix of.
Then the following statements are equivalent:(i) The plants.

yy denoteT$© and T X are simultaneously stabilized by«W¬<­®v¯ ¯ª 4ª(ii) There exists a finite subsetofsuch that (a)and (b) for each, bothandare simultaneously-stabilized by .

¬

y

T$©

TX

The proof is relatively easy, so that it is omitted.
Then we have another problem whether or not the simultaneous stabilizability over is equivalent to the simultaneous stabilizabilities over local rings of . We give theresult that they are not equivalent as follows.

T7©

Theorem 2 Let andfollowing statements:

T X be plants in vf'Eg . Consider the

T7© and T X are simultaneously stabilizable.
(ii) Therea finite subset ª ofsuch that (a)« ¬<­® ¯ exists
 and (b) for¬ each ¯ 4ª , both T$© and T X
are simultaneously-stabilizable.
(i) The plants

Then, (i) implies (ii). However, (ii) does not imply (i) ingeneral.

166

°

Proof: Because “(i) (ii)” is obvious, we will prove
only that (ii) does not imply (i) in general. To prove this,
it is sufficient to show an example that (ii) holds but (i)does not hold. As an example, we employ Anantharam’s
example [16]. He considered the caseZ
Z , where Z denotes the set of integers(This ring [17, pp.134–135] is isomorphic to Z
and is an integral domain but not a unique factorization domain. In fact,Z
has two factorizations,and). We let. Anandoes not admit
tharam [16] showed that a plant.
a coprime factorization but is stabilized by
We use these transfer functions. Let
and.
In the following, we first present the controller
parametrizations ofand , next show that they are notsimultaneously stabilizable and then show that they are simultaneously stabilizable overfor every.

±² 7³   N1´}² 7³  S´Oµ?  0? , NO³b6¸¹¶·¸  Y² 7³ 
S%Nº²  ³ <%º² 7³Y»s¤{
S8N6² 7³+ b
S ² 7³+b j b ¼ © 4S7Nu² 7³Y)
¼ X &<5 ² 7³Y+ b 
¼© ¼X
¬
¯ nª

½i ¼¾ denote the set of¼ © is given in [18],Controller Parametrization Let
all stabilizing controllers of .
The controller parametrization ofwhich is given as¼{b Ã ©5N2S5U² 7³Y ÃÃ bÅÄ © W&ÆÈ ÀÇ / (2)ÀÀÂ $SMNU² 7³Y ©  ÄÄÄÄ
Ä
Similarly, the controller parametrization of ¼ is given asXbÃ 45 ² 7³ Ã(3)½i ¼ X } Á À¿ <5UX ²  ³8 Ã  b ÄÄÄ X % ÆÈ ÀÇ /
À
ÀÂ
X ÄÄÄ
½i ¼ © $¿Á À

Non-Simultaneous-Stabilizability Suppose that there
exists a simultaneously stabilizing controller ofand .
Then, from (2) and (3), there exist and in such that

Thus the right hand side of (6) is greater than zero and less
Z. This implies that cannot bethan one for any
an integer and that the equation (4) of and does nothave a solution.
Hence we conclude that the plants
andare not siand
do not satmultaneously stabilizable, that is,isfy (i) of Theorem 2.

 X S X ¼©

ªÉ4Ð b  ¹

8Ã©

Ã©

¼©

¼
¼ XX

X

¬ ¯ ª«%¬S­®v¯ ¼¾© ¼ X b
¼X
¼-©
V,

Simultaneous Stabilizability overforFirst,
we let. Then.
andare and,
Because the denominators ofrespectively, bothandare in. Thusandare-stabilized by zero, which is a simultaneously
stabilizing controller.
. Observe thatcan be rewrite as
Considerandas. Thusandare coprime factorizations of over.
Letdenote the set of all-stabilizing controllersof . Then we have, by Youla-parametrization,,² 7³
¼¼X½i ¼  ¬

¼¾©

Ñ ¹
¼©
  <DNÏ² 7³
S ()¼$©V:X ¼X ¬ Ñ

b¼X
V,¹ 0+v
<+¼¾V:X X ½i ¼¾©< ÑÐSMNB¼ ©V:X ÃÃ ©+ ÃÃ ©$ ÃÃ ©im Ñ 0¤ÐÐ
½i ¼ X ÑÒ ÐSMNB¼ V:X X X + X  X mÑ00¤Ð/If ½ Ó¼¾©< ÑÔ ½i ¼  Ñ is not empty, then ¼¾© and ¼ are simulXX
taneously Ñ -stabilizable. Hence, consider the equation
ÃÃÃÃS5N¼ ©V:X ©  X 4SMNn¼ V:X X X  ©
Ñ . By the straightforward calculation, we see thatover
 Ã © Ã X %ÉD(. ¹  is one of solutions. Its simultaneously
Ñ -stabilizing controller is Õ N ²  ³Y+<vN ²  ³8 .
Thus, ¼¾© and ¼ are simultaneously Ñ -stabilizable.
X
Thus these ¼¾© and ¼ satisfy (ii) of Theorem 2. ThereX
fore, (ii) of Theorem 2 does not imply (i) of Theorem 2 ingeneral.

By the result of this section, we observe that the localglobal principle will not play the role to investigate the relationship between the simultaneous stabilizability and thestrong stabilizability of the multidimensional system.

¼© ¼X
Xb{Ã ©N1SMU²  ³ bÃ &5U²  ³
IV. O NE -S IDE C OPRIME FACTORIZATION
X(4)
$SMNU²  ³8 Ã © b <5U² 7³Y Ã X  bWithout considering the doubly coprime factorizability,Ã
the relationship between the simultaneous stabilizability
holds.
8© , © ,  X ,  X be integers with ©8©5NÉ² 7³Y© and the strong stabilizability of the multidimensional sysÃ Letand·
tem cannot be given yet. Alternatively, we loose the condiX X N²  ³ X . Then (4) can be decomposed into tion,
the real part and the imaginary part equations as follows:that is, we here give a assumption below.
Ê ¾ËmNBË©S N4¤8:©5N&.¤  b ¤8©< ¤{
Assumption 1 A plant admits one-side coprime factoriza(5) tion (at least one of right- or left-coprime factorization).
 b 8©} b  XX NË X ©NBË©<X  X ¤r/ X
From this assumption, we can use the following theorem.
By solving (5),  © can be expressed by  and  as follows:
X
X
Theorem 3 (Theorem 1 of [19]) If there exists a rightb N b  , OÌ8 N&.¤ , left- coprime factorization of the plant ToÖvf'Eg , then(6) the plant T  HDgEg  u× g{~.f)ØÙEgÅ the plant TzH f+E.f 8©i ³NË , X  b ¤8 X N b ¤ X , / and left-coprime factorizations.
X
XX
 +f E0× g~.f)Ø  has both rightRecall now that the variables  © ,  ,  have some inteX X
Thus, once we have Assumption 1, the plant 0TÉH
ger values. Even so, we here show that the right hand side orT  H    admits a doubly coprime factorization. To generof (6) cannot be integer. Now decompose each of the nudiscussion, we consider a block diagonal plantÚ Û Ü0Ý jour
merator and the denominator of (6) into two parts within alize
Tr.H7F+EC rather than ThH  or T  H    , and supparentheses:
pose,
loss of generality under Assumption 1, thatÚ Û Ü0Ý jTrwithout
.H F+EC  admits a doubly coprime factorization . Even
Sb MN b  ,X ÍN2<¹ 5b Ì8 X N4b ¤8 ,X  /so, we do not consider the doubly coprime factorizability of NË ,X ÍN2  ¤ X N ¤8 ,X 
T . If T admits a right-coprime factorization, then we apply
AB9e and ?B ¤ , so that T  HDgEg   admits a left-coprimebbThen we have ¤Î35N ¹  , Î b NË b , for every   Z also factorization as well as a right-one.
¤ÎÏ 9Ì8 X Nu¤8 ,X Î X ¤8 X N X ¤8 ,X for everyX  X  Z. Let us review the following theorem.
Ã©

Ã

167

Theorem 4 (Theorem 2 of [19])Let ½ijT$ andi½  Ú ÛÜÝ jTr.H7F'ECÐ denotethe sets of stabilizing controllers
Ú

Û

Ü
Ý
of the plants T andjT.HMFEC , respectively. Then thefollowing equation holds:½ijT$&)G g H g{EC  y OàÞ:áÙâß ß  8yu%½i Ú Û Ü0Ý jT.H F'EC )/Using this theorem, we have the following without theproof.

T f'EgÚ Û Ü0Ý jT.H Ú ÛÜÝ jTr.HM

. Suppose that
Proposition 1 Let be inadmits a doubly coprime factorization. Then is stabilizable if and only if
is stabilizable.

T

Now we apply the discussion above to the simultaneous
and
be in
. Suppose thatstabilizability. Let
andadmit doubly coboth
prime factorizations. Let ,,, and
be matrisuch that,
ces over, andfor matrices ,, , andover . Analogously the symbols , ,, and so on are introduced for.
Analogously to Proposition 1, we have the following.

Ú ÛÜÝ jT$©.HF'ECS T © Ú ÛÜT Ý X jT .HF+E C< f'Eg
X © Ql ©
Q©
v
R
Ú ÛÜÝ jT$©0.HF+EC<·Q©.RmRv©VYl X © ;lRm©V:X Ql ©
Q l © L © N R l © K © 3ãl L © Q © N K l © R © ãlL ©
Kl © L ©
K©
QRÚ ÛÜÝ jT .HF+EC< X X
Ql X
X

V. C ONCLUSIONSIn this paper, we have addressed the simultaneous stabilization problem. Primary goal of the current study is
to obtain some criterion of solving the simultaneous stabilization problem of the multidimensional system. We do
not obtain this yet. We have two investigation ways in the
future. One is to solve the conjecture given by Lin in [4],that is, to show that any stabilizable plant admits a doubly coprime factorization. If we can show it, then solving the simultaneous stabilizability can be recast as solving the strong stabilizability (as in [1], [2]). The other is
to show some criterion of solving the simultaneous stabilization problem without the coprime factorizability. If theconjecture will be unsuccessful, that is, if there exists a stabilizable plant of the multidimensional system that do not
admit a doubly coprime factorization, then because we will
not be able to use the local-global principle by the result of
this paper, we will need to employ the later way.

simultaneously stabiÚ ÛT$ÜÝ © jT$©..HT F+X ECare and Ú Û Ü0Ý jT X .HF+EC< are

Proposition 2 Plants
andlizable if and only if
simultaneously stabilizable.

“Only if” parts of both propositions are obvious. “If”
parts of both propositions are proved by using Theorem 4.
Thus the proofs are omitted due to the space limitation.
The following is an application of Theorem 5.4.3 of [1]to the above setting.
Lemma 1 Let

I> K l © R X N l L © Q X &J Q l © R X N R l © Q X / (7)Ú Û Ü0Ý jT7©0.HF+EC< and Ú ÛÜÝ jT .HF+EC< are simultaneThen
X
ously stabilizable if and only if there exists a matrix ä oversuch that IÅNOäJ is unimodular.
Provided that I is nonsingular, there exists a matrix ä
oversuch that IÏNäJ is unimodular if and only if
the associated system JÍIiVYX is strongly stabilizable. Thus
Lemma 1 enables us to recast the simultaneous stabilization
problem as the strong stabilization problem.

I

J

T©

TX

Proposition 3 Let and be as in (7). Thenandare simultaneously stabilizable if and only if there exists a
matrix over such thatis unimodular.

äIÅN2äJ

This is just a combination of Proposition 2 and Lemma 1.
Suppose that
andadmits right-coprime factorizations. Then by Theorem 3, bothandadmit doubly coprime factorizations. Then Let
and , , ,and , , , be as previously. Let
and as in (7). Now, by virtue of Theorem 3,
andare simultaneously stabilizable if and only if there exists a
matrix over such thatis unimodular.
Now we have the result that when a multidimensional
system admits one-side coprime factorization, solving thesimultaneous stabilizability can be recast as solving thestrong stabilizability.

T$©

T  HDgEg 
Q © X R ©  Q l © /././ Q X
I J
ä

TX

R X Q l X /././
IÅN2äJT © HDgEg  
T©

TX

168

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