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AUTOMATIC
TRANSACTlONS
ON
which, in general, is not zero. Without this term, the system(A.8)-(A. 10)
has an equihbrium at the originfm = 0, f = 0,e = 0, suitable for a local
stability analysis by linearization. After this analysis, the extent of local
stability properties should be tested by reintroducing the forcing term yw*e* as a perturbation. The linearization of (A.8)-(A.10) around the
origin leaves (A.8) unchanged while (A.9)-(A.10) becomes
[-1-3 [ =
j
y(w,hr+
A, e*Q bow : ]
[-;-I [
+
0
]
fm.
- yw*h,$
COhTROL.
AC-30, VOL.
NO. 10,0(3TOBER 1985
On Stabilization and the Existence of Coprime
Factorizations
V. ANANTHARAM
Abstract-Let X be an integral domain and 5 its qnotient field. We
show there are plants over 5 which have no stable coprime factorizations,
but can be stabilized in a stable closed loop. This answers a question
posed by Vidyasagar, Schneider, and Francis.
(A.12)
Since (A.8) does not depend on f or 8 and is u.a.s., the local stability
properties of the zero solution of (A.8)-(A.10) are determined by the
properties of the system
Our purpose is to develop an example which answers a question posed
by Vidyasagar, Schneider, and Francis [4]. Admittedly the example is
somewhat artificial from a system theoretic standpoint,
but it should prove
of assistance in subsequent investigations.
I.
PROBLEM
Let X be an integral domain and5 its quotient field. If P E S n X m
and
C E 5””“are such thatthe closed-loop system(Fig. 1) is stable, then the
input to error matrix
(Z”+PCy’
C(I,+PQ-l
If e, is small, the term ye*C can be treated as a perturbation and stability
analysis is performed without it, that is, on the system (1.1).
(I,+CP)-I
is such thatHe, E X ~ n + m ~ x ((by
n +definition
mf
X consists of stable scalar
plants-see [4]). It may be readily proved that the matrix
always
hasbotharightcoprimefractionalrepresentationanda
left coprime
fractional representation when the closed loop is stable [4].
The question posed in141 is the following. Is ir always necessary that
C and P individually have coprime factorizations when the closed
loop is stable? An answer is of great importance because the bane of
fractional representation theory is the question of existence of coprime
representations.
We construct an example which answers the question in the negative.
The key idea is that our integral domain X is not a unique factorization
domain.
[t i]
ACKNOWLEDGMENT
Discussions with K. Astrom, R. Bitmead, R. Kosut, C. R. Johnson,
S.S.Sastry, P. Ioannou, and L.Praly contributed tothe formulationof the
stability criterion in this note.
REFERENCES
B. D. 0. Anderson, “Exponential stability of linear equations arising in adaptive
identification,” IEEE Trans. Aufomat. Contr., vol. AC-22, pp. 83-88, 1977.
B. D. 0. Anderson. R. Bitmead, C. R. Johnson, Jr., and R. Kosut, “Stability
theorems for the relaxation of the strictly positive real condition in hyperstable
adaptive schemes,” in Proc. 23rd IEEE Conf. Decision Contr., Las Vega,
N V , 1984.
K. J. Astrom, “Analysis of rohrs counter examples to adaptive control,” in Proc.
22nd IEEE C o d . Decision Confr., San Antonio,
1983.
S. Boydand S. Sastry, “Necessary and sufficient conditions for parameter
convergence in adaptive control,” in Proc. 1984 Amer.Contr. Conf., San
Diego. CA, 1984.
J. K. Hale, Ordinary Dfferential Equations. Molaban.FL:Krieger.1980.
P. A. Ioannou and P. V. Kokotovic. Adaptive System wifh Reduced Models
(Springw-Verlag Series, Lecture Notes in Control and Information Sciences.Vol.
47). New York: Springer-Verlag. 1983.
-, “Robust redesign of adaptive control,” IEEE Trans. Automat. Confr.,
vol. AC-29, pp. 2M-211, 1984.
P. V. Kokotovic and B. D. R i d e , “Instabilities and stabilization of an adaptive
system,“ in R o c . I984 Amer. Confr. Conf., San Diego, CA. 1984.
R. L. Kosut, R. Johnson, and B. D. 0. Anderson, “Conditions for local stability
and robusmess of adaptive coneol systems,’’ in Proc. 22nd I€€E Conf.
Decision Confr., San Antonio, T X , 1983.
1. Krause, M. A h , S . Sastry, and L. Valavani, “Robustness studies in adaptive
control,” in Proc. 22nd IEEE Conf. Decision Confr., San Antonio, TX, 1983.
A. P. Morgan, and K. S. Narendra, “On the uniformasymptoticstabilityof
certain linear nonautonomous differential equations,” SIAM J. Contr. Oprimiz.,
vol. 15, p p . 5-24,1977.
K. S. Narendra and L. S. Valavani, “Stable adaptive controller designdirest
control,” LEEE Trans. Automat. Contr., vol. AC-23, p p . 570-583, 1978.
K. S. Narendra and A. Annaswamy, “Robust adaptive control,” in Proc.
Identificat. Contr. Workshop Flm‘ble
Space Structures, San Diego, CA, 1984.
C. E. Rohrs, L. Valavani, M. A h s , and G. Stein, “Analytical verification of
undesirable properties of direct model reference adaptive control algorithms,” in
Proc. 20th I E E Conf. Decision Contr., San Diego, CA, 1981.
-,
“Robustness of adaptive control algorithms in the presence of urnodeled
in Proc. 2157 IEEE Conf. Decision Conrr.. Orlando, FL, 1982.
TX.
dynamics.”
-FyZm+CP)-‘
II. THE EXAMPLE
+
Let X = 1 1 -51 = X[x]/(x2 5 ) where 2‘ is the ring of integers,
Z[x] the ring of polynomials
in x with integral coefficients,(x2 + 5) is the
ideal generatedby x2 + 5 € ax], and / means we form the quotient ring
(For all undefined terms see any good bookon algebra, e.g., [I]-[3].)
Then X is an integral domain because the ideal (x2 + 5 ) is prime. It is
also not a unique factorization domain, as can be verified from
Ourexampleisscalar.L.etp= (l+J=3)/2€5andC=(1-,/=3)/-2
€5. We verify that
Manuscript received February 10, 1983; revised March 5, 1985 and A p d 22, 1985.
This work was supported by the National ScienceFoundation under Grant ECS-8119763.
The author is with the Department of Electrical Engineering and Computer Sciences
and the Electronics Research taboratory, University of California, Berkeley, CA 94720.
0018-9286/85/1000-1030$01.00
0 1985 IEEE
*
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IEEE TRANSACTIONS OK AUTOMATIC CONTROL, VOL. AC-30, NO. 10,OCR )BER 1985
I
u9
+
Multiplying out gives the conditions
P
.
r
Fig. 1.
This has an immediatemoduletheoreticinterpretation.
To establish
contradiction, we wish to prove that the submoduleof ZZgenerated over
Zby the column vectorsof the 2 X 4 matrix above can nevercontain [:I,
whatever the values of a , /3.
Let us multiply the above equation
on the left by the Z-unimodular
I O
matrix [
,I. Since
Also, from [4, Theorem 4.11 it follows that, if we let
N=
1
-1-J-5
-2
-,
[:I
r
;] [:]
=
1
=l
3
-I-J-s
D=
[
1
a
a+3p
-6a-313
30
(2.10)
-p
(2.1 1)
20!+2p
i.e., about the submodule Spanned by its columns, over Z.
By forming O-linear combinations of the columns of the above
matrix,
one can check that the vectors [,“,I and [
generate the submodule
we are interested in. Thus, an equivalent problem is to show that it is
impossible to find integers rn and n such that
1 + G
I-J-5
[:I
we have the equivalent problem aboutthe integer matrix
and
-2
l
-d+aJ
-2
then the pair (N, 0)gives a right coprime fractional representation of
[
O
C
3.
We w
l
i now show that p has no stable coprime factorization (similar
argumentswork for c). First, we havetofmd
all possiblefractional
representations for p . Let
If CY and /3 have a nontrivial commondivisor, this is clearly impossible.
So we assume gcd(cy, B) = 1. Now, the only choices of rn and n making
thefirstrowzeroarern=kpn=-kawithsomekEZ.Ifk# f l , i t
row, so onceagainit is
appears as acommonfactorofthesecond
impossible to satisfy (2.12). So assume k = f 1 . We get
Multiplying we get the conditions
&(3fl2+2c@+2aZ)=1.
We may rewrite this as
which on simplification yield
2 p + f f z + ( a+ p y =
2~-2b+6d=O
It is easyto see that all possible solutions in integers
equations are given by
0, 1, - 1 )
Indeed, given any solution (a, b, c , d) let CY = b and
(2.13)
a,
of the above
m.CONCLUSION ( n i E A N S W R )
B E 2. (2.6)
= - d.
If X is permitted to be an arbitrary integral domain as in [4], it is
possible to stabilize plants which have no coprime factorizations.
Thus,
all possible fractional representationsof p are given by
P=
1.
But it is clearly impossible to find integers Q and which satisfy (2.13).
Thus, we have proved the impossibility
of satisfying (2.12). It follows that
p has no coprime fractional representation.
c=20+5d.
(a, b , c, d)=a(l, 1 , 2,0)+/3(3,
Zk
(ff+3/3)+crJ-5
(2ff+p)-pG
where CY,
/3 E Z,at least one nonzero.
Next, we prove that no such representation can be coprime over X.
Suppose the contrary. Then there exist integers u, u , w, x such that
ACKNOWLEDGMENT
The author would like to thank Prof. C . A. Desoer for his interest and
encouragement.
REFERENCES
M. F. Atiyah and I. G. MacDonald, Introduction f o Commutative Algebra.
Reading. MA: Addison-Wesley, 1969.
[2] T.W.Hungerford, Algebra. New York Springer-Verlag.1974, GTM 73.
[3] N . Jacobson, BeFic Algebra I. San Francisco, CA: Freeman, 1974.
[4] M. Vidyasagar, H. Schneider, and B. A. Francis,“Algebraic and topological
aspects of feedback stabilization,” IEEE Tram. Automaf. Contr.,vol. AC-27,
pp. 880-894, Aug. 1982.
[I]