CORRESPONDENCE
307
di un sistema di equazionidBerenzialiordinarie.”
Boll. Unione Ma#.Ilal.. vol. 11, PP. 510-514. 1956.
bl T. Yosbizawa. ‘Liapunov’s funcuons
and
boundedness of solutions,“ Funkciolaj K?amcioj. vol.
2. pp. 95-142, 1959.
161 F. Brauer, “ B o p d s for solutions of ordinary
differential equations. Proc. Atner. Math. Soc., vol.
it follows that
..
14. ~~.
nn. 36-43.
1963.
..
.
..._
Recalling (23), if wl(t)* for t > L , the
inequality ( 2 4 ) can be written in the form
171 G . Zames. “h’onlinear time-varying s y s t e m s
conditionsfor
L, boundedness.” Proc.3rdAnn.
AUerton Conf. on Circuit and Sysfem Theory (Urbana,
Ill.,1965).
Is1 I. W. Sandberg. ‘%me stability results related
to those of V. M. Popov, Bell Sys. Tech. J . . vol. 44,
pp. 2133-2148, Nozember 1965.
IS] G. Zames.
On theinput-outputstability
of
timevaryingnonlinear
feedback systems,” I E E E
T r a m . AuIumatic Cmzf701.vol. AC-11. pt. I, pp. 228238, April 1965; pt. 11. pp. 465476,aJuly 1966.
I101 P. P. Varaiyaand
R. Liu. Bounded input
bounded-output stability of nonlinear time varying
dserential systems.” S I A M 1.Confrol. vol. 4, no. 4.
nn.
698-104.
~~.
.. - . . 1966.
..___
1111
‘Bounded-input bounded-output stability for the Lur‘e problem,” IEEE Trans. Aufomatic
Control (Corresfiondeme). vol. AC-11. DR.
.. 745-746.
October 1966.
1121 A. Bergen, R. Iwens, and A. Roult. “On inputoutput stabilityof nonlinear feedback systems.” I E E E
T r a m . At+fmnatic Control(Co7respmzdence). vol. AC11, pp. 746-748. October 1966.
ua1 W. G. Vogt and J. H. George, ‘OnAizermran’s
conjectureand boundedness.’ I E E E Trans. Automatic Cmztrol (Correspondence), vol. AC-12, pp. 338339, June 1967.
1141 K. S. Narendra and J. H.Taylor, ‘Stability
of nonlineartime-varying systems” IEEE Trans.
Auimnalic Control (Conespondence). vol. AC-12, pp.
628629. October 1967.
is1
S l k a r e n d r i &d C. P. Neumann. ’Stability
of a class of differential equations with a sinole mouotone uonlinearity.’SIAM 1. Confrol. vol. 4ppp. 295t n a, M
2 . r 1066
-.-,
1161 G. Zames and P. L. Falb. =Stability conditions
with monotone and sloDe r e > i c t e d nonlinearities.”
~ ~ ~
presented a t t h e Conf. b n hiathematical Theory of
Control. Los h g e l e s . Calif., February 1, 1967.
11’1
‘On the stability of systems with monotone and odd monotone noulinearities,” I E E E Trans.
Atrlontafic Control (Correspondence). vol. AC-12, pp.
221-323, April 1967.
1181 G. Marchesini and
Tassinari. “Sull’imniero
di tecniche funzionali neUa detei%nazioi;del-rok:
portamentoasintotico e uei problemi disintesidi
sistemi di controllo nonlineari.” Alfa Frequenza. vol.
34, pp. 626-635. September 1965.
I191 K. S. Narendra and Y. S. Cho. ’Stability of
feedback systems containing a single odd monotonic
nonliineanty.” IEEE Trans. Autm~aticConfrol (Shorl
Papers). vol. AC-12, pp. 448.450, August 1967.
-.
.
~
The evaluation of the term in the righthand side of (25) is very easy if T.t71(s) is of
the form
xi
bis2 strictly Hurwitzian, and
with n>m,
all the poles p i simple. Then
K.
-
--
I
-.
In
this
correspondence, conditions for
asymptoticstability in the large of nthorderordinary differential equations with
time-invariant matrix coefficients are derived.
NOTATION
Let Rmand T+denote the real Euclidean
wdimensional space and the real interval
O < l < m , respectively. Consider a system of
m nth-order differential equations
X C n ) + B I X ( r l )+ B&n-Z)
+ ..
(1)
Bn-lX(I) B,X = 0
+
where XERm, X is an m vector, and B;,
i = l , 2,
, n, are m X m real constant
matrices. I t is assumed that (1) satisfies all
the requirements for uniqueness and exis;
. , X,o; to)
tence of solution ~ ( t X10,
which corresponds to
the
initial state
(to; Xl0, . , X,O) in the motion space
RmX T+,where the vectors Xz, Xa,. , X,,
represent the successive time derivatives of
the vector X I = X . Equation (1) is a vector
version for the system of real linear timeinvariant m nth-order differential equations
-.
.-
Xj(n)
I
~~~~
-1
-0
WO
where w0=2ir/tm, Bc=Bi’Wc(-pi), and
Bi’ are the residues of Wl(s) a t the poles p i .
I t is also possible to find a relation similar t o (25) if the poles of Wl(s) arenot
simple.
The accuracy of the approximation of
the integral (17) by the right-hand side of
( 2 5 ) depends on tm since the L1 norm approaches the LInorm as t,goes t o zero.
CoNnusIoNs
A sufficient condition is derived here for
the boundedness of the responses of a class
of nonlinear feedback systems to amplitude
limited signals. This condition depends on
the linear part of the system and on the
range of the derivative of the nonlinear
characteristic. An upper bound on the responses is then determined. On theother
hand, it is possible to find an interval, if it
exists, in which theamplitude of the responses is within a bound fixed a priori.
M.
-.
m
lpFk(-l)
+ * *
~ ~ ~m . ~ . . ,
+
n,g.pk= 0,
j = I,
.
(2)
*
,m.
kFl
Definition 1: The system of differential
equations (1) is said t o be asymptotically
, n)4
stable if the vectors X i ( i = l , *
as timet+=. The vectors Xi-10 imply that
every
component
of Xi, the
G=1,
m)+O as t - m .
Dejinition 2: The determinant formed,
similar to that of the Routh-Hurwitz determinant but with matrix coefficients [ B i ]
of (1) treated in place of real numbers, will
be called thematrix Routh-Hurwritz determinant.
-
a
,
Thearm 1
If the matrix coefficients [Bi]of (1) can
be simultaneously diagonalized by a nonsingular transformation, then positive definiteness of all minors Ai (i= 1, * , n ) of
the
matrix
Routh-Hunvitz determinant
are necessary and sufficient criteria
for
asymptotic stability of the system of equatidns(1).
Proof: Let the matrix [ L ] diagonalize
the matrices [ B i ] such that matrices [Ai]
are diagonal with the transformation,
-
Stability of a System of Linear
Differential Equations
Absfrucf-Conditions for asymptotic stability of an nth order linear timeinvariant
diierential equation of a particulartype with
matrix coefficients are shown to be similar
to those obtained through the Routh-Hurwitz inequalities, but wherein real numbers
and positivity are replaced by the matrix
coefficients and positive deiiniteness.
ACKNOWLEDGMEKT
Theauthor wishes tothank Prof. A.
Lepschy and Dr. A. Marzollo for the useful
IXTRODUCTION
discussions.
GIOVANNI
MARCHESIP~I
Seshu and Reed have mentioned the
School of Elec. Engrg.
conditions
under
which a second-order
University of Padua
linear time-invariant differential equation
Padua, Italy
with real constantmatrix
coefficients is
stable? Ezeilo established stability condiREFEREXES
tions for a system of 7t third-order nonlinear
differential equations of a particular class?
111 T. Y o s h i m . “Noteontheboundedness
of
solutions of a system of differential equations” Memoirs ojthe CoUege o j S c k z c e . Kyoto Imperial Unioersity.
ser. A, voL 28, pp. 293-298, 1953.
121
“Note on the boundedness and the ultimate boundedness of solutions of x‘=F(t. z),”ibid.,
vol. 29. pp. 275-291. 1955.
131 T. Sato.
‘Sur Yequation intCgrale nonlinhke
de Volterra,’ Composito Math.. vol. 11, pp. 271-290,
1953.
9 1 R. Conti. uSulla prolungabilita delle soluzioni
+
k=l
-.
~TB,
Pn
+- cotjir
.
+
Manuscript received January 16. 1968.
1 S. Sesbu and M. B. Reed, LinearGraphs
and
Eleclrical
Networks.
Reading, Ma.:
AddisonW
A4P”. ~
Wesley,
..P ~
1961.
~. ~
~ _ .
J. 0. C. Ezeilo, ‘n-dimensional extensions of
boundedness and stability qeorems for some third
order differential equations J . Mafh. A d . and
vel.
~ p p l . VOL
,
18. DD. 39-16,
june 1967.
f
i = 1,
[L]-’[Bi][L] = [A,],
.
* *
,A.
Then thepositive definiteness of the product
of matrices
{ [Bll[B21, .
, [Bll),
I I
i = 1, *
*
, 78
is equivalent to the positive definiteness of
the product of matrices
I [All[A*l,
*
-
7
[A211
since
Consider the equation with diagonal matrix
coefficients [ h i ] obtained by a transformation [X]
= [ L ] [ Y]which is equivalent, in
the sense of Liapunov transformation, to the
given system of equations (1). The modified
equation is
I'(n)
+
AlI'(*l)
+ . . . + An-lI'(')
fA,Y
= 0
(3)
which is a set of differential equations of the
fo m
yj(d
JUNE 1968
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,
308
+ lxjyj(trl) + . . . + ( n - 1 ) ~ ;
yj(l)
+-x]-)$
. . ., W Z .
= 0,
j = 1,
(4)
Let isi, i = l , . . , n, betheith-order
Routh-Humitzdeterminant minor corresponding to the single differential equation
in the yjth component of the m vector [ Y ] .
Thenthedeterminants
Gi, j = 1 ,
, M,
are the jjthdiagonal elements of the matrix
Routh-Hunvitz determinant Ai. For the determinants i8i to be positive it is necessary
and sufficient that the determinants Ai be
positive definite
Q.E.D.
I t may be noted thatthematrix
coefficients [Bi] may be symmetric or nonsymmetric. I t is known that any two symmetric
matrices
can
be
simultaneously
diagonalized if one of them is positive definite. I t then follows that the other coefficient matrices of (1) must be expressible as
functions involving positive integer powers
of these two matrices.
The conditions under which a set of nonsymmetric matrices can be simultaneously
diagonalized are notknown at present.
-
Example
Consider the following system of differential equations for stability:
In this new set of coordinates, the uncontrollable (unobservable) state variables are
easily identilied and removed. An example
is given.
h a t which
I t is well k n ~ w n [ ~ l * [ ~ la tsystem
is uncontrollable and/or unobservable can
be reduced to a zero-state equivalent system
of lower dynamic order by removing the uncontrollable and unobservable modes. This
problem of reducing the order of a system
has been considered by Silverman and
h'Ieadows,111*131Glass and D'Angel0,[41 and
others. It is the purpose of this correspondence to present a very simple procedure for
removing uncontrollable and unobservable
state variables of time-variabIe linear systems to yield a zero-state equivalent system
with the minimum number of state variables.
In earlier papers,[51*[61algorithmsfor constructing minimum-state realizations of
time-invariant
systems
were presented.
Some of these techniques will be applied to
the time-varying system problem.
The system is represented by the state
variable equations
[;I
5) For each uncontrollable state variable
i and
columnj from A , row i from B , and column
i from C t o obtain thereduced system
xi as identified jn step 4), dekte row
& = { A , , 8,, C).
Subsystem 8, is totally controllable2
[Qa(t)
[@oh(t) ;To']
(6)
is carried out using elementary row operations. The matrix TO'is the transpose of the
desired transformation matrix.
QJt)
where
=
[Po(t)iP*(t)i. . * iPn-1(t)I
Pt+l(t) = - A(t)PAt)
+A(t),
(2)
Po(t) = B(t)
and
[Ro(OiR~(t)i
* *
R . 4 ) = A'(t)&(t)
. iRn-l(t)]
+ fidt),
Example
The following example, considered previou~ly,[~1~[41
illustrates the use of the
algorithm. Consider system (1) ahere
(3)
-t+1
0
Ro(t) = C"
(7)
and where the prime indicates transposeand
the dot indicates time derivative.
An algorithm can now be stated.
of
Abstract-An algorithm for removing uncontrollable (unobservable) state variables
from time-varying systems is presented.
The algorithm makes use of the time-variable controllability (observability) matrix
which is augmented with the identity matrix
and transformed into Hermite normal form.
Manuscript received December 12, 1967. This
work was supportedin part by TexasInstruments
Inc.. and by theDept. of Defense.JointServices
Electronics Program under Grant AF-AFOSR-766-67.
(5)
Use will be made of the controllability and
observability matrices for the time-varying
system given previously.~3l~Ii1
These are as
follou~s:
+ B(t)u(t)
C(t)x(t).
Algorithm I
I t is known that the preceding coefficient
Given system (1) where A ( t ) , B(t), and
matrices can be diagonalized simultaneously.
C(t) are n Xn,~ z X mand
,
p X s matrices, reThe matrix Routh-Humitz test shows that
spectively, with possibly time-varying elethe given system of equationssatisfy the
ments, and where matrices A (t), B ( t ) , and
conditions for asymptotic stability.
C(t) togetherwiththeirfirst
x - 2 , n-1,
N. SREEDHARand tz - 1 derivatives, respectively, are cons. N. K40 tinuous functions, then any uncontrollable
Dept. of Mech. Engrg.
state variables in the system maybe removcd
Indian Inst. of Science
as follows.
Bangalore 12, India
1) Form the controllability matrix QJt)
as given in (2).
2) Augment Q.(t) with an n X n identity
matrix, i.e.,
Minimum-State Realizations
Linear Time-Varying Systems
e
(1)
k(t) = A ( t ) x ( t )
y(t) =
Qo(t) =
.
e}
A = (TC-'AT, - Tc-IFC)
8 = TC-'B
= CT,.
6)
and is zero-state equivalent to S.
Proof of this algorithm follows directly
from Theorem 5.1 of Nering,[B] Lemma 1 of
Silverman and h~Ieadoas,I11and Corollary
11.3.15 of Zadah and Desoer,[g] and is presented in Alberts~n['~], but it will not be
given here because of space limitations.
The algorithm for removing unobservable modes is the dualof Algorithm I,where
Qo(t) given in (3) is utilized and where the
transformation
where
19 8
+ [--12
411
coordinates, as those state variables corresponding to the all-zero Tom-sof Q c h ( t ) where
row i corresponds to statevariable xi. Transform the system ST { A , B , C} into the new
where
coordinate systems= {A, 8,
3) Transform Qc(t) into Hermite normal
(row-echelon) form ( Q c h ( t ) ) using elementary
row operations* and obtain the transformation matrix T,, which must be nonsingular
and differentiable, where
4) Identify the uncontrollable state
variables of the system, in this new set of
1 It m a s not a l w a y s be necesjaw to reduce Qc(t)
completely to Hermite form. but only to the point
where the all-zero rows of ,
Q are identified (see example).
The controllability matrix using (2) is
(8)
Now, using (4),
1 1-t
1 l+t
0
--t
0 . 1 0 0
2 . 0 1 0
-1.0 0 1
1
(9)
I t is noted that Qcn(t) has rank 2 everywhere and that statevariable XZ, in the new
coordinatesystem, is uncontrollable. Also
note that
Tc-l
2 See page 689 of Sil~wmanand Meadows[*lfor
the delinition of total controllability.