Ilchmann, Achim; Owens, David H.; Prätzel-Wolters, D. :
Sufficient conditions for stability of linear time-varying systems
Zuerst erschienen in:
Systems & Control Letters 9 (1987), S. 157-163
DOI: 10.1016/0167-6911(87)90022-3
Systems& Conlrol Letters 9 (1987) 157,163
North-Holland
Sufficient conditions for stabilitv
of linear time-varying systems
A. ILCHMANN
Inslitut lür Dynamischesysteme, uniuersität Bremen,2800 Bremen 33, li/est Germany
D.H. OWENS
Department of Mathematics, Uniuersity of Strathclyde, Glasgow Gl lXH, Scotland
D. PRATZEL-WOLTERS
lnstitut für Dynamische Systeme, IJniuersität Bremen, 2800 Bremen 33, Wesr Germanl,
R e c e i v e dl O O c t o b e r I 9 8 6
Revised 18 February 1987
Abstracl: In this paper we consider sufficient conditions for the exponential stability of linear time-varying systems of the form
*(t):
A(t)x(t ), 1 > 0. Stability guaranteeing upper bounds for different measuresof parameter variations are derived.
Keywords: Time-varying linear systems, Exponential stability.
l. Introduction
Stability analysis for time-varying linear systems is of increasing interest in control theory. One reason
is the growing importance of adaptive controllers for which the underlying closed-loop adaptive systenl
often is time-varying and linear.
In this paper we analyseexponential stability for systemsof the form i(r) : A(t)x(t), r > 0, where
l('):R++lR'x'is
p i e c e w i s ec o n t i n u o u s a n d u n i f o r m l y b o u n d e d . F u r t h e r m o r e f o r e v e r y / > 0 t h e
eigenvaluesofA(t)arecontainedinalefthalfplaneC-.:{seClResg-e}forsome€>0.However
this last condition is not strong enough to guarantee exponential stability. Additional restrictions on the
parametervariationsin A(.) have to be imposed.
In Section 2 we summarize different types of those sufficient parameter variation conditions including
the well known criteria of Coppel [2] and Rosenbrock [5] and two new conditions due to Kreisselmeier
[4]
and Krause and Kumar [3]. We give a new short proof of the Krause and Kumar result which was recently
published in this journal. However all mentioned conditions are qualitative results in the sensethat if some
measureof the parametervariation is'sufficiently small'the exponentialstability is ensured.
In Section 3 we derive explicit formulas for the parameter variations upper bound to guarantee
exponentialstability. Theseformulas involve some a priori knowledgeof
llA(t)ll and o(,4(l))2. Sufficient conditions for exponential stability
To derive stability results for linear time-varying systems of the form
r(/):A(t)x(t),
r>0,
( 2r )
it is usually a priori required that A(. ) belongs to the ser .V of all piecewisecontinuous ntatrix functions
l ( ' ) : R +- R n x '
0161-6911/81/$3.50 C) 1987, Elsevier Science Publishers B.V. (North-I{olland)
(22)
158
A. Ilchmann et al. / Sufficienl conditions
fctr stability
whrch satisfy:
t h e r ee x i s t sM > 0 s u c h t h a t I . q Q ) l l < M f o r a l l l > 0 ,
",:
t h e r e e x i s tas> 0 s u c h t h a to ( A ( l ) ) c C
{seClRe r.
The following definition of exponentialstabiLityis standard:
(2 3)
_o)
forall l>0.
(2 4)
2'l' Definition- A system (2-l) is called exponentiallystable if there exrst
Z, ^ > 0 such that
^
(
,
,
o
r
f o r a l l r > r o> 0 ,
ll+(/, /o) ll < l, e
.
)
w h e r eS ( . ,
d e n o t e st h e t r a n s i t i o nm a t r i x o f ( 2 . 1 ) .
In general A(')e-9l
is neither necessarynor sufficient for exponential stability.
2.2. Examples.(i) Coppel [2,p.3].Let
A(t):
cos/ stnrlf-t
s i n r c o s /J [ 0
-5lf cosl sinrl
-tl[ sinr .orrJ'
T h e no ( l ( l ) ) : t -11 for all / > 0 and a calculationof a fundamental matrix
shows that l(.) is unsrable.
(ii) wu [6].Let
A(t):
- ä + f s i n1 2 r
f c o s1 2 r I
+
- f s i n1 2 r j '
f c o s1 2 r
T h e n o ( l ( r ) ) : { 2 , - 1 3 ) f o r a l l r > 0 ; h o w e v e rr h e a s s o c i a t e ds y s r e m
i(t):A(t)x(t)
stable.
is exponentially
In order obtain sufficient conditions for exponential stability
additional restrictions for the variation of
the elementsof A(.) e g have to be imposed.
It has been shown that if ö > 0 is sulficiently small, then any
of the following conditions guarantees
exponential stability of (2.1):
l l , < ( r ) l<l ö f o r a l l r > 0 t 5 l .
l l A ( t r ) - A ( t r ) l l < 0 l l t r * t , l l f o ra l l r , , r , > 0
s u p l l A ( t + , ) - A ( r ) l l < ö f o r s o m eä > 0 .
(2.5)
[Z,p.Sl.
(2 1)
O<r<h
,<(.) is continuous,
lll(.) llis uniformlybounded
a n dt h e r ee x i s t sT > 0 s u c ht h a t
' 0 " ' l l Ä ( , \ l l d 1< ö r
"/t n
(2 6)
for ail ro> 0.
(2 8)
condition (2'7) is a consequence
of rheorem 3.2 (iii) in Section3 of this paper and is lessrestrictive
,
than the similar conditionin Lemma 3 of
[4]:
.lim sup ll A(t + r) l(/) ll : 0 for all lr > 0.
r-@
0<r<lr
Furthermore (2'8) is less restrictive than the criterion
in [3] which requires the integral inequality of (2.g)
for all "- To and some ro > 0. In fact condition
12.8) can be proved much shorter following the ideas of
Rosenbrock'sproof for condition (2.5).
'Simpler'proof
of the Krause and Kumar condition. Let A(.)eg,l(.)
c o n t i n u o u sa n d u n i f o r m l y
bounded. Supposethe integral condition in (2.8) is satisfied
for some Z'> 0, ö > 0. Then the set
,n(e),: e Ä* llll(.) , r}
ll
{r
A. Ilchntann
is a union of open intervalssincel(.)
s e t s1 c R , t h e n b y ( 2 . 8 ) ,
et al. / Sulficient
r59
conditions for stability
is continuous.If -!?(1) denotesthe Lebesguemeasure of measurable
g((to,l0 + f n "r(u)) .e < öZ
)
for all lo> 0.
T ' t r u sJ f ( ( r o , / 0 + Z ) n q ( r ) ) - 0 a s ö - 0 . L e r n o w
V(x, t) ': xrR(r)x
( 2.e)
wnere
,@
R (' r ) , :
I
,n'
Jg
(r)s
a"lrt)s6r.
(2.10)
Similar to Rosenbrock's[5] proof we show that if ö is sufficiently small, then Z(x, /) is a Liapunov
function for (2.1). Since ,4(.) is bounded there exist (cf. Brocketr [1,p.203]) cr, r ' r > 0 s u c h t h a t
c , 1 , ,< R ( t ) z - c r I , . F u r t h e r m o r eb e c a u s eR ( t ) A ( t ) + l r ( t ) R ( l ) : - I , w e o b t a i n
.
f @
n(r): l
r " ' t ' r ' I n ( r ) r ( / ) + l r ( / ) R ( r )e] r { ' ) ' d s
(2.11)
and for e sufficiently small,
v ( x ( t ) ,r ) : ( x ( r ) , [ Ä ( , ) - r , , ] ' ( r ) )
< - j t t x ( r ) l l ' .- + v ( x ( t ) ,
S i n c el l l ( / ) l l < K *
t(x(t), t).K
t)
t o rt € r n ( e ) .
forsomeK*>0,
l l x ( r )l l ' = { r ( x ( r ) ,
for some K> K*. Since 9(rnQ\-
r)
f o rr e r , ( e )
0 as ö - 0, there exists for ö sufficiently small o > 0 such that
., Z(x( s). s )
ds < -r,;(l - rn)
I _,#
J r "Z ( x ( s ) , s )
for all r > to+ T,
and this provesexponential decayingof the solutions of (2.1). !
As an existence condition (2.8) is not really an improvement of (2.5). The above proof shows that if
ll l( ) ll is assumedto be uniformly bounded there always exists sufficiently small ö > 0 such that rhe
auerage parameter variation condition (2.8) implies that l(')
satisfies Rosenbrock's criterion. The
conversedirection is obvious. Hence,viewed as existenceconditions (2.8) and (2.5) are equiualent.Inorder
to show that (2.8) is lessconseruatioethan (2.5), upper bounds for the integral in (2.8) should be derived
which guaranteeexponential stability of (2.1). In the next section we determine such quantitative boun{s
for the ö's involved in (2.5) (2.1).
3. Upper bounds for parameter variations
In order to prove the main result of this paper we need a lemma and some preparatory forrnulas:
3 . 1 . L e m m a [ 2 ] . S u p p o s e A ( . . ) e 9 , i n p a r t i c u l lal A
r (t)ll<M.Thenforeueryee(0,2M),
o+')o
l l e A ( ' tl'l < ( 2 M / E ) ' - 1 . {
lor alt o, / > 0.
i60
A. ILchmann et al. / Sufficient conditions for .srahility
For füed /0 c R ", (2.7) can be rewritten in the form
A(t)x(t) + IAQ) - A(t))x(t),
*(t):
and for x(to):
/ > 0,
xo € R' its solution is given by
e A ( t ü ( t - t o \ x o[ '+ s o r , , x ,' , [ . a ( r ) - A ( t ) ] x ( s )
,(/):
(3.1)
(3.2)
ds.
t ,r,
Thus for A(-) eJZ Coppel's Lemma yields
'o)
ll"(l) ll ( r. e(-o+')(' ll"oll
* n ', l [ ' e ( - d + € ) ( / - lrl)- a ( r ) - A ( t )
'o
l ll lx ( s ) l l d s
for / > /6,
(3.3)
where
n,.: (2M7u),
,.
(3.4)
Applying Gronwalls's Lemma to (3.3) gives
ll"(r) ll =n. "*o[1-a+e)(r-lo) + n,['ll,t(r)_iA(t) 1a
1rl ll'oll
l
"
r
n
l
3.2. Theorem. SupposeA(-)e9.
*(r):A(t)x(t),
forailr> ro.
(35)
Then
r>0,
(3.6)
is exponentially stable if one of the followirtg conditions holds for all t > 0'.
( 1 )a > 4 M .
(u) l(.) is piecewisedifferentiableand
o4n-2
llÄ(t\ll <6< =]2n-l274an'a'
( n ) F o r s o m ek ) 0 , l e
(0,1), a>2Mq
+((n-
I)/k)logq
l lA ( t+ , ) - A ( r )l l- < 8 . 4 ' '( " - r M n + +
,::?-
and
l o sr ) .
(iv) a > n - 7 and for some 4 e (0, 1),
,,A(t+h)
i*ll#
r/'\
l l < ö . 2 r ' - ' ( a - 2 M 4+ ( n - l ) l o er r ) .
P r o o f .( i ) S i n c e l l - a ( s ) - A ( t ) l l
< 2 M V s , 1 6 > 0 , i n e q u a l i t y ( 3 . 5 ) i m p l i e sf o r s o m ef t > 0 a n d t e ( 0 , 2 M ) ,
l l " ( r ) l l ( r c .e [ - o + ' + K e 2 M ] ( tl-l txo( )r o ) l l . n . " [ u * n . 2 M - 4 M - lhl r] ( l o ) l l
The function
f :(0,2Mf -p,
s*)s+ rcu2M*4M-h,
- h , T h u s t h e r ee x i s t se e ( 0 , 2 M 1 s u c h t h a t
i s c o n t i n u o u sa n d f ( 2 M ) :
,f(r) < 0.
(ü) We prove that V(x, t) as defined in (2.9) is a Liapunov function of (3.6). Its time derivative along
solutionsof (3.6) is
Ä
:
f i v t r t 1 ) ,/ ) " ' ( r ) [ - 1 ,+ Ä ( / ) l x ( / )
A. [lchmann et al. / Sufficient condirions
for srability
16]
We have to show that
n(r)=f,,
forallr>O.
ej)
Applying Coppel's Lemma to (2.11) and (2.10) we obrain
".r,
"\'''
s'2R
ll(r)llllr(1)ll
e - ' a* r". ") ds d
\;/
2{n-r)
I
" * . , . 6 r 1 ':5r ( 2 ! ) 4 ( 4 - r ) /
\'"
-= -r tf f\ !
t J
J/ *o. r ,
i
e
\
I
\2(_arf)/ö
l r-r
l l, R ( r | <
J6
and thus (3.7) holds if for some e e (0, a),
-
|
F
u.t\m)
\ 4 { a - l )
(o-.)':,s(e).
It is easilyverified that g(-) achievesits maximum at eo: ak/(k+
Zd4n
1) on (0, a) and
2
'
t'(, (2M)o<'
l)
s { F . .l :
( i i i ) B y ( 3 . 5 )w e h a v e f o r e v e r yI € N , k > 0 a n d t e [ t o + t k , t o + ( / + 1 ) k ] ,
l l " ( t ) l l : 6 . " r ( r - r o - , o ) l l r ( l o+ / f t ) l l
s K . e 7 ( / - r o - l k ) n . " r ( k ) l l x ( l o + ( / - 1 ) f t:)nl l3 e y ( / - 1 0 - ( r/ ) * ) l l r ( r o + ( / _ 1 ) k ) l l
where 1:: -a * e * rc.ö.Thus
r o sK . + Y ( / - 1 0 )
ll"(t) ll s K.e/
llr(rn) ll
Becauset - to> /k we obtain
1"(r)tr= ,.."*p{(+J!* ,)r,- ,,))1"(ro)
1.
Consider
log rc,
Ä
los(2 M /t\'t Y:-T
It sufficesto find e <2M,
log rc^
( e- a ) + - - ' <
|
*e*rc.ö-a'
k > 0 such that (log rc,)/k +y < 0 or equivalently,
r,6,
respectively
| |
U<ö<-la-r
r(,\
lop.,(- - - i j \l .
k
l
Howevertoeveryee(0,a)existsk*>0suchthatd-€-(1ogrc,)/k*>0andweobtarnexponential
stability of (3.6) for everyl(.) for which
s u p l l A ( r + r ) - A ( r ) l l< a .
osr:k"
Now (iii) follows with 4 : e/2M.
Ll,
Kr\t-t-
l o er c - t
>O
O--,/
762
A. Ilthmann
et al. / Suflicient tonditions for stability
(iv) Assume
l l / ( /r rh) - / ( / ) l l < Ä
ll-
ll
for e\ieryft > 0. Then by (3.5)we have
"'hödh)
l l r ( r ) l l. n , . * p { { - a r p l ( t - t , , 7t n , 1 , ,
tt*t,ri
- , . ,. ^ n ( ( . . ' 1 K
a ) { ,- , , ,t ) t t* t , "t t l
!'"
F o r I e [ 1 , , ,l o + 1 ] w e o b t a i n
l l r ( l ) l l - ( K . e r ( l - r nl )l r ( r n ) l l
[ / 0 + 1 , t o + l + 1 ] , l e N a r b i t r a r y , w e c o n c l u d e a s i n t h e p r o oo ff ( i ü ) ,
whereT:e-ct*]rc.ö.Forle
t . + Y x l1 0 )
llr(rn)ll
It,(l) ll! r. e(r"B
andlogrc.*y<0if
')
u . r ( * ) "' ( " - , - ' or yr () "
Then (iv) follows with 4 : e/2M.
!
Note that the proof of (iii) presentsa short proof of Lemma 3 in Kreisselmeier[4].
If additional information on the exponentialdecay of e'(')" is known, the bounds derived in Theorem
3.2 can be simplified as follows:
and let
3 . 3 . T h e o r e m .S u p p o s e
A(')ef
'"
lle't'r"ll < n e
f o r s o m eK , o ) 0 .
for all l, o > 0,
Let ß,:r/n.Thenthesystem
;(r):A(t)x(t),
/>0,
is exponentiallystable if one of the following conditions holdsfor all t > 0:
(i)llzt(r)ll<M<+ß.
(ii) l(')
is piecewisedifferentiableand
t t Ä Ql )l= ö . 2 8 ' .
(lit) There existsh > 0 such that
s u p 1 A ( t + r ) - a ( 1t )< 8 < B _ \ i Y
o=a<h
( i v )s u pl l a ( r+ n ) . a ( ' ) l l .u < z ß- 2 l o sn .
i,'oll
h
ll
K
The proof is similar to that of Theorem 3.2.
A. Ilchmann
I
et al. / Sulficient
conditions for stability
163
Rererences
[1] R.W. Brockett, Finite Dimensional Linear.S,fs/erru(John Wiley, New York, 1970).
[2] W.A. Coppel, Dichoromiesin Stability Theory,Lecture Notes in Mathematics No.629 (Springer, Berlur-New York, 1978).
[3] J.M. Krause and K.S.P. Kumar, An altemative stability analysis framework for adaptive control, Syslens Conrrol lstr. ? (1986)
t9-24_
[4] G. Kreisselmeier,An approach to stable indirect adaptive control, Automotica 2l (9) (1985) 425-437.
[5] H.H. Rosenbrock, The stabiüty of ünear time-dependent control systems, Internat. J. Electr. Conrrol 15 (i) (1963) 73 80.
[6] M.Y. Wu, A note on stabihty of linear time-varying systems, IEEE Trans. Automat. Conrrol 19 (1974\'162.