80
FEBRUARY
IEEE TRANSACTIONS
CONTROL]
ON AUTOMATIC
1973
Time-Domain Criteria for the Lz-Stability of
Nonstationary Feedback Systems
&I. K. SUNDARESHAN
AND
hi. A. L. THATHACHAR
Fig. 1. The feedbacksystemunderconsideration.
of linear andnonlinear timeAbstract-Criteria for the L-stability
varying feedback systems are given. These are conditions in the
time domain involving the solution of certain associatedmatrix
Riccati equations and permitting the use of a very general class of
Lroperators as multipliers.
I. INTRODKCTIOX
The problem of deriving criteria for the &-stability of systems
containing a linear time-varying operator
and a memoryless nonlinearity in cascade, i n a negative feedback loop, is not amenable to
simple treatment, owing tothe difficulty in obtaining positivity
conditions for rime-varying operators. I<ecently, n’illems [ l ] , [2],
a-5vie11 R X Estrada and newer [3], have obtained conditions for the
positivity of systems with a statespace description. in terms of an
associated matrix lliccati equation. Here, we follow this approach
and give ronditions for the positivity and stability of time-varying
systems. The poaitivity conditi~~n
obtailled here is the same as that
given ill [ 3 ] ,but the method of derivation is felt to be simpler and
more direct and 1135 been inspired by the work of \Tillem:: [ l ] , [2].
Although the altalyris iscollcerlled with finite-dimensional singleinput single-outputsystems, the extension to the general case is
straightforward.
tctIT,andzerootherwise;~(.)EL.=)~~(.)EL~,’dTTE+].
Note that H E EA =) H has finit.e gain and HI,Hz E EA =) H =
HIHz has finite gain.
H E Q E n 81is said to be positive ( e ) [strongly positive ( e ) ] if the
and
inequality (UT(. 1, YHT(. 1) 2 (UHT(.), U H T ( . )), ‘plu H ( . E Ls
‘pl T E R+holds with e = 0 [ e > 01.
System: The syst.em (Fig. 1) is described by t,he input-out,put
relations e l ( . ) = u l ( . ) - w2( . ), e!( . ) = UZ(.)
WI(.) wit,h w l ( . ) =
Gel( .), G E 6~ n 6 r and WP(. ) = Se?( .), :V E %o.w, i.e., S:Lnc +
L?. 3 S T ( . ) = n ( x ( . I), 0 I
x ( . ) n ( ~ ( . )I
) s x 2 ( .1, V x ( . 1 E L e ,
and n(.) is odd and monotone nondecreasing. Kote that. S = E,
the identit,!. operator in I,,,, yields a linear system more general
than that considered in [ 4 ] .
Problem: Given that u1(. ), z c n ( . ) E LPand el(. ), e ? ( .) E LZe, find
conditions on G which ensure e l ( . ), e ? ( .) E L.
The method of solution to this problem by factoring the open
loop into two positive ( r ) operators, one of which is strongly positive
( e ) and has finite gain, is by now well established, and the introd u d o n of “multipliers” to render flexibility to this approach is well
known [ 4 ] .
+
111.
11. PROBLEN
FORMULATIOX
,\‘otation and Definitions: Here, detailed definitions will be omitted
as these can be found in an earlier paper by the authors [4]. Some
new notation will be introduced.
Let $ E denote the class of linear causal operators H in L?, with an
ext.ernal (input-output) description, i.e., H E ZPE =) 3 a map h :
K+ X R +
K I s w h t h a t y ~ ( f )= H ~ c x ( t )
h ( t , i ) z c ~ ( dr )i , V
U H ( . ) E L e , where U H ( . ) : R + -+ R is the input to H and y ~ .):
(
R + + K is t.he output of H and h ( t , ~ =
) 0, V T > t .
=io”
+
&EA
C 6E
3 H E &Ed -}J=
-
L=
(h.(t,r)(2dt d l
is finite.2 Kote that H E
=) H : Lz + L? [’i].
Let. 91denote the class of linear causal operators H in Lr, aith an
internal (statespace) description, i.e., H E 61=) 3 AH( ) : R+
R n X K”, ~ H ( . ) : R+- Rn, C H ( . ) : R +
I En,and d ~ ( . ) : & ++ R,
such that H is described by t.he dynanlical equations
--t
Lemma: An operator H E
dH(t)
MAIN
QIM
RESULTS
is positive ( e ) if: a)
> 0,
‘d t
E R+,
(3.1)
and b) there exists a real symmetric bounded nonnegative-definite
n X n matrix RH(^) satisfying the Riccat.i equation
kH(t) = ;dH-’(t)[cH(t) - RH(t)bH(t)I [ b H ’ ( t ) R H ( t )- CH’(t)]
+
- [ R H ( ~ ) A H ( ~Aa’(t)Rx(t)l.
)
(3.2)
H E Qr,w is strongly posit.ive ( e ) if there exists an e > 0 such that
(3.1) and ( 3 . 2 ) are satisfied wit.h d a ( t ) replaced by [ d a ( t ) - E].
Proof:
a ) Positiuify ( e ) of H: It is required t.0 prove that
( u d . ) ,Y
H ~ . ) )2
0,
v
UH(-)
E Ln. and
‘pl
E R+. (3.3)
+
i H ( t ) = A H ( ~ ) z H ! t )b H ( t j t t H ( f ) ; y H (=t j c H ’ ( f ) z H ( f )
+
where
XH(.):R+
+ R n i s the state of H and U H ( . ) ,
dH(t)ieH(t)
IJH(
(2.1)
.) defined s
above.
c 61 3 H E t p I y = ) (2.1) is a minimal representation of H.
I t is aell known [ 5 ] that H is uniformly reachable and uniformly
observable =) H E 61.u.
H E QE is said to have finite gain if
+
+
( $ z ~ ‘ ( t ) [ k ~ (Rt a) ( t ) A d f ) A ~ ’ ( t ) R ~ ( t ) l z ~ ( t )
= SOT
isfinite
[ET(.)
is the truncation of x ( .) defined b>- z ~ ( t =
) z(t),
+ ra‘(t)RH(t)be(t)uH(t)-
U H ( ~ ) C H ’ ( ~ ) T H (~ )d H ( t ) U z r 2 ( t ) }
dt,
(3.5)
Manuscript reeeiwd July 3, 1972: revised September 13, 1972.
l l l e authorsarewiththeDepartment
of ElectricalEngineering.Indian
Institute o f Sclence. llangalore 12. India.
1 K denotes t i l e real nrlml~ers;K - ,the nonnegarive real numbers: and Rm. the
~r-dirnen~i~nal
Eurlidean space.
2 A s aninterestingaside.
i t may I!e nutedtllatsuchoperators
are termed
“€lill~ert-8cl1mi~l~
operators” in the matllematics literature [ i .p. 541.
since H E 61.u.
If the RHS 5 0, then (3.4) is satisfied. Wewill show this by
proving that, if RH(!)is a solution of (3.2), t.he integrand on the
RHS I
0. Kow, the integrand on t.he RHS of (3.5) is
81
TECHNICAL NOTES AND CORRESPONDENCE
<&H‘(t)[kia(t)
+
+ R ~ ( t ) A a ( t+) A ~ ’ ( t W ~ ( t ) l z ~ ( t )
[zH’(t)Rrr(t)bH(t)UH(t)- U H ( ~ ) C H ’ ( ~ ~ H ( ~ )
sup
a~(.)ELzc
- dH(thLH2(t)l.
(3.6)
Now evaluate the supremum on the RHS by differentiating this
with respect to u H ( f )and setting it equal t,o zero. Thus, on solving,
we get.
u x ( t ) = $dH-’(t)[sH’(t)RH(t)bH(t)-
(3.7)
CH‘(t)TH(t)l
and, because of (3.1), this U H ( ~ ) ,for which the supremum is attained,
exist,s.
Substit,utingin (3.6) and simplifying, we have
RHS of (3.6)
=
+
+
& ~ ’ ( t ) ( f ? ~ ( t ) RH(f)AB(t) AH’(t)Ra(t)
REFERENCES
+ %dH-l(t)[RH(t)bH(f)- cx(t)l [ b x ’ ( t ) R d t )
J. C. WiUems. “Least squares stationary optimal control and the algebraic
Riccati equation,” I E E E Trans. Automal. Contr. (Special Issue a n L i n e a r
Quad:ptic-Guussian Problem). vol. .4C16, pp. 621-634. Dec. 1971.
DissiDativedvnamicalsystems-Partsand
I 11,”Electron.Syst.
Lab.; Mass.-Inst. Technol., CambLidge. Tech. Rep., Nov. 1971.
stabi1it.y of systems with a
R. F. Estrada and C. a. Desoer Passivity and
state representation,” Int. J. Cohr., vol. 13. pp. 1-26, 197.1.
K. Sundareshan and XI. A. L. Thathachar, “L?-stablllty of l i n y timvarying
systems-Conditions
involving
noncausal
multipliers,
IEEE
Trans. Automat. Conlr.. vol. .4C17, pp. 501-510, +ug,. 1972.
R. E. Kalman, P. L. Falb. and M . A . Arbib, Toptcs In Mathematical System
Theory. ?Jew York:McGraw-Hill. 1?,69.
H. D. Albertsonand R. F . \Vomack Minimum-stat.erealizations of linear
t.ime-varying systems.” I E E E Trans.’Aufomat.Confr. (Corresp.), vol. AC-13.
pp. 305-309,June 1968.
N. I. hkhiezerand I. M. Glazman. Theory of LinearOperatorsinHilbert
Space, vol. 1. .New York: Ungar, 1961.
-.
- cx’(t)l J S H ( t )
=
0,
M.
because of (3.2). Hence, the desired result follows.
b ) Strong posiiivity ( e ) of H : It,is required to prove that
(WIT(.
1, Y
H ~)).
v u H (.) E LZr,
LHS
=
1, w
-
1) 2 0,
r
T E R+, and for some
+d
(uH~
- ),( C H ’ ( * ) ~ a r )( .
~ )UET(
(
)) -
= (UHT(.1, BHT(. )X
where @x(.) =
CH’(.)ZH(.)
E
> 0.
E(UHT(.
(3.8)
1, U H T ( . 1)
+ [ d ~ ( --)
E]uH(.).
Hence, (3.8) holds if H is positive ( e ) with respect to the new output
g H ( . ), Le., with d ~ () .replaced by [dx(.) - e ] .
Q.E.D.
Using t,his lemma,the proofs of the following stability theorems are
straightforward.
Theorem 1 (Linear System): If t.here exists anoperator flri E
n QrM such that ~ l f - 1 E 68.4, the composition L = M G E
8l.U and if the following condit,ions:
a)
dL(t) >
E
> 0,
v t E R’,
ddt)
> 0, v t E R+;
(3.9)
b) t,here exist
real
symmetric
bounded nonnegat,ive-definite
matrices R L ( ~and
) R.u(t), solut.ions of the Riccati equations,
B.L(~)=
$ [ d ~ ( t)
- R s ( f ) b ~ . ( t ) l [ b ~ ’ ( t) Rc~~ (’ (f t)) l
E]-~[CL(~)
+ A ~ ’ ( t ) R d t ) l (3.10)
- [Rdt)Adt)
I
t,his doesnot pose a serious problem,since, L being a linear operator,
startingwithanarbitrary
realization of L(i.e., L E O r ) , it is
possible to arrive at a minimal realizat,ion t.hrough well-established
comput,ational algorithms 161. However, if one were to follow t.his
approach,certain
precautionary measures need be taken.Note
that, since a minimal realization would result in certain ext.ra st,ates
to be removed, which would not, consequently, influence t.he stabilit.y
conditions, it is necessary t o ensure a priori that these statesdo not
contribute,by themselves, toinstability. One sufficient. condit.ion
guarant.eeing this, for example, would be t,o require that f l f and G
are globally asymptotically st.able, which implies limt,,
s,w(f) = 0
and limt,,
m ( t ) = 0. It. is to be emphasized t.hat t,he stability
theorems should be applied only after reducing L to a minimal form.
and
B s f ( t ) = ;d.w-’(t)[cAr(t) - R.w(t)b~(t)l
[b.w’(t)Rx(t) -
~.w’(t)l
+ A~r‘(t)R.ri(t)l (3.11)
- [R.ri(t)A.dt)
are satisfied, then t.he syst,em described by Fig. 1 is LTstable for all
G E EA n 6r.w and N = E.
Theorem 2 (AVonZinear System): If there exists an operat.or L M E
89~4n bru such t,hat,X - 1 E $ E A , L = MG E 8 r u , and the following
conditions:
Comments on “A Simplified Irreducible
Realization Algorithm”
R. D. GUPTA
.4ND
F. W. FAIRMAN
In the above paper,’ Chen and Mital have provided a theorem
t,hat enables a reduct.ion in the size of the matrices used in Ho’s
algorithmforminimal
realizat,ion. This is achieved by utilizing
theadditional informationcontained in the degres of t.he least
common denominators of the rows and columns of thetransfer
function matrix G(s). Theorem 1’ is shown here tobe asimple
consequence of known resu1t.s for t.he realizat.ion of single-input
multi-output. and single-outputmulti-inputsystems.
The proof
following.
given here isbased on Lemma 4 [I], which is restated as the
L a m : If G(s) is a st,rictly proper rational matrix and t.he monic
polynomial y ( s ) is the least. common denominator of t.he entries,
gij(s), in G(s), t,hen ~ ( s is
) the minimal polynomial of the minimal
realization of G(s).
Consider a single-output multi-input. system. The transfer function matrix G(s) can then be written as(using the aut,hors’’ numbering of equations where possible)
0
G(s) = [ql(s),gz(s); . .,gp]
=
L ~ ~ L I S - ~
(1)
i=l
a) d ~ ( t >
)
E > 0, tt t E R+;
b)there exists a realsymmet,ric bounded nonnegative-definite
matrix R L ( ~ solution
),
of (3.10); and
c) J!I = p~
z, E R’, z E
3 &7) = z(t I 4 v ) l dzJ < P ,
+
are satisfied, then the syst.em described by Fig. 1 is &stable for all
G E EA n ~ I and
M
M E XOU.
Remurk: The condition of the theorems that L = MG E 6rJr
Manuscript received July 31, 1972. This work was supported by the National
needs some explanation. Alt.hough M,G E 61 =) L = MG E 8 1 , ~Research
Council of Canada.
R. D. Gupta is a CommonwealthScholaratQueen‘sUniversity,Kingston.
it does not follow, in general, that, M,G E 6 r . v =) L E Q r u . But Ont.,
Canada, on leave from lladhav Engineering Col!ege, Gwalior,.India.
A simple realizaticn of L may be obtained by defining r L ( t ) = [rG(t)zx(t)l’
vcW = u,w(l), ~ ( t =) u d ) , and YL(O = u d t ) .
J
F. \V. FairmaniswiththeDepartment
of Llectrlcal Engineerlng,Queen’s
University.Kingston,Ont..,Canada.
1 C.-T. Chen and D. P. Mltal, I B E E T r a n . Automat. Conlr. (Short Papers),
vol. AC-17, pp. 535-537, Aug. 1972.