674
IEEE TRANSACTIONS ON .IUTOMIATIC CONTROL, DECEMBER
1973
the interval (O,t] or, equivalently, the triple ( A , B , C ) is controllable
E C, the rank of Q( A,B,C)equals n where
if, for each
+
Q(A,B,C) =
The center matrix in (4)will be positive semidefinite if and only
if conditions a), b), and c) are satisfied [ l l ] .
I n addition, ( 3 ) will be a.ymptotically stable if 6-l(xt)# 0 along
any nontrivial solution of (3) [9].
To show asymptotic stability, note that the
solution of ( 3 )possesses
the representat.ion
[Q1',...,Q1n,Q*2,...,Q~n,.'',QnnI
and
911
=
C
=
AQP
+ BQ,-'&
QF
forj = Oorj> k
= 0,
(Le., cont,rollable in the sense of Kirillova and Churakova [6] ).
The algebraic crit.erion for conlplete conbrollability given in
Definition 1is equivalent t.o [7]
X(t,o,+)
X ( t - s)CC'X'(t - s) ds
rank
=
n
DeJninition 2: System (1) is point,wise complete on the interval (O,t]
or, equivalent.ly, the pair ( A , B ) is pointwise complete if the rank of
R(A , B ) equals n where
X'(t
S_O,
X(t -
s -
r)BB'X'(t
-
s -
T )
=
1'
(5)
exp (- AT)CC' exp (- A'T)
T > 0,
u ( x ( t ) ) = -C'W-'x(t)
where
2x'(t)W'-'x(t)
+
I:,
Improved Conditions for the L?-Stability
of Nonstationary Feedback Systems
CC' W-1.
A straightforward calculation shows that verifying that V ( x ) L'
Lyapunov functional for t.he closed-loop system
+ BX(t -
is equivalent to verifying that
t-,(xt) = 2x'(t)Wx(f)
+
T )
S
(2)
ll~
AI'x(t)
+ B'X(t
~
11. I(. SUNDA4RESHASA X D 11. A . L.THATH.4CHriR
Abstracf-Concerning the Lrstability of feedback systems containinga linear time-varying operator,some of the stringent restrictions imposed on the multiplier a s well as the linear partof the
system, in the criteria presented earlier, are relaxed.
x'(s)Qx(s) ds
The authon' paper [ l ] presents new criteria for the Lrstabiliry of
linear and nonlinear systems containing a linear time-varying operator G in L2, in an otherwise timeinvariant negative feedback loop.
is a Lyapunov functional for
i(t) =
(7)
x'(s)W-'QW-'x(s) ds
is t.he Lyapunov functional associated with the st.abilized syst.em.
(See [9] or [ l o ] for detailsregarding Lyapunov functionals and
their derivatives.)
.Prooj: Since ( A , C ) is controllable, W > 0. Let A^ A A -
X([) = A^X(t)
T ) B $ 0.
REFEREXCES
+ CC'
exp (- As)CC' exp ( - A ' s ) ds,
-S-
(6 1
111 T u . E. Osipor."Stabilization
of controlsystemswithdelays."
Di-feerentid'nve L-rarneniyn. rol. 1 . no. 5 , pp. 605-618. 1965.
121 D. It-,,
Ross. "Optimal control ofs,-stems describedby dicerential-difference
equations. Ph.D. dissertation.Dep.Elec.Eng..StanfordCniv..Stanford.
Calif..1967.
linear differentialdelay
[3] G . J. Kazaroff."Stabilityandstahilizationof
. systems." I E E E Trans. Automat.Contr. (Tech.XotesandCorresp.),vol.
AC-18. pp. 317318, June 1973.
F u n k . Ekrac., vol. 11,
[A] D. L. Lukes. "Stahilizability and optimal control."
pp. 39-50. 1968.
[5] D . L. Kleinman. ";\n easy u.ey t o stabilize a linear constantsystem."
I E E E Trans. dufomat. Contr. (Corresp.). vol. XC-15. p. 692. Dec. 1970.
161 F. XI. Kirillova and S. V. Churakova. "The problem of controllabilityof
linear systems with after efiect." Diflerential'nye 1.rarnrniya. vol. 3. no. 3.
pp. 436-446. 196i.
Kith time-delays." in
[7] D. H. Chyungand E. E. Lee."Optimalsystems
Proc. 1966 IF.4C C o p g r .
[ S ] T . 000.J. I.amssskx. and T . Sawarapi. "On t h e controllahility of systems
with a time-var>-ing delay."Int. J . Contr.. 1.01. 14. no. 5 . pp. 975-987. 1 9 i l .
[9] X. X. Krasox-skii. Stability of Motion. Stanford.Calif.:StanfordUniv.
Press.1963.
[ l o ] J . Hale. Functiond Di-ferentif Equations. Xew X-ork: Springer.1971.
[ l l ] E. Irrein$er and A . Jameson. Conditions for nonexativeness of partitioned
matrices I E E E T r a n s . A7domaf. Cordr. (Tech. Sores andCorresp.).vol.
AC-17. pp. 14i-148. Feb. 1972.
then (1) is stabilized by the cont.ro1law
V(Xt) =
d8.
Hence the asymptotic stabilityof (3) is etablished R-hich, in turn,
establishes the asymptotic stabilityof (2).
This complet.esthe proof.
* den0t.a the Penrose-Moore pseudoinverse.
' Theorem: If ( A , C ) is controllable and ( A , B ) is pointwise complete,
and the conditions a ) Q 2 0,b) B'Q*Q, and c) Q - 4 WB'Q* B W 2 0
are satisfied where
=
T)B'Q(S)
R ( i , B ) = 12.
ds = n
f o r O < t < 03.
In the following theorem,
W
-
X'(t - s) is the fundamental solution associated with (2).
Assume that (6) is not satisfied. Then the controllability ofAt.he
triple ( i , B , C ) is contra$cted since ( A , C ) controllable =) ( A , C )
controllable =) rank Q ( A , B , C ) = R .
Assume that ( i )is not satisfied. Then the pointwise completeness
of ( 2 , B ) is cont.radicted since ( A , B ) pointKise complete =) rank
(i.e., pointwise completein
the sense of Ono, Tamasaki,and
Sawaragi [8] ).
The algebraic
criterion
for point.wise completeness given in
Definition 2 is equivalent. t o [SI
and
S
X'!t - s)C $ 0
R ( A , B ) = [B,AB,...,A"-'B]
Q
X(t -
Substit,ut.ion of ( 5 ) into (4) clearly implies p,(xt) # 0 along a
nontrivial solution provided
f o r o < [ < 03.
rank
x(f)+(o)+
-
7).
i3 )
3Ianuseript received April 27. 1973.
Differentiating Vl(xr)along the solutions of (3)yields the quadratic
The authors are withtheDeoartment
form
India.
560012,Institute~of~Science.
Bangalore
of ElectricalEngineering.Indian
675
TECHNICAL NOTES A N D CORRESPONDENCE
These criteria permit a new class of linear causal operat.ors X in Lfe
with nonst.ationary kernels to be used as multipliers. However, the
criteria in [I] require G, M, and X - 1 t.o belong t o a class $9~4of
linear causal operators in L e defined by the following condition: if
H E Q m , then Hz(f) =
~ ( ~ , T ) x d( rT )tC x(-)E Lpe, R7it.h the
) 0 +f T > t and
kernel sat.isfying the twin condit.ions h ( t , ~ =
I h ( t ? ~ dt
) ) d~r < a.I t has of late been noticed that t.he latter int,egral
condition renders the class EA to be of limited use in practical situations, since it excludes even t.he simple convolutions. The purpose
of this correspondence is to remove this stringent requirement from
t.he criteria of [ I ] .
A careful study of the st,abilitycriteria of [ l ] reveals that t.he
condition G, 31, and 31-1 E $EA is imposed only to ensure that
these are operat.ors in Ln with finite gain. Let us now define cert,ain
new classes QEB and QBC of linear causal operators H in L,, as follows.
1) if H E *RE,
then
.fr
b
Hz(t)
=
where F = F(t), and w
=
E[w(t)]= 0, E[w(t)wT(r)]
= Qs(t -
dj
rj+l
> t,
=
wj+l =
Q(f).
< .. . < t . 5
~ T by
+ vi
(2 1
Ntj+lJj)Tj
-I;
+
wj+l
*(tj+l,g)w(n)dg
(7)
where f w j ) is a discrete white-noise sequence with covariance f A,G)),
(1)
'$ T
AjXj
=
where zj is an mj vector, xj = r(tj),and { v,) is a zero-mean whitenoise sequence t,hat, isstatisCically independent of w(f) and r(@.
The problem of interest in this correxpondence is that of obtaining
filtered estimates of x. By introducing the state t,ransit,ion matrix
corresponding to F ( t ) , we can replace ( 1 ) by
~ ( ~ , T ) zd~
( T+f
) X(.) E Lz,,
h(f,T) = 0
=
Q
T),
The model is observed at. discrete times ti, lo < tl
.fr
I=
w ( t )is a whit.e-noiseprocess with
A&'
(21
+ 1)
=
Jy"
~(tj+l,u)Q(u)OT(tj+l,u)
du.
(3)
Evaluation of the @ and A
, matrices generally adds computational
complexit,gto thefiltering problem.
Motivation for conversiy from the continuous formulation ( 1 ) t.o
2) if H E QEC, then H i s defined by ( 1 ) and (2) as earlier, with the thk discrete formulat.ion (1) is that t.he latt.er problem may be solved
additiona1,condition
using square-root filtering techniques (cf. [SI). These techniques are
generally more accurate than is the discret.e Kalman filt,er. In this
correspondence it is shown that t,he Dyer-MeReynolds (D-XI)
square-root filter can be applied directly to the cont.inuous-discrete
system (1) and (2). By writing diflerential equations for t.he D-3f
It may be noted that H E QEB implies I:H~(t)ll6 KIII exp ( - k ~ t )
data array matrix, one circumvents the problem of computing
@ ~ ( t ) '(where
:
denote the &norm and * denotes convolution),
and A U + which are used by thediscrete square-root filter.
and hence H is an operator in Lz wit.h finite gain. Further, H E QEC
implie's (see Dunford and Schwartz [2]) ! , ; H z ( - ) 6
' / ( S l L \ T ? ) : /I%(-):],
T H E COKTISCOVE-DI~CRETE
D T E R - ~ f c R E n o L D s FILTER
and hence H is an operator in Lo with finite gain. With t.his, it is now
The D-1.1 filter [ 11 consists of t.he data array (R,d)where RTR =
simple to realize that the stabilitycriteria [ I , theorems 1 and 21 hold
(COY (a))-l and d = Rr*, X* being the minimum variance estimate of
in t0t.o with the operaton G, ~11,and ~1j-lpernlitt.ed to be members
x (time arguments are suppressed for notat.iona1 simplicity). This
of the class 8 9 ~U ~Q E B U QEC.
filter is most useful for problems that. involve state estimat.es a t
REFEREXES
relatively fen t,imes, for example, after a quantity of nleasurements
[ l ] hI. K. Sundareshan and M . -1.L. Thathachar, "Timedomain criteria for the
have been processed. Except at. these times, it is possible t o deal only
Lystahility of nonstationaryfeedbacksystems,"
IEKt.' Trans. Aulomaf.
mith t.he data array (4)and (5). Each time an estimate is required,
Contr., x.01. AC-18. pp. 8C-81. Feb. 1973.
[Z] N. Dunfordand J. T. Schwarts. Linear O p e r a t o r s , part 1. New Tork: it is necesary to solve a matrix equat.ion Rx = d, and each t,ime t,he
Interscience, 1958, p. 518.
covariance of t.he estimate isrequired, one must compute P =
R-lR-T.l When estimatesand/or
covariances are required at
numerouspoints,
the computationalburden
becomes excessive.
The algorithm follows.
A Square-Root Data Array Solution of the
Propagalion t < t 5 tj+,:
and Ih(t,r)l
and KZ,
6
K I
-
exp (-%(t
for some positive constants KI
(3)
7))
.I,
Continuous-Discrete Filtering Problem
d
- [Rjd] = [ -RF
di
GERALD J. BIERMXN
Abstract-The Dyer-McReynolds [ 11 discrete square-root filtering algorithm is extended to accommodatecontinuous dynamics.
Differential equations are given to represent the time evolution of
the iilter data array. These equations are nonlinear, but it is shown
that the nonlinearities act to enhance thestability of the solution.
where r = -4RQRT.
The differ_ential equation
[ X l d ] = [Rj/dj].
is
+ rR!rd]
init.ialized a.t times
ti by
t = tj
L7pdating (at time tjll): Choose an orthonormal t.ransformation
Tj+lsuch t.hat
INTRODUCTION
In this correspondence we consider the linear dynamic model
d
--5 =
clt
F-5
+w
+
+
l ) R y - T ( j 1) and the mat,rix on the
where cov ~ j += ~Rp-l(j
right is upper triangular. Comput.ation of the triangularization (5)
is not discussed here. References [ l ] ,[ 3 ] - [ 5 ] explain in detail how
the computation is performed.