782
IEEE T I M N S - A ~ O E ; ~ON AUTOELTIC
a s special cases of a randomsampling
scheme.
Recently, JuryandKanlenumerated
and classified the various methods
for
studying linear sampled-data feedback systems in xhich the width of the sampling
pulses is not negligible. The purpose of
this correspondence is topoint out. that
work has been done [l] on an extremely
general type of sampler which opens and
closes for random time intervals, so that a
sampler is characterized by two interval
probabilitydensityfunctions;
one for the
random time interval for which a sampler
is closed andthe ot,her forthe random
time interval for d i c h a sampler is open.
The advantage of suchaformulation
lies
in the fact that all det.erminiitic types of
sampling become special cases [2] of the
general theory, since deterministic sampling
schemes give rise to interval density functions which consist of Dirac delta functions.
As a matter of terminology, the generalized randomsamplerhas
been called a
randomgate. This is consistent withthe
terminology used in electronics where a
gate is simplya device, which when triggered admits its input as its output.Anot.her
familiar example is that of an ideal diode.
Theoutput. of an ideal diode consists of
the positive portions of itsinput signal.
Thus,the
word sampler is generally reserved for an impulse sampler.
Looking throughthe long list of references presented byJuryandKan,lit
is
apparent
that
the
design of optimum
multivariable
reconstructors
has been
largely neglected. However, in the case of
multiple random samplers with stationary
randominputs,the
optimum reconstruct.ion of multiple randomly-sampled signals
can be found in [a].
In time-multiplexing problems, multiratesampleddata systems are oftenencountered. When t.he widths of the sampling
pulses are
not
negligible, the analysis
becomes unwieldy. Thesituation
where
the random samplers or gates are triggered
by a master or parent sampler has been
considered [4] together with the optimum
reconst,rrction of suchrandomlysampled
signals. Bearing in mind that deterministic
time-multiplexing problems are simply
special cases of random time-multiplexing
problems, it is clear that.the t.heory of
random sampling is widely applicable t o
the stat.istica1 design of variomtypes of
sampled-dat,a systems.
I n conclusion, it is worth stressing that
since deterministic sampling schemes form a
subset, of random sampling schemes, fut.ure
research in the
area
of sampled-data
systems would no doubt tend toconcentrate
on the latter.
S. G. Loo
Dept. of Math.
Atonash University
Clayton, Vic. 3168, Australia
REFERENCES
[l] S. G . Loo, “Spectral a n a l p of randomlygated
random
signals
Internatl. J.
CoFpoZ, vol. 6. pp. 22b-239, September
1Yti7.
,
“Spect.ra1 densities,, of mated staI n b m t Z . J.
tionary
random
signals
Conlrof: vol. 7 pp. 73-79 ’January 1968.
- Optimbm linear multivariable systams‘ withrandomly-sampledsignals,” J.
Franklin Inst.,
287, pp. 471-481.
June 1969.
“Spectral propertiesandoptimum
reco&uction
of .rando,@y gated
st,aI E E E Trans.
tionary ra.ndom sLgals
Automatic Control (Shdrt Papers). vol.
AC-14, pp. 564-567, October 1969.
1701.
COATROL,
DECEMBER
1969
for all u’,u2 and for all zio and to. Otherwise, it is controllable. If ( 3 ) does not hold
thenthe system
for any i, i = l,---,n,
is completely controllable.
The proof of this lemma is obvious. I n
particular, u2 can be considered as a result
of a perturbation in cont.ro1 ul. This is
similar tothe continuous variation considered by BBlanger. u 2 is defined as the
following
U * ( t ) = u’(t)
+q(t).
(4)
Lemma 2
Controllability and Sensitivity
Abstract-Controllability is defined with
respect to sensitivity of performance index
for variations in control.
I.
The state zi of system (1) is uncontrollable if and only if the firstvariation
6Vii of Vij vanishes for all ui and for all
z i p and h. However, it&odd
be noted
that the first variabion vanishes if uf is an
extrema1 control for the index of performance considered. So one should choose a
nonextremal cont.ro1 to testthe
controllability.
The proof of this lemma is also obvious.
INTRODUCTION
111. LIKEAR
SYSTEMS
The problem of sensitivity of opt.ima1
controlsystemshas
received considerable
attention in recent rears [l]. Most of the
papers in thisarea dealwith sensitivity
with respect to paramet.er variations. Some
work has been done regarding sensitivity
wit.h respect to control variations. BBanger
[2] discusses two problems: the computatiou of performance index variation of an
optimalcontrol syst,em dueto a control
variation, andthe tolerances that. should
be placed on the control to assure that the
target manifold will be reached. The
formeraspect
is considered in this correspondence, and the relation with controllability of a system is shown.
The linear system considered is of the
form
2(t) = A(t)x(t)
z(t0) = 1:o.
STATE
COXTROLLABILITY
(1)
If the control is u2((t),given by (4), then
‘ ( t ) = z‘(&) -k
The system considered is
Vi(Zi0,Ui) =
1:
Li(X.) dt. (2)
The subscript i refers to a particular
stateandthe
superscript j represents a
given cont.ro1vector. The interval [ h , T ] is
finite.
Lemma 1
The state zi of system (1) is uncontrollable if and only if
vir
= vc2
Manuscript received July 8, 1969.
L:
& ( ( t , T ) B ( 7 ) 7 ) ( 7 ) dT.
(7)
(1)
where 1: is the state n vector and u is the
control m vector. A posit.ive scalar function
Li(zi) of state zi is chosen and an index
of performance V i j is defined BS
vij P
given by
T W O LE?.iXAS O N
2 = f ( x , u ) , x(&) = 1:o
(3)
(5)
The solution of ( 6 ) with control ul(t) is
given by
z z ( t ) is
11.
+B(t)U(l),
The respective costs with controls d ( t )
and u2(t) are given by
v~Y =
1:
Li(z2) dt
(Sa)
and
Substituting for xi1 and x? from (6) and
(7) into (Sa) and(8b)and
with little
manipulation, we obtain
783
CORRESPONDEKCE
Partial Fractiwn Expansion of a Special
Tmnsfer X&ix
CONSTRUCTION O F A CONTROLL4BLE
REALIZATION
Therefore,
The method given' uses Z ( s ) e q a n d e d
into its partialfractions
Consider a matrix M(s) of the special
form
Z(s)
If xi is uncontrollable, by Lemma 2,
+ ... + X
6TTG =
0.
(11)
From a well-known lemma of calculus of
variations [SI, (11) implies
/I:
[ @ ( t , ~ ) B 7( r( 7) ) ] idr
=
0.
IV. CONCLUSION
It is shown t,hat controllability is equivdenttothe
fireborder sensitivit.. of an
index of performance depending on a
particular st.ate.
8. SARMA
SAHJESDR.4 x. S l N G H
Dept. of Elec. Engrg.
Indian Institute of Science
Bangalore 12, India
REFERENCES
[11 31. Sobral, "Sensitivity in optimal control
systems," Proc. I E E E , 001. 66, pp. 16141652, October 19I3,S.
[ 2 ] P. R . BBlanger, Some aspects of control
tolerances and
first-order
sensitivity in
optimal control systems," I E E E Trans.
A u t o n ~ ~ t iControl.
c
vol. 4C-11. .
DII.
* 7;-83.
January 1966.
'
131 G . A . Bliss, Lectures on Calculus of V a r i a Lions. Chicago, Ill.: University of Chicago
Press, 1961.
~
h ( s ) . g ( s ) / ( s - X)'
(4)
where h ( s ) is a polynomial column and
g(s) a polynomial row of degrees less than T.
-4 Taylor expansion gives for Z ( s )
( s - X)'
It requires the computation of the independent. rows in the matrices
(12)
Equation (12) implies uncontrollability of
state xi.
v. 1'.
]
=
Then, these rows are utilized to express t.he
matrices M i as a sum of a number of
matrices wit.h rank 1 (i.e.,product, of a
column by arow).
This form is rat.her
difficult to obtain and a direct method as
explained in [l] is to be preferred.
Using t.he Leibnitz formula, (3) and (4)
are the sameif and only if
hence, the direct realization by (6) of a
transfer matrix mitten as (4).
Trayfeer Matrix Corresponding to a
Jordan. Bloc
COJIPREHEXSIVE
REALIZATION
According to Smith and Frobenius [2],
any polynomial mat.rix can be splitinto
Let. the system be defined by
;i: =
Fx
+ Gu
R
(2 1
y = Hx
where F is assumed to be an upper Jordan
bloc r X r: J(X,r). The corresponding
transfer matrix k given by
Z(S) = H ( s I - J(X,?))-'G = HBG
with [ 2 ]
I
5 =
- J@,r))-I
(SI
1
1
( s - X)
( s - X)'
~~
...
1
~
( s - X)'
h ( s )' g k ( s ) .
U(s) =
k=l
These quantities for the various h(s) and
g(s) have proved to immediat.ely provide
an irreducible rea.lization for Z ( s ) [3], [4].
But. the act,ual compuhtion of these terms
is rather difficult., and it k easier to deal
with the columns (orroas) of M ( s ) in
spite of the requiredreduction procedure.
Let m r ( s ) be the columns of M ( s ) , and
Put
hk(s)
=
mn(s),
gk(s)
=
[ O m .-010.
- -01
with 1 in the kth column. Therefore,
m
M(s) =
Comments on "Irreducible Jordan
Form Realization of a Rational
Matrix"
. In a recent paper', S. P. Pandaand
C. T. Chenhavestudied the construction
of an irreducible realizat.ion for a transfer
matrix of the form
Z(s) = M ( s ) / ( s - X)'
.
0
(1)
where M ( s ) is an n X m-polynomial matrix.
The solution 60 this problem has already
been given in [I] using, as in t.he previous
paper, a two step procedure:
servable) realization;
2) reduction of this model by eliminating
the possible unobsewable
(uncontrollable)parts.
1969.
1 S. P..Panda and C. T.Chen, I E E E T r a n s .
Autonzatzc Control (Short Papers), vol. AC-14.
pp. 66-69,
February
1969.
0
(s
-
1
1
( s - X)
Consider now the general form of (2),
with P = dia.g [Jl,Jp, -,JR]and H and
G partitioned according to F. The transfer
matrix is
--
Z(S)
-
Writing down the columns of H and r0-m
of G as
H ( s I - F)-'G
R
R
Hk(sI - Jk)-'Gk
b=1
1) construction of a controllable (ob-
h,Ianuscript received June 5 ,
hk.gk.
k=l
HLFkGn.
=
(7)
k=1
Hence, using (7) and (6), a comprehensive
realization of (1) is easily obtained as the
composition of each elementary r X r realization of transfer matrix (4).
REDITTIONOF THE REALIZATION
results in
The method proposed byPandaand
Chenl is based on the computat.ion of a
transformation m a t r k T that commutes
with t.he Jordan form of matrix F and
allows the elimination of the unobservable
parts of the realization. This global transformation depends on a number of param-