RGA

1. 1966 CORRESPONDENCE 133
where
From ( 2 ) and (6) wecomputetheesti-
mate of s,,, given (butnot Ai,tobe
= {estimator of the form (6! {
.{ LUNV estimate of ~ " - 1 1
= R,,,n-~G?~. (15)
Similarly,we calculate the error variance
d, = L[& - L ) & - &)TI
as givenby (11), notingfrom (2j and (3)
that
Tocompletethetheorem, me dehnethe
conditional random variable A:,= ( . f n - ? z ) ,
and introduce the dataat time t, in the form
A& =zn-..I 2 r L =-.I "A?". Since
it fo!lows (Theorem l j that t_he LUNY
estimate A?,*, given 5,-1 and A&, only de-
pends uponA$.. Thus
EquaLions (9) and (12j follow from (16)
It is interesting to note that the forms
of the estimation and covariance equations
changewhenoneseekstoobtainthecon-
tinuousestimatorbylettingthetimebe-
inPfeiifer [j] it is shownthat the differ-
tween data samples go to zero. In particular,
ence equation for the discrete estimate con-
verges to a linear differential equation which
cont?ins thedataderivativebutnotthe
data itself. The derivationof the continuous
case does not depend upon introducing the
notion of continuouswhitenoise,indeed,
(4)and (5) make no sense if F(t, s) contains
a Dirac delta function.
CARLPFEIFFER
Space Guidance Theory Group
Jet Propulsion Lab.
Pasadena, Calif.
REFERESCES
[l] R. E. Kalman. "A new approach to linear filtering
and nrediction Droblems." 1.Basic EnRra. DD. 35-
i j - k a r c h 1960.
R.' E. Kalman, and R. S. B X ~ .*Kevnresults in
linearfilteringandpredictiontheory, J. Basic
Engrg., pp. 95-10?, March 1961.
linearpredictionandfilteringtheory;.Research
R. E. Kalman.Kewmethodsand,resultsin
Tech. Rept. 61-1. June 1961.
InstituteforAdvancedStudy,Baltimore. Md.,
C. G . Pfeiffer. "Seouential estimationoi correlated
_ _ ..
.. ~
. .
ctnrhastirvariables.' l e t ProuulsionLab.. P a s -..~-...~~ ~ ~ ~ ~~~~
IS] C. G. Pfeiffer. Continuousestimation of x-
dena.Calii.. Tecb Rept,-32-4k. July 196j.
quentiallycorrelatedrandomvariables."Jet
PropulsionLab.,Pasadena.Calii.,Tech.Rept.
[6] J. I,. Doob. Slochaslic Processcs. S e w Vork: !Vile?.
32-524, October 1963.
1953.
On a New Measure of Interaction
for Multivariable ProcessControl
IKTRODUCTIOS
Thiscorrespondencedescribes an inter-
actionmeasureformultivariablecontrol
developed to overcome theoretical and prac-
ticaldeficiencies of thematrixrepresenta-
tion as a toolfordesign of systems. A lin-
earized,time-invariant,multivariablepro-
cess described bya square gain matrix input-
outputrelation is assumed.Simpleexten-
sions of themeasureandits use to handle
dynamics,nonlinearity,andnonsquare
matrices exist. It is usually better, however,
to use the unmodified measure to determine
the initial design structure of control allow-
ingdynamicbehaviorandnonlinearityto
determine later details.
The control problem discussed is shown
inFig. 1 with mj and c; reconnectedwhere
the process is described by the matrix-vector
relation Ac=@Awz with a a processgain
matrixwhoseelementsareindicated as
+if. However, the matrix by itself is a poor
tool for classifying control properties of the
system. I t is highlydependentonscaling
(see Property 2) and choice of units. For in-
stance, the two matrices
10 10 10 10
(0 10) and ( 9 10)
having radically- differentcontrolcharac-
teristics, can be scaled by the same change
of units to apparently similar matrices
(: I:) and lo)
0 9 1
Scalingbecomesmoreconfusingwithlarge
systems.Conventionalproperties of ama-
trix, including norms, the determinant and
the eigenvalues also depend on scaling or on
theordering of rows andcolumnsinthe
matrix.Andthepropertiesthataremost
relatedtocontrolsuch as the eigenvalues
are also most dependent on variable scaling
and ordering,makingthemunsuitable as
measures of process structure.
Fig. 1. Controlledmultivariable system x%ith a n
input-outputpairdisconnected fromcontrol.
Theterminteractionarosebecausein-
dustry has often found it desirable to control
the multivariable process as if it were made
up of isolated single variable processes. The
resultingloops"interact."Xninteraction
measureattemptstoanswerthequestion:
Map 26, and October 8. 1965.
hianuscript received October 16. 1964; revised
How is themeasuredtransferfunctionbe-
tween a given manipulated variable mrz, and
a given controlled variableci affected by the
completecontrol of all othercontrolled
variables, as in Fig. l ?
DEFINITION
The measure taken to answer this ques-
tion is theratio of twogainsrepresenting
first the process gain in an isolated loop and,
second, theapparent processgainin that
sameloopwhenallothercontrolloopsare
closed. In the first case, the gain between ci
and mj is [(Aci/amj)IAwzk =O when k # j ]
=4;i, the openloopgain.Inthesecond
case, thesteady--stategain is
[(Aci/Aml)IAck = 0 when 0 k # i] = l/+ji-l,
where &,-I is anelement in theinverse
matrix.Noticethattheabstractionrepre-
sentedbythesegainsdoesnotdependon
the structure chosenfor the contro:lerex-
cept that it be effective in enforcing steady-
state values of c equal to thedesired values.
The ratio of these gains defines an array-11
with elements:
p..A ,$,..b.-1
i>- ' 1 . 1 % .
Thisdefinition is relatedinstatementand
purposetothe"conditionnumber"sug-
gested by von Neumann and Goldstine [2].
Otherquitedifferentmeasureshavebeen
proposed 131-[5]. Theauthorwas re-
centlyreferredtorelatedwork of Fiedler
and Ptak [6].
PROPERTIES OF THE MEASURE
The following properties are easily shown
1) A n y row or column sums to one.
2) The measure of a matrix is invariant
underscaling.(Scaling of a matrix
correspondstothemultiplication of
thematrixbytwogeneraldiagonal
matrices D' and D so that the scaled
matrix becomes 0'=D'aD.)
3jlThe only effect of altering the order
to be true:
of rowsorcolumnsin 0 is to intro-
duce the same alteration of order in
1M.
Measures much larger than one imply
a "nearly" singular gain matrix.
Thesubmeasure of an effectively
isolated subprocess is the sameas the
measure of thesubprocess. All other
elements of rows andcolumns corn-
mon to the submeasureare zero.
The measure shows up in calculations
of the changes introduced ina control
system caused by changes in process
parameters(andnonlinearit>-);for
instance, the relation
dpij = p<t(d4<1/+ii+d41<-'/4ji-').
The measure is an approximate mea-
sure of its own sensitivity.
Properties 7 and 8 relatetointeraction
induced dynamic behavior. Both properties
are generallytrueand holdrigorously if
integralcontrolactiondominates all other
process and control dynamics. This is not a

2. 134 IEEETR.WSACTIONS ON AUTOMATICCONTROLJANUARY
trivialcasesinceitcorresponds to ''loose''
or "conservative" control and sets the tenor
of practical control.
7 j The transfer function between ci and
m, measuredasinFig. 1 withall
other loops closed will benonmini-
mumphase or unstable if gij is
negative.
8) .-Itwo-by-twoprocesscontrolledby
twonegativefeedbackcontrollers
set loosely as beforein a minimum
loopsystemmustbestable if con-
trollers are assigned to variable pairs
with positive measure. The same sys-
tembutbasedonnegativemeasure
d l bestable only if one loop is in
positivefeedbackandthenonlvfor
certainratios of loopgains xvith the
negativefeedbackloopclosed.Sim-
ilar care is required in more complex
systems. Detailed generalizations can
be proven for three-by-three processes
and,therefore,for all systemsin
Ix-hich two-by-tn-o or three-b>--three
subprocessesdominate.
EX.UPLESASD DISCLXIOS
Figure 2 shows five examples of gain
matrices of two-by-twomultivariablepro-
cesses alongwiththemeasurecalculated
fromthem. In Fig. 2, (a),(b),and(c)
are an>- nonzero numbers and6 is a nonzero
number of absolutevaluemuchlessthan
one.Figure2(a)and(b)bothshow pro-
cesseswithnointeraction.Theprocess of
Fig. 2(c) givesa measure which also appears
toshownoninteractionas x-ould any tri-
angular matrix or matrix obtained by per-
muting rows orcolumns of atriangular
matrix.The scaling
scales Figs. 2(c) to2(a) in the limit as 01 ap-
proaches zero. Any two loop control system
appliedstablytoanydynamic process
whosegain matrix is that of Fig.2(a) will
alsobestable -hen appliedtoFig.2(cj.
Decoupling for this class of effective1)- non-
interacting process amounts to simple feed-
forwardand is alsoneverdestabilizing.
Stability is unchanged because in each case
no new loop is introduced by the interaction.
The process of Fig. 2(d)showsalmost
identical effects of eachmanipulatedvari-
ableoneachcontrolledvariable.Forthis
reason,independentcontrol of eachcon-
trolled variable is difficult to achieve prac-
tically. Sear singularity is shown by Prop-
ertv 1. Existence of negativemeasure is a
necessan-characteristic of nearlysingular
systems by Properties 4 and 1. Other prob-
lemsaremadeclearfromthemensureby
Properties 6, 7, and 8.
I n contrast to Fig. 2(d), Fig. ?(e) shows
aprocess in whicheachmanipulatedvari-
ablehaslargebutdifferingeffects on each
controlledvariable.Controlactions on in-
dependent tu; donot cancelone another,
and the system is moreeasil>-andinsensi-
tivel>- controlledLvith or without decoupling.
The character of thematrix is reflectedin
aninteractionmeasurehavingpositive
values less thanone.
Fig. 2. Table of gainmatrices a i t h theirinteraction
processes.(c) an interactingprocess whose inter-
measuretables for: (aiand(b)noninteracting
action is iree oi feedback paths, (d) a highly inter-
acting process with considerable control difficult,-.
and (e! a highly interacting process. vhose inter-
action iseasily decoupled.
Co~cLusros
The basiccharacter of thesetypes of
interactioncarriesoverintoa-by-npro-
cesses. The measure cat1 serve as a design
tool to selectpreferredprocesses andto
specify the control structure once a process
is selected. This control structureis specified
by a one-to-onepairing of thecontrolled
andmanipulatedvariablesas a basis for
control. Each pair may be closed in a single
loop in a minirnum loop s p t e m or more re-
fined decoupling may be used. In any case,
theprocedureamountstopicking a pre-
ferred principal diagonal to the matrix. The
examplesandpropertiessuggestthatthe
measurecorrespondingtothepairedvari-
ablesbepositiveandas close tooneas
possible. Numbersnegativeormuchlarger
than one are to be avoided and large nega-
tive numbers are particularly undesirable.
This design procedure is hardly complete
but it is simple to use, and it usually gives a
uniquedesignevenwhenstatedinsuch
qualitative terms. It is supported by Prop-
erty 8 as well as the intuitive argument and
hascorrelatedperfectly uithstandardde-
signs of actualindustrialprocesses.The
author hopes to be able to free the support-
ingexperimental data forpublication a t
some time i n the future.
EDGARH. BRISTOL
Foxboro Company
Foxboro, IIass.
REFERESCES
[l] F.B. Hildebrand, Mt-lhods nf.4pplit-d J i a f h r ~ ~ ~ a f i c s .
[ 2 ] J. ran Seumann and H. H. Goldstine. Xumerical
EnnlevoodCliffs. X. J.: Prentice-Hall, 1952.
in-erting of matrices of highorder. Proc. Amt-r.
Mafh. Sac.. vol. 2: ~p.~188-202.1951.
[3] 11. D. hIesaro-lc. A measure of interaction
and its application to control problems.' Systems
ResearchCenter.CaseInstitllte of Technoloy?-.
[41 R. Brocl;ett. "The control of linearmultivariable
Cleveland.Ohio,Rept. A-6-60.
systems. P1l.D. thesis. Case Institute of Technol-
[jl P. C. Clliu and. C . R. Webb. ".Analog computer
oxy. Cleveland. Ohio. 1962.
studv of amultl-ariablecontrolsvstem. Coelrol.
161 M . Fielderand V. Ptak. 'On matrices xvirh now
rX3. no.49. pp. ii-80.
cipalminors." Chekhoslmalskii.fale~nalicheskii
positiwoff-diagonalelementsandpositi-eprin-
Zharnal, vol. 12, PP. 382-400. 1962.
Dead Beat Response of Higher-
Order Servo Systems by Single
Switching Operation
Smithsuggested a method [I] for ob-
taining dead beat responseof lightly damped
second-orderservosystemstostepsignal
inputs. 111this method, the input command
has to be suitably controlled, which can be
effected bysingleswitchingoperation[2],
[3]. By such controlof the input signal it is
possible toobtaindeadbeatresponse of
higher-ordersystemsprovidedthesystems
havesuitabletransferfunctions.Thepur-
pose of thepresentcorrespondence is to
investigatethesetransferfunctions of
higher-ordersystems.
Co:lsider a closed loop transfer function
of annth-ordersystemas
bo--
hnSn+bn.lSn-'+ . . . +bS +bo
(1)
Ivhere the coefficients h are all real constants
and n arepositiveintegers.Thetransform
of the output response C(S),for a unit step
escitation function and for zero initial con-
ditions is
11-eshall assume that the roots of the equa-
tionB(S) =O areall distinct. LetS1S2.. .S,
bethe n distinctroots.Then C(S) canbe
expanded in the follo-ing form
wherethenumerators of thepartialfrac-
tions in (3j areconstants.Theoutput re-
sponse Cit) is given by
~ ( , t )= 1 + .A;exp C-S,~). (4)
The expression for the qth derivative of the
output can be written as
i-L
Sincetheinitialconditionsarezero, it fol-
l0n-s from (5) that the following relation is
satisfied
n
SPAi = 0 (6)
r - 1
for all valuesof q lying between 1 to n -1.
Sow, if at anyfinite time, all the deriva-
tives for the previous values of q are to at-
tain zero values simultaneously. all exponen-
tialtermsin (5) musthavethesame
value at that time. Since the roots are dis-
tinct, this can occur when the real parts of
all therootsareequal,that is, when the
roots lie on a lineparallel to the imaginary
axis of the complex plane. However, for ob-
tainingdeadbeatresponsebythesingle
switchingoperation it is required that all
the previous derivatives should attain zero
valuesbythetimethe firstderivative
Manuscript received August 2, 1965.