IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-26. NO. I. FEBRUARY I98I
Multivariable
FeedbackDesign:concepts
for a ClassicallModern
Synthesis
JOHN C. DOYLE eNo GUNTER STEIN. MEMBER.
TEEE
Abtt l4ct-Tbb poper pnesetrtsr prrcdcd OeOg perspecOveon muld_
vrrlabh feedbrdr cootol problemc. It rcvlews tte b.slc kore-foe&oct
deflgr ir tbe lrce of ure.rtalndes-and genenllze ltrwr sirgbltrDnt,
sltrgt€.otrQut(SEO) statencnts and consfalnts of the d6lg! probl€m to
muldlrput, EuldouQut (MIMO) cas€. Tt?o mJor MIMO deslg! ap
prooch€ssrt tten evahrrted In the context of th* re$lts.
I.
Ir+rnooucrroN
Tr HE last two decades have brought major developments in the mathematical theory of multivariable
linear iime invariant feedback systems.Theseinclude the
celebratedstate spaceconcept for systemdescriptionand
the notions of mathematical optimization for controller
synthesis [U, [2]. Various time-domain-basedanalytical
and computational tools have been made possible by
these ideas. The developments also include certain generalizations of frequency-domain conceptswhich offer analysis and synttresistools in the classicalsingle-inpu! 5ingleoutput (SISO) tradition [3], [4]. Unfortunately, however,
the two decades have also brought a growing schism
between practitioners of feedback control design and its
theoreticians.The theory has increasingly concentrated on
analytical issuesand has placed little emphasis on issues
which are important and interesting from the perspective
of design.
This paper is an attempt to expressthe latter persp€ctive and to examine the extent to which modern results .
u1smsaningfulto it. The paper beginswith a review of the
fundamental practical issue in feedback design-namely,
how to achieve the benefits of feedback in the face of
uncertninties.Various types of uncertainties which arise in
physical systemsare briefly described and so-called ..unstructured uncertainties" are singled out as generic errors
which are associated with all design models. The paper
then showshow classicalSISO statementsof the feedback
design problem in the face of unstructured uncertainties
can be reliably generalized to multiinput, multioutput
(MIMO) systems,and it develops MIMO generalizations
Manuscript rcccived lrdarch !], 1980; revised Octobcr 6, 19E0.This
rcscarch was supported by the ONR uader Contract N0001+75-COl.l4.
!y the__QpE undcr Contract ET-78-C-01-3391,and by NASA undei
Grant NGL22ffi-124.
and Rescarch Ccntcr, Honcywc[ Inc,,
. .J. C. Do.yle-isrit:q th-:eSys-tems
Mim-^eapolis,MN 55413 and thc University of California, Bdkebi Ci
94720.
. -9. St€iq.is -yt! -99- 9yrr"--! aqd Rcscarch C.catcr, Honcywell, Inc.,
Minn,capolis, MN 55413 and thc Departmcnt of Elcctrical Enchl,crini
?nd. Com-puterScienccs, Massachusctts Institutc of Tcchnolofr, CamI
bridgc, MA 02139.
of the classicalBode gain/phase constraints[5], [6] which
limit ultimate performanceof feedbackin the face of such
uncertainties.SeveralproposedMIMO designprocedures
4re 6;4mined next in the context of the fundamental
feedbackdesignissue.Theseinclude the recent frequency
domain inverse Nyquist anay (INA) and characteristic
loci (CL) methods and the well-known linear-quadratic
Gaussian (LQG) procedure. The INA and CL metlods
are found to be effective, but only in special cases,while
LQG methods, if used properly, have desirable general
features. The latter are fortunate consequencesof
quadratic optimiiation, not explicitly soughtafter or tested.
for by tle tleoretical developersof the procedure.practitioners should find them valuable for design.
II.
Frrosecr Fur.rpeurvrers
We will deal with the standard feedbackconfiguration
illustrated in Fig. l. It consistsof the interconnectedplant
(G) and controller (iK) forced by commands (r), measurement noise (4), and disturbances(d). The dashed
precompensator(P) is an optional elementusedto achieve
deliberate command shaping or to repres"ol 4 lelrrnity
feedback system in equivalent unity feedback form. All
disturbancesare assumedto be reflectedto the measured
outputs (y), all signals are multivariable, in general,and
both nominal matlematical modelsfor G and K are finite
dimensional linear time invariant (FDLTI) systemswith
transfer function matrices G(s) and K(s). Then it is well
known that the configuration, if it is stable, has the
following major properties:
I ) Input-Output Behaoior:
y:GK(I+GK)-t(r-rr)+(r+GK)-td
(l)
A
e: r-y
: (/+ GK)- |(r - d) + GK(I + GK)- t
n.
2) System Sensitioity [7]:
AH"r - (I+G,K)-tAH"t.
(2)
(3)
In (3), A.F/.,and Aflo, denote changesin the closed-loop
system and changesin a nominally equivalent openJoop
system, respectively,causedby changesin the plant G,
i . e . ,Q ' : G + A G .
Equations (l)-(3) summarizethe fundamental benefits
and design objectivesinherent in feedbackloops. Specifi_
cally, (2) shows ttrat the loop's errors in the presenceof
0018-9286
o r98I rEEE
/81 /0200-0004$00.7s
STEIN: MULTIVARIABLE FEEDBACK DESIGN
DoYI-E AND
which show that rpturn difference maenitudes approximate the loop gains, o-[GK), whenever these are large
-+i P I
compared with unity. Evidently, good multivariable
L --J
feedback loop design boils down to achieving high loop
gains in the necessaryfrequencyrange.
Despite the simplicity of this last statement,it is clear
Fig. l. Staadard fecdback configuration.
from years of researchand desigrractivity that feedback
is not trivial. This is true becauseloop gainscannot
c o m r n a n d s a n d d i s t u r b a n c e s c a n b e m a d e . . s m a l l ' ' bdesign
y
difreturn
arbitrarily high over arbitrarily large frequency
made
inverse
be
the sensitivity operator, or
r"li"g
"small,"
that
they must satisfy certain performance
(3)
shows
Rather,
and
ranges.
ir"o"ioperator, (I+GK)-t,
limitations. A major performance
design
conditions,
and
same
tradeoffs
loop sensitivityis improved under these
commandand disturbance
concerns
for
example,
tradeoff,
G' doesnot stray too far from G.
error reduction [10].
noise
versus
sensor
smallness
of
reduction
error
For SISO systems,the appropriate notion
is evident in (2).
two
objectives
these
between
The
conflict
for the sensitivity operator is well-understood-namely,
range
frequency
large
values
over
a
o_[GK(i<^r)]
Large
o," ,.qoir. that the complex scalar [l +g(i@)k(itl)l-l
also
they
However,
d
small.
r
and
to
due
make errors
have small maCnitude,or converselythat l+g(ia)k(i@)
"passed
is
noise
this
large
because
ttre
to
where
due
co
make errors
4
have large magnituds,for all real frequencies
i.e.,
commands,disturbancesand/or plant changes,AG, are through" over the samefrequencyrange,
SISO
of
significant. In fact, the performance objectives
- ' q- l' r.
( 9)
y:GK( io) lt+ cx( iu) l
feedUact systemsare commonly stipulated in terms of
explicit inequalities of the form
large loop gains can make the control activity
(4) Worse still,
Var(c,r6,
ps(<o)<ll+sjio)k(j@)l
(variable u in Fig. l) quite unacceptable.This follows
from
whereps(<o)is a (large) positive function and arsspecifies
tle active frequencYrange'
u : K I I +G K I - ' Q - n - d )
This basic idea can be readily extended to MIMO
(10)
d).
xG-t(iot)(r-qproblems through the use of matrix norms' Selectingthe
*pa"ttul nonns as our measureof matrix size,for example, Here we have assumed G to be square and invertible for
the correspondingfeedbackrequirementsbecome
convenience. The result'ng equation shows that commands, disturbances,and sensor noise are actually amplil
.
l
r
1
;f (r+ G(ja)K( jo))-' ) small
fied at u whenever the frequency range significantly exceedsthe bandwidth of G, i.e.,for <osuchthat 6[G(i<,r)]<<I
or conversely
we get
(5)
ps(<.r)
<o-lt+G(io)x(iot)l
(l l)
q [ c - ' ( ; r ) ] = - , - ^ l- > r .
for the necessaryrange of frequencies.The symbols6 and
q in theseexpressionsare defined as follows:
One of the major contributions of modern feedbackthej-r !
!S1:
ory is the developmentof systematicproceduresfor con,.i-l
:
(6)
3
max
A1
ol
lll.r ll
*l,l,.cf
ducting the above performance tradeoffs. We are refer:.in
llxll-l
ring, of course,to the LQG theory [11] and to its modern
Wiener-Hopf frequency domain counterpart [12]. Under
(7)
ol A1 2 min ll,4.rll:
^^le"q")
reasonableassumptionson plant, disturbances,and perll:ll-l
formance criteria, these proceduresyield efficient design
compromises. In fact, if the tradeoff between
where ll'll is the usual Euclidean norm, tr['] denotes
command/disturbance error reduction and sensornoise
eigenvalues,and [']* denotes conjugate transpose'The
error reduction were the only constraint on feedback
two o's are calledmaximum and minimum singularvalues
design,practitioners would have little to complain about
of ,4 (or principal gains [4]), respectively,and can be
with respect to the relevance of modern theory' The
calculatedwith availablelinear systemsoftware [8]' More
problem is that theseperformancetrades are often overdiscussionof singular values and their properties can be
shadowedby a second limitation on high loop gainsfound in various texts [9].
namely, the requirement for tolerance to uncertainties'
Condition (5) on the return difference I + GK can be
Although a controller may be designed using FDLTI
interpreted as merely a restatementof the common intui'
"tight" loops yield good models,the designmust be implementedand operatewith
tion that large loop gains or
a real physical plant. The properties of physical systems,
performance.This follows from the inequalities
in particular tle ways in which they deviate from finite'
(
8
)
dimensional linear models, put strict limitations on the
o-lcKl- I <q[ I+GK]<o-[G,(] + I
d
I
I
I
I
)
IEEETRANsAcrroNsoN AUToMATTC
coNTRoL, vor. lc-26. No. l. rrsnurnv
frequency range over which the loop gains may be large.
In order to properly motivate theserestrictions,we digress
in SectionIII to a brief descriptionof the types of system
uncertaintiesmost frequently encountered.The manner in
which theseuncertaintiescan be accountedfor in MIMO
design then forms the basisfor the rest of the paper.
lggl
This statementconfines G' to anormalizedneighborhood
of G. It is preferable over (12) because colpensated
transfer functions have the same uncertainty representation as the raw model (i.e., the bound (13) applies to GK
as well as to G). Still other alternative set membership
statementsare the inverse forms of (12) and (13) which
confine (q)-' to direct or normalized neighborhoods
a b o u tG - r .
III. UNcrnrervnrs
The best choice of uncertainty representation for a
specific FDLTI model depends,of course,on the errors
While no nominal design model G(s) can emulate a the model makes. In practice,
it is generally possible to
physical plant perfectly, it is clear that somemodelsdo so represent some
of these-errors in a highly structured
with greaterfidelity than others.Hence,no nominal model parameterized form. These
are usually the low frequency
should be consideredcomplete without some assessment error components. There
are always remaining higher
of its errors. We will call these errors the ..model unc€r- frequency etrors, however,
which cannot be covered this
tainties," and whatever mechanism is used to express way. These are caused
by
such effects as infinitethen will be called a "representation of uncertainty."
dimensional electromechanicalresonances[16], [17], tim6
Representationsof uncertainty vary primarily in terms delays, diffusion processes,
etc. Fortunately, the lessof the amount of structure they contain. This reflectsboth structured representations,(12)
or (13), are well suited to
our knowledgeof the physical mechanismswhich cause representthis latter class
of errors.Consequently,(12) and
differencesbetween model and plant and our ability to (13) have becomewidely
used "generic,'uncertainty reprepresentthesemechanismsin a way that facilitates con_ resentationsfor FDLTI
models.
venient manipulation. For g;ampl€, a set membership
Motivated by these observations, we will focus
statement for the parameters of an otherwise known throughout the rest
of this paper exclusively on the effects
FDLTI model is a highly structured representationof of uncertainties
as representedby (13). For lack of a
uncertainty. It typically arises from the, use of linear better name, we will
refer to tlese uncertainties simply as
incremental models at various operating points, e.g., "unstructured." We
will assumetbat G,in (13) lsneins 4
aerodynamic coefficients in flight control vary with flight strictly proper FDLTI
systemand that G, has the same
environment and aircraft configurations, and equation number of unstable
modesas G. The unstable modes of
coefficients in power plant control vary with agini, slag G'and G do not
need to be identical, however,and hence
buildup, coal composition,etc. In each case,the amounts Z(s) may
be an unstable operator. These restricted asof variation and any known relationshipsbetweenparam_ sumptions on
G' make exposition easy. More general
eters can be expressed by confining the parameters to perturbations (e.9.,
time varying, infinite dimensional,
appropriately defined subsets of parameier space. A nonlinear) can
also be covered by the bounds in (13)
specific exampleof such a parameterizationfor the F_SC provided
they are given appropriate "conic sector,' interaircraft is given in [13]. Examplesof less_structured
repre- pretations via Parseval's theorem. This connection is desentationsof uncertainty are direct set membershipstate_ veloped
in [4J, [5] and will not be pursuedhere.
ments for the transfer function matrix of the model. For
When used to represent the various high frequency
instance,tle statement
mechanisms mentioned above, 1fuelquading functions
/-(o) in (13) commonly have the propertiesillustrated in
G'( i r) : G(,r<,r
) + AG ( i r)
Fig. 2. They are small (<<l) at low frequenciesAnd increase to unity and above at higher frequencies.The
with
growth with frequency inevitably occurs because phase
o [ A C ( i & , ) ]1 t " ( o )
Vo)0
(r2) uncertaintieseventuallyexceed-r 180degreesand magnitude deviations eventually exceed tle nominal transfer
where /"(.) is a positive scalar function, confines the
function magnilud$. Readerswho are skepticalabout this
matrix G' to a neighborhoodof G with magnitude /,(co).
reality are encouraged to try a few experiments with
The statementdoes not imply a mechanismor struiture
physical devices.
which givesrise to AG. The uncertainty may be causedby
It should also be noted that the representationof uncerparameter changes,as above, or by neglecteddynamics,
tainty in (13) can be used to include perturbation effects
or by a host of other unspecifiedeffects.An alternative
that are in fact not at all uncertain. A nonlinear element,
statementfor (12) is the so-calledmultiplicative form:
for example, may be quite accurately modele4 but becauseour designlsshniquescannot deal with the nonlinG'( jr) - | I + t( iot)] c(,r")
earity effectively,it is treatedas a conic linearity [14], U5].
As another example,we may deliberately chooseto ignore
with
various known dlmamic characteristicsin order to achieve
ol L( j0,)) 1l-(o)
Vco) 0.
( l3) a simpler nominal design model.
'7
AND STEIN: MULTIVARIABLE FEEDBACK DESIGN
form of the standard Nyquist criterion,r and in MIMO
cases,they involve its multivariable generalization [18].
Namely, we require that the encirclement count of the
map det[.I+GK(s)], evaluatedon the standardNyquist
D-contour, be equal to the (negative)number of unstable
open loop modesof GK.
Similarly, for requiremetl 2) the number of encirclements of the map det[1+G'K(s)] must equal the (negative) number of unstable modes of G'K. Under our assumptionsot G', however,this number is the sameas that
of.GK. Hence,requirement2) is satisfiedif and only if the
number of encirclementsof det[/+G'K(s)l remains
unchangedfor all G' allowed by (13). This is assurediff
det[1+ G'K] remains nonzero as G is warped continuously toward G', or equivalentlY,iff
o< q[ /+ | r+.21s;] c(J)^K(J)
]
LOGFREOUETICY
Fig. 2. Typical behavior of multiplicativc perturbations.
Another important point is that the construction of
l-(<o) for multivariable systemsis not trivial' The bound
'#to-". a singleworst caseuncertainty magnituds attlicable to all channels. If substantially different levels of
uncertainty exist in various channels, it may be necessary
scale the input- output variables and/ or aPply
transformations[15] in such a way
,frequency-dependent
thai /- becomesmore uniformly tigbt. These scale factors
and tiansformations are here assumed to be part of the
;,nominalmodel G(s).
ry.
Fnnosecr DrsrcN IN rr{E Fecr or
UNsrnucrr-rnno UNcent lrvrrrs
Once we specify a designmodel, G(s), and accept the
existenceof unstructured uncertaintiesin the form (13)'
the feedback design problem becomesone of finding a
'
. . compensatorK(s) such that
is
i,,,., t)-the nominai feedback system, GKII+GKI-I,
i:.stable;
2) the perturbedsystem,G'K[I+G'K7-t, is stablefor
'q;'
, ;'l'' all possible
G' allowedby (13); and
^-'l'
o-t ---.^--^-^^
^L:^^4:.
^+i.fi-;
f n , all
n l l possible
nncsihlo
are .satisfied
performance objectives
3)
Jor
, G' allowedby (13).
All three of these requirementscan be interpreted as
...
!l- frequencydomain conditions on the nominal loop transfer
i$iilr matrix, GK(s), which the designermust attempt to satisfy.
i
Stability Conditions
: ':
', The frequency domain conditions for requirement l)
are, of course, well known. In SISO cases,they take the
(14)
for all 0(e(1, all r on the D-contour, and all Z(s)
satisfying (13). Since G' vanisheson the infinite radius
segmentof the D-contour, and assuming,for simplicity,
that the contour requires no indentations along the j<oaxis,2(14) reducesto the following equivalentconditions:
0 < q[ 1 + G( i o) K( j ot)+ eL ( i @)GUor),K(ia, ) ] ( l s)
for all0(e ( l,0(c.t(o,
a n da l l L
<+o< s[/+ LGK( I+cK) - ' ]
(16)
for all 0 ( r.r( m, and all I
e alcxlr+c,()-'] < | / t^(o\
(17)
for all 0 ( o( co.
The last of these equations is the MIMO generalization
of the familiar SISO requirement that loop gains be small
whenever the magnitude of unstructured uncertainties is
large. In fact, whenever /.(c,l)>>I, we get the following
constraint on GK:
olcK(i,,,)l<r / t^(ot)
( l8)
for all <^rsuchthat /-(<,r)>>1.
We emphasizethat theseare not conservativestability
conditions. On the contrary, if the uncertainties are truly
unstructured and (17) is violated, then there exists a
perturbation l,(s) within the set allowed by (13) for which
the system is unstable. Hence, these stability conditions
impose hard limits on the permissible loop gains of practical feedbacksystems.
Perjonnance Conditioru
Frequency domain conditions for requirement 3) have
already been described in (5) in Section II. The only
lsce aay classical cotrtrol tcxt.
2If iadcntations arc rcquirc4 (la) and (f7) Fust hold in thc limit for
aU ,ron tni-i"ac"tcd patf, as thi rddius of indcntation is tokcn to zcro.
rEEETRANsAcrroNsoN AuroMATrc coNTRoL,
vor.,qc_26, No. l, FEBRUARvlggl
modification needed to account for unstructured uncertaintiesis to apply (5) to G'instead of G. i.e..
COI{OITION
e ps( olt+ tcxlr+GK)-tlo_lt+cxl
#b
<qlc.Ku',)l
f
.
p r < s LI + ( I + L ) G K I
*
i,
( le)
DITlON
fo11[ cosuch that I^(<o)( I and glcK(j6)l>>t.
(17)
This is the MIMO generalization oi another familiar
SISO design rule-namely that performance objectives
can be met in the face of unstructureduncertaintiesif
the
nominal loop gains are made sufficiently large
to com_
Fig. 3. Thc dcsign tradeoff for GK.
pensate for model variations. Note, however,
that finite
solutionsexistonly in the frequencyrangewhere/_(co)(
l.
stability and performanceconditlns derivid above
illustrate that MIMO feedback design problems
do not generally not tle same. They can be analyzedwith equal
differ fundamentally from their SlSO'counterparts.
In ease,however,by using the function o_[I+(KG)-rl in_
t1 in
both cases, stability must be achieved nominally
and steadof rII+(GK)1ZO;.This can-be readily veri_
assuredfor all perturbationsby satisfyingconditions (17) fied by evaluating the encirclement count
of the map
and (18). Performancemay tlen be optimized by satisfyl det(I + KG) under perturbationsof the form :
C, G( I + L)
ing condition (19) as well as possible.What distinguishes (i.e., uncertaintiesreflectedto the input).
The mathemati_
MIMO from SISO design conditions are the functions cal stepsare directly analogousto (15)_(lg)
above.
used to expresstransfer function ,,size,,,Singular values
Classical designerswill recognize,of course, that the
replace absolutevalues.The underlying concepts
remain differencebetweenthesetwo stability robustnessmeasures
the same.
is simply that each uses a loop transfer function appropriate for the loop breaking point at which robustneis
We note that the singular value functions used
in our
statementsof design conditions play a design role
is being tested.
much
like s1*1"" Bode plots.-TheA[i+Cfl fuo"tion
in (5) is
the minimunx return difference magnituAeof
the closed_
V. TneNsrnn FuNcrroN LnsterroNs
loop system,g[GK]in (g) and otcKlin (tg)
are minimum
and maximumloop gains,and AtGK(I+'Gkl-rt
The feedback design conditions derived above are pici<fa is
the maximum closedJoopfrequinry response.
Thesecan tured graphically in Fig. 3. The designermust find a loop
all be plotted as ordinary frequency dependent
functions transfer function matrix, GK, for which the loop is nomii_
in order to display and analyzi the ieatures
of a multivari- nally stable and whose maximum and minimum singular
able design.Such plots will here be called o_plots.
values clear the higb and low frequency ..disign
One of the o-plots which is particularly significant
with boundaries" given by conditions (17) and 1fD. ffre nigh
regard to design for uncertainties is oUtainea
by inverting frequency boundary is mandatory, while the low frequency
condition (17),i.e.,
one is desirable for good performance. Both aie in_
fluenced by the uncertaintybound, t^(r).
l^(r)<
:o _ ft+Gx(j a ;)-tf (20)
The o-plots of a representativeloop transfer matrix are
o-[GK(r
+ Gn-'l
also sketchedin the figure. As shown, the effective bandwidth of the loop cannot fall much beyond the frequency
for all0{ar(oo.
or, for which l-(q,): l. As a result, the frequency range
The function on the right-hand side of this expressionis
over which performanceobjectivescan be met is explicitly
an explicit measureof the degreeof stability (or stability
constrained by the uncertainties.It is also evident from
robustness)of the feedbacksystem.Stability is guaranteed
the sketch that the severityof this constraint dependson
for all perturbationsZ(s) whosemaximum sirrgrlu, values
the rate at which o_lcKl and o[GKl are attenuated.The
fall below it. This can include gain or phase changes
in steeperthesefunctions drop off, the wider the frequency
individual output channels, simultaneous changis
in range over which condition (19) can be satisfied. Unforseveralchannels,and various other kinds of perturbations.
tunately, however, FDLTI transfer functions behave in
In.effect, o_[I+(GK)- 11is a reliable multivariablegen_
such a way that steep attenuation comes only at the
eralization of SISO stability maryin concepts (e.g.,
of small o[I+GKl values and small o[.f*
dependentgain and phasemargins).Unlike the :lp_:lr:
(GK)- f l valueswhen o[Gi(] and.o[GKlxl. This means
l1191encf
SISO case, however, it is important to note that o[1+
that while performanceis good at lower frequenciesand
(Gr() - tl measurestolerancesfbr uncertaintiesat the piant
stability robustnessis good at higher frequencies,both are
outputs only. Tolerancesfor uncertaintiesat the inpui are
poor near crossover.The behavior of FDLTI transfer
9
STEIN: MULTIVARIABLE FEEDBACK DESIGN
D6YLE AND
limitation on
functions,therefore,imposesa secondmajor
systems'
the achievableperformanceof feedback
SISO TranslerFunction Limitation
For SISO cases,the conJlict betweenattenuation rates
anciioop quality at crossoveris again well understood'We
know that any rational, stable, proper, minimrrm phase
loop transfer function satisfies fixed integral relations
betweenits gain and phasecomponents.Hence, its phase
anglenear crossover(i.e., at valuesof <.rsuch that lgkuor)l
erl) is determineduniquely by the gain plot in Fig. 3 (for
-a-g=lgeD. Various expressionsfor this angle were derived by Bode using contour integration around closed
contours encompassingthe right half plane [5, ch. 13' l4].
One exPressionis
''
a
Qs*" arglsk(i"")]
:+l:*
tnlsk(i@(v\.)l:tnlsk(io")l
., er)
sinh y
where v:ln(ot/t's"), <,r(z):o"expz. Since the sign of
sinh(z) is the sameas the sign of z, it follows tlat f"o" will
be large if the gain lg&l attenuatesslowly and small if it
attenuatesrapidly. In fact, qgt" is given explicitly in terms
of weighted average attenuation rate by the following
alternateform of (21) (alsofrom [5]):
Qg*c:Q^*"*Qp"
(Q^*"
(2s)
and thereforeaggravatesthe tradeoff problem. In fact, if
l0r"; it too large,we will be forced to reducethe crossover
frequency. Thus, RHP zeros limit loop gain (and thus
performance)in a way similar to the unstructured uncertainty. A measureof severity of this added limitation is
which can be usedjust like l-(a) to constrain
ll-p(j")|,
phasedesign.
minimum
a nominal
If g(s) has right half-plane poles, the extra phase lead
contributed by these poles compared with their mirror
imagesin the left half-planeis neededto provide encirclements for stability. Unstable plants thus also do not offer
any inherent advantageover stable plants in alleviating
the crossoverconflict.
M ult ioariable Generalization
The above transfer limitations for SISO systemshave
multivariable generalizations;with some additional complications as would be expected.The major complication
is that singularvaluesof rational transfer matrices,viewed
as functions of the complex variable J, are not analytic
and therefore cannot be used for contour integration to
derive relation such as (21). Eigenvalues of rational
matrices, on the other hand, have the necessarymathematical properties.Unfortunately, they do not in general
relate directly to the quality of the feedbackdesign.(More
is said about this in SectionVI.) Thus, we must combine
the propertiesof eigenvaluesand singular values through
the bounding relations
es*"=+
f-ry(r"*tnS)a,. Q2)
The behavior of qrr" is sigaificant becauseit defines the
magnitudesof our two SISO design conditions (17) and
(19) at crossover.
Specially,when lg&l- l, we have
o _ l A l l<^ [ r ] l < a l . q l
(26)
which holds for any eigenvalue,tr;, of the (square) matrix
A.T\e approachwill be to derive gain/phase relations as
of I+GK aridI+(GK)-I and
in (21) for the eigenvalues
to use tlese to bound their minimt'm singular values'
Since good performance and stability robustnessrequires
The quantity n*Qs*" is the plrasemargin of the feedback singular valuesof both of thesematrices to be sufficiently
system.Assuminggk stable, this margin must be positive large near crossover,the multivariable system'sproperties
for nominal stability and, according to (23), it must be can then be no better than ttre properties of their eireasonablylarge (=l rad) for good return difference and genvaluebounds.
stability robustnessproperties.If. riQrr" is forced to be
Equations for the eigenvaluesthemselvesare straightvery small by rapid gain attenuation,the feedbacksystem forward. There is a one-to-one colTespondencebetween
will amplify disturbances(ll+g&l<<l) and exhibit little eigenvaluesol GK and eigenvaluesof I + GK such that
uncertainty toleranceat and near orc.The conflict between
I,l/+Gi(]: I +I,[G,<].
(27)
attenuation rate and loop quality near crossoveris thus
clearly evident.
Likewisefor .f+ (GK) t ;
It is also known that more generalnonminimum phase
andf or unstable loop transfer functions do not alleviate
r,[r+1cr;-']:r + -!
(28)
this conflict. If the plant has right half-plane zeros, for
r,[Gr(]
example,it may be factored as
Thus,when lIr[GK]l : I for somel, and {o:rd", we have
(24)
g(s):rn (s)p(s)
lr+sftt:lt+(g/c)-rl:t1""(?)lQ3)
where n(s) is minimum phaseandp(s) is an all-pass(i.e.,
Vo.) The (negative)phaseangle of p(s) relp(io)l=l
ducestotal phaseat crossover,i.e.,
lI,[/+ c/r]l: ;r,fr+ 1cr;-'];
:'l''(*)l
@)
oN AuroMATIccoNTRoL,vol. rc-26, xo. l, resnulnv l98l
IEEETRANsAcrtoNs
Since this equation is exactly analogousto (23) for the
scalar case,and since ll,,l bounds g, it follows that the
loop will exhibit poor propertieswheneverthe phaseangle
(z*0r.") is small.
In oider to derive expressionsfor the angle 4^," itself,
we require certain results from the theory of algebraic
functions t201-I261.The key conceptsneededfrom these
referencesare that the eigenvalues1,,of a rational, proper
transfer function matrix, viewed as a function of the
complex variable s constitute one mathematical entity,
tr(s), called an algebraicfunction. Each eigenvaluetr, is a
branch of this function and is defined on one sheet of an
extended Riemann surface domain. On its extended domain an algebraic function can be treated as an ordinary
meromorphic function whose poles and zeros are the
system poles and transmission zeros of the transfer function matrix. It also has additional critical points, called
branch points, which correspond to multiple eigenvalues.
Contour integration is valid on the Riemann surface domain provided that contours are properly closed.
In the contour integral leading to (21), g&(s) may
thereforebe replacedby the algebraicfunction, l(s)' with
contour taken on its Riemann domain. Carrying out this
integral yields severalpartial sums:
r,U'")l]
:: [* > Imlr,(1"(r)l-ul
r,
) +^,,
y
sinh
i
,
'
-
-
@
i
(30)
where each sum is over all branchesof tr(s) whosesheets
are connectedby right half-plane branch points. Thus the
eigenvalues{tr,} are restricted in a way similar to scalar
transfer functions but in summation form. The surnmation, however, does not alter the fundamental tradeoff
betweenattenuationrate and loop quality at crossover.In
fact, if we deliberatelychposeto maximizethe bound (29)
by making o" and f^," identical for all i, then (30) imposes
the samerestrictionson multivariable loops as (21) imposes on SISO loops. Hence, multivariable systemsdo not
escapethe fundamentaltransfer functibn limitations.
As in the scalar case,expression(30) is again valid for
minimum phase systemsonly. That is, GK can have no
transmissionzeros3in the right half-plane. If this is not
true, the tradeoffs governedby (29) and (30) arc agavated becauseevery right half-plane transmission zero
adds the samephase lag as in (25) to one of the partial
sums in (30). The matrix GK may also be factored, as in
(24), to get
of multivariable nonminimum phasenessand used like
/-(<o) to constraina nominal minimum phasedesign.
VI.
Mwrrvnmesrn DsslcN sY MoorRN
FnreusNcv DouxN Mnrnoos
So far, we have describedthe FDLTI feedback design
problem as a design tradeoff involving performanceobjectives [condition (19)], stability requirementsin the face
of unstructureduncertainties[condition (17)]' and certain
performancelimitations imposed by gan/phase relations
which must be satisfiedby realizable loop transfer functions. This tradeoff is essentiallythe same for SISO and
MIMO problems. Design methods to carry it out, of
course,are not.
For scalar design problems, a large body of well"classical control") which
developed tools exists (e.g.,
permits designersto conshuct good transfer functions for
Fig. 3 with relatively little difficulty. Various attempts
have been made to extend thesemethodsto multivariable
design problems. Probably the most successfulof these
are the inverseNyquist array (INA) [3] and the characteristic loci (CL) methodologies[4]. Both are based on the
idea of reducing the multivariable design problem to a
sequenceof scalarproblems.This is done by constructing
a set of scalar transfer functions which may be manipulated more or lessindependentlywith classicaltechniques.
In the INA methodolory, the scalar functions are the
diagonal elementsof a loop transferfunction matrix which
has been pre- and post-compensatedto be diagonally
dominant. In the CI- methodology,the functions are the
eigenvaluesof the loop transfer matrix.
Basedon tlre designperspectivedevelopedin the previous sections,thesemultiple singleJoop methods turn out
to be reliable designtools only for specialtypes of plants.
Their restrictions are associatedwith the fact that the
selectedset of scalar designfunctions are not necessarily
related to the system'sactual feedbackproperties'That is,
the feedback systemmay be designedso that the scalar
functions have good feedbackpropertiesif interpreted as
SISO systems,but the resulting multivariable systemmay
still have poor feedbackproperties'This possibility is easy
to demonstratefor the CL method and, by implication,
for the INA method with perfect diagonalization' For
thesecases,we attempt to achievestability robustnessby
satisfying
t ^ ( , ) < ; r , [ r + 1 c r ) - ' ] l : 1 1+ l / I , ( c r K ) l ( 3 2 )
for all i and 0(ro(o
GiK(s):M(s)P(s)
and similarly, we attempt to
( 3 1 ) achieveperformanceobjectivesby making
where M(s) has no right half-plane zeros and P(s) is an
Analogousto the scalar
all-passmatrix fr1-s;f1s;:L
case,o-(/-P(s)) can be taken as a measureof the degree
3For our purposcs,traasnission zcros-[41]-q:9--values.i such that
dcttG(J)K(r-)1j0. Degcncratcsystcmswith dct[GrK]:0 for all 'r are not
ieiiusc thcy canaot mcct condition (19) in Fig' 3.
of ilticit
rs(ol)
< lr,[ r+ cK]l: ll +I,(clK)l
=ry
l- l^(a)
(33)
foralliand0(o(co.
As discussedin SectionV, however, the eigenvalueson
the right-hand sides of these expressionsare only upper
ll
MULTIVARIABLE FEEDBACK DESIGN
DOYLE AND STEIN:
los (t)
Fig. 4.
o-plots for /+G-r.
[G from (3a).]
bounds for the true stability robustnessand performance
conditions (20) and (19). Hence, o-lI+(GK)-t1 and/or
o_ll+ GKI may actually be quite small evenwhen (32) and
(33) are satisfied.
An ExarnPle
Fig.5. stabirityrcgionsror
{r.i?
;]}
" [Grron(34).]
ity robustnessis given in Fig. 5. This figure shows stability
"gain space" for tle compensator K(s):
These potential inadequaciesin the INA and CL meth- regions in
an unstableregion
ods are readily illustrated with a simple example selected diag(l +kbl+/<r). The figure reveals
point. The INA
design
nominal
to
the
proximity
in close
specifically to highlight the limitations. Consider
becausethey
tools
design
reliable
not
are
and CL metlods
presence.
its
to
designer
fail to alert the
This example and the discussion which precedes it
should not be misunderstoodas a universal indictment of
This system may be diagonalized exactly by introducing the INA and CL methods.Rather, it representsa caution
constant compensation.Let
regardingtleir use.There are various types of systemsfor
prove effective and reliable. Condi(35) which the methods
satisfy can be deduced fiom
systems
tions which tlese
must have tight singular
they
(32) and (33)-namely
This
includes naturally diagovalue/eigenvalue bounds.
"normal"
nal systems,of course, and also the class of
systems[28]. The limitations which arise when the bounds
"
*
'
G:uGU-t:l
t g o l are not tight have also been recognizedin [4].
n l.
We note in passingthat the problem of reliability is not
unique to the INA and CL methods. Various examples
can be constructured to show that other design apIf the diagonal elements of this G xe interpreted as
"single-loop-at-a-time" methods
proaches
such as the
independentSISO systems,as in the INA approach, we
cornmon in engineeringpractice and tridiagonalization
could readiiy conclude that no further compensationis
approachessuffer similarly.
necessaryto achieve desirable feedback properties. For
6lampl€, unity feedbackyields stability margins at crossover of + oo dB in gain and greater than 90 degrees in
VII. Murrrvenresrr DssrcN VH LQG
phase.Thus, an INA designcould reasonablystop at this
-t:/'
A second major approach to multivariable feedback
Sincethe diagpoint with compensatorK(s): UU
We have
onal elementsof G are also the eigenvaluesof G, we could design is the modern LQG procedure [lU' [12]'
with the
also be reasonablysatisfiedwith this designfrom the CL already introduced this method in connection
reduction
tradeoff between command/disturbance error
point of view.
and sensor noise error reduction. The method requires
entirely
an
to
leads
Singular value analysis, however,
noise, comdifferent conclusion.The o-plots for (/+G-r) are shown that we select stochasticmodels for sensor
mean
in Fig. 4. Theseclearly display a seriouslack of robustness mands and disturbances and define a weighted
goodness
for
the
with respect to unstructured uncertainties. The smallest squareerror criterion as the standard of
comFDLTI
value of o is approximately 0.1 neat u:2 rad/s. This disign. The rest is automatic. We get an
model G(s)
nominal
the
stabilizes
which
K(s)
p"ooto.
l^(2):
as
means that multiplicative uncertainties as small
; 0.1 (avl}Vo gain changes,=6 deg phase changes)could (under mild assumptions) and optimizes the criterion of
produce instability. An interpretation of this lack of stabil- goodness.All too often, of course' the resulting loop
G(s)=t,*fu
| -ff,*' ;3:.r] (34)
,:l'u;] '-':l-l -t]
i+
ol
I o *t)
vor' lc-26' No l' rrgnuenv
IEEETRANsAcrloNsoN AUToMATIccoNTRoL,
t2
l98l
perforAs noted eadier, the first three functions measure
uncertainto
mance and stability robustnesswith respect
(i) in Fig' 6)'
ties at the plant outputs Qoop-breakingpoint
and robustand the sicond three measureperformance
nesswithrespecttouncertaintiesattheplantinputQoop.
*Li=o;+gu+(!.
generally
[t.uti"g point (tt) in Fig. 6)' Both points are
in design'
significant
-T;;
other loop-breakingpoints, (i)' and (ii)" are also
shown in the figure. Theseare internal to the compensator
they
Fig. 6. LQG fecdbackIooP'
una tfr"t"fote havelittle direct significance'However'
rebe
have desirableloop transfer properties which can
unacceptable
entirely
proPerare
transfer functions, GK or KG,
to the propertiesof points (i) and (ii)' The
of Fig' 4' lated
,"i"r, "*"-ined against the design constraints
ties and connectionsare these'
w,u,.ttrenforcedtoiteratethedesigrr_adjustweigbts Fact l: The loop transfer function obtained by breakstochastic disin the performance criterion, change the
LQG loop at point (i)' is the KBF loop transfer
etc' There are ing the
dynamics'
irrrbuoce and noise models,add
function CQK..
frustration sets in
breakso many parametersto manipulate that
Fact2: thJ loop transfer function obtained by
and theoretito
made
be
(t)
G{('.It
is
point
qrlrny "ia the schismbetweenpractitioners
at
loop
i"g in" LQG
-CLXrpointwise
9an
in s by designingthe LQR in
"i*t L""o.es easierto understand'
up"ptoo"ft
controilers
I-$C
o!
"sensitivity
recovery" proceduredue to
Fortunately, such designiterations
witi a
last few years ."aorduo"a
have becomi easier to iarry out in the
of these con- Kwakenaak [29].
breakbecausethe frequency domain propertiesFact 3: The loop transfer function obtained by
new results
transfer
loop
LQR
the
(ii)'
is
are bettei undlrstood' Someof the key
point
;Ji;*
at
LQG loop
with respect ing the
a1s s11--arized below and their significance
K"QB.
function
controllers
to Fig. 3 are discussed.For our purposes'LQG
loop transfer function obtained by break4:iLe
Fact
specialinternal
*itLl
to
compensatott
FDLTI
oi"oiAio"ty
LQG loop at point (tt) is t(G'.It can be made
6 and is well i"g tn"
Fig'
in
in
is
shown
KBF
the
structure
designing
This
by
s
in
poi"t*ise
structure.
(KBF) de- up?io*n k"oa
filter
to
Kalman-Bucy
due
a
procedure
of
consists
It
recovery"
known.
oritt u..robustness
nominal model "f"ord*""
the
of
realization
space
state
a
for
signed
Stein [30].
disturbance Doyle and
c?ri, i""fuai"g uU upp"ttded dynamics for
3 can be readily verified by explicit evaluaand
I
Facts
is
and 4 take
ptI*.t"t, comiands,-integral action' etc' The model
tion of the transfer functions involved' Facts 2
( 3 7 ) moreelaborationandaretakenupinalatersection.They'
x€R', ue R^
must be
*:Axi Butt;
also require more assumptions'Specifically' G(s)
Fact 4'
fot
mlr
Fact2'
lor
-i"rt tt" phasewith m)r
y:Cx*tl;
YeR'
n'unes
the
Also'
both'
for
and hence,G(s) must be square
;r"*i i"i y redvery" and "robustness recovery" are oveily
and satisfies
(38)
G("):CO(s)B
,,Full-stateloop transfer recovery" is perhapsa
restrictive.
that
d:CO(s){
tetter name for both procedures,with the distinction
(ii)'
points
to
other
the
(t)'
and
one applies to points (t),
with
(rr)'.
'
we can
(3e)
tn" significance of these four facts is that
@ @ ( s()s: I ^ - A ) - l
.
feedfull-state
a
on
a"G i6c toop transfer funcrions
a
noise
with
white
usual
adequately
the
them
The symbols { and 4 denote
Uaci Uasisand tlen approximate
design
a1f
state
Kt
bV
full
(i),
the
pro*r.".. The filter's gainsare denoted
recovery procedure. For point
ilt :?t"
(i'e" its
by
estimates by i. The state esti:nates are multiplied
must b; ione with the KBF design equations
to
gains'
equations'
(LQR)
LQR
the
d'
full-state linear-quadratic regulator
Riccati equation) and recovery with
which drive the plant and while for point (ii), full-state design must be done with
commands
contiol
ptoA"""
the
'are
also fed back internally to the KBF' The usual condi- tl" t-qn'equations and recovery with the KBF' The
tions for well-posednessof the LQG problem are as- mathematicsof thesetwo options are, in fact' dual. }Ience,
in furt]er
suned.
we will describeonly one option [for point (ii)]
interlater in
of
used
functions
and
In terms of previous discussions,the
detail. Results for tie other are stated
and
difference'
"ri io nig. 6 are the loop transfer, return
our examPle'
stability robustnessfulctions
GK, I,+GK,
I,+(GK)-t,
and also their counterParts
KG, I^+KG,
I^+(KG)-r'
Full-State I'ooP TransferDesign
The intermediate full-state design step is worthwhile
prop€rbecauseLQR and KBF loops have good classical
":.
DOYLE AND
IJ
STEIN: MULTIVARIABLE FEEDBACK DESIGN
few years This implies [35] that
ties which have been rediscoveredover the last
LQR caseis that LQR
l31-t331. The basic result for the
r^ + r - |(io;)l> t / z
o_l
matrices
transfer
ioop
t-
v
U
r
v
;-
I
)
(45)
Hence, LQR loops are guaranteedto remain stable for all
unstructured uncertainties(reflected to the input) which
satisfyt^(a)10.5. Without further knowledgeof the types
satisfythe following return difference identity [32];
of uncertaintiespresent in the plant, this bound is the
*Rl r- + r(lco)]
+
T(
greatestrobustnessguaranteewhich can be ascribedto the
t
ia)]
I ^
v0(co( m (41) regulator.s
:R+ | naQot)Bl'lnolir\nl
While it is reassuringto have a guaranteeat all, the
matrix,
weighting
control
the
standard
is
>0
bound is clearly inadequatefor the requirements
R:Rr
I^<0.5
where
weighting
state
(20) with realisticl^(a)'s.In order to satisfy
the
corresponding
is
condition
of
a31dHTH:Q>0
of
size
generality,
be
H
can
(20)
loss
of
in LQR designs,therefore,it becomesnecWithout
condition
matrix.
(41)
with
(7)'
(6)
and
definitions
manipulate the high-frequencybehavior
the
Using
directly
to
essary
(mxn) [34].
can be derived from known
behavior
that
This
implies
Z(s).
of
R:pI
regulator as the scalarp tends
properties
of
the
asymptotic
to zero l2gl,l47l, [48], [34], [36]. The result neededhere is
IHaB
o,lI^ + r(i"\):\A,L I + - ( HaB)*
tfiat under minimum phase assumptionson H@8, tJl,e
)
LQR gains K" behaveasymptotically as [29]
r(s) 3 r<"o(s)B
(40)
:Vt *:r,[1nor;*HaBl
)
( co.
V0 ( <^r
r+
|"?tuo(io)Bl
where W is an orthonormal matrix. The LQR loop transfer function, T(s), evaluated at' high frequencies,
(42) s:jc/l p with c constant,is then given by6
This expressionapplies to all singular values o, of I(s)
and, hence,specificallyto o and o-.It governsthe performanceand stability robustnesspropertiesof LQR loops.
Wheneverg.[f ]>>l, the following approximation of (42)
shows explicitly how the parameters p and .FI inlluence
(s):
(43)
We can thus choosep and H explicitly to satisfy condition
(19) and also to "balance" the multivariable loop such
that o[7] and o[Z] are reasonablyclose together.4This
second objective is consistent with our assumption in
SectionIII that the transferfunction G(s) has been scaled
and/or transformed such at /-(co) applies more or less
uniformly in all directions.This is also thejustification for
consideringcontrol weighting matrices in the form R:pI
only. Nonidentity R's are subsumedin G as GRt/z.
Robustness
Properties[Conditions(17) and (20)l
It also follows from (42) that the LQR return difference
alwaysexceedsunity, i.e.,
sll^+Z(jco)]> I
V0(to(o.
r{,i, / | p ) : I p x "(ict - Vi A)-' B
(47)
-+l{HB/jc.
this means that the
Since crossoversoccur at o,lrl:1,
frequency
of the loop is
maximum (asymptotic)crossover
PerJormanceProperties (Condition I 9)
o,lr1i "1loo,ln o1iu1nl / fi .
(46)
t/ p K"--WH
(44)
alt mav also bc ncccssarv to appcnd additional dynamics. Ia ordcr !o
must tetrd !o €
cxample, olHABl
achic,vc z]aro st€ady statc cirors,i6r
8s ar+O. This may rcquirc additional integratioas ia thc plant"
,"-*:d[
HB)/{
p
(48)
As shown in Fig 3, this frequency cannot fall much
beyond co;, where unstructured uncertainty magnitudes
approach unity. Hence, our choice of fl and p to achieve
the performanceobjectivesvia (43) are constrainedby the
stability robustnessrequirementvia (a8).
Note also from (47) that the asymptotic loop transfer
function in the vicinity of crossoveris proportional to | / ot
(- I slope on logJog plots). This is a relatively slow
attenuatioo rate which, in view of Section V, is the price
the regulator pays for its excellent return difference properties. If l^t(") attenuatesfaster than this rate, further
reduction of o" may be required. It is also true, of course'
that no physical system can actually maintain a l/a
characteristicindefinitely [6]. This is not a concern here
since I(s) is a nominal (design) function only and will
5Tbc r-<0.5 bound turos out to bc tigbt for pure Sait changcs,i,c.,
0.5<+6 dg, which is identical to rcgulato/s cclcbratcd guarantocd sain
marsin t331.The bound is conscrvative if thc unccrtainties are known to
bc pirc'pnisc changcs,i.c., 0.5<+*30 dcg which fu t63 than tbc known
-rtiO dcg guarantcc[33].
trnir-ftciri"
Utoitttig processis appropr-late for thc so-callcd..F-oi:
casc 136l with full nnk HB. Morc gcneral vcnilons ot ())) wrur ra$(
tHBi<;n are dcrivcd in ['141.
14
rEEETRANsAcrroNsoN AUToMATTccoNTRoL, vor. ec-26, No. l, FEBRUARvl9gl
later be approximatedby one of the full-state loop transfer recoveryprocedures.
Full-State Loop Transfer Recooery
as the target LQR dynamics satisfy Fig. 3's constraints
(i.e., as long as we do not attempt inversion in frequency
ranges where uncertainties do not permit it). Closer inspectionof (50)-(56) further showsthat thereis no dependence on LQR or KBF optimality of the gains K" or Kr.
The procedure requires only that Kp be stabitizing and
have the asymptotic characteristic 6r. Thus, more general statefeedbacklaws can be recovered(e.g.,pole placement), and more generalfilters can be usedfor the process
(e.g.,observers).7
As describedeadier, the full-state loop transferfunction
designedabove for point (ii)' can be recoveredat point
(ii) by a modified KBF design procedure. The required
assumptionsare that r) m and that C@8 is minimum
phase.The procedurethen consistsof two steps.
l) Append additional dummy columns to .B and zero
row to K" to make COB and r("O.8 square (rxr). CQB
An Exanple
must remain minimum phase.
2) Design the KBF with modified noise intensity
The behavior of LQG design iterations with full-state
matrices.
loop transfer recovery is illustrated by the following abstracted longitudinal control design examplefor a CH4Z
E(€€r):luo + qran']a1r-";
tandem rotor helicopter. Our objective is to control two
measured
outputs-vertical velocity and pitch attitudcE(\'f):Nod(t-r)
by manipulating collective and dilferential collective rotor
where M6, No are 1fosn6minal noise intensity matrices thrust commands. A nsmins,l model for
the dynamics
obtained from stochastic models of the plant and 4 is a relating thesevariablesat 40 knot airspeedis
[a5]
scalarparameter.Under tlese conditions,it is known that
-o.u
the filter gains K, have the following asymptotic behavior
0.005 2.4 -32
0
.
1
4
d
l
- 1.3 -30
as q-+m [30]:
0.4
r
Ixr.-nwul'r,
(4e)
Here lV'is another orthonormal matrix, as in (46). When
this K, is used in the loop transfer expression for point
(r'i), we get pointwiseloop transfer recoveryas q-+co, i.e.,
rK(s)G(s):K"[O-t+BK"+KrC7-'K]C@B
(50)
:c[6-or, Q,+ coxr)-'.6] KrceB
(51)
:x"oxr(t,+cdxr)-tcon
(s2)
--+r"o,a(co
a)-tcon
(53)
=KpB(1,+KpB)-'
. lc on14+K"or)- I] -' ceB
(54)
: {r"o,a1cos)-t}coa
(55)
:K"98.
(56)
In this seriesof expressions,(D was used to represent the
matrix (sI,-A+BK)-r,_(49)
was used to get from (52)
to (53), and the identity QB:@B(I+K"O.8)-t was used
to get from there to (5a). The final step shows explicitly
that the asymptoticcompensatorK(s) [the bracketedterm
in (55)l inverts the nominal plant (from the left) and
substitutes the desired LQR dynamics. The need for
minimum phaseis thus clear, and it is also evident that
ttre entire recovery procedure is only appropriate as long
u-:l 00
0.018 - 1.6
0
1
t.2
0 ]'
-i'tuJ,
.13:3!
I o . r + - 0 . 1 2I
,:[3I 3 ,?,]'
Major unstructured uncertainties associated with this
model are due to neglected rotor d5mamicsand unmodeled rate limit nonlinearities. Theseare discussedat greater
length in t46]. For our present purposes,it, suffices to note
that they are uniform in both control channels and that
I^(ot)) I for all co) l0 rad/s. Hence, the controller bandwidth should be constrainedas in Fig. 3 to ro"-o ( 10.
Since our objective is to control two measuredoutputs
at point (i), the design iterations ut'lize the duals of
(40)-(56). They begin with a full state KBF design whose
noiseintensitymatrices,E(€€r):11161r-r;
and,E(qf )
:p/d(r-r),
are selectedto meet performanceobjectives
at low frequencies,i.e.,
qlrlxqlcarl/{p
>ps,
(s7)
while satisfying stability robustness constraints at high
frequencies;
o"-* :o-[ CI ] / | p ( l0 r/s.
(58)
?Sti[ morc gcnclally, thc modificd
KBF proccdure will actually r+
covsr full-statc fccdback loop transfcr functions at any point, ul, ii thc
systcE for which CO.8l is m-ininum phasc t30l.
l5
MULTIVARIABLE FEEDBACK DESIGN
DOYLE AND STEIN:
L5
classical SISO approachesto this issue can be reliably
generalizedto MIMO systems,and has defined the extent
to which MIMO systemsare subject to the same uncertainty constraintsand transferfunction gain/phaselimitations as SISO ones.Two categoriesof designprocedures,
were then examinedin the context of theseresults.
There are numerousother topics and many other proposed design procedureswhich were not addressed'of
assignments'
course.Modal control,[42] eigenvalue-vector
methods
geometric
field
of
entire
[19] are
the
and
[43]
prime examples.Thesedeal with internal structural properties of systemswhich, though important theoretically,
ceaseto have central importancein the face of the inputoutput nature of unstructured incertainties.Hence, they
were omitted. We also did not treat certain performance
objectivesin MIMO systemswhich are distinct from SISO
systems.Theseinclude perfect noninteraction and integrity. Noninteraction is again a structural property which
losesmeaningin the face of unstructureduncertainties'[It
is achievedas well as possibleby condition (19)'l Integrity, on the other hand, cannot be dismissedas lightly' It
"orr""*. the ability of MIMO systemsto maintain stability in the face of actuator and/ot sensor failures' The
singular value conceptsdescribedhere are indeed useful
for integrity analysis.For example,a designhas integrity
with respectto actuator failures whenever
FULL STATEKBF
v
I.
I-
d
l-
ts
le
:
)r
:s
0.I
l.O
10.
100.
q[1+(,(G)-']>
I
vc,r.
(59)
F N E Q U I T CRY/ 5
Fig. 7. Full-statc loop traasfer recov€ry.
e
rt
t-
ts
rf
;e
")
For the choice f =8, (58) constrainsp to be greater than
or equal to unity.s The resulting KBF loop transfer for
p:lls shownin Fig.7. For purposesof illustration,this
' function will be consideredto have the desired high gain
, propertiesfor condition(19),with low gainsbeyondor: l0
,t ior condition (20).eIt then remains to recover this function by means of the full-state recovery procedure for
point (i). This callsfor LQR designwth 2:9o+qzcCr
and R=Ro. Letting Qo:0, Ro:/, the resultingLQG
transferfunctions for severalvaluesof q ate also shown in
Fig. 7. They clearly display the pointwise convergence
, propertiesof the procedure.
,s
VIII.
r)
rC
CoNcrusroN
This paper has attempted to present a practical design
petspeciilneon MIMO linear time invariant feedbackcontrol problems.It has focusedon the fundamental issuefeedback in the face of uncertainties.It has shown how
8If Cf (or IfB) is singular, (58) or (48) are still valid in lhe oonzcro
directions.
eibi tunction should not bc considercd final, or coursc' Bcttcr bali ""4" ""a grcatcr gain at low.frequcncies via appcndcd
i t""" Ltro"
itrtcgraton would bc?esirablc in a serious dcsip.
it
This follows becausefailures satisfy l^(l'Moreover
via
designed
laws
can be shown [37] that full-state control
Lyapunov equations,as opposedto Riccati equations,as
in Section VII, satisfy (59). It is also worth noting that
integrity properties claimed for design methods such as
INe ana Clsuffer from the reliability problem discussed
in SectionVI and, hence' may not be valid in the system's
natural (nondiagonal)coordinate system'
The major limitations on what has been said in the
paper are associatedwith the representationchosen in
Section III for unstructureduncertainty' A single magnitude bound on matrix perturbationsis a worst caserepresentation which is often much too conservative(i'e', it
may adrnit perturbations which are structurally known
ooito occur). The use of weightednorlns in (8) and (9) or
selective transformations applied to G (as in [39]) can
alleviate this conservatismsomewhat, but seldom completely. For this reason'the problem of representingmore
structured uncertaintiesin simple ways analogousto (13)
is receivingrenewedresearchattention [38]'
A second major drawback is our imptcit assumption
that all loops (ail directions)of the MIMO systemshould
have equai band-width (g close to d in Fig' 3)' This
"rro-ptioo is consistent*ittt u uniform uncertainty bound
but will no longer be appropriate as we learn to represent
more complexuncertainty structures'Researchalong these
lines is also proceeding.
#
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Johtr C. Doyle reccived thc S.B. and S.M' degrc€s in electrical enginccring from the MassachusettsInstitutc of Technologt n lW.
Hc now divides his time betwcen consulting
work for Honeywell Systems and Research
Center in Minneapolis, MN aad graduate studies in mathcmatics at thc University of California at Berkclcy. His research interest in the
arca of automatic control is in developing theory
that is relcvant to the practical problcms faccd
by engineers.
Gunter Steln (S'66-M'69) receivedthe B.S.E.E
degrec from General Motors Institute, Flint, MI'
and the M.S.E.E.and Ph.D.degrcesfrom Purdue
University, West Lafayette, IN, in 1965 and
1969,respectivelY'
Sincc 1969 hc has been with thc Honcywell
Systemsand RescarchCetrtcr, Minn€apolis, MN.
As of January l97,be alsohas hcld an Adjunct
Profcssor appointmcnt with the Departmcnt of
Electrical Engineering and Computcr Scienccs,
Massachus€tts Institute of Technolory' Cambridge. His research interests are to makc modern control theoretical
methods work.
Dr. Stein is a membcr of AIAA Eta Kappa Nr.r,and Sigma Xi' Hc has
scrved as Chairman of thc Tcchnical Committcc on Applications (19741975), IEEE Tnrxsecnoxs oN AItroMATtc CoNrRor" and as Chairman
of the l4th Symposium on Adaptivc Proccssas.