LennartLjung
Department of Electrical Engineering
Linkping University, S-581 83Linkping, Sweden
WWW: http://www.control.isy.liu .se
Email: ljung@isy.liu.se
2000-03-07
REG
LERTEKNIK
AUTO
MATIC CONTR
OL
LINKÖPING
Report no.: LiTH-ISY-R-2220
Forthe IFACSymposium on System Identi cation, SYSID2000,
SantaBarbara, CA, July 2000
Technicalreportsfrom theAutomatic Controlgroupin Linkpingare available
by anonymous ftp at the address ftp.control.isy.liu.se. This report is
DESIGN
Lennart Ljung
Divisionof AutomaticControl,LinkopingUniversity,
SE-58183, Linkoping, Sweden, email: ljung@isy.liu.se
Abstract: Model validation and estimating the size of a possible model error is a
central aspect of System Identi cation. In this contribution we discuss the model
errorconceptsandmodelerrormodelingforcontroldesign.Ofspecialinterestishow
to makeuse of periodic inputs, and how to deal with non-linear errormodels. The
discussionislimitedto SISOmodelsandstabilityrobustnessissues.
1. INTRODUCTION
Inthiscontributionweshallconsider the
estima-tionoflinearmodels,ofthetype
y(t)=G(q;)u(t)+H(q;)e(t) (1)
Thisisofcourseastandardtopicintheliterature,
e.g.,(Ljung 1999a). Theparameter vector will
be estimated using a prediction error method,
leading to an estimate ^
N
with corresponding
transferfunction estimate
^ G N (q)=G(q; ^ N ) (2)
Thisestimatewill thenbeused todesigna
regu-lator
u(t)=K(q)(r(t) y(t)) (3)
sothat theclosed loopsystem
y(t)= K(q) ^ G N (g) 1+K(q) ^ G N (q) (4) behaveswell.
Theproblemisthat ^
G
N
willonlybean
approxi-matedescriptionofthetruesystem.Theextensive
literature on robust control design, e.g., (Zhou
et al. 1996), (Skogestad and Postlethwaite1996)
deals with thequestion to ensure good behavior
oftheclosed loopsystem,despiteinaccuraciesin
themodel ^
G
N .
When the model has been identi ed from data,
the uncertainty of the model. The interplay
be-tweentheestimationandthedesignprocesseshas
receivedsubstantialattentionoverthepastyear,
e.g.,(SmithandDoyle1992),(Kosutetal.1992a),
(RanganandPoolla1996),(Skelton1989),(Gevers
1993),(Kosutet al.1992b),(SmithandDullerud
1996),(Poollaetal.1994).
Part of this problem is to estimate, in reliable
way, the model error or model uncertainty, not
assuming that the true system canbe described
withinthechosenstructure(1).See,e.g.,(Ninness
and Goodwin 1995), (Ljung 1999b) Then it is
mostlyassumedthatthetruesystemislinear,and
givenby atransferfunction G
0
(q),and thatthe
modelerroriscapturedbyestimatesofthesizeof
thefrequencyfunction
~ G N (e i! )= ^ G N (e i! ) G 0 (e i! ) (5)
In this contribution we shall consider the more
general situation of possibly non-linear model
errors,anddiscusstherolesthatanexplicitmodel
errormodelwillplayinsuchaset-up.
The discussion is con ned to
single-input-single-output systems/models and to stability
robust-nessissuesonly.Thisis asimpli cationanddoes
2. MODELVALIDATIONANDMODEL
ERRORMODELING
To validate a model is to confront it with facts
aboutthesystem.Anidealsituationisthatthere
is prior knowledge about the system which can
becompared withthe model's properties. Often,
however,theonly\facts"thatareathandarethe
datathemselves.Itisdesirablethatthedataused
forvalidation arenotthesameasforestimation,
a separate \validation data set" should be used.
It then boils dow to comparing what the model
thinkstheoutputshouldbewithwhattheactual
measured output turned out ot be. That is, the
residuals, the model left-overs, should be
exam-ined:
"(t)=y(t) y^
N
(t) (6)
Herey^isthesimulatedoutput
^ y N (t)= ^ G N (q)u(t) (7) ^ G N
isthenominal model, whichmaybetheone
estimated in (2), be may alsobechosenin other
ways.
Ifanoisemodel ^
Hhasbeenestimateditisnatural
toconsidera lteredversion
" F (t)= ^ H 1 N (q)"(t) (8)
andsubjectthatto theanalysis below.
2.1 Classical Correlation AnalysisofResiduals
One of the most basictests, (Draper and Smith
1981),is to computethecorrelation betweenthe
regressors, in our case the past inputs, and the
residuals: ^ r N ( )= 1 N N X t=1 u(t )"(t) (9)
Itis customaryto plottheseestimatesasa
func-tion of and compare with theirstandard
devi-ations to check if they are signi cantly di erent
from zero.If not,wehavenot tracedany
signi -cantin uenceofuin",sowecannotsaythatthe
model ^
Ghasnot pickedupall thein uence ofu
ony.(Notethedoublenegation:wearenotsaying
that\ ^
Ghaspickedupall...").Itisconvenientto
form '(t)= u(t 1) ::: u(t M) T (10) h M N = 2 6 4 ^ r N (1) . . . ^ r N (M) 3 7 5 = 1 N N X t=1 '(t)"(t) (11)
Under theassumption that " iswhite noisewith
variance,hhasanormaldistributionwithzero
meanandvariance=NR
N ,where R N = 1 N N X t=1 '(t)' T (t) (12) M N = 1 N k N X t=1 '(t)"(t)k R 1 N (13)
will in this case have a 2
distribution, and the
familiar 2 -test M N < (14)
isbasedonthis.Theresultistypicallypresented
as a plot of the autocorrelation of the residuals
andaplotof (9).SeeFigure1.
Notethatotherkindofdependencescanbetested
quiteanalogouslybyletting'(t)beother,
nonlin-ear,functions ofpastinputs:
'(t)=f(u t
) (15)
2.2 CorrelationAnalysisasModelErrorModeling
0
5
10
15
20
25
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Correlation function of residuals. Output # 1
lag
−25
−20
−15
−10
−5
0
5
10
15
20
25
−0.2
−0.1
0
0.1
0.2
0.3
Cross corr. function between input 1 and residuals from output 1
lag
Fig. 1. Traditional residual analysis: Auto- and
cross-correlation functions with uncertainty
regions.
Theinformationfromthecrosscorrelation
analy-sisbetween"andu,canalsobeinterpretedasan
implicitFIRmodel forthetransferfunction ~
Gin
"(t)= ~
G(q)u(t)+w(t) (16)
from u to ". For control purposes, it is much
more e ective to present the (amplitude)
fre-quency function of this model error model, with
uncertaintybounds asinFigures 2{4.Thedata
usedin these guresaresimulatedfromasecond
orderARMAXmodel.Itisclearthatconventional
model validation corresponds to increasing the
modelcomplexityuntilthemodelerrormodelhas
uncertaintyboundsthatincludezero(asinFigure
4), since then there is no clear evidence that ~
G
is not zero { the estimated model is then not
falsi ed. But it is also clear that the two plots
together;themodelandits\sidekick",themodel
errormodel,canbeusedforcontroldesign,evenif
themodelisfalsi ed.LookatFigure3.According
well beused forcontrol designiftheinformation
inthelowerplotistakenintoaccount.
10
−1
10
0
10
1
10
2
10
−4
10
−2
10
0
10
2
10
−1
10
0
10
1
10
2
10
−4
10
−2
10
0
10
2
Fig. 2. Upper plot: Amplitude Bode plot of a
rstordermodelwithestimateduncertainty
bounds. The true system is also plotted.
Lowerplot:Themodelerrormodelcomputed
asa20:thorderARXmodelfromutoy ^ Gu
10
−1
10
0
10
1
10
2
10
−4
10
−2
10
0
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2
10
−1
10
0
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1
10
2
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−4
10
−2
10
0
10
2
Fig. 3. As in previous gure, but second order
ARXmodel
10
−1
10
0
10
1
10
2
10
−4
10
−2
10
0
10
2
10
−1
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0
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1
10
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−4
10
−2
10
0
10
2
Fig. 4. As in previous gure, but second order
ARMAXmodel.Herethemodelerrormodel
containszeroinitsuncertaintyregion,which
meansthatthetopmodelisnotfalsi ed.
GAINS
3.1 TheGeneralForm
A model error model is a description of how u
a ects ".We willtypicallynotbeinterestedin a
detailed suchdescription{that shouldrather be
doneasamoredetailednominalmodel.Primarily
wefocusonbounds onthegainof such amodel.
To go well together with linear control design,
we shall work with combinations of frequency
weightings and unstructured, unknown models
withbounded(estimated)gain.Thatmeansthat
thegeneralstructureofthemodelfromuto"can
bedepictedasinFigure5.Formallywehave
w " ~ g mem u W 1 W 2
Fig.5.Themodelerrormodel
" F (t)=W 1 2 (q)"(t); u F (t)=W 1 (q)u(t) (17) " F (t)=g~ mem (u t 1 F )+w(t) (18) Here, W 1 and W 2
are given linear lters,
super-scriptt 1denotesallthesignalvaluesuptotime
t 1,andg~
mem
is anuncertainmodel. The
bot-tomlineisthat,withsomeamountofcon dence,
basedonthemeasureddata,andpossiblyonprior
information,weshouldbeabletosaythatitsgain
isboundedby1: k~g mem k 1 <1 (19)
Normalization can always be achieved using W
1
andW
2
.Weshallshortlydiscusshowtoestimate
suchabound.
Themodel~g
mem
cancome indi erentshapes.It
couldbe
Linear Time Invariant. In this case we can
take either W
1 orW
2
to be unity, since all
theblocksinin Figure5commute.
LinearTimeVarying.
GeneralNon-Linear.
3.2 Estimating the Gain of the Model Error
Model: DistinguishingModel Errorfrom Noise
Theestimationtaskistodeterminethe ltersW
i
andestimatethegainofg~
mem in " F (t)=g~ mem (u t 1 F )+w(t) i.e.estimate sup t sup t k~g mem (u t )k ku t k
to determine this gainwithout prior knowledge.
Even in the linear case, an arbitrarily thin
res-onance peak will require an arbitrarily long
se-quence of data, due to theuncertaintyprinciple.
After having considered a nite data record, no
guarantee canbe given for the \true"gain. The
estimate of the gain will always re ect the data
thathavebeenmeasured,possiblyinconjunction
withpriorknowledge.
A rstfundamental problemistodistinguishthe
model error ~g
mem
from the noise contribution
w. Loosely speaking, the noise w would be that
contribution to " that would not change if u is
changed.Thisproblemisaddressedin(Smithand
Dullerud 1996) by looking at trade-o s between
modelerrorsizeandhardamplitudeboundsonw.
From astatistical perspective,it ismorenatural
totryand\correlateout"w.Therearetwobasic
methodsforthat:
Useaperiodicinput,andeliminate/reducew
byaveragingovertheperiods.
Build a(parametric) model of ~g
mem
in (18)
assuminguandwtobeindependent.
3.3 UsingPeriodicInputs
Suppose the noise contribution is additive as in
(18)andthat uisperiodicwith periodP. Then,
afteratransienthasdiedout,alsoy^
N
(t)(see(7))
will be periodicwith thesameperiod. Moreover,
thepartof y that originatesfrom theinputwill,
under weak assumptions, also be periodic with
periodP.Letydenotetheoutputoveroneperiod,
obtained by averaging over all the periods, and
similarlyfor" F (t): " F (t)=W 1 2 (q)(y(t) y^ N (t)); t=1;:::;P
From(18)wenowhave
" F (t)=~g mem (u t 1 F )+w(t)
where w will tend to zero asN tends to in nity
(moreandmoreperiodsareaveragedover).
A simpleunder-bounding estimateof thegainof
~ g mem is \ k~g mem k 2 = P P t=1 " 2 F (t) P P t=1 u 2 F (t) (20)
A natural choice of lter is W
1 (q) =
^
G (q) (the
nominalmodel).Thismakes
u F (t)=y^ N (t) (21) Moreover,W 2 canbechosenas W 2 (q)= ~ H(q),a
\noisemodel"fory(t) y^
N (t), y(t) y^ N (t)= ~ H(q)(t) (22)
with (t) beingwhite noise. That will makethe
numeratorin(20)assmall aspossible.Notethat
thismakes \ k~g mem k 2 = P P t=1 [W 1 2 (q)(y(t) y^ N (t))] 2 P P t=1 ^ y 2 N (t) (23)
a measure of the proportion of \unexplained"
output variation.Thishasacloserelationshipto
themultiplecorrelationcoeÆcientusedinresidual
analysis,page33in (DraperandSmith1981).
3.4 GeneralLinearBlackBoxModelErrorModels
The model error model concept gives us more
freedom in investigating the residuals than the
classical residual correlation test, which
essen-tiallyemploysaFIRmodel.Amoregenerallinear
model "(t)= ~ G(q)u(t)+ ~ H(q)w(t) (24)
could improve the estimate of the error model,
sinceamodelofthedisturbanceisused.
Insteadofparametriclinearmodels,wemayapply
spectral analysis to try and extract any linear
in uence of uon ",(Kosut 1986),(Stenman and
Tjarnstrom2000).Inanycase,thereisaclose
re-lationship between the Blackman-Tukeyspectral
analysisestimateofthistransferfunctionandthe
oneobtainedbyaFIR-model.
Once we have a speci c, linear model of the
in uence from uto ", along with an uncertainty
(con dence) region, the error model in Figure 5
is directly de ned:Take W
2 =1 and W 1 as ~ G N +
. Here is the estimated variance error (at a
certain con dence level), so that the amplitude
curve of W
1
corresponds to the upper curves in
(thelowerplotsof)Figures2-4.Ifnononlinearity
hasbeendetectedin theerrormodel,this choice
ofW
i
inFigure5givesacorrectdescriptionofthe
erroranduncertaintyassociatedwiththenominal
model.
3.5 Non-linear Model Error Models
Intheliterature,mostmodelerrordiscussionsas
well as the identi cation-for-control approaches
aredealtwithinasettingwhere\thetruesystem"
isahighorderlinearmodel,andthemodelsareof
lowerorder.In practicaluse,itis ofcourse more
common that the model errors are ignored
non-linearities rather than unmodeled linear
dynam-ics. From a model error perspective, this simply
meansthatweshould testnon-linearmodels:
" F (t)=g~ mem (u t 1 F )+w(t) (25)
neural network NNFIR model, cf (Sjoberg et al. 1995): " F (t)=~g mem (u F (t 1);:::;u F (t m);)+w(t) (26)
Thenumberoflaggedinputscanbechosen
rela-tivelysmall here,likem=5orso.Toappreciate
thesize (gain) ofanyestimated non-linearity(in
particularforcontrolapplications)itisnaturalto
usethesup-norm
k~g mem k 1 = sup u1;:::;um j~g mem (u 1 ;u 2 ;:::;u m ;)j^ 2 u 2 1 +:::+u 2 m (27)
Then alsodetermine theworstcase valueof this
gaininaproperlychosencon denceregionforthe
estimate:^ k~g mem k=sup ^ 2 k~g mem (u(t 1);:::;u(t m);)k^ 1 (28)
4. ROBUSTCONTROLDESIGN FOR
UNCERTAINSYSTEMS
4.1 LinearModelErrors
Once a model with its model error uncertainty
is delivered, the question is how to design a
controller that will stabilizethe systemrobustly.
Bythiswewouldmeanthatthechosencontroller
should stabilize allmodels inthe \region"de ned
bythe nominalmodelandthe model error model.
Incasewehaveusedalinearmodelerrormodel,
this region is easily depicted in the frequency
domain. It will look likea strip in the Bode, or
Nyquistplot.SeeFigures2-4,i.e.
G2G=fGj jG(e i! ) G nom (e i! )j<(!)g (29)
How to achieverobust stability forsuch aset of
modelsiswellknown:ChoosearegulatorK,such
thatthecomplementarysensitivityfunction
T = G nom K 1+G nom K (30)
islessthantheinverserelativemodelerrorbound:
jT(e i! )j< G nom (e i! ) (!) ; 8! (31) H 1
techniques canbeused to determine ifsuch
aKexists,forgivenG
nom
and.See,e.g.(Zhou
etal.1996).
4.2 Frequency Weighted Non-linearModel Error
Model
The error model (18) corresponds to a closed
loop block diagram as in Figure 6. This can be
y ~ g mem G nom u K W 1 W 2
Fig. 6. Block diagram of the feedback loop with
model error
rearranged to be seen as feedback between the
non-linear part of the error model ~g
mem and KW 1 W 2 =(1+KG nom
)(keepingin mindthat we
onlyconsiderSISOmodelshere).Supposethatwe
normalizeW
1 W
2
sothatthegainofthenon-linear
partissubjectto k~g mem k 1 <1 (32)
The small gaintheorem tellsus that stability is
assuredif W 1 (e i! )W 2 (e i! )K(e i! ) 1+K(e i! )G nom (e i! ) <1 8! (33)
Comparingwith (31)werealizethat we just can
considerthesetofpossiblesystemdescriptionsto
belinearandgivenby
G2G=fGj jG(e i! ) G nom (e i! )j<::: (34) <W 1 (e i! )W 2 (e i! )g
By stabilizing any linear model in this set, i.e.,
achieving(31)for=W
1 W
2
,wehavealsomade
thelinearcontroldesignrobustagainstnon-linear
model errorsof the type (18).(Considering only
stabilityrobustness.)
5. ESTIMATINGNON-LINEAR FREQUENCY
WEIGHTEDERRORMODELS
Whendealingwithnon-linearmodelerrormodels
forcontroldesign,theoptionoffrequency
weight-ing isessential.As anexample,supposethetrue
systemis ~ y(t)= 1 (q 1)(q ) u(t) (35) y(t)=f(~y(t)) (36)
forsomestatic,linearlybounded, nonlinearityf.
Suppose,forsimplicity,thatthenominalmodelis
G
nom
(q) = 1=(q 1)(q ) so the model error
isthisnominalsystemfollowedbythestatic
non-linearityf(y) 1.Suppose that kf(y) 1k<1.
Thegainofthismodelerrormodelisin nite,due
tothein nitegainofthelinearpart.So,thiserror
modeldoesnotdomuchgoodforcontroldesign.
IfwenowpicktheweightingsW
1
(q)=1=(1 q),
weightingW
2
in (33)is no majorproblem, since
thedenominatoralsohasin nitegainatzero.
NotethatitisonlytheproductW
1 W
2
thata ects
therobustnesstest,whilethesplitintoW
1 andW
2
mayhavea substantial in uence onthe norm of
~ g
mem
.Thisgivesanextradimensiontothemodel
error modeling. In loose terms the weightings
should be chosen so as to minimize k~g
mem k 1 whileletting W 1 W 2
haveashapethat suits(33).
A morecomprehensive discussionof howto deal
with this freedom has to be deferred to another
occasion.Somehintsare, however,asfollows:
Using W
2 =
^
H (the nominal noise model)
reducesthe poweroftheoutput of the
non-linearblock as much aspossible, givingthe
possibilityofasmallergainofthenon-linear
part.
Using W
1 =
^
G (the nominal model), will
focustheerrormodelontherelativeerror.
6. CONCLUSIONS
Building linear models may be deceptive. Given
any data set,of arbitrary length, wecan always
build a linear model, that will pass standard
residual analysis validation tests. The reason is
that any (non-linear, time-varying) system from
whichquasistationaryinput-outputsequencesare
collected,hasalinearsecondorderequivalent.We
cannotdistinguishthetruesystemfromthislinear
model, using just secondorder tests.This means
that the standard linearidenti cation setup will
deliveramodelwithuncertainty
^
G
where the(variance)error will tendtozero as
moredata are processed. Subjecting the data to
themodel-error-modeltreatmentdescribedinthis
contributionwillgiveamodel
^
G; =W
1 W
2
where will tend to zero, only if non-linearity
testsshownosignofnon-linearities.Otherwise
will re ect aremaining uncertainty/model error.
However,ifwejust consider stabilityrobustness,
itwillbesafetotreatthesystemasalinearone,
withthegivenuncertainty, evenifnon-linearities
havebeendetected.
7. REFERENCES
Draper, N.R. and H. Smith (1981). Applied
Re-gressionAnalysis,2nded.. Wiley,NewYork.
Gevers,Michel (1993).Towards ajointdesign of
identi cationandcontrol?.In:Essayson
con-trol:Perspectivesinthetheoryandits
appli-cations (H L Trentelman and J C Willems,
Kosut, R., M. K. Lau and S. P. Boyd (1992a).
Set-membership identi cation of systems
with parametric and nonparametric
uncer-tainty.IEEETrans.AutomaticControl
AC-37,929{941.
Kosut, R.L.(1986).Adaptivecalibration:An
ap-proach to uncertainty modeling and on-line
robust control design. In: Proc. 25th IEEE
Conference on Decision an Control. Vol. 1.
Athens,Greece.pp.455{461.
Kosut,R.L.,G.C.GoodwinandM.P.Polis(Eds)
(1992b). Special Issue on System
Identi ca-tionforRobustControlDesign,IEEETrans.
AutomaticControl,Vol37.
Ljung,L.(1999a).SystemIdenti cation-Theory
for the User. 2nd ed.. Prentice-Hall. Upper
SaddleRiver,N.J.
Ljung, Lennart (1999b). Model validation and
model errormodeling. In:The
Astrom
Sym-posiium on Control (B. Wittenmark and
A. Rantzer, Eds.). Studentlitteratur. Lund,
Sweden.pp.15{42.
Ninness, B. and G. C. Goodwin (1995).
Es-timation of model quality. Automatica
31(12),1771{1797.
Poolla,K.,P.P.Khargonekar,A.Tikku,J.Krause
and K.Nagpal (1994). A time-domain
ap-proachto model validation.IEEE Trans.on
AutomaticControl AC-39,951{059.
Rangan, S. and K. Poolla (1996). Time-domain
validation for sample-data uncertainty
mod-els. IEEE Trans. Automatic Control
AC-41,980{991.
Sjoberg, J., Q. Zhang, L. Ljung, A. Benveniste,
B. Delyon, P.Y. Glorennec, H. Hjalmarsson
andA. Juditsky(1995).Nonlinearblack-box
modeling in system identi cation: A uni ed
overview.Automatica31(12),1691{1724.
Skelton, R. E. (1989). Model error concepts in
controldesign.Int.J.Control49,1725{1753.
Skogestad, S.and I.Postlethwaite(1996).
Multi-variableFeedbackControl.JohnWiley.New
York.
Smith,R.andG.E.Dullerud(1996).
Continuous-time control model validation using nite
experimental data. IEEE Trans. Automatic
ControlAC-41,1094{1105.
Smith, R. and J.C. Doyle (1992).Model
valida-tion:aconnectionbetweenrobustcontroland
identi cation. IEEE Trans. Automatic
Con-trolAC-37,942{952.
Stenman,AndersandFredrikTjarnstrom(2000).
A nonparametric approach to model error
modeling. In: Preprints of the 12th IFAC
Symposium onSystemIdenti cation. To
ap-pear.
Zhou, K., J. C. Doyle and K. Glover (1996).
Robust and Optimal Control.Prentice-Hall.
