NASA Technical Reports Server (NTRS) 20040100772: Validation of an Experimentally Derived Uncertainty Model

11 Pages·0.64 MB·English

by  NASA Technical Reports Server (NTRS)

VALIDATION OF AN EXPERIMENTALLY DERIVED
UNCERTAINTY MODEL
(cid:3) y z x
K.B. Lim, D.E. Cox, G.J. Balas and J-N. Juang
The results show that uncertainty models can be obtained directly from system identi(cid:12)cation data by using
a minimum norm model validation approach. The error between the test data and an analytical nominal
model is modeledas a combination of unstructuredadditive and structuredinput multiplicative uncertainty.
Robust controllers which use the experimentally derived uncertainty model show signi(cid:12)cant stability and
performance improvements over controllers designed with assumed ad hoc uncertainty levels. Use of the
identi(cid:12)ed uncertainty model also allowed a strong correlation between design predictions and experimental
results.
1 Introduction
this method de(cid:12)nes the smallest subset of plants about
the nominal which validates the given experimental
In many engineering problems, a linear, time-invariant
data, hence the name, minimumnorm model validation
(LTI) and (cid:12)nite dimensional model, while satisfying
(MNMV).
known physical relationships, is still an approximation
This paper is an attempt to demonstrate and vali-
of a true plant. The need for robust control is often
date the MNMVapproach formethodicallyconstructing
due to the corruption of the measurement data by the
uncertainty models for a real application. As part of a
secondary e(cid:11)ects of measurement noise, external dis-
validation,the uncertainty weights identi(cid:12)ed are used in
turbances, nonlinearities, and possible time variations.
redesigningthe controllerandcomparingtheexperimen-
Although strictly speaking, robust control theory for
talclosedloopperformancetocontrollersthatassumead
nonlinear,time-varyingsystems shouldbe appliedunder
hoc uncertainty levels. The feasibility of the method is
the above circumstances, it is currently not available.
investigated by applyingit to the Large Angle Magnetic
However, it has been shown that LTI-based robustness
Suspension Test Fixture (LAMSTF) [6]. The LAMSTF
theory can handle a class of time varying and nonlinear
is an experimental testbed located at NASA Langley
uncertainties or e(cid:11)ects via conic sector theory [1, 2, 3].
Research Center for precision pointing control studies
ThehopeisthatasmallsetofLTIplantswillbesu(cid:14)cient
in support of large gap magnetic suspension technology.
to describe these secondary e(cid:11)ects.
The LAMSTF system is open loop unstable, hence sys-
Amethodhasbeenproposedrecently[4,5]forcalcu-
temuncertainty identi(cid:12)cation(UID) is performedclosed
latingthesmallestnormofthedi(cid:11)erencebetweentheraw
loop.
systemidenti(cid:12)cationdataandthepredicted valuefroma
An analyticalmodelas derived in[7,8,9]isused as
given nominalmodel. In particular, this minimumnorm
the nominal model in this study although an identi(cid:12)ed
is calculated in closed form (to within a linear matrix
modelcanbeobtainedandusedasanominalmodelfrom
equation) and holds for a general nominal/uncertainty
the same UID data. In principle, it should not matter
structure inlinearfractionaltransformation(LFT)form.
which nominalmodel to use so long as the set of plants
Amongall modelvalidatingplants about the nominal,
canbe described withoutstretching the uncertainty size.
Hence, any nominal model provided it is not too far
away from the true unknown system could be used.
(cid:3)Research Engineer, Guidance & Control Branch, Flight Dy- For example, an observer based system identi(cid:12)cation
namics&ControlDivision,MS161,NASALangleyResearchCen-
(ID) technique [10, 11, 12] may be used to construct a
ter,Hampton,VA 23681,[email protected]
yResearch Engineer, Guidance & Control Branch, Flight Dy- singlebest nominalmodeland the residualdiscrepancies
namics&ControlDivision,MS161,NASALangleyResearchCen- between the raw system ID data and the nominalmodel
ter,Hampton,VA 23681,[email protected] are bounded by a structured uncertainty connection,
zAssociateProfessor,Departmentof AerospaceEngineering&
assumed a priori[13, 14]. This wouldyield a P-(cid:1)model
Mechanics, 107 AkermanHall, 110 Union StreetS.E., University
of Minnesota,Minneapolis,MN 55455,[email protected], for robust control design directly fromempiricaldata.
MemberAIAA Thepaperisorganizedasfollows. Section2summa-
xPrincipal Scientist, Structural Dynamics Branch, Structures rizesthemethodusedinthedeterminationofuncertainty
Division,MS230,NASA LangleyResearchCenter,Hampton,VA
models by MNMV approach. In Section 3, the LAM-
23681,[email protected],FellowAIAA
1
STF system is described brie(cid:13)y and the experimental fromanominalmodelareusedtodevelopunconservative
con(cid:12)guration and UID parameters are described. This frequency dependent structured uncertainty models.
is followed by results from UID experiments. Section To account for the discrepancies between the avail-
4 describes how a series of controllers are designed, able measured outputs and feedback signals and their
testedandthencomparedintermsofstabilityrobustness estimates from a nominal model, a priori knowledge of
and disturbance rejection performance. Conclusions are possible sources of uncertainties in the system are used.
given in Section 5. Figure 2 shows how a nominal model, P22 and an un-
certainty connection structure can be used to de(cid:12)ne the
LFT parameterized set ofplants with output noise. The
2 Uncertainty ID Algorithm
direct connection between the structured uncertainties,
y u
(cid:1),totheoutputandinputmismatch,e ande isshown.
The general form of the model structure is given in
These errors are de(cid:12)ned by
Figure 1. Using a bounded, but unknown structured
D
w’
D ~n
h x
ey - y~ P11 P12
x h H S S P21 P22 u~
P11 P12 w K(z) S - S eu
u P21 P22 y’ S y
r
n
Figure 1: A priori model structure of plant in standard y S True u S r
form plant
f
uncertainty model, a set of plants are de(cid:12)ned that K(z)
validates the availableinput-output data which contains
the deviation or scatter about the nominal. The input-
output relationship can be written as Figure 2: Con(cid:12)guration for system UID.
0
y =Fu(P;(cid:1))u+Hw (1)
(cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27)
y
e y y~
where y, u,and w0,denote the output, inputand output e:= u = (cid:0) (5)
e u u~
noise. The upper LFT is de(cid:12)ned by
(cid:0)1 It is important to note that the above errors are the
Fu(P;(cid:1))=P22+P21(cid:1)(I(cid:0)P11(cid:1)) P12 (2)
residuals that remain after a single best model (cid:12)t. This
errortimehistoryisusuallydiscardedinstandardsystem
P denotes the augmented plant. The important point
ID applications. However, this residual error is precisely
is that if the nominal P22 is known, the rest of the
the data used in generating uncertainty models. It is
augmented plant can be constructed from a priori as-
clear that this error is composed of errors due to model
sumptions on the uncertainty structure. Note that H is
mismatchand errors due to (cid:12)ltered noise. Note that the
the (cid:12)lter modelfor the noise and we let the uncertainty,
error expression in the estimated output for closed loop
(cid:1), to belong to the set of structured uncertainty D, i.e.,
ID is the same as in the open loop [4].
(cid:1)2D [13].
The algorithm used to identify uncertainty bounds
Let the overallstructured uncertainty be de(cid:12)ned by
is given in [4] for open loop and in [5] for the closed
the block diagonalmatrices
loop case. In both cases, the norm of the smallest
(cid:1)=diag((cid:1)1;...;(cid:1)(cid:28)); (cid:1)j 2Cmj(cid:2)nj (3) structured uncertainty that validates the available ID
data at each frequency is found, i.e., a minimumnorm
and the set of all block diagonal and stable, rational modelvalidation.
transfer function matrices be given by It is assumed that the controller dynamics, K, is
(cid:8) (cid:9) known and the plant inputs, u 2 Rnu(cid:2)1, and outputs,
D = (cid:1)((cid:1))2RH1 : (cid:1)j(so)2Cmj(cid:2)nj;8so 2C(cid:22)+ (4) y 2 Rny(cid:2)1, are measured while the external command,
where(cid:28) andC(cid:22)+ denotethenumberofuncertaintyblocks rha2veRdnirm(cid:2)e1nissiosnesle(cid:17)ct2edC. nT(cid:17)haen(cid:12)dc(cid:24)tit2ioCusn(cid:24)siwgnhaelrsein Figure 2
and the closed right-half plane, respectively [13]. We
consider the class of problems where the uncertainty X(cid:28) X(cid:28)
connections to the nominal and the plant inputs and n(cid:17) = nj; n(cid:24) = mj (6)
outputs are given. In the next section, these deviations j=1 j=1
2
The error in equation 5 is given by 3 Uncertainty ID of LAMSTF
(cid:0)1
e=eo(cid:0)R21(cid:1)(I(cid:0)R11(cid:1)) M12 (7)
A detailed description of the LAMSTF facility and the
where open-loopdynamicpropertiesofthemagneticsuspension
system is given in [7]. Earlier studies on system ID and
R11 = Fl(P;K) (8)
controlforLAMSTFinclude[15,8,9,16]. ForLAMSTF,
(cid:0)1
R12 = P12(I (cid:0)KP22) [I K] (9) themodeluncertaintyisduetoerrorsinthelinearization
(cid:20) (cid:21)
I (cid:0)1 about the equilibriumstate, an inaccurate knowledge of
R21 = (I (cid:0)P22K) P21 (10)
K thespatialdistributionofthemagnetic(cid:12)eld,errorsinthe
(cid:20) (cid:21)
(cid:0)1 (cid:0)1 sensorsystemhardware,anderrorsattheplantinputdue
(I (cid:0)P22K) P22 (I(cid:0)P22K)
R22 = (cid:0)1 (cid:0)1 (11) to induced eddy currents.
(I (cid:0)KP22) (I (cid:0)KP22) K
(cid:26) (cid:27) (cid:26) (cid:27)
y r
eo = (cid:0)R22 (12) 3.1 Analytical Model
u n~
(cid:26) (cid:27)
r An analytical model is derived in [7, 8, 9] and reviewed
M12 = R12 n~ (13) onlybrie(cid:13)yhere. Figure3showaschematicoftheLAM-
STF system. It basically consists of (cid:12)ve electromagnets
Notethateo istheresidualfromnominal(cid:12)twhen(cid:1)=0.
which actively suspend a small cylindrical permanent
De(cid:12)ne the above residuals at the discrete frequencies
magnet. The cylinder is a rigid body and has (cid:12)ve
j!iT
(cid:10)=(z1;...;zn!); zj =e (14) independent degrees of freedom, with motionin the roll
axis being both unobserved and uncontrolled. Reference
by takingthe discrete Fouriertransformofboth discrete
time signals and systems.
^
b
Under conditions which ensure that a model vali- 3
datingsolution exists, a solutionto the MNMV problem
is summarized as follows. A lower bound on the i-th b^
1
component uncertainty at frequency zj is given by ^
b2 x
i
(cid:27)(cid:22)((cid:1)i)=k(cid:1)ik2 (cid:21) kkxyikk22 (15) n^3
when xi 6=0, and n^ n^2
1
x:=M12+R11(w+(cid:30)); y :=w+(cid:30) (16)
which are partitioned
(cid:0) (cid:1) (cid:0) (cid:1)
1 (cid:28) 1 (cid:28)
x=col x ;...;x ; y =col y ;...;y (17)
in a conformal manner with respect to (cid:28) uncertainty Figure 3: LAMSTF Con(cid:12)guration
blocks. The word \col" denotes a columnvector formed
[8] gives a detailed derivation of both the linear and
from its arguments. Let N denote the null space of P21
nonlinear models. The followingprovides a synopsis for
and (cid:30)2N. It has dimensionn(cid:24)(cid:0)ny and is spanned by
the linear modelused in the control design.
the last n(cid:24)(cid:0)ny columnsof V. Let
By de(cid:12)ning the state vector
+ y
w=P21(I (cid:0)P22K)eo (18)
T
(cid:24) =(!2;!3; (cid:12)2;(cid:12)3; v1;v2;v3; x1;x2;x3) (21)
The singular value decompositionis given by
P21=USV(cid:3) (19) the linearizedperturbed motionabout theequilibriumis
wmhaetrreiceUs a2ndCSn2y(cid:2)Rnyny(cid:2)annd(cid:24) isVa2fullCrna(cid:24)n(cid:2)kn(cid:24)diaagreonHalermmaittriiaxn. given by (cid:14)(cid:24)_=A^(cid:14)(cid:24)+B^(cid:14)(cid:26) (22)
The pseudo-inverse is given by where the detailed expressions for A^ and B^ are given
+ + (cid:3) in [8]. Note that these analytical expressions depend
P21=VS U (20) on many physical constants including a series approxi-
Acomponentuncertaintywiththislowerboundhasbeen mation of the magnetic (cid:12)eld distribution. In fact, this
i
shown to exist. Thus the minimumnorm bound (cid:14) for (cid:12)eld distribution and its frequency dependence due to
each uncertainty block can be computed from Eq.(15). eddy currents are believed to be the primary source of
For robust control design, the minimum bounds can model errors. The variables !i, (cid:12)i, vi, and xi denotes
be overbound by a stable, realizable low-order transfer ith angular velocity of cylinder with respect to body
functions for each uncertainty block. frame,ith Euler parameterrelative toinertialframe,ith
3
101
Analytic Eigenvalues Degree-of-Freedom
0:00(cid:6)0:95i z-axis 100
0:00(cid:6)7:97i x-axis,(cid:18)y
(cid:6)9:77 y-axis Tur
(cid:6)57:80 (cid:18)z 10−1
(cid:6)58:78 (cid:18)y;x-axis
10−2
Table 1: Eigenvalues of ten state analytic model 100 101 102
101
translational velocity and displacement of the centroid
100
respectively. Six optical sensors detect in plane and out
of plane motion, and provide an over-determined set of
measurements for position in x;y; and z and rotation in Tyr10−1
pitch and yaw. The six optical sensors and (cid:12)ve control
10−2
coils yield a fully controllable and observable ten-state
system whose eigenvalues are shown in Table 1. 10−3
Thesystem’sdynamicsare dominatedbythe unsta- 100 101 102
freq (rad/sec)
ble pitch and yaw modes. These modes, called compass
(cid:14)
needle modes, result from the magnetic (cid:12)eld being 180
Figure 4: Singular values of input sensitivity (top) and
out of phase with the cylinder’s axial magnetization at
closed-loop transfer function across plant (bottom).
the unstable equilibriumpoint. Foradetaileddiscussion
of the physical signi(cid:12)cance of all modes, the interested
reader is referred to [7].
at these low frequencies is an optimistic expectation of
the actual response. The magnitudes of the input and
outputssignalsarelargestatfrequenciesnear60rad/sec.
3.2 Uncertainty ID Experiment
The output response to the UID signal is attenuated
The UID input test signal used consists of a frequency at higher frequencies. This means that an identi(cid:12)ed
weighted, zero mean, white noise random signal for all uncertainty will be limited in accuracy at both low and
(cid:12)ve inputs, r (see Figure 2). This was generated by high frequencies
(cid:12)lteringawhitenoisesignalwithstandarddeviationof1
(Ampere) through a fourth order low-pass Butterworth
3.3 Uncertainty Model
(cid:12)lterwithbreakfrequencyof60Hertz. Asecondrandom
signal with a bandwidth of 2 Hertz but with a standard It is known that in the LAMSTF system, there are
deviation of 5 (Amperes) is added to the (cid:12)rst wider several sources of uncertainties which include errors in
bandwidth signal. This second signalis used to alleviate the linearizationabout the equilibriumstate, inaccurate
the lack of power at low frequencies which mayresult in knowledge of the spatial distribution of the magnetic
a poor model at these low frequencies. The total time (cid:12)eld, errors in the sensor system hardware, and errors
oftheexcitationsignalis 40:96seconds corresponding to at the plant input due to induced eddy currents. For
13
2 discrete time points at 200 Hertz samplingrate. example, obtaining accurate analytical models of eddy
Due to the closed loop coupling, the external UID currentsusingrecentlydevelopedsophisticatedcomputer
signal introduced at the plant input is modi(cid:12)ed ac- code ischallengingespeciallywith multipleeddy current
cording to the unknown true input sensitivity transfer circuits with complex geometries [17].
function matrix. Based on a nominal model, the top In this study, we consider a combination of ad-
plot in Figure 4 show the input sensitivity matrixTyr = ditive and structured input multiplicative uncertainty
(cid:0)1
(I (cid:0) Fu(P;(cid:1))K) while the bottom plot shows the to describe the deviation of a nominal model. The
closed loop transfer function across the plant Tyr = reason for choosing this uncertainty structure is to give
(cid:0)1
(I (cid:0) Fu(P;(cid:1))K) Fu(P;(cid:1)). Of course the true plant a su(cid:14)cient degree of uncertainty freedom to permit a
denotedbyFu(P;(cid:1))isnotknownsothatthebestmodel model validating solution. Of course in general, the
before the test is used. Figure 4 also show that the UID selection of the uncertainty connections or structure is
input is attenuated both at the input and output of the still an open issue.
plant at low frequencies. This is due to the inherent Figure 5 shows the assumed connections for the
disturbance rejection property of the controller used in uncertainties. Theboundsonthisstructureduncertainty
the experiment. Note that the maximumsingular value can then be experimentallydetermined by the minimum
4
D D 1
10
add mult
S Nominal S
Sensor voltages LAMSTF Input currents Model100
minal
No
Figure 5: Assumed uncertainty structure. Unc,
Add 10−1
norm modelvalidationmethod. The P (cid:0)(cid:1) connections
are given by
(cid:26) (cid:27) (cid:20) (cid:21)(cid:26) (cid:27)
(cid:24)add (cid:1)add 0 (cid:17)add −2
(cid:24)mult = 0 (cid:1)mult (cid:17)mult (23) 10100 101 102
freq (rad/sec)
8 9 8 9
< (cid:17)add = < (cid:24)add =
(cid:17)mult =P (cid:24)mult (24) Figure 6: Additive Uncertainty: Predicted (dot), uncer-
: ; : ;
y u tainty (cid:12)t (solid), maximum(dash) and minimum(dash-
dot) singular value of nominalmodel
where the structured multiplicativeuncertainty is
(cid:1)mult =diag((cid:14)1;...;(cid:14)5); (cid:14)i 2C (25)
Figure7showsthecalculatedminimumnormmulti-
plicativeuncertainty(dot)andthecorrespondingsecond-
The augmented plant for the diagonal multiplicative
order (cid:12)t (solid line) for each input channel. The multi-
uncertainty is
2 3
plicative uncertainty levels are generally larger at low
0 I I
P =4 0 0 I 5 (26) frequencies and roll o(cid:11) with increasing frequencies.
Input channel number 1 shows largest uncertainty
I G G
levels of up to about 90 % but at frequencies near
The above variables have the following dimensions:
crossovertheuncertaintylevelsofallinputmultiplicative
6(cid:2)1 5(cid:2)1 5(cid:2)1 5(cid:2)1
(cid:24)add 2 C , (cid:24)mult 2 C , (cid:17)add 2 C , (cid:17)mult 2 C , uncertainties are about 10 % or less. However, each
6(cid:2)5 5(cid:2)1 6(cid:2)1
(cid:1)add 2C , u2R , and y 2R . channel appears to have di(cid:11)erent break frequencies for
The uncertainty bounds are calculated at discrete
the roll o(cid:11). For example, channel 1 having larger
frequencies which are linearly spaced. A white noise
uncertainty levels at lowfrequencies than channel 2 rolls
(cid:0)6
signal having a spectral density of 10 is assumed for
o(cid:11)atabout2Hzascomparedtoabout10Hzforchannel
the presence of measurement noise. Since the number of
2. Notealsothatthese experimentallybaseduncertainty
uncertaintychannels(q =11)isgreaterthanthenumber predictionsareverydi(cid:11)erent whencomparedtoassumed
ofoutputs (ny =6),P21 isrectangular anditsnullspace constant uncertainty levels. The consequence of this
will be of minimum dimension 5. This means that the
di(cid:11)erence is evident fromexperimental results.
uncertainty bounds can be further reduced if this null
space freedom is utilized. However, this freedom is not
considered in this study. 4 Performance Validation
Figure 6 shows the calculated minimum norm ad-
ditive uncertainty at discrete frequency points over the The main objective of the control design is to stabilize
wholerangeofNyquistfrequency. Thecurve (cid:12)twasper- the highlyunstable open loopequilibriumcon(cid:12)guration.
formedinteractivelyusingthe(cid:22)(cid:0)Tool [14]drawmagrou- Subject to closed loop stability, reasonable disturbance
tine using a stable, rational,(cid:12)rst-order transfer function rejection performance across the plant is desired. Of
(solid line). The maximum(dashed line) and minimum course the above objectives are complicated by the
(dash-dot line) singular value response of the nominal presence of modelerror inaccuracies which are painfully
plant are shown for comparison. It is seen that at evident during experiments. Hence, robust disturbance
frequencies belowapproximately7Hz,the uncertaintyis rejection is sought which requires a de(cid:12)nition of the
lessthantheminimumsingularvalueresponseindicating uncertainty set for whichthe performance isguaranteed.
that the additive uncertainty levels are very smallwhile Uncertainty in the uncertainty model itself is a real
athigherfrequencies(above7Hz),thepredictedadditive dilemma which motivates this study which is a contin-
uncertainty e(cid:11)ect becomes more important. At frequen- uationofanearlier study [9]. Ofcourse nowwehavethe
cies beyond 60 Hz, predicted uncertainty is unreliable bene(cid:12)t of a new tool [4, 5] to more realistically capture
because the ID input signal is bandlimited. modeluncertainty.
5
Input Pi Zi ki Contr Wunc Wy (cid:22)des (cid:22)ID kkyrkk22 (cid:27)(cid:22)(Tyr)
1 -7.915(cid:6) 12.26i -19.80(cid:6) 381.5i 1.127e-03
C01 .001 1 .17 5.44 U
2 -22.57(cid:6) 44.16i -14.92(cid:6) 430.2i 1.377e-03
C02 .1 1 1.89 2.89 0.29 2.69
3 -14.89(cid:6) 17.74i -15.78(cid:6) 420.5i 1.294e-03
C03 .2 1 3.22 3.18 MU
4 -13.92(cid:6) 19.96i -14.01(cid:6) 418.3i 1.243e-03
C04 .3 1 4.57 3.48 MU
5 -27.32(cid:6) 49.56i -30.91(cid:6) 427.1i 1.115e-03
C11 IDed 1 1.56 1.56 0.19 1.40
Table 2: Parameters for multiplicativeuncertainty (cid:12)t. C022 .1 .1 1.41 5.63 U
C023 .1 .01 1.25 22.5 U
C12 IDed .1 1.29 5.85 0.42 5.67
C13 IDed .01 1.23 28.5 0.84 26.4
4.1 Controller Design
Table 3: (cid:22) Control laws tested, U=Unstable,
Figure 8 shows the interconnection used in the design of
MU=MarginallyUnstable.
H1 and (cid:22) controllers [13]. The robust performance is
de(cid:12)ned by the principal gains such that
(cid:2) (cid:3)
(cid:0)1
(cid:27)(cid:22) Wy(I (cid:0)Fu(P;(cid:1))) Fu(P;(cid:1)) (cid:20)1;8! 2[0;1) (27) The (cid:12)rst set of control designs, C01,C02,C03, and
C04, is based on assumed constant uncertainties at four
where the set of plants de(cid:12)ned by the uncertainty,
di(cid:11)erent levels :001, :1, :2 and :3, respectively. The
second control design, C11, is based on an identi(cid:12)ed
(cid:1) = blk(cid:0)diag((cid:1)add;(cid:1)mult) (28)
uncertaintymodelandusesthesameunityoutputweight
(cid:1)add = (cid:1)^addWadd; (cid:27)(cid:22)((cid:1)^add)(cid:20)1 (29) as the (cid:12)rst set. A thirdset ofcontrollers, C022 andC023,
(cid:1)mult = (cid:1)^multWmult; (cid:27)(cid:22)((cid:1)^mult)(cid:20)1: (30) are based on the assumed uncertainty level of :1 which
wasusedtodesigncontrollerC02butwithsmalleroutput
andtheinputmultiplicativeuncertaintyhasthediagonal weightsof:1and:01respectively. Thefourthandlastset
form of controllers, C12 and C13, uses the scaled down output
weights with the identi(cid:12)ed uncertainty weights.
(cid:1)^mult = diag((cid:14)^1;...;(cid:14)^5) (31)
Wmult = diag(wm1;...;wm5) (32)
4.2 Performance Comparison
The additive uncertainties are (cid:12)tted with the following In this subsection, we evaluate the validity of the iden-
(cid:12)rst-order stable s-domaintransfer function ti(cid:12)ed uncertainty model by comparing predicted robust
stability (RS) and robust performance (RP) to actual
(cid:0)3(s+3380:7)
Wadd =3:7708(cid:2)10 I5(cid:2)5 (33) experimental results.
(s+60:615)
The multiplicative uncertainties are (cid:12)tted with the fol- 4.2.1 Stability Robustness
lowingsecond-order stable transfer functionsofthe form
In the (cid:12)rst set of control designs only controller C02,
(s+Zi)(s+Zi(cid:3)) which assumes uncertainty level of :1, was stable. Con-
wmi =ki(s+Pi)(s+Pi(cid:3)); i=1;...;5 (34) troller C01 was violently unstable while controllers C03
and C04 were marginallyunstable, i.e., it built up oscil-
and the poles, zeros, and gains of the transfer functions lationsslowlytoeventuallygooutofrangeofthesensors.
are given in Table 2. The uncertainty weight for both In contrast, a single control design, C11, which is based
additive and multiplicative case are subsequently dis- on the identi(cid:12)ed uncertainty model,was stable and gave
cretized using the Tustin approximationof a continuous good performance without trial and error.
(cid:12)lter. The output performance weight, Wy, is chosen to Based on the set of plants de(cid:12)ned by the nominal
beaconstantdiagonalmatrixandequalforallsixoutput and identi(cid:12)ed uncertainty model and the unity output
channels. weight, the predicted RS, nominal performance (NP),
To validate the experimentally derived uncertainty and RP were calculated for all nine controllers resulting
model, four sets of (cid:22) controllers are considered. Table toFigures9through17. By comparingthepredicted RS
3 shows the four sets of controllers which were designed (solidline),thefourcontrollersthatwereactuallystable,
and tested. In all cases, the analytical model is used as C11,C12,C13,C02, have the four best predicted RS levels
the nominalmodel. The (cid:12)rst two sets of controllers are with respect to the identi(cid:12)ed uncertainty.
based on unity output weights while the next two sets The two marginally unstable controllers, C03,C04,
assume much smalleroutput weights of :1 and :01. have slightly worse predicted RS levels than the least
6
stable controller, C02 among the four stabilizing con- Clearly, if the set of plants assumed is in question, so
trollers. This suggests that the true system uncertainly is the reliability of the predicted robust performance.
o
level, Wunc, lies in the narrow margin between C03 (or For instance, (cid:22)des for controller C01 is the smallest at
C04) and C02, i.e., :17(see Table 3)but when implemented,the closed loop
system for this controller resulted in the most unstable
Wunc o Wunc system among the nine controllers tested. On the other
(cid:20)Wunc (cid:20)
(cid:22)Wunc([Fl(P;C03)]11) (cid:22)Wunc([Fl(P;C02)]11) hand, the best RP predicted by (cid:22)ID is controller C11 at
(35) 1:56. When tested, this controller actuallygave the best
The most violently unstable controller, C01, corre- performance out of the nine controllers.
sponds to the worst RS prediction as shown in solidline In order to experimentally validate the RP predic-
in Figure 9. This controller design practically ignores tions, a system ID experiment was conducted on the
RS by assuming Wunc = :001(cid:2)I10(cid:2)10. Note that since disturbance to output path of the closed loop system.
stabilitywasattained in the four controllers in spite ofa Identifyingthesysteminthispathallowsthecalculation
violationof RScondition (ofless than unity) by a factor of the worst case response, i.e., the maximum singular
ofupto2atlowerfrequencies, the identi(cid:12)eduncertainty value over all frequencies. A wideband uncorrelated
model is slightly conservative and the RS condition is disturbance input was added to each of the control coils
only a su(cid:14)cient condition for true stability. However, a and measurements recorded from the system’s sensors.
larger violation of RS condition as in controllers C022, This data was used with the OKID algorithm [10] to
C023, and C01 (see Figures 14,15, and 9) results in an generate state-space models of the closed-loop system’s
unstable closed loop system. disturbance path, denoted as Tyr. The maximumsingu-
Motivated by the actual stability of controller C02, lar values were then calculated to obtain the identi(cid:12)ed
controllers C022 and C023 were designed with reduced worst case response subject to system ID limitations.
output weights which resulted in an improvementin the This maximum singular values were directly compared
predictedRPintermsof(cid:22)des. Iftheassumeduncertainty to the predicted (cid:22)ID bounds for robust performance.
of :1 is an accurate model, then reducing the output Figures 18, 19, 20, 21 show these comparisons for all
weight should lead to improved RS. Instead, controllers four stabilizingcontrollers over all frequencies.
C022 and C023 were unstable when implemented. This ForthecontrollerpairC02andC11,whichhavegood
instabilityishoweverconsistentwiththepredicteddegra- RP,theexperimentalsystemhadworst-caseperformance
dation of RS when evaluated with the identi(cid:12)ed uncer- just below the predicted (cid:22)bound, as shown inFigure 18
tainty model as shown in Figures 14 and 15. In stark and 19. This supports the earlier observations involv-
contrast, controllers C12 and C13 with reduced output ing robust stability, namely, the identi(cid:12)ed uncertainty
weight were stable although the disturbance rejection model was found to be slightly conservative so that the
performance is poor as intended. This low performance predicted (cid:22)isexpected tobeabovetheidenti(cid:12)edexperi-
but robust stability for controllers C12 and C13 can be mentalworstcaseresponse. Theconsistencybetweenthe
seen fromFigures16and17. Namely,thepredicted poor identi(cid:12)edexperimentalworstcaseresponseandpredicted
RPisdue topoor disturbance rejection atlowfrequency worst-case performance (based on identi(cid:12)ed uncertainty
while good RS level is maintained. This means that the model) makes a strong case for the \accuracy" of the
identi(cid:12)eduncertaintymodelused incontrollersC11,C12, uncertainty model. The second pair of controllers, C12
andC13,displaysapropertyofan\accurate"uncertainty and C13 were dominated by RS constraints (see Figure
model, namely the RS condition should not depend on 16, 17) and had poor predicted RP. However, even in
the choice of output weight. these casesthere isstillaremarkablecorrelationbetween
thepredicted worstcase performance,andthemaximum
4.2.2 Robust Performance singular values of the identi(cid:12)ed disturbance path, as
shown in Figures 20 and 21.
In Table 3, (cid:22)des denotes the designed (cid:22) with respect
The last two columns in Table 3 show the ratio
to the particular uncertainty model (assumed or iden-
of signal 2-norms between the wideband white noise
ti(cid:12)ed) and output weights. The value, (cid:22)ID denotes the kyk2
disturbance and the outputs, , and the maximum
predicted (cid:22) based on the identi(cid:12)ed uncertainty model krk2
singular value of an identi(cid:12)ed model,Tyr. FromTable 3
and the (cid:12)xed unity output weight. Thus, the di(cid:11)erences
itcan be seen that forallfourstabilizingcontrollers,the
between (cid:22)des and (cid:22)ID values are solely due to the
followingrelation is satis(cid:12)ed:
di(cid:11)erence in both the uncertainty model assumed and
the output weight. Therefore, (cid:22)des equals (cid:22)ID only for kyk2
C11. krk2 (cid:28)(cid:27)(cid:22)(Tyr)(cid:25)(cid:22)ID; (36)
Recall that a predicted RP is meaningful only with
respect to the performance de(cid:12)nition and the set of Thepredictedandmeasuredworstcaseresponsematches
plantsde(cid:12)ned bythenominalandanuncertainty model. approximatelywhiletheyclearlyboundaparticularratio
7
of signal 2-norms. in IEEE Transactions on Automatic Control and
1997 ACC.
[6] Groom, N.J., \Description of the Large Gap Mag-
5 Conclusions
netic Suspension System (LGMSS) Ground-Based
Experiment," Technology 2000, NASA CP-3109,
This study demonstrated the use of a recently proposed
Vol. 2, 1991,pp.365-377.
algorithm for determining uncertainty models directly
from system uncertainty identi(cid:12)cation test data. Over- [7] Groom,N.J.,andBritcher,C.P.,\Open-LoopChar-
all, there was a strong correlation between actual and acteristics of Magnetic Suspension Systems Us-
predicted robust stabilityand performance. Because the ing Electromagnets Mounted in a Planar Array,"
LAMSTF testbed is highly unstable in open loop and NASA-TP 3229,November 1992.
is sensitive to model errors, these results represent a [8] Lim,K.B.,andCox,D.E.,\RobusttrackingControl
signi(cid:12)cant experimental demonstration of the identi(cid:12)ed of a Magnetically suspended Rigid Body," 2nd Int.
uncertainty modeland subsequent robust controller per- Symp. on Magnetic Suspension Technology, 1993.
formance. Hence we conclude that the results validate
[9] Lim, K.B., and Cox, D.E., \Experimental Robust
the identi(cid:12)ed uncertainty model.
Control Studies on an Unstable Magnetic Suspen-
Robustcontroldesignbased onthe identi(cid:12)eduncer-
sion System," 1994 American Control Conference.
tainty model produced signi(cid:12)cantly better performance
in terms of stabilityand robust performance. Use of the [10] Juang, J-N., Phan, M., Horta, L.G., Longman,
identi(cid:12)ed uncertainty model also signi(cid:12)cantly improved R.W., \Identi(cid:12)cation of Observer/Kalman Filter
predictability. This reduces the need to tweak around Markov Parameters: Theory and Experiments,"
both performance and uncertainty weights in a robust Journal of Guidance, Control, and Dynamics,
control design and subsequent application for real sys- vol.16,No.2,March-April 93, pp.320-329.
tems to obtain satisfactory performance. [11] Juang, J-N., Phan, M., \Identi(cid:12)cation of System,
The robust stability and performance results in- Observer, and Controller from Closed-Loop Exper-
dicate that the identi(cid:12)ed uncertainty was not overly imental Data," Journal of Guidance, Control, and
conservative. However,thenullfreedomavailableforthe Dynamics, vol.17,No.1,Jan-Feb 94, pp.91-96.
su(cid:14)ciently parameterized uncertainty freedoms,was not
[12] Juang, J-N., Applied System
used to reduce the minimum norm uncertainty bound.
Identi(cid:12)cation, Prentice-hall, Inc., Englewood Cli(cid:11)s,
Althoughtheexperimentalresults giveninthisstudyare
New Jersey, 1994, chapter 8.
based on a particular structure of uncertainty, namely,
a combined additive and unstructured input multiplica- [13] Stein, G., and Doyle, J.C., \Beyond singular values
tive uncertainties, the system uncertainty identi(cid:12)cation and loop shapes," Journal of Guidance, Control,
methodologyapplies to the general LFT framework. and Dynamics, vol.14,No.1,Jan-Feb 91, pp.5-16.
[14] Balas, G.J., Doyle, J.C., Glover, K., Packard, A.,
and Smith, R., (cid:22)-Analysis and Synthesis Toolbox,
References User’s Guide, The MathWorks, Inc., Natick, MA.,
1995.
[1] Safonov,M.G.,Stabilityandrobustnessofmultivari-
[15] Groom, N.J. and Scha(cid:11)ner, P.R., \An LQR Con-
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[2] Zames, G., \On the input-output stability of time- 1990.
varying nonlinear feedback systems- Part I," IEEE
[16] Huang,J-K,andCox,D.E.,\ClosedloopsystemID"
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[17] Britcher, C.P., and Groom, N.J., \Eddy current
[3] Waszak, M.R., \A methodology for computing un-
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modelvalidatinguncertainty,"Submittedforreview
8
100 RS (−), NP (−.), RP (−−) for C_01
6
1
c
n −2 5
U10 e
ult Valu4
M ar
−4 ul
10 g
0100 101 102 m Sin3
10 u
m2
2 axi
c M
n −2 1
U10
ult 0
M 0 1 2
−4 10 10 10
10 Freq (rad/sec)
0 1 2
10 10 10
0
10
Figure 9: Predicted robust stability (solid), nominal
3
c performance (dash-dot), and robust performance (dash)
n −2
U10 for controller C01.
ult
M
−4
10
0 1 2 RS (−), NP (−.), RP (−−) for C_02
10 10 10
0 3
10
4
c 2.5
Un10−2 alue
ult ar V 2
M ul
g
10−4 Sin1.5
0 1 2 m
010 10 10 mu 1
10 axi
M
5
c 0.5
n −2
U10
ult 1000 101 102
M Freq (rad/sec)
−4
10
0 1 2
10 10 10
Figure 10: Predicted robust stability (solid), nominal
freq (rad/sec)
performance (dash-dot), and robust performance (dash)
for controller C02.
Figure 7: Input Multiplicative Uncertainty: Predicted
RS (−), NP (−.), RP (−−) for C_03
(dot), uncertainty (cid:12)t (solid). 4
h h
D^add addWadd D^mult multWmult ular Value3
x x Sing2
add mult um
y y m
w Wy S G S Maxi1
u 0
0 1 2
10 10 10
K S Freq (rad/sec)
r Figure 11: Predicted robust stability (solid), nominal
performance (dash-dot), and robust performance (dash)
for controller C03.
Figure 8: Interconnection for robust controller design.
9
RS (−), NP (−.), RP (−−) for C_04 RS (−), NP (−.), RP (−−) for C_023
4 30
25
ue3 ue
al al
V V20
ar ar
ul ul
g g
Sin2 Sin15
m m
u u
m m10
Maxi1 Maxi
5
0 0
0 1 2 0 1 2
10 10 10 10 10 10
Freq (rad/sec) Freq (rad/sec)
Figure 12: Predicted robust stability (solid), nominal Figure 15: Predicted robust stability (solid), nominal
performance (dash-dot), and robust performance (dash) performance (dash-dot), and robust performance (dash)
for controller C04. for controller C023.
RS (−), NP (−.), RP (−−) for C_11 RS (−), NP (−.), RP (−−) for C_12
2 6
5
ue1.5 ue
al al
V V4
ar ar
ul ul
g g
Sin 1 Sin3
m m
u u
m m2
Maxi0.5 Maxi
1
0 0
0 1 2 0 1 2
10 10 10 10 10 10
Freq (rad/sec) Freq (rad/sec)
Figure 13: Predicted robust stability (solid), nominal Figure 16: Predicted robust stability (solid), nominal
performance (dash-dot), and robust performance (dash) performance (dash-dot), and robust performance (dash)
for controller C11. for controller C12.
RS (−), NP (−.), RP (−−) for C_022 RS (−), NP (−.), RP (−−) for C_13
6 30
5 25
e e
u u
al al
V4 V20
ar ar
ul ul
g g
Sin3 Sin15
m m
u u
m2 m10
axi axi
M M
1 5
0 0
0 1 2 0 1 2
10 10 10 10 10 10
Freq (rad/sec) Freq (rad/sec)
Figure 14: Predicted robust stability (solid), nominal Figure 17: Predicted robust stability (solid), nominal
performance (dash-dot), and robust performance (dash) performance (dash-dot), and robust performance (dash)
for controller C022. for controller C13.
10