A real time adaptive controller was designed and tested successfully on a fourth order laboratory dynamic system which features very low structural damping and a noncolocated actuator sensor pair. The controller, implemented in a digital minicomputer, consists of a state estimator, a set of state feedback gains, and a frequency locked loop (FLL) for real time parameter identification. The FLL can detect the closed loop natural frequency of the system being controlled, calculate the mismatch between a plant parameter and its counterpart in the state estimator, and correct the estimator parameter in real time. The adaptation algorithm can correct the controller error and stabilize the system for more than 50% variation in the plant natural frequency, compared with a 10% stability margin in frequency variation for a fixed gain controller having the same performance at the nominal plant condition. After it has locked to the correct plant frequency, the adaptive controller works as well as the fixed gain controller does when there is no parameter mismatch. The very rapid convergence of this adaptive system is demonstrated experimentally, and can also be proven with simple root locus methods

Cannon, R. H., Jr.

Chiang, W. W.

English

NASA Technical Reports Server

'5 N85-31205 
SELF-TUNING ADApTlvE CONTROLLER USING 
W. W. C w  and R. B. CIIMOa, Jr. 
Stanford University 
Palo Alto, CA 94304 
ONLINE FREQUENCY IDENTIFICATION 
ABSTSACT 
A real-time adaptive controDn bu b e e m  designed ud tested smcceddly 00 8 fourth order laboratory 
dynamic system which feat- very low strrrcturl damping and a mon-cobdcd -r pair. The 
controller, impkmented in a digitd miaicomp.ta, couistr'd a stsk estimator, 8 set of state kedbuk e. and 8 Requency-- (FU) for d time pMmtter idecrtilutbm. Tk FLL cam dckct 
parameter and its counterput in the state estimator, and corrcct the eathator puuncter in d time. The 
adaptation dgwithm cam comct tk com?roikr error ud stabihe the system for mocc tLu 50$6 *uirtion 
in the plant natural frequency, compared ritk 8 10% M t y  m u g b  in kqac8q vuhtkm hrahd-gain 
controller havig the SuDe palormuce at the nomid plant conditiin. After it b a  kcLcd to the eomct 
plant freqaency, the adaptive controkr works aa well u the bdqpb eolltrolkr does d e 8  there is no 
parameter mismatch. The is dcmo.strrtd -, 
a d  can ab0 be proven with simpk rootbcos methods. 
the closed-loop natural freqaency of the system S i  coatrogsd, t.kmhte tb mbatcb betreerr.-t 
rapid c-e of tkb drpti*e 
I. INTRODUCTION 
A controller using Kalmu 5lter and hn state Wb8ck usually has good pertornuncC, provided a 
very accurate model of the plant is known. Bot such coltrdkn ue very Seuit ie  to parameter rui.ti, 
especially when the plant has very low inherd damping, and rhem the masor is not colocated with the 
actuator. 
A twodisk laboratory model, consisting of two inertia dias coanccted 8 torsion rod, rkb bas a 
structural damping of 0.004, and with separated seasor ud actortor locatio- WM constmcted to krt mct.l 
adaptive controller designs. The form of the quatiom of moth of the model is known due to the eaee of 
analysis of the lumped system; but the bek of u c m r a k  knowkdge about the n a t d  structural fmquency 
during controller design corresponds to a phnt parameter u a c t r k i o ~  or rui.tmn; and this mcertaiuty is 
what the adaptive controller handles. 
It has been proposed by Kopf, Brown, h4arsh (Ref.1) and Mac& (Mt)  to use a Phm: ;.0eLd-L00p 
to implement tuned damping and notch dltered command toque, so that the ftedback control force a z x  the 
structural frequency can be adjusted propedy according to the natural frequency of the plant, Bosca%ul 
and Cannon (Ref.3) have impkmented such a kind of controller for the tw&k experimental aystem. 
Under the same match project, a dillennt a p p d  using a Frequency-Locked-Loop (FLL) to Mente 
the plant bequency waa developed. Thi- oaper d w r i b a  in detail how the FLL identides the Inknown plant 
parameter and updates the controller in red time. 
145 
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II. DESCRIPTION OF THE TWO-DISK PLANT AND FIXED-GAIN CONTROUEbt 
The plant to be coutrolled b a mechanical system which consists of two h0riCent.l steel dish connected 
by a vertical elastic steel rod. The two dish u e  supported by bearinga which d b w  rotational motion ody. 
A low-friction DC motor is attacked to the bwer disk, and RVDT sensor detects the angular position of 
the upper disk. 
If structural damping is negkted*, the state eqiiation of motion of thb system can be e x p d  a~ 
where tl  and ts are the position state3 of the rigid body mode aod the stmctural osciIl8tion mode respec- 
tively, ts and t4 are rates of those states respectively; w,, ia the natural frequency, J is the tokl moment of 
inertia of the two dish, and Y b the control toque from the DC motor. 
The Sensor output b 
# = 21 + 2s: 
A fintsrder high-paw !Utet with 100 Hz cutoff frequency is used to Merenthte the position semr output 
and provides the pseudo-rate of the top disk. 
. 
(2) 
If all the parameten of the plant are known accurately, a0 LQG design (Ref.4) rill result in a set of state 
feedback gains C for regulation and estimator gains I ,  b r  state estimrrtion. However, if the p h t  n a t d  
frcqacncy w,, is not known by the controller designer, and a value We is aaed m the estimator, the stability of 
the whok system has to be analyzed by augmenting the system state eqrutha  with thosc of the estimator 
states, and finding the modd frequencies and dunpimp of the system (Ref.5) 
Using the same penalty weightings for control effort and state errors, an LQG design produces Merent 
feedback gains C and L for dserent natural frequencies w, of the plant. Analysb shows that the stabity 
of the whok sipstern is lesa sensitive to those feedback g a k  tbu to the parameter we used in the estimcrtor, 
since an error in the latter parameter corruponds to a modeling error, w h i k  ~ k b i o ~  in the dormer ones 
comspond to ditlcrent weighting in the LQG design procar. In the experiment d e s c n i  here, feedbreit 
gains C and L are chosen for the nomind plant frequeoc;, and are kept constant m order to demonstrate 
the adaptation of the controller by correcting we in the estimator. 
. h m  the analysis of the augmented system state quatiom, the kqueney We of the most unstable 
c.6sd-loo~ mode can be found as a function of w,, and we, if .II other puuneten u e  kept constant. Thir 
'UdCth. 
we = J(we,wm), (3) 
ulll &t f the closed-loop performance of the adaptation proeas, and has to be taken into account in the 
design yrocess. The twodisk model has a nomind frequency of 13.3 rad/see, md the function described t 
equatio 1 (3) can be shown approximately na in Fig. 1, can be appmxhakd as 
We - Wn = ( W a  - We)  + 0.6 (4) 
for /wI - we/  < l . b r a d / ~ ~ .  
m. FREQUENCY IDENTIFICATION USING FRXQUENCY-LOCKED-LOOP 
A Phasc-Cocked-Loop (PLL) WM initidly p r o p d  to be d to detect the vibration frequency. PLLs 
have been used widely in locking onto high-frequency signals in electrical engineerbg applications, but it 
- 
It is actually 0.001 
146 
is only beginning to be uaed h locking onto lor-frequency &nab in mceh.nical systems. A P U  bos tba 
ability to identify the phrw and frequency of a signal eoatuninattd by a nktively large amount of nok .t 
other trtguencies. Several signal components at Merent trepuencke can be ideatfled by using w e d  PLLa. 
The traditional PLLs are nonlinear elements for which the perforname is hucd to analyse and predict; 
and they have Limited locking ranp  due to their nonlinearity? Besides, PLLs ace more aeeositive to the 
phase than to the frequency of their driving signal, which makes them olrsgitabk for frequency ideatibeatior 
because the identieeation will be disturbed by the phose in the e e m r  syld every time a new mition 
command or an external disturbance i s  applied to the system, even though a PLL haa i den tW the comet 
plant frequency already. 
A modification is made to a PLL to eliminate ita sensitivity to phase in the input signal and make the 
input/output relation bear in a larger tracking range, so that it works better for frquetcy identitleation, 
while retaining the other virtues of PLLs. The dnal product, caUed a lkquency-Loeked-hp (FLL), ia 
shown schematically in Fig. 2, and its input/output relation can be seen from the tnoetioaal block diagram 
in Fig. 3, where w, is the frequency of the input &pal and wo ia the output signal - the frequency detected by 
the FLL. Also shown in the same block diagram ue w,, the starting osciliation frequency; Au, the cometion 
on the output; and we,, the error of the output of the F'LL. 
The character of the block G(4) can be chosen arbitrarily by the designer cia bng M it c.0 update the 
output frequency of the FLL accorcliig to its error 0,. If a h p k  integrator 0 is choscn M the element 
C(r),  then the FLL will have a pole at -K where 
Parameters a and b should be determined with the folbwing restriction 
In the present case, 
and the linear sear& range is choscn to be 
wm = 13.snn/Sce, 
The pole loeation 8 = -K should be determined m the molt of a compmmhe ketrcen speed of nsponae 
and noise rejection, at the nomind locking frequency range. In thio case, the parametera of the FLL ut 
chosen rn 
to work in the range of 1 to 3 Hz. 
o=6.0, 6=4.0, C=#).O, w Kz1.67, (9) 
With parameters chose0 M above, the block diagram in Fig. S can be aimpUded to the t d r  hroction 
Fig. 4(s) ahma the test molt of the FLL output when the frequency of the $pa& signal is changed stepwiscly. 
The mpow for r m d  input change (the ha t  change in Fig. 4(a) ) io similu to the step mponee of a dnt- 
order dlter with pole at -K, M ahown io Fi. 4(b). The rerponse for a larger input change (the w-ond 
147 
change in Fig. 4(a) ) experienced some nonlioeuity at the beginnkg beczrrse its internal strocton b mot 
linear; however, the FLL still tracked the input signal and provided the comet output in a mmondde time. 
IV .  CORRECTION OF PARAMETER ERROR IN TIlE CONTROLLER 
Because eigenvalues are properties of the system, they are independent of the instantaneous d u e  d 
state variables and are indueneed only by changes of parameters. The rehtion between we aud we, m shown 
in Eqn. 4, CM be expressed am in Fig. 5. Using the Uemnce between (u,, - 0.6) and we to mpdate - 
thmuKh the integrator $ - the parameter w, in the controller, the dared loap dyalmirr of the parameter 
variation, identfieation, and correction can be expremed ua in Fa. 6. The ~ i c r i S t i c  equation of the 
closed parameter adaptation loop is 
or, 
a(# + K )  + (0  + 2K)B = 0, 
which can be written in Evan's form as 
The root locus of Eqn. 13 vs. the positive value of R with K = 1.67 is shown in Fig. 7, and tbe value cf 
H = 9.9 is chosen obviously to maximihe the adaptation d e .  The change of the dope io Fig. 1 cornponds 
to a variation in the gain in Eqa. 4, and Eqn. (11) C.D be m&ed M 
rH K 
+ (8  + X ) ( r  + K) = O, 
where 2 > r > 0 , and the mot b u s  shown in Fig. 8 quarantees the stability of the rystem over the range 
of the gain V .  
Any sensor measurement, controller state variable, or linear combidion thereof can be &asen as the 
input signal to drive the FLL, so long as the signal contains the modal traquency of merest (the k g e r  the 
better!!. The error between the sensor rate and the estimate of it is &own to drive the FLL, since there is 
less error signal if all parameters in the controller are correct. 
The FLL must be turned off if its input s i g d  is too s d l ,  in order to reject the Muence h m  random 
noise. 
A PDP-11/23 minicomputer was wed to implement the controller and the FLL at 2S HG sample rate. 
The test results of thio adaptive system an summerid in the dotbring section. ' 
V. EXPERlMENW RESULTS 
Fig. 9 rhows the n a t d  0rCill.tiOn d the uncontrdled disk system. The kquency d o e e h t b n  in 
2.11 Hs. with 0.004 damping. (The bng-period motion b cwcd becaase the disk b Long fkom the 
ceiling with a long steel wire to reduce the ud.l t h t  01 beuir;p. "him mod. t a p p h a t e d  u a rigid 
body mode in the controller design adyds.) 
Fig. 10 shows the step nrponse of 8 nonulapthe control system designed with the LQC method. The 
response is  very good (Fig. 10) when there b no modcling error in the controlbr d a i p .  However, u Fig. 11 
shows, the system becomes unstabk when there is 10% modeling error in frequency in the designing of the 
nonadaptive controller. 
148 
When the FLL is used in the adaptive control, the system e m  detect and eomct a controkr'r p M m e k t  
error of 50% or more ia frqaency. Fign. 13 (a) through (f) show the KpIor ontpot in Werent teata. 'fke 
instability due to the initid parameter erm L ahown when the control syatem waa juat tuned oa, urd tke 
system was then stabilized after the uhptatios algorithm had corrected tbe controlkr'r error. The initid 
turn on of the control system aad the time rbeo podtion eommlada u e  &aged an mar&& on thow 
recordings. 
Fig. 13 shows the comparison of the impulsive disturbance nspoose, between the nonadaptive controller 
with no modeling error and the adaptive one after its parameter e m r  h a  been corrected. The compuison 
shows almost no diaercnce betweep their perfomamea. 
VI. DISCUSSON 
(A) Frequency-Locked-Loop 
The F1.t is a nonlinear element, but its input/output relation b almoat hear. It behaves linearly for 
40% changes ID input r i p d  frequency, and rtin work for loosC chow in frequency in the nonlinear mn. 
The test recorded in Fig. 4 attests to the dteuaaion above.. The hear rage  can 1. chosen by aekting 
parameters pnperly. 
,The FLL still worka when the amplitude of its input si& h, aa weak aa two quantiution intermla of 
the A/D ccnverter, if it L free of noise and biru, but in mal rpptiUtbns it mast be t m e d  od at d &vel 
of input signd to reduce the effect of noiac. 
The FLL can idetitifv the plant characteristic in a s d  rindow ol the &cqaency rpectrum, 10 tba: the 
effects of other parts of the ayrtem dynsmier do not have to be wten into aecoan4 if they are rot  critical 
to the overall performance. It em only detect modes that ere either only slightly damped or unatabk, since 
they can provide oscillatory sign& for detection; however, heavily damped modes ue u a d i y  robuat to 
parameter uncertainty and dw't need adaptive control 
(B) Parameter Error Correction Loop 
The parameter e m r  correction acheme w be determined by root-bcua analyam, or even by the LQG 
method, siuce the FLL haa a linear chluaeteristie. 
Fig. 12 shows some small-smplitnde vibration brrilding up due to the WI d r i p d  to lock the FLL, but 
the pvameter estimate emc  wm soon corrected and vibration s u p p d .  
By examining the reaponre to command h g e  and to dbtnrb.nees, it b band that the Se&lhning 
Adaptive Controller behaved dmwt the aune a8 tbe c o m t  h e d  o p t h d  controller, except tor bhe few 
cycles of vibration at the beginning when the pvnmcter error WIU being corrected. 
It is better to use the c m r  of an eatimated aenaor output to drive the FLL, mince it is undisturbed by 
the control force during a new command cbange if the model is correct. 
Both the identiaeatioa and error cometion are running in red time while' the controller ia  doinc Its job. 
Any change in the plant can be tracked and adapted to rapidly. 
VU. CONCLUSION 
The uae of FLL in identifying ryrtem vibration hquency and adapting controllet parameters b promis- 
ing, AH kinds of eootrolkn, roch u K a l m u ~  alter U I ~  state feedback, band-pur, or notch dlten CUI bave 
their parameter enom corrected in a rimilu way. It L expected that ayrtem with II).DY vibration modea can 
149 
be handled with several FLLI. 
ACKNOWLEDGMENTS 
The research wporttd upon here wm spomred by the National Aeronautics and Space AdmSistntbm 
through the Langley Research Center and the Jet Propubion Laboratory. The  author^ appreciated the 
advice and kelp from D m  E. Rosenthd who was doing ibe other part of t!ie research under the M m e  NASA 
cont :act. 
REFERENCE 
(1) Edward H. Kopf, Thomas K. Brown and Elbert L. Muah, 'Flexible Stator C O I A ~  on the GJileo 
Spacecraftp, paper 79161, M S / W  Astrodyaamica SpecwiSt Coafereacc, rroviacetorn, MASS/ Jane 
25-27, 1979. 
(3) Dan E. Roscntbal and Robert H. Cannon, Jr., "Experiincntr with Adaptive Control of F!&k SLne- 
tures", to be published in the Journal of The Astronautical SC~BCCI, 1984. 
(4) h. E. Brywn, Jr. 6 Y-C Ho, 'Applied Optimal Control", Hemisphere Publishing Co., IQ'lb. 
(5) Daniel B. DeBra, 'Estimatbn in Satellite CoatroP, Proeecdiop of the Nmth Intermti - . A  Symposium 
on space Technology and Seknee, Tokyo, 1971. 
150 
-3 -2 -1 
Odt : (na/scc) 
151 
rP U 
" L 
152 
. 
153 
154 
-Re 
H =99 
(0  = -sa) 
-1 L 
U 
I\ 
.I 
n 
-E  -K 
? = l  
I 
Fig. 8 Variation of P o k  of the C l o d  Parunttet Loop .*mu t, the Ch.ogbg Sbpc io Fig. 1. 
155 
156 
157 
T.O. C.C. C.C. 
158 
T.O. : initial turned o.. 
C.C. : step command &agec. 
Fig. 12 Step Response of the Adaptive Cortmllcr with FLL Detcctbg W kde)hg Emorb Pk.r hcq.cncl: 
(8) 80 ern?. 
(b) -lO%enor. 
(c) +25% error. 
(e) - S o % e m .  
(2) +50% error. 
(d! -25%err0r. 
159 
Fig. 13 Comparison of the Impobirt Disturbance Rcrp0.r~ betveen (a) tbe Nonadaptive Controller with No 
McrdcIing Error snd (b) the Adaptive One .ikr Its Parameter Error H.s Been Corrected. 
160 


Self-Tuning Adaptive-Controller Using Online Frequency Identification

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