Robustness of Continuous-time Adaptive Control Algorithms Inthe Presence of Unmodelede Dynamics

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TRANSACTIONS ON AUTOMATIC CONTROL, VOL.C-30,O. 9, SEPTEMBER985 88 1

l Notes and Correspondence

Note to he Reader:The following paper has spurred much discussion

the adaptive controls communityor a numberof years now. Becauseof

subsequent esearch it generated, acommentarywas elt to be

ate. Hen?, at the end of this paper, read on for some insight

by Karl Astrom.

RAY DECARLO

Robustness of Continuous-Time Adaptive ControlAlgorithms in the Presence of Unmodeled Dynamics

E. ROHRS,ENAALAVANI,ICHAEL

ATHANS, A N D GUNTER STEIN

Abstruct-This paper examines the robustness properties of existing

hms to unmodeled plant high-frequency dynamicsrable output disturbances. I t is demonstrated that there exist

dynamic ystemwhich

and parameter errors. The

of the existence of such infiite-gain operators is

1) sinusoidal eference inputs at specific requencies and/or 2)

idal output disturbances at any frequency (including de),can cause

gain to increase without bound, thereby exciting the unmodeled

dynamics, and yielding an unstable control system.is concluded that existing adaptive control algorithms as they

presented in the literature referenced in this paper, cannot be used

nce in practid designs where the plant contains unmodeled

instability is likely to result. Further understanding isto ascertain how the currently mplemented adaptive systems

the^ theoretical ystems tudiedhere and how further

I. INTRODUCTION

This paper reports the outcome of an investigation of the stability andhe

of unmodeled high-frequency dynamics and persistent unmea-output disturbances. Every physical system has such (parasitic)

the cross-

frequency and require the control system to contain adequate gainAlso, every control system mustbe able to operate in

unstable. In nonadaptive linear controldesigns the presence of

upon the closed-loop stability issue.

It should be stressed that he existing adaptive control algorithms, whento control an unknown linear time-invariant plant, result in a highly

andime-varying losed-loop ystemwhose tabilitydoes

he external inputs (reference inputs and disturbance inputs).the stability of the adaptive closed-loop

Transactions Editorial B oard. This work was supported byNASA Ames andManuscript received May IO 1983; revised December 10, 1984. Paper recommended

rch Centers under GrantNASAlNGL-22-009-124, by the Office of NavalGrantONRIN00014-82-K4582 h’R 606-003). and by theNational

e Foundation under Grants NSFIECS-8210960 and NSFIECS-8206195.C. E. Rohrs is with he Department of Electrical Engineering, University of Notre

otre Dame, IN 46556 and Tellabs Research Laboratory, South Bend. IN 46635.

L. Valavani, M.Athans, and G. Stein are with the Laboratory for Information andof Tecbnology , Cambridge, MA 02139.

design and to inquire about its global tabilityproperties. This has

provided the motivztion for the research reported in this paper.

Due to space limitationswecannotpossiblyprovide n hispaper

analyticalandsimulationevidenceofallconclusionsoutlined n the

Abstract. Rather, we summarize the basic approach only or a single classof continuous-time algorithms that ncludes hose of Monopoli [14],

Narendra and Valavani [l ], and Feuer and Morse [2]. However, the same

analysis techniqueshave been used to analyze more complex classes of1)

continuous-time adaptive control algorithms due to Narendra, Lin, and

Valavani [3], both algorithms suggested by Morse [4], and the algorithms

suggested by Egardt qwhich include those of Landau and Silveira [6],

and Kreisselmeier [19]: and (2) discrete-time adaptive control algorithmsdue to Narendra and Lin [22], Goodwin, Ramadge, and Caines [5] (the

so-called dead-beatcontrollers), and thos t developed nEgardt [ l q ,

which include the self-tuning regulator of Astrom and Wittenmark [181

and that due to Landau [20]. The thesis by Rohrs [151 contains the full

analysis and simulation results for the above classes of existing adaptive

The end of the 1970’s marked significant progress in the theory of

adaptive control, both in t e r n of obtaining global asymptotic stability

proofs [1]-[7 as well as inunifying diverse adaptivealgorithms, the

derivation of which was based on different philosophical viewpoints [8],

Unfortunately, the stabilityproofs of all these algorithmshave n

common a very restrictive assumption. For continuous-time implementa-

tion thii assumption is that the number of poles minus the number of zeros

of the plant, i .e. , its relative degree, is known. The counterpart of thisassumption for the discrete-time systems is that the pure delay inhe plant

is exactly an integer number of sampling periods and that thii integer is

known. It is also assumed that an upp r bound for the number of poles in

the lant is known in both the continuous-time nd iscrete-time

formulation. Finally, global parameter convergence equires hat the

inputs satisfy a “sufficiently rich” condition.

The restrictiverelative degree assumption, n turn, isequivalent to

enabling the designer to realize for an adaptive algorithm, a positive real

error transfer function, on which all stability proofs have heavily hingedto-date [8]. Positive realness implies that the phase of the system Cannotexceed 9 0 ” or all frequencies, while it is a well-known fact that models

of physical systems become very naccurate n describing actual planthigh-frequency phase characteristics. Moreover, for practical reasons,

most controller designs needto be based on models which do not contain

all of the plant dynamics, in order to keep the complexity of the required

adaptive compensator within bounds.

Motivated from such considerations, researchers inhe field inthe early1980’sbegan investigating the robustness of adaptive algorithmso violation

of the restrictive (and unrealistic) assumption of knowledge of the plant

order and its relative degree. Ioannou and Kokotovic [IO] obtained error

bounds fordaptivebselversnddentifiersnheresencefunmodeled dynamics, while such analytical results were harder to obtain

for reduced-order adaptive controllers. The first such result, obtained by

Rohrs et ai.11 11, consists of “linearization” of the error equations, under

the assumptionhat the overall ystemsntsinal pproach to

convergence. Ioannou and Kokotovic [12] later obtained ocal stability

results in the presence of unmodeled dynamics, and showed thathe speed

ratio of slow versus ast unmodeled)dynamicsdirectlyaffected the

stability region. Even he local stability results of [12]anonly be attained

when the speed ratio is small, i.e., when the unmodeled dynamics are

much faster than the modeledpartof the process. Earlier simulationstudies by Rohrs et al. [13] had already shown the high sensitivity of

adaptive algorithms to disturbances and unmodeled dynamics, generation

of high-frequency control inputs and ultimately instability. Simple root-

locus type plotsfor the linearized system n [ l I ] showed how he presenceof unmodeled dynamics could bring about nstability of the overall

algorithms.

191.

0018-9286/85 /0900-0881 01 .~ 985 IEEE



882 IEEE TRANSACTIONSN AUTOMATICONTROL, VOL.C-30,O. 9, SEPTEMBER 1985

system. It was also shown there that he generated frequencies n headaptive loop depend nonlinearly n the magnitudes of the reference input

and output. Kosut and Friedlander [23] have also tried to define the class

ofplantperturbationsunderwhich daptive ontrollerswilletainstability.Theyhave also found severe limitations as in [ l l ] , on the

reference input and output characteristics for which the linearized error

equations are stable.The main contribution of this paper is in showing that two operators

inherently present in all algorithms considered (aspart of the adaptation

mechanism) have infinite gain. As a result, two possible mechanisms ofinstability are isolated and discussed. It is argued that the destabilizing

effects in the presence of unmodeled dynamics can be attributedo either

phase (in the case ofigh-frequencynputs), or, primarilyain

considerations (in the case of unmeasurable output disturbances of any

frequency, including dc), which result in nonzero steady-state rrors. The

latter factsmost isconcertingor the performance f daptive

algorithms since t cannot be easily dealt with by additional filtering,givenhat ersistent isturbance fnyrequency can have

destabilizing effect.Our conclusion s hat the adaptivealgorithms as published n the

literature are likely to produceunstablecontrol ystems if they are

implemented on physical systems directly as they appear in the literature.

The conclusions stem from the results of this paper which show unstable

behavior of adaptive systems when these systemsre confronted with twopremises that cannot be ignored in any physical control design: 1 ) there

are always unmodeled dynamics at sufficiently high frequencies (andt is

futile to try to model these dynamics); and2) the plant cannot be isolatedfrom unknown disturbances (e.g., 60 Hz hum) even though thesemay be

small.The original version of this paper was presented as [24]. Since that

time, there has been a great deal of research generated on the robustnessissues raisedhere. In particular, ithas come to appear that some problems

presented in his paper may possibly be overcomeby sufficient excitation.The following idea of the usef sufficient excitation can e traced back at

least to [25]. By using sufficient excitation the nominal adaptive system

can be made exponentially stable. Sincehe system is exponentially stable

there is some modeling error and some disturbances for which stability

wll bemaintained. It shouldbenoted,however, hat heamount ofmodeling error or the amount of disturbance for which the adaptive

system can maintain stability may be extremely small.

The issue of sufficient excitation is ot considered here. It is a complex

issue which will roduce many papers of its wn. However, the following

results pertaining to sufficient excitation should be noted. Since the inputswhich cause destabilization n this paperactuallyprovide the usual

measure of sufficient excitation it is clear from these results that not just

sufficientexcitation sneededbysufficient ow-frequencyexcitation.

Ioannou and Kokotovic [lo] used the term “dominantly rich excitation”in their investigationof the concept. Since the publication of [24], Krause

ef ai. [26] have analyzed the infinitegainoperatormoredeeplyandextended the analysis of this paper n a significant way o nclude theeffects of the richness of input in the stability analysis. The results of the

analysis of [26] show that the input must not onlybe sufficiently exciting

to produce parameter convergence in the nominal system, but that theinputmustbedominantly ichenough to overcome the destabilizingeffects discussed here. Thus, sufficient excitation should not be viewedsa panacea which creates robust adaptive controllers but as a stabilizingeffect in a delicate engineering tradeoff.

In Section 11the infinite gain of the operators generic o the adaptationmechanism sdisplayed.Section III contains the developmentof wo

possible mechanisms or instability thatarise as a result of the infinite gain

operators. Simulation esults hat how hevalidityof heheuristic

arguments in Section ID are presented in Section IV. Section V containsthe conclusions.

n. THE ERRORMODEL. TRUCTURE FOR A REPRESENTATION

ADAPTIVE ALGORITHM

The simplestrototype for amodeleferencedaptiveontrolalgorithm in continuous-time has itsrigins to at east as far back as 1974,

in the paperbyMonopoli [141. This algorithmhasbeenproved

asymptotically stable only for the case when the relative degree of the

Model

Fig. 1. Controller structure of CA1 with additive output disturbance d(r ) .

plant is unity or at most two.The algorithms published by Narendra and

Valavani [ l ] and Feuer and Morse [2] reduce to the same algorithm for

the pertinent case. This algorithm will henceforth be referred to as CAI

(continuous-time algorithm No. 1).

The following quations ummarize the dynamical quationshat

describe it;see

also Fig.1 .

The equations presented herepertain

to thecase where a unity relative degree has been assumed. In the equations

below, r(t ) is the (command) reference input and the disturbance d t) n

Fig. 1 is equal to zero.

plant:

auxiliary

variables:

model:

control

input:

output

error:

parameter

adjustment aw:

nominalcontrolledplant:

error

equation:

u ( f ) k T ( f ) w ( f )k*‘(t)w(r) +C T f ) W f )

= k* ‘ f ) W f ) +a f )

+ - - [ T I .*B*(s)* SI A T ( f ) W ( f )

In the above equations the following definitions apply:

k f ) k* + &(f)

where k* is a constant 2n vector.



TRANSACTIONS ON AUTOMATIC COhTRO L, VOL. AC-30, NO. 9 SEPTEMBER 1985 883

NOMINALLY CONTROLLED

PL ANT d ( t )

Fig. 3. Block diagram of a generic adaptive controller.

Kas)6k&- I p n - 2 + k;*( , -2)~”-~+. +kzl

k:i is the ith component of kz.

K ~ s ) ) k , ~ s ” - ’ + k ~ ” - , ) s “ - 2 +.. + k J

k is theith component of k;. In the preceding equations we have

[3]-[5], with P s)

g the characteristic polynomial for the state variable filters andthe parameter misalignment vector. In (8) g*B* s)/A* s)epresents

Awere

zero, Le., if a constant control law k = k* were used. Under

Bds) divides P s), hen k* can be chosen [l], such that

g*B*(s)/A*(s)can only getclose to g, As)/A, ) as the feedback structure of he controller

The first term on the right-hand side of (9) results from such a

(1 1) were satisfied, (9) reduces to the familiar

form that has appeared in the literature [8] for exactg. For more details the reader is referred to the literature cited in

section as well as to [15].

Fig. 2 represents, in block diagram form, the combination of parameter7) and (9).

In general, existing continuous-time algorithms [I]-[9] can be classi-

groups labeled here and in 1151 as CA1, CA2, CA3, CA4,, for continuous-time algorithms 1 , 2 , 3 , 4 .Fig. 3 represents a

d error structure which can be particularized to describe theany one of the existing adaptive algorithms, both n

by its discrete analog) in discrete time as well. In

3, the forward loop consists of a positive real transfer function, while

h contains the infinite gain operator(s). In the figure, thesystem input 2(t) is synthesized according to

ri(r)=P(t)Cp(t), (t)l

f t ) F ( s ) D [ H f ) , u t)le t)lI

C is a linear time-invariant system representing an observer or

an auxiliary state generator

Fig. 4. The infinite gain operators of CA2, CA3, and CA4.

M isusually a memoryless map, often the identify map

D is a linear time-invariant system analogous to C nd ften

identical to it< s) a stable ransfer function, often the identity

k( t ) , i t ) parameter error vectom related by &(to = m t ].

The above-mentioned four classes of adaptive algorithms have i

common the error feedback loop structure and the basic ingredients of the

parameter update mechanism, i.e., multiplication-integration-multiplica-tion, which forms the feedback part of the loop and is shown [15] to

constitute the infinite gain operators present in all existing adaptivealgorithms. These operators are hown in Fig. 4 as they apply specifically

to CA2, CA3, CA4. Analogously, the discrete-time algorithms are

classified into three distinct groups, DAl, DA2, D M . The four classes

mentioned n the preceding differ in the specific parameterization that

realizes the positive real transfer function in the forward path and in theparticular details of the parameter adjustment laws [choice of C, D , M,

F(s)].For example, in the simplest class of continuous-time adaptivealgorithms (CAI) under perfect modeling, the forward transfer function

represents the reference model transfer function, which *e controlledplant is able to match exactly, by assumption 1 1). Unfortunately, when

unmodeled dynamics are present, the controlled plant can only match thereference modelonly up to a certain frequency range and, thus, the

forward loop transfer function which represents the nominally controlled

plant and determines theerror dynamics, loses its positive realnessproperty. This, in conjunction with the infinite gain operator(s) in thefeedback loop, can bring about instability.

IU.THE INFINITE GAIN OPERATORS

A . Quantitative Proof of Infinite Gainfor Operators of CAI

The error system in Fig. 2 consists of a forward linear time-invariantoperator representing the nominal controlled plant complete with unmo-deled dynamics g*B*/kFA*, and a time-varying feedback operator. It isthis feedback operator which is of immediate interest. Theoperator,

reproduced inFig. 5 for the case where w is a scalar and r = 1 , is

parameterized by the function w t) and can be represented mathematicallyas

W ) Gwtn[e(t)lgo+w ( t ) j‘ w(7)e(7) d7. (12)



884 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC- 30, NO. 9, SEPTEMBER 1985

Fig. 5. Infinite gain operator of C A I.

Also

{lle(t)ll;2}2=a2r sin h o t d t s a 2 T .‘0

Therefore,

In order to make the notion of the gain of the operator G, ( t ) (.) precise, Ila(t)lli2we introduce the following operator theoreticoncepts. Ile Oll;,

truncated normDefinition I: A functionflt) from [0, ) to R is said to be inL k if the

[( I 2 )2b2c2os2Q a2e4cos2@I

l l f t o l l ~ 2 ~ (:?(7) d7 ’” (1 3) 24 T1l2

+ P - K 2 P - K , T - K o12

is finite for all finite T.

L 2 into functions in Lk is defined as

Definition 2: The gain of an operator Glf t ) ] ,which maps functions inT-. me3

and, therefore, G , for w as in (15) has infinite gain.

IlGV(t)lllL

J7f)€L, Ilr t)lll, .

In addition to the fact hat the operator G,(t) from e t) to C(t) has

(14) infinite gain, the operator ff,, from e(t) o ) in Fig. also has infinitelGll= SUP

gain. This operator is described byTE O.

If there is no finite number satisfying (14), then G is said to have infinitegain.

Theorem 1: If w ( t ) is given byTheorem2: The operatorHN,) ith w ( t ) given in (15) has infinite gain.

w(t ) = b + c sin wot (15) Proof:Choose e(t)= Q sin wot as before.hen [(to = Hyo[e(t)]sgiven by (18). Proof of infinite gain for this operator then follows n

Proof: The proofconsists of constructingasignal e@) , uch hat Remark I : Both operators G , and H , will also have nfinite gain forfor any positive Constants b, C , wo the operator of (12) has infinite gain. exactly analogous steps as in Theorem 1 and is, therefore, omitted.

.,vectors ~ ( t ) ,ince the operators’ nfinitegainscan arise from any

(16) component of the vector w ( t ) .

is unbounded.Let e(t) = a sin (wot + 6) ith an arbitrary positive constant, 6 n

arbitrary phase shift, and wo he same constant as in (15). These signals

produce

w(t)e(t) ab(sin wof cos 4 +COS o sin 6)

+ uc sin (sin wot cos Q +cos mot sin +) (17)

1

2= +- QC cos . t +constanterms + periodicerms (18)

C(r) = G,(t)[e(t)] Go+ w ( t ) j‘ w ( r ) e ( r ) dr

B . Two Mechanisms of Instability

In this section, we use the algorithm CA1 to introduce and delineatetwo mechanismswhichmay cause unstablebehavior n headaptive

system CAI, when t s mplemented n the presence of unmodeled

dynamics and excited by sinusoidal reference inputs or by disturbances.The argumentsmadeor CAI re also validorother lassesof

algorithmsmentionednection m u fa t k mutandis. Since the

arguments explaining nstability are somewhat heuristic n nature, heyare verified by simulations. Representative simulation resultsare given inSection IV.

In order to demonstrate the infinite gain nature of the feedback operator

of theerror system ofCAI in SectionII it is assumed that component of

w(t)has the form

w i ( t )= b + sin Wot (26)

== (ubc+ac2 sin w o f ) t + constant erms + periodic erms.and hat the error has the form2

(19) e(t)=a sin wot+Q). (27)

- K ; T ) l / 2 (20) havedistinctsinusoidsatacommon frequency, the operator GW of (12)

Next, using standard norm inequalities, we obtain from (19)The arguments of Section indicate that, ife ( t )and a componentof w ( f )

Ilc(t)ll;2, - cos Q - c2t sinI1

2 and the operatorHKfI)f (25)will havenfiniteains. Two possibilities for

e( t) , w ( t ) to have the forms of (26 ) and (27) are now considered.

e.g.,

with Ki ainiteonstant.ow Case I : If the referencenputonsistsfinusoidndonstant,

1

2abct cos 6+- ac’t cos Q sin wot (21)

r(t)=r,+r2 in od (28)

[ 2 b 2 c ~ x x 2 02c4cos’ Q

where rl and r2are constants, then the plant output y(t) will contain a

24 constanterm and ainusoid at frequency wowith an arbitrary phase shiftT 3 - K 2 P + K , T + K o (22)

( a 2 b 2 c ; y 2@ a2c4 cos2 6 If the controlled plant matches he model at dc but not at the frequency

l l ~ o I l ; ’ ) 2 ~ + T’ -~~ K~T - K ~ . wo, he output error24



ON AUTOMATICONTROL, VOL. AC-30, NO. 9, SEPTEMBER 1985 885

icontain a sinusoid at frequency w0. Thus, the conditions for infinitein the feedback path of Fig. 2 have been attained.

Case 2: If a sinusoidal disturbance, d(t), at frequency ~0 enters the

as shown in Fig. 1, the sinusoid will appear in w t)through

(3) in the presence of an output

wYi( t )=-@(t)+d(t) ] ; i = l 2, n .

(6) when an output disturbance is present

e(t)v(t) + d t) Y M ( ~ ) . (31)

I

P S)(30)

sinusoid present in d t) will also enter e(t) hrough (31). Thus, the

e(t)and w ( f )will contain sinusoids of the same frequency and the

H,,,) and GWcr)ill display an infinite gain.

1) Instability Due to the Gain of the Operator G, of 12): TheG, of (12) isnotonlyan infinite gain operator but its gain

such a manner as to allow arguments using linear

as outlined below.

Assume, initially, that the error signal is of the form of 27), i.e., a

~ 0 .ssume also that a component of w(t) s of the(26), i.e., a constant plus a sinusoid at the same frequency wo as

The output of the infinite gain operator GWcof (12), as givenby

consists of a sinusoid at frequency 00 with a gain which increases

0 rads (i.e., dc) and other

w0, Le., O t ) = 112 acZt in mot cos 6 + other terms.The infinite gain operator manifests its large gain by producing at the

a sinusoid at the samerequency, w0 as the input sinusoid butBy concentrating on the signal

u0 ndviewing the operator GWcr)s a simple time-

n with no phase shift at the frequency coo,. and very smal l

will be able to come up with a mechanism

2, where GMoconsists of the

g*B*(s)/ka*(s), f the

2, has less than k 180 phase shift at the frequency W O ,

if the gain of G, ,)were indeed small at all other frequencies, then theG,,,) at wo would not affect the stability of the error loop. If,

g*B*(s)/kTA*(s) oes have 180 phase shiftwo the combination of this phase shift with the sign reversal will

GWio,hereby

GWfr).he sinusoid will then

G,) grows, until

combined gain of Gdr) nd g*P(s)/k7A*(s) xceeds unity at thewo. At this point, the loop itself will become unstable and all

s will grow without bound very quick (as the effects of the unstable

G,, compound).

Since the infinite gain of GWcr)an be achieved at any frequency wo if

has f 180 shift at any frequency, the adaptive system

to instability from either a reference nput or a disturbance.Thus, the importance of the relative degree assumption, which

g*B*(s)/kFA*(s)s sbictly positive

is seen. The stability proof CAI hinges on the assumption thats strictly positive real and that G%,) s passive, Le.,

J + + w I ~ ~ ) dt=o. (32)

h properties of positive realness and passivity are properties which aref the gain of the operator involved. However, it is always

only aon the gain of the plant at high frequencies.

for a large class of unmodeled dynamics with relative degreeor greater, the operator g*B/kTi*, will have f 180 phase shift at

ency and be susceptible to unstable behavior if subjected to

2) Instability Due to the Gain of the Operator H , of 25): In the

the situation was examined where the amplitude ofe(t) grew with ime due to a positive feedback

loop. In this subsection, we explore the situation

where the sinusoidal error e@ , is not at a frequency where it will grow

due to the error system but, rather,when there exist persistent steady-state

errors. Such a persistent error could easily arise when an output sinusoidaldisturbance d(t)enters as shown in Fig. 1, causing the persistent sinusoid

directly on e(& through (31), and w(t) through 30).

Assume that a component of w(t)contains a sinusoid at frequency wo as

in (26) and that e(t)contains a sinusoid of the same frequency. Then the

operator H&, has infinite gain and the norm of the-output signal of thisoperator, ), increases without bound. The signal k(t)will take the form

of (18), repeated here

1

2t) Eo + - act cos ++constant terms + periodic terms.

From the second term one can see that theparameters of the controller,defined in (lo), i.e., k ( t ) = k* + @), will increase without bound.

If there are any unmodeled dynamics at all, increasing the size of the

nominal feedback controller parameters withoutboundwill cause theadaptive system to become unstable. Indeed, since it is the gains of the

nominal feedback loop that are unbounded, the systemwillbecomeunstable for a arge class of plants including all those whose relative

degree is three or more, even if no unmodeled dynamics are present.It should be noted that the arguments of this section have assumed that

the reference input or the disturbance is purely sinusoidal. Since the

adaptive system is nonlinear and the principle of superposition does notapply, he presence of a sinusoidal component n a signal willnot

necessarily give rise to the same effects. However, all that is needed forthe argument given toapply s that a dccomponent result from a

correlation between the output and the error. The dc component in the

correlation will produce parameter drift. Thus, any disturbance whosespectrum shows frequency peakmg will produce instability.

IV.SIMULATIONESULTS

In this section the arguments for instability presented in the previous

The simulations were generated using a nominally first-order plant withsections are shown to be valid via simulation.

a pair of complex but highly damped unmodeled poles, described by

2 229y t)=- .

s f l ) (s2+30S+229)W I

and a reference model

(33)

(34)

The simulations were all initialized with

k,(O) = - .65; k,(O)= 1.14 (35)

which yield a stable inearization of the associated error equations. For the

parameter values of ( 3 9 , one finds that

g*B*(s) 527

-- -A *(s) s3+3 Is2+259s+ 527 .

The reference input signal was chosen basedupon the discussion of

Section III-B-2

r(f)=0.3+1.85 sin 16.lt, (37)

the frequency 16.1 rad/s being the frequency at which the plant and the

transfer function in 42), i.e., g*B*(s)/k?A*(s),have 180 phase lag. A

s m a l l dc offset was provided so that the linearized systemwouldbeasymptotically stable. The relatively large amplitude, 1.85 of the sinusoid

in (37) was chosen so that the unstable behaviorwould occur over a

reasonable simulation time. The adaptation gains were set equal to unity.

A . Sinusoidal Reference Inputs

Fig. 6 shows the plant output and parameters k,(t) and k d t ) for theconditions described so far. The amplitude of the plant output at the



886 IEEE TRANSACTIONSNUTOMATIC CONTROL, VOL. AC-30, NO. 9, SEPTEMBER 1985

90, I

-9.0 I 1 II0 2 4 6 8 IO 12 14 16 18 20

T I M E

-2.0 II

,I J

0 2 4 6 8 IO 12 14 16 18 20

T I M E

-121 I 1

0 2 4 6 8 IO 12 14 16 I8 20

T I ME

Fig.6. Simulationof CAI with unmodeled dynamics and r ( t ) = 0.3 + 1.85 sin 16.1 t .(System becomes unstable.)

critical frequency (wg = 16.1 rads) and the parameters grow linearly

wth time until the Imp gain of the error system becomes larger thanunity. At this point in time, even though the parameter values are well

within the region of stability for the linearized system, highly unstable

behavior results.

Fig. 7 shows the results of a simulation, ths time with the reference

r t )= 0.3+2.0 sin 8.0t . (38)

This simulation demonstrates that, if the sinusoidal input is ata frequencyfor which the nominal controlled plant does not generate a large phase

shift, the algorithm may stabilize despite the high gain operator.Similar results were obtained for the algorithms described in [3], [4],

[6],7], [ 9 ] , but are not included here due to space considerations. Thereader is referred to [15] for a more comprehensive set of simulation

results, in which instability occurs for various sinusoidal inputs.

input

B. Simulations with Output Disturbances

The results in this subsection demonstrate that the instability mecha-

nism explained in Section III-B-1 does indeed occur when there -is an

additive unknown output disturbance at the wrong frequency, entering thesystem as shown in Fig. 1. In addition, the instability mechanism ofSection HI-B-2 which will drive the algorithms unstable when there is a

sinkoidal disturbance at anyfrequency, is also shown to take place. Thesame numerical example is employed here as well.

Instability Via he Phase Mechanism of Section III-B-I: In thiscase, CA1 was driven by a constant reference input

r t)=2.0 (39)

with an output additive disturbance

d(t)=0.5 sin 16.1t. (40)

The results are shown in Fig. 8, and instability occurs as predicted.Instability Via the Gain Increase Mechankm of Section 111-8-3:

Fig. 9 hows the results of a simulation of CA1 that was generated with

r=2 0

0 2 4 6 8 IO 12 14 16 18 20

Fig. 7. Simulationof CAI with unmodeled dynamics and <t) = 0.3 + 2.0 sin 8.0 t .No instability observed.)

T I M E

l8

0 8 16 24 32 40 48 56 642

' 5

8

-2410 8 16 24 32 4 0 48 5 64 72

T I M E

@I

Fig. 8. Simulation of CA1 with highly damped unmodeled dynamics ( t ) = 2.0 and

d( r ) = 0.5 sin 16.1 f. System becomes unstable.)



TRANSACTIONSN AUTOMATIC CONTROL, VOL. AC-30. NO , SEPTEMBER 1985 887

8

- 4 -

- 80 80 160 240 320 40080 560

T I M E

(a)

30

20 - k, I -

kY

.20 7 -

- 3 00 80 160 240 320 4 0 0 480 560

T I M E

(b)

9. Simulation of CAI with highly dampea unmodeled dynamics, fit = 2.0 and

d(z) = 0.5 sin 8 t . (System becomes unsrable.)

disturbance was changed to

) = O S sin st. (41)

of increasing amplitude, which is characteris-of instability via the mechanism o f Section III-B-1, is not seen in Fig.

What is seen is that the system becomes unstable by the mechanism of

III-B-2. While the output appears to settle down to a steady-stateky parameter drifts away until the point where the

9 in order to maintain scale.)

The most disconcerting part of this analysis b that none of thems analyzed with constant set point reference inputs have beeno counter this parameter drift for a Sinusoidal disturbance at

frequency triedIndeed, Fig. 10 shows the results of a simulation run with reference

r=0.0 (42)

constant disturbance

d=3.0. (43)

simulation results show that the output settles for a long time withky increases in magnitude until instability

adaptive algorithm shows no ability to act even as aare output disturbances.

However, inorder odrive the system unstable with a constant

as it took to drive the

h a higher frequency sinusoid, the magnitude of the

must be much larger in the constant disturbance case. Thedisturbance must be larger because the nominal control system

a larger gain at dc and thus better disturbance rejection. The time it

- 4 1 ' ' ' '0 80 160 240 320 400 480 560 640 720 800 880 960 1040 1 1 2

TIME

(a)

16

- 3 2 ~ ' ~

0 80 160 210 320 400 480 560 640 720 800 880 960 I040 1120

(b)

T I M E

Fig. 10. Simulation of CAI with highly dam@ unmodeled dynamics, fit = 0 0 ndd t) = 3.0. (System becomes unstable.)

takes for the system to go unstable is inversely proportional to the square

of the magnitudeof the part f disturbancestill present at the utput of the

closed-loop controlled system.In the previous simulations the unmodeled dynamics which were used

were highly damped. Fig. 11shows the results of a simulation of a plantwith less well damped unmodeled dynamics at a somewhat lowerfrequency and a smaller disturbance. The plant used n the run is

described by

The reference input was again

r(t)= 2.0. (45)

The disturbance used was

d(t)=O.l sin 8t . (46)

In this case the parameter drift is slower than in Fig. 9. However, the

parameters need not drift as far to cause instability due to the less benignurnodeled dynamics in ths case, so the system exhibits unstable behavior

in approximately the same amount of time.It is important to understand that the parameters will drift and cause

instability in these cases no matter how s m a l l the disturbances. If the

disturbance is made s m a l l by filtering, the parameters will drift slowly and

the system will take a long time to become unstable but it will indeedbecome unstable if the disturbance persists long enough. Also a

sinusoidal disturbance at any frequency will cause parameter drift.It appears that the only way to counteract the drift cause by sinusoidal

disturbance is to swamp out the effects of the disturbance with probingreference inputs. These inputs must be more than sufficiently exciting.Further research is needed to quantify the effects of probing signals and

develop practical guidelines as to their use.

V. CONCLUSIONS

In this paper it was shown, by analytical methods and verified bySimulation results that existing adaptive algorithms as described in [I]-141, [ 6 ] , i'J, [19], have imbedded in their adaptation mechanisms infinite



888 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30, NO. 9 , SEPTEMBER 1985

-48 I0 BO 160 240 320 400 480 560

(a)

T I M E

-60 80 160 240 320 400 480 560

T I M E

(b)

Fig. 1 I. Simulation of CAI aith poorly damped unmodeled dynamics. fit = 2.0 andd(t) = 0.1 sin 8 t. System becomes un stable.)

gain operators which, in the presence of urnodeled dynamics, can cause

the following:

instability, if the reference input is a high frequency sinusoidinstability, if there is a sinusoidal output disturbance at any

frequency including dc.

While the first problem c n be alleviated by proper limitations on the

class of permissible reference inputs, the designer has no control over theadditive output disturbances which impact his system. Sinusoidal distur-bances are extremely common in practice and can produce disastrous

instabilities in the adaptive algorithms considered.Suggested remedies in the literature such as low-pass filtering of plant

output or error signal [ 2 6 ] , [7l, [21]wi not work either. It is shown in

[151 that adding the filter to he output of the plant does nothing to change

the basic stability problemas discussed in Section II-B. It is also shown in

[I51 that filtering of the output error merely results in the destabilizinginput being at a lower frequency.

Exactly analogous results were also obtained for discrete-time al-

gor i thms as described in [ 5 ] , [ lq , [18], [20] and have been reported in

r151.Finally, unless something is done to eliminate the adverse reaction to

disturbances at any frequency in the presence of urnodeled dynamics, theexisting adaptive algorithms should only be considered as alternatives toother methods of control in situations where stable control c n be

guaranteedby possibly switching to known stable backup controllers or by

implementing other “safety nets.” A deeper understanding of theproblems presented here may ead to increased understanding of what

“safety nets” are applicable and reasonable.

REFERENCES

[I] K. S. Narendra nd L. S. Valavani, Stable adaptiw controller design directcontrol,” IEEE Trans. Automat . Contr. ,vol. AC-23. p p . 570-583, Aug. 1978.

[2] A. Fwer and A . S. Morse, “Adaptive control of single-input single-output linear

systems,” IEEETrans. Automat.Contr., vol. AC-23, pp. 557-570,Aug.

1978.K. S. Narendra, Y.H Lm. nd L. S. Valavani.“Stableadaptivecontrollerdesign. Pan II:Proof of stability,” IEEE Trans. Automat. Contr., vol. A C-25,

A. S. Morse, “Global stability of parameter adaptive control system s,” IEEEpp. 440-448, une 1980.

G. C. GoodwnP. J. Ramadge, and P. E. Caines. “Discrete-time multivariableTrans. Automat. Contr., vol. AC-25, pp. 43 34 40 , June 1980.

adaptive control,” IEEE Tram. Auttomat. Contr., vol. AC-25, pp. 449-456,June1980.I. D. Landau nd H. M.Silveira, “A stability heoremwith pplications to

adaptive control.” IEEE Trans. Automar. Contr., vol. AC-24, pp. 305-312,Apr.1979.B. Egardt. Stability nalysis of continuous-time daptive ontrol system s,”SIAM J . Confr. Optimiz., vol. 18, no. 5 , pp. 540-557, Sept. 1980.L. S . Valavani,“Stabilityandconvergence of adaptivecontml algorithms: A

survey and some new results,” Proc. JACC Conf.. San Francisco, CA Aug.1980.B. Egardt,“Unification of somecontinuous-timeadaptivecontrol chemes,”IEEE Trans. Automar. Contr.. vol. AC-24, no. 4, pp. 588-592. Aug. 1979.P. A. Ioannou and P . V. Kokotovic, ”Erro r bound for model-plant mismatch inidentifiers and adaptiveobsetvers,” IEEE Trans. Automat . Confr. , ol. AC-27.

C. Rohrs, L. Valavani, M. Athans, and G. tein,“Analyticalverification of

pp. 921-927, Aug. 1982.

undesirable properties of d i r e c t model reference adaptive control algorithms,”Mass. Inst. Technol.. Cambridge, LJDS-P-1122, Aug. 1981; also in Proc. 20thIEEE Conf . Decision Contr. . San Diego, CA, Dec. 1981.P. Ioannou and P. V . Kokotovic, “Singular pemrbations and robust redesign of

adaptive control ,” Proc. 21st. IEEE Conf . Decision Contr., Orlando, FL,Dec.1982, pp. 24-29.C. Rohrs, L. Valavani, and M. Atbans, “Convergence studies of adaptive controlalgorithms, Part I: Analysis,” in Proc. IEEE Conf. Decision Contr., Albuquer-

R. V. Mo nop li, “Model eferenceadaptivecontrolwith an augmented errorque, NM, 1980, pp. 1138-1141.

signal,” IEEE Trans. Automat. Contr., vol. AC-19, pp. 474-4 84, O d . 1974.

dissertation, Dep. Elec. Eng . Comput. Sci., Mass. Inst. Technol., Aug. 1982.C. R o b , “Adaptive control n he presence of unmcdeled dynamics,” Ph.D .

P. Ioannou, “Robusmess of model reference adaptive schemes with respect tomodeling errors.” Ph .D. dissertation, Dep. Elec. Eng ., Univ. Illinois at Urbana,

B. gardt, “Unification of some discrete-time adaptive control schemes,” IEEERep. DC-53, Aug. 1982.



C-30,O. 9, SEPTEMBER 1985 889

K. . M m nd B. Wittenmark, “A self-tuning regulator,” Automatica, no. 8 ,

Trans, Automat. Contr.. vol. AC-25, no. 4, pp. 693-697, Aug. 1980.

p p . 185-199, 1973.

G Kreisselmeier.“Ada ptive control via adaptiveobservationandasymptoticfeedbackmatrix synthesis,” IEEE Trans. Automat . Contr. . vol.AC-25, pp.717-722, Aug. 1980.I. D . Landau “An extension of a stabilityheorem pplicable to adaptive

I. D. Landau and R. Lo r n o , “Unification of discrete-time explicitmodelcontrol,“ IEEE Trans. Automat . Contr. ,vol. AC-25, pp. 814-817, Aug. 1980.

K.S. Narendra and Y.H. Lin “Stable discrete adaptive control,” IEEE Trans.reference adaptive control design,”Automatica. no. 4, pp. 593-611, July 1981.

R. L. Kosut and B. Friedlander, “Performance robustness properties of adaptive

Automaf. Confr.,vol. AC-25, pp. 456-461, June 1980.

control systems,” in Proc. 21st. IEEE Conf. Decision Contr., Orlando, FL,

C . E. Rohrs, L. Valavani, M. Athans, and G tein, “Robustness of adaptivecontrol algorithms n the presenceof unmodeled dynamics,.” inProc. 21sr. IEEEConf. Decision Contr., Orlando, FL, Dec. 1982, pp. 3-11.B. D . 0 nderson,“Exponentialconvergence andpersistent excitation,” in

J . Krause, M.Athans, S. S. Sasuy, and L. Valavani, “Robustnessstudies n

h o c . 2Jst IEEE Conf. Decision Contr., Orlando, FL, Dec. 1982, pp. 12-17.

Dec. 1982, pp. 18-23.

inmagnitude to overcome destabilizingeffects from high-frequency

inputs or noise.

The heuristic arguments given above are supported by analysis and

simulations n [2] and [3]. Differentialequationswhichapproximatelydescribe the dynamics of the adaptation gains are derived using averagingmethods. It is shown that these equations have an equilibrium line when

the reference is a step and that the parameterswdrift along this line

when there are disturbances. It is also shown that the equilibrium is a

point for sinusoidalcommand ignals. The equilibriumand ts local

stability depend on the frequency of the reference signal.

It may justifiably be questioned if the properties of the reference signalcan be postulated. If this cannot be one the excitation of the input signalcan be monitored. If the signal is not properly exciting, perturbations can

be added, as suggestedbydualcontrol theory, or the adaptationbe

switched off 4]. Variations of such schemesare found in several practical

adaptive systems [ 5 ] . They may be viewed as implementations of thecommon sense rule “don’t fit a model to bad data.”

adaptive control,” in Proc. 22nd IEEE Conf. Decision Conrr., San Antonio, In closing I would like to thank you, Charles, for sticking your neck outTX, DE. 14-16. 1983, pp. 977-981. as a young Ph.D. and challenging “the adaptive establishment.” I have

personally enjoyed our discussions. I have learned a lot from them andfrom trying to understand what happens in your simulations.

A Cornmentaw on the C. E. Rohrs et al. Paper“Robustness of Continuous-Time Adaptive Control

of Unmodeled Dynamics”

KARL OHAN ASTROM

versions of the above paper have been presented at several1980. These presentations have certainly contributed

makingthe sessions on adaptive control at the CDC Conferences

and fun. The discussionshavealso nspireda otofworkon

o our

It is therefore with great pleasure thatsee

above paper is finally publishedo that the discussions can reach a

was demonstrated byRobs that a simpleMRAS with two parametersbehave peculiarly when applied to a system with unmodeled high-

the reference signal is a step, then the system can

driven unstable by adding a mall sinusoid with proper frequencyo the

r by adding measurement noise of any frequency. The paperthe observed henomena y emonstratinghat there are

with infinite gain in the adaptive loop. It is then pragmatically

this will generically lead to difficulties.

my opinion that this argument does notapture all the aspects of the

I would therefore offer another explana-of the observed phenomena. The key elements of the argument are

be determined reliably unless the input signal isexciting of the appropriate order [ l ] . In thepresenceof

dynamics t s also critical hatexcitation sachieved bythe proper frequency range. From this viewpoint the source of

is that the reference signal is a step which is persistentlyThis means hatonly one parametercan be

two parameters are adjusted they may end

In the presence of unmodeled dynamics the

so large that the system becomes unstable.difficulties can be overcome if he nput sasinusoidwhich s

exciting of order two. Two parameters can hen be deter-Unmodeled ynamicswill ot ause ny ifficultiesf the

reference signal is sufficiently low and sufficiently large

R F N X E N C E S

[l] K. J. Astr6m and T. Boblii, “Numerical identification ofLineardynamic systemsfrom normal operating records,” in Theory of Self-Adaptive Control Systems,P. H. ammond, Ed. New York Plenum, 1966.

121 K. I . Astnjm, “Analysis of Rohrs counterexample o adaptive control,” in Proc.22nd IEEE Conf. Decision Contr., SanAntonio TX, pp. 982-982, Dec. 1983.

[3] - “Interactions between excitation and m o d el e d dynamics in adaptivecontrol,” in Proc. 23rdIEEE Conf. Decision Contr., Las Vegas,NV, pp.1276-1281.

[41 - “LQG self-tuners,” n Adapt ive Systems in Control andSignalProcessing, Proc. IFAC Workshop, San Francisco, June 1983), Landau et al.E d . New York: Oxfordb 1984.

[ 5 ] B.Wittenmark and K. . Astrom, “Practical issues in the implementation of self-tuning control,” Automatica, vol. 20, pp. 595605. 1984.

On the Stability of Parallel MRAS with an AEF

BULENT KERIM ALTAY

Absstruct-The stability of the parallel model reference adaptive system(MRAS) with an adjustable error fiiter ( A m is discnssed. It is shownthat thenull solutions of the parameter error vector and output error are

asymptotically stable if the parameter update recursion employs time-varying adaptationgainmatrix.Theanalysisproceedswith imple

algebra, eliminates the strict positive realness SPR)machinery, and does

not resort to Lyapnnov’s direct method and hyperstability.

I. INTRODUCTION

In parallel M U S , the adjustable system is described by the difference

equation

25, 1985. LManuscript r eceived Octo ber 15, 1984.

uthor iswith

rhe Department ofAutomatic Control. Lund Institute of The isithhe hp mentf ~ l ~ h dnd ~ l ~ ~ ~iddle, Lund, Sweden. East Technical Universicy, Ankara, T urkey.

0018-9286/85/0900-0889 01.00 985 IEEE

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Robustness of Continuous-time Adaptive Control Algorithms Inthe Presence of Unmodelede Dynamics