A Robust Direct Adaptive Controller - CiteSeerX

IEEE TRANS.ACTIOKS ON AUTO!viATIC COSTROL. VOL. AC-31. NO. I I . NOVEMBER 1986

1033

A Robust Direct Adaptive Controller PETROS A. IOANNOU

AND

Abstrucr-This paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics. The algorithm is designed based on the reducedorder plant, which is assumed to be minimum phase and of known order andrelativedegree, but is analyzedwith respect to the overall plant which, due to the unmodeled dynamics, may be nonminimum phase and of unknown order and relative degree. It is shown that if the unmodeled dynamicsaresufficiently small in thelow-frequencyrange,thenthe algorithm guarantees boundedness of all signals in the adaptive loop and “small” residual tracking errors for any bounded initial conditions. In the absence of unmodeled dynamics, the residual tracking error is shown to be zero.

I. IKTRODUCTIOK ECENTLY. several attempts have beenmade to formulate and analyze the adaptive control of plants with unmodeled dynamics. In [ I]-[8] it is shown that unmodeled dynamics or even small bounded disturbances can causemost of the adaptive control algorithms to go unstable. These instability phenomena are being investigated by linearization [4]. [6], [7]. Lyapunov-like techniques [3]. [5]. [8], [9]. conic nonlinearity [lo]. [ll]. averaging methods[6],[13]-[20].andsimilar analytical tools [12]. [21]. 1271. The modification ofadaptivealgorithmsforcounteracting instabilities andimprovingrobustness with respecttobounded disturbances and unmodeleddynamicsisveryimportant for applications. It received the attention of several researchers who came up with several approaches for robustness. In the case of bounded disturbances. the basic idea of most of the modifications is to prevent the instability by eliminating the to guarantee pure integral action of theadaptivelawsand boundedness of all signals in the adaptive loop. In [I]. [2]? 131, [22]-[24], a dead zone is introduced which consists of switching off adaptation when thetrackingerror is within a selected threshold. As stated by the authors. the main drawback of this approach is that several terms which depend on the unknown plant parametersandtheboundsforthedisturbancesare needed in A conservative calculatingandimplementingthedeadzone. bound for the dead zone leads to large tracking errors. evenwhen the disturbances are small or zero. The knowledge of an upper bound Mo for the norm ofthe desired controller parameter vector is used in 1251, 1261 to retard adaptation whenever the norm of the estimated parameters exceeds M o . In contrast to the dead zone. this modification guarantees boundedness andzero residual tracking errors when the disturbances are removed. A umodification. i.e., an adaptive law with the extra term - u0. u > 0 is used in 151. [S] for the case of plants with relative degree one. The importance ofthis modification is that no a priori knowledge is required to design u. However, as in the case of the dead zone, zero residual trackingerrorscannotbeguaranteed when the

R

Manuscript received July 8.1985;revisedFebruan, 25, 1986. Paper recommended by Associate Editor, R. R . Bitmead. This work was supported by the National Science Foundation under Grant ECS-8312233. The authors are with the Department of Electrical Engineering-Systems, Universit) of Southern California, Los Angeles. C A 90089-0781. IEEE Log Number 8610813.

KOSTAS S. TSAKALIS

disturbancesareremoved.Anotherapproach[27], [28] for handling disturbances requires the reference input signal to have enough frequencies for the measurement vector to be persistently exciting in order to guarantee exponential stability and therefore boundedness in the presence of disturbances. As shown in [28] the disturbance should besmall relative to the excitingsignal, otherwise the disturbance can counteract excitation and may lead to instability. When unmodeled dynamics are present, global stability cannot be guaranteed by simply eliminating the pureintegral action of the adaptivelaws.The unmodeled dynamicsactasanexternal disturbance in theadaptivescheme which can no longerbe assumed to be bounded. Despite this difficulty, however, several local results havebeenobtained in the literatureforadaptive schemesappliedto plants whosemodeledparts are minimum phase and of relative degree one and whose unmodeled parts are due to fast and stable parasitics. In [8], [9] it is shown that the umodification guaranteestheexistence of a “large”region of attraction from which all signals are bounded and thetracking error converges to a “small” residual set provided the amplitude and frequency content of the reference input signal is away from the parasitic range. In theabsence of parasitics. however,the residual trackingerror may benonzero. To avoid nonzero u is trackingerrors in the ideal case,thedesignparameter modified [29] so that it is zero whenever the normof the controller parameters is below Mo. Local robustness results as a consequence of persistence of excitation through the reference input signal are shown in [ 101: [ 141, [ 171. Global stability results are obtained in [ X ] , [30] for a wide class of unmodeled dynamics by establishing that the linear time-invariantpart of the adaptive loop is strictly positive real and by using persistently exciting signals [30] or a modified adaptive law 1251. The robustness results of are applicableto 181-[lo], [14]: [17],[25],[30],however, continuous-time plants whose dominant parts are minimum phase and of relative degree (n*) one and whose unmodeled parts are stable. The extension of these results tothe general case of H * > 1 without any further modifications of the structure of the controller and adaptive laws is not clear at this stage. Interesting robustness results are obtainedby Praly in [31]$[32] for discrete-time plants whose modeled parts may have arbitrary but known relative degree and whose unmodeled parts arestable. by introducingthe idea of a normalizing signal to bound the modeling error. and a projection which keeps theparameter estimatesboundedand within a chosensphere. If thedesired controllerparametervectoris within thesphere.thenglobal stability is guaranteed in the sense that for any bounded initial conditionsallthesignals in theadaptivelooparebounded. In another approach 1331. the idea of normalization together with a relative dead zone and a projection are used in the adaptive law to achieve robustness fora discrete-time algorithm. The approach of [33]requiresboundsfortheunknown plant andcontroller parameters as well as for the plant zeros. A similar approach as in [33]is used toshowrobustness in thecase of an indirect continuous-time adaptive controller 1371. It is shown that if the unknown parameters of the plant lie in a known convex set throughout which no unstable pole-zero cancellation occurs, then the use of normalization together with a suitably designed relative deadzoneguarantees stability in thepresence of small plant uncertainties. In [31]-1331, [37] the class of unmodeled dynamics

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AC-31, IEEE AUTOMATIC VOL. COhTROL. TRANSACTIONS ON

considered are dueto additive plant perturbations which are stable and sufficiently small at allfrequenciesand/orare sufficiently small relative to the normalizing signal. In this paper we propose a direct adaptive control algorithm which is applicable to continuous-time plants with arbitrary relative degree. The algorithm is designed for the reduced-order plant which is a w m e d t o be minimum phase and of known relative degreeandorder. but appliedto the full-order plant which.due to the unmodeled dynamics. may be nonminimum phase and of unknown order. The class of unmodeled dynamics of both additive and considered is a generaloneandconsists multiplicative plant perturbations.Theadditive plant perturbafrequencies but the tions are assumed 10 be small at all multiplicative onesare reqtlired to be small only at the lowfrequencyrange. \4'e show that for this class of unmodeled dynamics the proposed algorithm guarantees boundedness for all signals in the closed loop and residual tracking errors which are small in the mean. In the absence of modeling error, the algorithm guaranteeszero residual trackingerror.Therobustness of the algorithm is achieved by using a robustadaptivelaw which employs a normalizing signal m and a - UB term a5 in [34], in such a way that the boundedness of the controllerparameter vector B and 19can be established, irrespective of the boundedness of the 5ignals in the adaptive loop. The only u priori information required forthe implementation of the algorithm is an upper bound for the norm of the desired controller parameter O* as in [26], 1291. [31]. 1321 and a bound on the stability margin of the poles of the unmodeled dynamics. Our contributions relative to the work of 13I ] . 1311. [33]. [371 andprevious work are the following. i) We gave a complete (n* 2 1 ) solution to the robustness problem of continuous-timedirectadaptivecontrol.Such a problem was the center of many discussions andtreatnlents during the recent years [5]-[16]. 1231. [28], [30],13-1.1. i,i) The class of unmodeled dynamics considered includes both additive and multiplicative type of perturbations and is similar to that considered in robust control with known parameters. iii) We use the a-modification in the adaptive law which led to different but less restrictive assumptions than those made in [33],

WI.

iv) Theproposedalgorithmrequiresno persistently exciting signals and no positivity condition needs to be checked. The paper is organized as follows. In Section I1 we give the model of the plant and state the control objective. The structure of the proposed algorithm is given in Section 111. In Section IV we analyze the robustness properties of the algorithm with respect to unmodeled dynamics and state our result. The results are demonstrated in Section v using a simple example.

S3: the sign of k,, and the values of m, n are known. Without loss of generality, we shall assume that k p > 0. For the unmodeled part of the plant we assumed that: S4: AI@) is a strictly proper stable transfer function, SS: &(s) is a stable transfer function. S6: a lower bound po > 0 on the stability margin p > 0 for which the poles of Al(s p ) , &(s - p ) are stable is known. Remnrk 2.1: It is worth noting that assumptions SI to S3 are standard in most adaptive control systems. The minimum-phase assumption for G&) is needed in any reference model following whether it is adaptive or not 139). However. SI to S3 do not imply that the full-order transfer function G(s) is minimum phase nor that the order o r relative degree of G(s) is known. Remark 2.2: We should note that small p guarantees 1 &( ju)l to he small in the low-frequency range. However. since A2(s) IS allowed to be improper for n* = n - m > 1 . I p a z (j w ) I may be large at high frequencies, i.e.. ~

lp&(jw)I-~

~

.?o=A+~o+bou+Alq ~))0=A~qo+~b~i

(2* 4) (2.5)

by using a first-orderlow-pass filter to filter anystrongly observable parasitics in the plant output. In (2.4)-(2.6) -yo E Rno is the slow state, qo E R"'0 is the parasitic state, Re h(AJ < 0 and 1 % E > 0 is thesingularperturbationparameter.From (2.4)-(2.6) it is clear that

Consider the single-input single-output (SISO) plant

p = E,

Go(s) = c,'(sI - Ao)-'bo, and

Az(s) =

where G(s) is strictly proper; G&) is the transfer function of the modeled part of the plant: pAl(s), pAz(s) is an additiveand a multiplicalive plant perturbation,respectively.Forclarity of presentation and without loss of generality, both Al(s), A:(s) are rated by the same positive scalar parameter p. For the modeled part of the plant

(2.3)

even when p is very small. Remnrk 2.3: The only a priori informationrequiredabout A,@), A:($ is a lower bound on the stability margin p of their poles. In the case of unmodeled fast dynamics, p is of O ( ~ / E ) where 0 < E < 1 and therefore a lower bound Po in S6 can be easily found. Remark 2.4: Weshouldnote that theparameter p in (2.1) rating the plant uncertainty Al(s), &(s) is not an artificial one. For example. consider a SISO linear time-invariant plant with n slow ("dominant") and u z fast ("parasitic") stable modes. Considering the slow eigenvalues to be O( 1). the fast eigenvalues are of at least O(l/t), where E is smallunknown positive scalar. Such a plant can be represented in the singular perturbation form [35], 151. I81

where

11. PLANTAND THE CONTROL 0mcl-m

as (wI+=

sc,'

[adj ( s Z - A o ) ] A I ( ~ ~ Z - - A 2 ) - 1 b C; [adj (sZ- Ao)] bo

The stability of &(s) follows from the stability of A 2 and the assumption that Go($ is minimumphase.Similarly, the unmodeled fast dynamics in (2.4)-(2.6) can be expressed asan additive perturbation of the form pAl(s), where A,@) is a strictly proper stable transfer function whose poles are of O(l/p), by using an additional low-pass filter to filter the input u as shown in [34]. The adaptive control problem can be stated as follows. Given the reference model

(2.2)

we make the following assumptions: SI: Z&) is a rnonic Hurwitz polynomial of degree m( In 111

S2: R,(s) is a monic polynomial of degree n,

where D,,,(s)is a monic Hunvitz polynomial of degree n* = n and r ( f )is a unifomily bounded reference input signal, design an adaptive controller so that for some p* > 0 and any p E [0, p * ) the resulting closed-loopplant is stable and the plant outputy 177

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IOANNOU AND TSAKALIS: ROBUST DIRECT ADAPTIVE CONTROLLER

of W,(S - qo) andthe where qo > 0 is suchthatthepoles qd are stable and 62 is a positive constant. eigenvalues of F Since W&) and (sZ - F ) havestablepolesandare to be designed. (3.8) canalwaysbesatisfied if p o in S6 is known. Hence. the only a priori information used in_the adaptive law III. STRUCTURE OF THE ADAPTIVE CONTROL SYSTEM (3.3), (3.6) is an upper bound for 110*11 and-llO*ll, respectively. The input u andoutput y are used io generatea (2n - 1)- Our analysis in Section IV shows thatMo or Mocan be as large as possible but less than O( 1 / p ) . Since in most applicationsp is small dimensional auxiliary vector as [34]choosing Moor l i i o large we can always satisfy Mo > )1O*11 or d l = F w l + q u , &=FW2+qy (3.1) $io > Ile*ll max andstillremainawayfrom the O( 1 / p ) range. The design parametersh,, 62, uoare arbitrary fixed where F is a stable matrix, ( F , q) is a controllable pair, andw' = constants which, as our analysis shows, may affect the conver[UT, UT, y ] . The input to the plant is taken as gence rates of the signals in the adaptive loop. The important m and the u = e T W + cor (3 . a terms in the adaptive law are the normalizing signal switching a-modification. In contrast to our earlier results 1291, where OT(t) = [OF, O?(f), O,(r)] is a (2n - 1)-dimensional [34], the switching of u from 0 to uo is modified so that u is a continuous function of IlOll, a condition required in our analysis. co(t) is afeedforwardparameter controlparametervectorand An important property of the normalizing signalm which is used scalar. Thecomplexity of theadaptivelawforadjustingthe in our stability analysis is given by the following lemma. e(t), co(t) is determined by theprior controllerparameters Lemma 3.1: Consider the system knowledge of the gaink, of the reduced-order plant.We first deal with the simpler case, when kp is known, and later consider the z = W ( S )u (3.9) case when kp is unknown. For clarityof presentation, the proof of stability is also given separately for the two cases in Section IV. where W(s) is a strictlyproperstabletransferfunctionwhose Case i (kpknown): With no loss of generality we can assume poles pl satisfy that k, = k, = 1. This implies that co = 1 in (3.2).The equation for the adaptive law to adjust the parameter vector 0 is h 0 + h 2 s r n i n \Re (pl)l (3.10)

tracks the outputy , of (2.8) as closely as possible forall possible plantperturbations AI@), A2(s) satisfying S4, S 5 , S6. In the following sections we present the design and stability of such a controller.

+

(a, m)

J

and U(t) 5 lu(t)l + ly(t)l constant cI > 0 such that

+ m(t),V t 2 0. Then there exists a (3.11)

where E , is a term which depends on initial conditions and decays exponentially to zero with a rate at least as fast as exp (-&of). TheproofofLemma3.1isgiven in Appendix B. In the followingsection we will useLemma 3.1 in establishingthe stabilitypropertiesandrobustnessoftheproposedalgorithm (3.1)-(3.8) with respect to unmodeled dynamics.

case ii (kp unknown): The equation for the adaptive now given by

IV. ROBUSTNESS ANALYSIS

Let us now apply the algorithm (3.1)-(3.8) to the full-order plant (2.1). The question we need to answer is the following. Is there a class of unmodeled dynamics, i.e., a p* > 0 such that for each p E [0, p * ) the plant (2.1) with the controller (3.1)-(3.8) is stable for any possible perturbationAI(s), A&) satisfying S4,S5, where u is as defined in (3.3) with 8 , Mo replaced by 8 and f l o , S6. By stability we mean that for any bounded initial conditions respectively, and any uniformly bounded reference input r(t), all the signals in the closed loop remain bounded. We answer the above question by e=[OT',co, 4 5 0 ] T , F=diag (r, 7 , 71, ! = K T , y m , { I 7 the following theorem. Theorem 4.1: Assume that r, r are bounded. Then there exists E ~ = ~ ~~ = e -~ ~ - ~v + C~ o ~ ,~n - v +yoo.= ~WAS)^^^ ~ ~ (3.7) ~ a p* > 0 such that for each p E [0, p* ) all the signals of the fullorder closed-loop plant (2.1) with the controller (3.1)-(3.8) are and signals m , v , rare as given in (3.4). (3.5).In this case two bounded for any bounded initial conditions. Furthermore, there additional scalar parametersco, $o need to be adjusted: otherwise. existsaconstant y1 > 0 andasmallconstant E suchthatthe the form of the adaptive laws is the same as in Case i. tracking error el = yo - y , belongs to the residual set Remark 3.1: The structure of the controller (3.1)-(3.7) can be shown to be similar to that in [36]by choosing L(s)in [35]as L(s) = W ; I(s). However, the adaptive laws are different due to the umodification and the normalization of the augmented error el or FI by nzr when used in the adaptive law. In (3.3)-(3.7), h l , uo arepositivedesignparameters, Mo > ~ p y l + F ,V t o Z O , T>O . (4.1) IIO* 11, A?o > Il8*ll max X@) where By = [ O r T , e;', 0: ] and = [e* ', c: , $,* ] are the desired controller Corollary 4.1: In the absence of modeling error (i.e., when p parameter vectors definedin 1361. The parameter h0 is designed so = 0 ) the algorithm (3.1)-(3.8) guarantees boundedness aswell as that convergence of the tracking error el to zero. For clarity and ease of presentation we give the proof for the 60+625min [ P O , 401 (3.8) case of k, -= 1. In Appendix D we show how the proof of

'

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Theorem 4.1 andCorollary 4.1 forthecaseof kp unknown follows from the proof for k,, = 1 . Proof of Theorem 4.1 (k,, = I): Using a result in [36], where it is shown that a constant vector 8" exists such that forf3 = f3* the transfer function Go(s) of the modeled part of the plant together with thecontroller (3.1),(3.2) matches that of the reference model (2.8), we show in Appendix A that for kp = 1 (i.e., c; = 1) the plant output may be expressed as Y = w,(s)[QTw+r1+pq,

q=A(s)u

(4.2)

where

arbitraryconstant.Thus,theadaptive law (4.6) guaranteesthe boundedness of Q and therefore 8 and the smallness in the mean for ( Q r { ) ? / m z ,lQTw I / m and 08 T$I for small p and irrespective in the of theboundedness of m , {, w . q or anyothersignal adaptive loop. In order to complete the stability analysis we need to analyze thestabilitypropertiesofthe output errorequation (4.4) and establishboundedness of theremainingsignals in theadaptive loop. Consider the following nonminimal state representation for (4.4) obtained in Appendix A. i.e.. for c; = 1 P = A , e + b , d T ~ + p b , l q I + p b c 2 ~ 2e l. = h r e + p q l (4.12)

A ( s ) = W,(s)A,(s)[l - O f * ( s I - F ) - ' q ] + A , ( S ) [ I + w,(s)(e,*+e:*(sz-~)-lq)l

(4.3)

is strictly proper due to the fact that G(s),A,(s) are strictly proper and Go(s) has the same relative degree as W,(s). From (2.8) and (4.2) the measured tracking error el(t) can be expressed as

el = W,(S)Q

The auxiliary error signal

E],

+ pq.

(4.4)

used in the adaptive law,

is then

given by

where q1 = Al(s)u, 9, E &(s)u, A , is a stable matrix, &(S) is proper, and the polesof A,(s - p o )are stable. In order to analyze (4.12) we consider the positive definite function

m2 W=kleTPe+2

(4.13)

where kl > 0 is an arbitrary constant to be chosen and P 0 satisfies

=

PA,+AJP= - I .

Pr > (4.14)

The time derivative of W along (4.12) and (3.5) satisfies El=

wm(S)~Tu+erw,(s)w- w,(s)eTw+pL?.

(4.5)

Notingthat 8' W,(s)w - W,(s)f3 = Q T W,(s)w - W,(s)Q' w . we have that = 4T{ + pq and therefore the adaptive law (3.3) can be expressed as

(4.6) In order to analyze (4.6) let us choosethepositivedefinite function 1 v=6Tr - I + . (4.7) 2 The time derivative of V along (4.6) is given by

Using assumption S6, (3.8) and applying Lemma 3.1 to 17 in (4.2) it follows that (71/ m is bounded. Hence, (4.8) can be written as

W= - k l ~ ~ e ( ~ 2 + 2 k l e 7 P b , ~ T w + 2 p t k , e T ~ ( b , I ~ I + b , 2 ~ ~ -60mz+61t?7(/u\+ I)'(

+ 1).

(4.15)

Since 8 is bounded, we can use (A9)-(A12) of Appendix A to show that IuI + l y ( 5 y,lle(l + ys + p791q11for some positive llOll, Iy,,,I and constants yi, ys, y9 which depend on the bounds for llun,11, where w, is a bounded signal defined in (A 1 1). Using S6, (3.8) it follows from Lemma 3.1 and Remark B1 of Appendix B that lqlI /m,172 1 /m are bounded: therefore (4.15) may be written as

li's -klIIe(lz+klyloIIe)((QT(??1-60,2

+ ~ o r n + P 1 ~ ~ e ~ ~ m + p k ~ ~ l ~ ((4.16) ~e~~m+ where 7 1 0 = 2IIPb,11, PO = 6 d 1 + y d , $1 = YII = 2JJPll(llbc~llTl + lIbc2II'Y2), -79 = Y P - ~ and ~ 71,% are bounds for IqA/m and ( q z /l m , respectively. Noting that klyIo(((el(/ ~~W)(~QTw~/@ 5 ) v'K&(l$ITwl/m)W W where P i = y;,[min X(P)] - I (4.16) can be rewritten as

> 0 for ()f3)1 > MO where y2 is the bound for 1 q I 2 / r n 2 . Since dTQ and d l T + = 0 for llf3ll < Mo, ri < 0 whenever V > V, for some fixedconstant Vo > 0. Hence, [38] V andtherefore Q are uniformly bounded and none of the signals in the closed loop can (4.9) we growfasterthananexponential.Furthermore,from obtain

where gl = sup,, I V(fo)- V(fo + T)I and g, = y2/2. The expression (4.10) implies that for small p, (&r{)2/m2 and oOTQ are small in the mean. Using (4.10) we show in Appendix C that

V t r t o r 0 (4.11)

where p , y3 to

76

are positive constants and

EO

E (0, 11 is an

(4.17) By choosing kl = SP:/So, po = min (60/8/31yll,60/86~T,) and adding and subtracting the term - P W on the right-hand side of (4.17). we have that for each p E [O. p o l

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I0.4NNOU AND TSAKALIS: ROBUST DIRECT ADAPTIVE CONTROLLER

Choosing 0 = min I l / 2 ~ ~ PS o~] and ~ , letting 20i/SO, we have that w5

-pw+p,

y

7

01 = v'kl/31,p4 =

W+P4.

(4.19)

We analyze the stability properties of (4.19) by considering the system ?4f0=

where q2,q3,q4, eo are positive constants which depend on the bound for I/m and on the constantsy3,-y4, y6, eo of (4.11) and Cis small for small eo. Using (4.29) in (4.28) integrating and letting T 03, the proof is complete. Proof of Corollary 4. I (kp = 1): Setting p = 0 the tracking error equation (4.4) and adaptive law (4.6) become -+

-pwo+pj Id wo + P 4 m

with Wo(to)= @'(to). The homogeneous part of

Since l / m is bounded, it follows from (4.11) that

(4.20) (4.20) is

w o=

el = W,.r(s)+Tw (4.21)

(4.30) (4.31)

and therefore The time derivative of thepositivedefinitefunction 4 T r - 1 ~ /along 2 (4.31) is now given by Vt?fo>O.

(4.32)

(4.22)

Using (4.11) in (4.22) we obtain that the equilibrium Wo = 0 is exponentially stable if

Y =

which implies that I/ is bounded and that

(4.23)

Fix eo to be in the range 0 < eo 5 min ( ( / 3 / 4 & ~ ~ ) ~1)~and p , take

Then for each p E [ 0 , ,u* )

Since $J is bounded, it followsfromLemma Cli), vii) in Appendix C that d/dt(4Tl/n7)2is bounded and therefore (4T{/ m ) 2 is uniformly continuous. From LemmaC2 Appendix C, d T 4 is alsoauniformlycontinuousfunctionoftime.Hence,using Barbalat's lemma [42, p. 2101 it follows from (4.33) that (b'l)*/ m'+uBr~-)Oast-tmwhichimpliesthat~To/m-tOast~ m (see Theorem C2in Appendix C). Since from Theorem 4.1, m is bounded for any p E [01p*), we have that --t 0 as t 03. Hence, in view of (4.30) it followsthat e, goestozero asymptotically. Remark 4. I : For clarity and ease of presentation the exponentially decaying terms E , , appearing in the bounds in Lemma 3.1 to and LemmaC 1 of Appendix C, are assumed to be small enough be incorporated in the corresponding constants of the bounds in most parts of the proof of Theorem 4.1 and Corollary 4.1, Since from (4.7). (4.9) 4, 0 will still be bounded even if E , is included in the constant yz none of the signals in the closed loop can grow faster than an exponential. Therefore, since for f Ito and some finite to I0 the E , terms are small and none of the signals can become unbounded for any t E [0, to] the E , terms do not affect thepropertiesofthealgorithm given by Theorem 4.1 and Corollary 4.1. Remark 4.2: It is worth noting that the results of Theorem 4.1 are also applicable to plants with nonlinear perturbations such as -+

(4.25)

wo

Hence, = 0 is exponentiallystableandtherefore Wo(t)is bounded.Using the comparisontheorem[38,pp. 571 W(t) is boundedandtherefore n7, e arebounded.Boundednessof n7 implies that all the signals in the adaptive loop are bounded. In order to obtain a bound for the residual tracking error. we write the following minimal state representation for (4.4):

eO=A.,eo+b,,,4rw, el=hieo+pq

(4.26)

where eo E R '*, A,,,is a stable matrix, andhL(s1 - A , ) -Ib,, = W,,,(s).Using the variation of constantformula [41, p. 451, it follows that

Iel(t)lsP~IIeo(to)llexp [-qI(t-to)l +P6

['

-

)$'(7)w(7)1

exp [-ql(t-7)1 d7+&

(4.27)

Io

where q l ,pS,0 6 are positive constantsand 0,> 0 is the bound for (71. Hence,

(4.28)

i=Ax+bu+pbq,

(4.34)

Y=cx+Pql+g(x, u , t )

(4.35)

where g(x, u , t ) is a nonlinear function (not necessarily bounded), as long as I g(x, u , t )I In7 Ipkg for some constant kg > 0 and tlt L 0. Remark 4.3: The restriction for i. to be bounded required by Theorem 4.1 is needed only in the case ofkp unknown. However, such a restriction can be avoided by prefiltering r using a low-pass first-order filter before use in the adaptive algorithm. Remark 4.4: The a-modification usedin (3.3): (3.6) switches u from 0 to uo whenever l l B l l or l l e l l exceeds Mo or A&. An qlternative approachis to use a differentu for each element of0 or 0. The stability results and proofs presented will not change with this approach,eventhoughsimulationsindicatethatsuchan

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approach may havebeneficialeffects on thetransient of the adaptive algorithm. Remark 4.5: Theproposedadaptivealgorithmguarantees p E [0, p * ) andsome p* > 0 which boundednessforany depends on the parametersof the plant. Since these parameters are unknown p* cannot be calculated. However, the results obtained are qualitatively the same as those that could have beenobtained if the parameters of the modeled part of the plant were known. For example, if we set 8 = 8*, i.e., 4 = 0 and co = c$ we have from (A6) and (A4) that

-501

I

0

(4.36)

50

I

100 TIME

I

150

I

200

llall

+

where F2(s) = 0: O,*'(sZ - F ) - I q . Since the zeros of Go($ are stable, a sufficient condition for stability for any Al(s), A2(s) satisfying S4, S5 is that p E [0, p*) where

i

b* = inf I P ( j w ) l ;

6k

0

/

V

41 0

and p* > 0 due to the fact that1/ m(s)is proper and the zeros of F?'(s) are located in Re [s] < 0. The comparison between p* and &* is not clearatthisstagebecause p* is calculatedusing conservative bounds for the normof parameters and time varying signals in thestaterepresentations of the plant andadaptive algorithm when ,G* is obtainedfrom frequency a domain condition.

I

50

I

100 TIME

I

150

I

200

V. DISCUSSION AND EXAMPLE

In this sectionwe demonstrate the effectiveness ofthe proposed algorithm by applying it tothefollowingnonminimumphase plant: TIME

where1 > p > 0. Because of the unstablezero,noneofthe modelreferenceadaptivecontrolalgorithmsareapplicableto (5.1). In our approach, however, the unstable fast zero can be considered as aperturbation pAz(s) = p of the multiplicative form and the adaptive algorithm can be designed based on the dominant part of the plant y = l/s(s - I)u, which is minimum phase, rather then (5.1). The reference model is chosen to have the same relative degree as the dominant part of the plant, i.e., 1

Y m = ( s +I)(s+2)

Fig. 1.

Bounded response, but large transients due to large initial parameter error.

the transient and time of convergence improves considerably as indicated in Fig. 2 for 8(o) = [ - 3, 6, - 81 '. In all casesthenormalizingsignalplaysanimportantrole of the duringthetransient by guaranteeingslowadjustment controller parameters where appropriate. The smooth adjustment of the controllerparametersreducestheexcitationofthe unmodeled dynamics by the control input. and therefore improves the accuracy of adaptation. The price paid. however, for robustness and global stability is that the signals in the adaptive loop may have abad transient andlong time of convergence when the initial conditions. especially the initial parameter vector, are far away from the desired ones and the plant is initially unstable. However. such a tradeoff is realistic and should be expected in every robust adaptive control algorithm.

With this choice of reference model the perturbation ps in (5.1) violates two of the most crucial assumptions for stability of model reference adaptive control; the relative degree and the minimum phase assumptions for the overall plant. We apply the proposed algorithm (3.1)-(3.5) to (5.1) by choosing F = - 1, q = 1, 6o = 0.7,61 = 1, Mo = 20, uo = 0.1, r(t) = 10 sin 0.9 and y = 10. VI. CONCLUSIONS The simulationresults for p = 0.02 are presented in Figs. 1 and 2 In this paper a direct adaptive controller for continuous-time for different initial valuesof the parameter vector 8. When 8(o)is far from the desired parameter vector 8* = [ - 4, 8, - 101 so plants is presented and shown to be robust with respect to stable that the closed-loopplant is initially unstable at f = 0 the transient unmodeled dynamics of the additive and multiplicative form. The is as controller is designed for the reduced-orderplantwhich of the signals have large magnitude and take longer to converge shown in Fig.1for O(o) = [ - 2, 1, -41'. However. the assumed to be minimum phase and of known order and relative algorithm still converges and guarantees boundedness and small degree, butis appliedtotheoverallplantwhich,duetothe tracking errors. When the initial condition for8(o)is closer to 8* presence of unmodeled dynamics, may be nonminimum phase and

IOAKh'OlJ ROBUST AND TSAKALIS:

1039

DIRECT ADAPTIVE CONTROLLER

as these perturbationsare sufficiently small relative to the normalizing signal. The effect of certain design parameters in the adaptive loop, which for stability can be arbitrary: on the convergence rates and error bounds as well as the comparison of the proposed robust adaptive controller with robust fixed controllersare topics for future research.

-51

-10

APPENDIX

DERIVATION OF THE PLANT OUTPUT EQUATIOK ANI) N O ~ " M A L STATEERRORREPRESEKTATIOK

I.

0

A

I

I

50 TIME

IO0 EquationOutput

50 TIME

100

The plant equation (2.1) may be written as

111-

10.2 -

1 0 -

m

I

0

T

where F,(s) = 8; (SI - F ) -- Iq, F2(s) = O;'(sZ 8:. Using (A'I) .in ( A have l ) we

40t

+ G o ( s ) A 2 ( ~+) dl (s) l

OO

o

50

x 100

TIME Fig. 2.

- F ) -Iq -I-

Bounded response and €04 transient for small initial parameter' error.

of unknown order and relativedegree. It is shown thatif the unmodeled dynamics are sufficiently small in the low-frequency range, then the closed-loop plant is stable in the sense that for any bounded initial conditions all the signals in the closed-loop are bounded. Furthermore, it is shown that the residual tracking error to zero whenthe unmodeled is small in themeanandreduces dynamics disappear. The controller has the same structure as the one that would be used in the caseof known plant parameters. The controller parameters are adjusted using an adaptive law with a normalizing signal and a a-modification. An important property of this adaptivelaw is that itguarantees bounded controller parameters and bounded speed of adaptation irrespective of the in the closed loop.Theonly behavior of theothersignals additional a priori informationrequiredfor the design of the of the desired algorithm is an upper bound Mo forthenorm ) 1 O * 1 1 and a bound onthe stability controllerparametervector margin of the poles of theunmodeleddynamics. Our analysis allows Mo to be largebut away from the rangeof O(l/p). Since in most applications [34] p is small. Mo can be chosen large enough to satisfy Mo > 1 / O * 1 1 and still be awayfrom the O(l/p) range. It is clearfromouranalysisthattheproposedalgorithm will also guarantee stability in the presence of bounded disturbances and possibly unbounded nonlinear time varying perturbations as long

Using the relationship

1

11.

co*Go(s)

rvn,(s ) = 1 -F,(S)-FFz(S)Gn(S) '

643)

(A41

established in [36]. [39], in (A3) have we y=,

1

w,,,(s)[dTW+cgrI+p~(S)z~

(A5)

CO

where 4(s)=A1(s)+I / c ~ W , ~ ( s ) F z ( ~ ) A l ( s ) + ( l / c ~ ) W , , ( s ) A ~ ( s ) [l -Fl(s)].A n alternative expression can be obtained by substituting for u = { 1 - Fl(s)- F.(s)Go(s)- pF:(s)[A,(s)+ Go(s)42 (s)]}-l(+Tu+cor) in ( A 5 ) , i.e..

Nonminind State Error Representation The plant (2.1) can be expressed as

1040

IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. .4C-3 1. NO. I I . NOVEMBER 1986

where q(s) is an arbitrary polynomial of degree n* - 1 with the roots of q(s - p o ) stable. Since Go(s)A2(s) is strictly proper. it follows that Go(s)q(s) is strictly proper and &(s) _= Az(s)/q(s)is proper.Furthermore,using (S6) the poles ofAz(s - p o ) are stable. From (A7) we can write the following state representation: P=AX+bu+phf/:, Y=hTX+pqI

(A8)

where ( A , 6 , h 3 is-a minimal state representation of Go(s), ql(s) = Al(s)u, q2(s)= A2(s)u and 6depends on b and the coefficients of q(s). The input u may be expressed as

for some positive constants 65, A6. Finally. comparing (B6) to (B5) and noting that rn(t) L 61/6~, we obtain (3.11)wherecI is a ~ = ~ ~ w + c ~ r + e ~ ' ~ ~ T+ x +ep o, , * q~l . ~ ~(A9) + e ~positive h constant and er an exponentially decayingtermthat depends on the initial conditions and on exp ( - 6or). Defining the augmentedstate Y, = [ x T ,ur, u']' as in [36] it Remark BI: Lemma 3.1 also holds if t = w(s)U,where W(s) follows from (A8). (A9). (3.1) that is proper and U satisfies I U ( 5 f i 3 m for some A3 > 0. The proof of this statement follows directly from the proof of Lemma 3.1 by noting that Z = W(s)U + dU where W(s)is strictly proper and d is a constant. APPENDIX C

where A , is a stable matrix, A,, b,, h, are as defined in [36] and bCl = [ b T ,qTOf, q T I T ,bC2= [6', 0 , O l T . Since hT(s1 A,) -Ib, = l / c f W,,,(s)(see[36]) we canwrite the following nonminimal state representation for W&):

THESMALLNESS IN THE MEANOF Theorem CI: If

where g,, g2 are positive constraints, T bounded. then

APPEKDIX B Proof of Lemma 3.1: Let

.?=AX+ bU;

(B1)

=C S

(6 /rn

> 0, t

2 to 2 0 and

C$

is

with p = 2 -w*+I), y3to y6 are positive constants and eo E (0, 11. In the proof of Theorem C1 we will use several lemmas which are stated and proven below. Lernrna C1: There exists positive constants g3 to g;, a,, i = 1 , 2, n* 1 and Ti; i = 0, 1, 2, n* suchthat

+

. - a ,

e . . ,

be a minimal state representation of (3.9). Then

I

-r

z = c exp [At]x(O)+c

0

exp [ A ( t - ~ ) [ b U ( dr. i)

(B2)

From (3.10)it follows that there exists a constant a > 0 such that IIexp [AtlIlsa exp [ - ( S O + $ ) In view of (B3)andtheassumptionthat (B2) yields

i]

sa exp 1-6,tl. U 5 (uI

+

(B3)

( y (+

rn, lul

v) - 5 g : + e / ,

m

, n * + I , and

where 6;, h4 are positive constants. Moreover, from the definition of rn rn(r)=m(O)exp

where E , is an exponentially decaying term due to nonzero initial conditionsanddenotesthe ith derivativeof {. Proof: The proof of i), ii), iii) follows directly from (3.8) and Lemma 3.1 by noting that

(-~5~7)