ROBUST CONTINUOUS TIME ADAPTIVECONTROL BY PARAMETER PROJECTION �Sanjeev M. Naiky P. R. Kumary B. Erik YdstiezNovember 29, 1990AbstractWe consider the problem of adaptive control of a continuous time plant of arbitraryrelative degree, in the presence of bounded disturbances as well as unmodeled dynam-ics. The adaptation law we consider is the usual gradient update law with parameterprojection, the latter being the only robustness enhancement modi�cation employed.We show that if the unmodeled dynamics, which consists of multiplicative as well as ad-ditive system uncertainty, is small enough, then all the signals in the closed loop systemare bounded. This shows that extra modi�cations such as, for example, normalizationby a specially constructed signal, or relative dead zones, are not necessary for robust-ness with respect to bounded disturbances and small unmodeled dynamics. In thenominal case, where unmodeled dynamics and disturbances are absent, the asymptoticerror in tracking a given reference signal is zero. Moreover, the performance of theadaptive controller is also robust in that the mean-square tracking error is quadraticin the magnitude of the unmodeled dynamics and bounded disturbances, when bothare present.�The research reported here has been supported in part by U.S.A.R.O. under Contract No. DAAL 03-88-K-0046, the JSEP under Contract No. N00014-90-J-1270, and by an International Paper Fellowship forthe �rst author.yDepartment of Electrical and Computer Engineering and the Coordinated Science Laboratory, Universityof Illinois, 1101 West Spring�eld Av., Urbana, IL 61801, U.S.A.zDepartment of Chemical Engineering, Goessmann Laboratory, University of Massachusetts, Amherst,MA 01003, U.S.A. Please address all correspondence to the second author. A longer version of this is [20],to appear in the proceedings of the 1990 Grainger Lectures on Adaptive Control held at the University ofIllinois, Sept. 28-Oct. 1, 1990. 1

1 IntroductionRecently, there have been many attempts to study the adaptive control of plants withbounded disturbances and unmodeled dynamics. In his pioneering work, Egardt [5] showedthat even small bounded disturbances can cause instability in adaptively controlled plants.He further demonstrated that modi�cation of the adaptation law by projecting the param-eter estimates, at each time instant, onto a compact, convex set known to contain the trueparameter vector, however provides stability with respect to bounded disturbances.Proving the stability of an adaptive control system in the presence of unmodeled dynamicsis however more di�cult, and researchers have proposed various additional modi�cations tothe adaptation law to analyze and bound the e�ects of unmodeled dynamics. For example,in [8] [9] [10], Praly introduced the device of using a normalizing signal in the parameterestimator, while other modi�cations have included �-modi�cation [12] [14], normalized dead-zone [11], etc. An excellent uni�cation of these various modi�cations and results can be foundin Tao and Ioannou [18].In this paper, we show that Egardt's original simple modi�cation of parameter projectionis in fact enough to provide stability with respect to small unmodeled dynamics as well asbounded disturbances. Our main results are the following:(i) A certainty equivalent adaptive controller, using a gradient based parameter estimatorwith projection, ensures that all closed loop signals are bounded, when applied to a nominallyminimum phase continuous time system with bounded disturbances and small unmodeleddynamics (Theorem 9.1).(ii) In the absence of unmodeled dynamics and disturbances, i.e. in the nominal case, theerror in tracking a reference trajectory converges to zero (Theorem 10.1). When unmodeleddynamics as well as bounded disturbances are present, the mean-squared tracking error isquadratic in the magnitude of the unmodeled dynamics and bounded disturbances (Theo-rem 11.1). Thus the adaptive controller provides robust performance in addition to robustboundedness.While our work thus shows that a simple modi�cation is su�cient to ensure robust2

boundedness and robust performance, and some early modi�cations may have been proposeddue to the limitations of proof techniques used, we feel that it is nevertheless important forfuture work to compare the various modi�cations on the basis of the amount of robustnessprovided, the resulting performance as a function of unmodeled e�ects, transient response,etc., as well as the complexities of the modi�cations themselves.The key stimulus for our work here is the recent paper of Ydstie [19], which showed thatparameter projection in a gradient update law is su�cient for ensuring the boundedness ofclosed loop signals for a nominally minimumphase, unit delay, discrete time plant with sometypes of unmodeled dynamics as well as bounded disturbances.The continuous time systems studied here give rise to several additional issues such as�ltering of signals, parametrization of systems, di�erentiability considerations of signals,augmented errors, etc., which motivate various changes, and allow us to establish stabilityfor nominal plants with arbitrary positive relative degree, as well as for a class of unmodeleddynamics which is larger than those considered earlier, for example in [21],[14] or [19]. Forinstance, unlike Ydstie[19] we do not require the true plant to be stably invertible; only thenominal plant is assumed to be minimum-phase. Additionally, in contrast to [14] we alsoallow the unmodeled dynamics to be nonlinear or time-varying, and do not require di�eren-tiability of either the bounded disturbance, which is lumped together with the unmodeleddynamics in our treatment, or the reference input.The rest of the paper is organized as follows: Section 2 introduces the system and ref-erence models. In Section 3, we reparametrize these models, and describe, in Section 4, theadaptive control law. Our analysis starts in Section 5, where we show that all closed-loopsignals in the system are bounded by a particular signal m(t). In Section 6, we introduce asignal z(t) de�ned through a \switched system" which overbounds m(t). In Section 7, weshow that the �ltered signals are comparable to z over certain bounded intervals of time. Toapply these results to the stability analysis of the closed-loop system, in Section 8 we presenta nonminimal representation of the closed-loop system, which is then used in Section 9 tocomplete the boundedness analysis by showing that a certain positive de�nite function of thesignal z(t) and the non-minimal system state error e(t) is bounded. In Section 10, we show3

that asymptotic tracking is achieved in the nominal case, and in Section 11 we establish amean-square robust performance result. In Section 12 we present simulation examples toillustrate the results. Finally, Section 13 presents some concluding remarks. Some necessarytechnical results are collected in Appendices A and B. A preliminary version of the resultspresented here is contained in [21].2 System and Reference ModelsConsider the single-input, single-output system,y(t) = B(s)A(s) (1 + �m�m(s))u(t) + v(t); (1)where B(s)A(s) is the transfer function of the modeled part of the plant, �m(s) represents themultiplicative uncertainty in the plant, and v(t) represents the e�ect of additional additiveunmodeled dynamics as well as bounded disturbances.We will make the following assumptions on the nominal model of the plant:(A1) A(s) = sn +Pn�1j=0 ajsj , and B(s) =Pmj=0 bjsj, 0 � m < n.(A2) B(s�p0) is Hurwitz for some p0 > 0, and bm � bmin > 0. We will denote by nr := n�m,the relative degree of the nominal plant.We will make the following assumptions on the unmodeled dynamics and bounded distur-bance of the plant:(A3) The multiplicative uncertainty, �m(s), is a transfer function with relative degree greaterthan or equal to (1� nr) such that �m(s� p0) is stable.(A4) The additive unmodeled dynamics and disturbances give rise to a signal v(t) whichsatis�es jv(t)j � Kvm(t) + kv + kv0exp[�pt], where m(�) is de�ned by,_m(t) = �d0m(t) + d1(ju(t)j+ jy(t)j+ 1); m(0) � d1d0 , (2)All the constants are positive. Furthermore 0 < p � d0, and d0 + d2 < p0 < 2d0 + d2, forsome d2 > 0. The term kv0exp[�pt] above allows for the e�ect of initial conditions.Example. A class of unmodeled dynamics, possibly nonlinear, satisfying (A4) is,v(t) = f(t; g1(t) �H1(s)y(t); g2(t) �H2(s)u(t))4

where f is any a�ne function or any non-linear function satisfying jf(t; x1(t); x2(t)j �k1jx1(t)j + k2jx2(t)j + k3, g1(t); g2(t) are bounded time-varying signals, and H1(s);H2(s)are strictly proper transfer functions such that H1(s� p0) and H2(s� p0) are stable.The goal of adaptation is to follow the output of a reference model given byym(t) = Wm(s)r(t) (3)where Wm(s) is a stable transfer function with relative degree nr, and r(t) is a referenceinput. We will suppose that jr(t)j � kr1;8t � 0 and jym(t)j � kym;8t � 0.3 Parametrization of System and Reference ModelsWe now reparametrize the system and reference models so that they are in a form moresuitable for the development of an adaptive control law. To do this, we need to �lter theinput and output signals.Let a be a positive number satisfying a > d2 + 2d0. We de�ne the \regression" vector1, T := ( y(s+ a)2n�m�1 ; : : : ; y(s+ a)n�m ; u(s+ a)2n�m�1 ; : : : ; u(s+ a)n�m );We will presently show that there exists a \parameter vector" � = (�1; : : : ; �2n)T such thatthe system (1) can be represented as,y(t) = T (t)� + v0f(t) + �2d0(t); �2n = bm (4)where T (t)� represents the nominal part of the system, �2d0(t) represents the e�ect of initialconditions arising from the �ltering operations2, and v0f(t) represents the e�ect of unmodeleddynamics and bounded disturbances.1All of the results of this paper continue to hold if instead of the �lter 1(s+a)2n�m�1 , we use the �lter1(s+a)n�m�1(s) , where �1(s) = (s + a)�01(s) with �01(s) being monic, of degree (n � 2), and with all rootshaving real parts less than or equal to �(2d0 + d2).2Here and throughout, �q(t) will denote a signal which satis�es the following properties:(i) j�(i)q (t)j � c0exp[�qt]; i = 0; 1; where �(i)q (t) denotes the i�th derivative of �q(t).(ii) �q(t) � 0 when initial conditions are zero.When the value of q is unimportant, we will sometimes drop the subscript on �.5

We will also reparametrize the reference model (3) asym(t) = 1(s+ a)nr r0(t) + �2d0(t) (5)where r0(t) := (s + a)nrWm(s)r(t). Note that r0(t) is well de�ned since the relative degreeof Wm(s) is nr. Further, r0(t) is bounded since r(t) is bounded. We shall directly supposefrom now onwards that r0(t) � kr;8t � 0.To see the existence of a � for which (4) holds, let F (s) be a monic polynomial of degree(nr� 1) and G(s) a polynomial of degree less than or equal to (n� 1), such that A(s)F (s)+G(s) = �(s), where �(s) := (s + a)2n�m�1. Then, from (1), taking into consideration thee�ect of the initial conditions introduced by the �ltering operation 1�(s), we have,�(s) �G(s)�(s) y(t) = F (s)A(s)�(s) y(t) + �2d0(t)= F (s)[B(s)�(s) u(t) + �mB(s)�m(s)�(s) u(t) + A(s)�(s) v(t)] + �2d0(t):Thus, if � = (�1; : : : ; �2n)T is de�ned from the coe�cients of G(s) and B(s)F (s) by, �n(s +a)n�1 + : : :+ �2(s + a) + �1 = G(s); �2n(s + a)n�1 + : : : + �n+2(s + a) + �n+1 = B(s)F (s),and v0f(t) = A(s)F (s)�(s) v(t)+�mB(s)F (s)�(s) �m(s)u(t); then (4) is satis�ed. We note that since F ismonic, �2n = bm.4 The Adaptive Control LawWe will use the control law, �T (t)�̂(t) = r0(t); (6)to implicitly de�ne the input u(t), where �(t) := ( y(s+a)n�1 ; : : : ; y; u(s+a)n�1 ; : : : ; u)T , and �̂(t)is an estimate of � that we shall presently specify. Note that the \regression" vector (t)de�ned earlier satis�es (t) = 1(s+a)n�m�(t).The adaptive control law (6) is a \certainty equivalent" control law, since if �̂(t) = � in(6), then the nominal part of the system (4) tracks ym(t) since,y(t) = T (t)� + �2d0(t) = 1(s+ a)n�m [�T (t)�] + �2d0(t)= r0(t)(s+ a)n�m + �2d0(t) = ym(t) + �2d0(t):6

For parameter estimation, we shall use the gradient update law with projection,_̂�(t) = Proj[�̂(t); � (t)ea(t)n(t) ]; k�̂(0)k �M; �̂2n(0) � bmin; (7)where � > 0, ea(t) is the \augmented" error,ea(t) = y(t)� ym(t) + 1(s+ a)n�m [�T (t)�̂(t)]� T (t)�̂(t);n(t) := � + 2n�m�1Xi=1 ([ y(t)(s+ a)i ]2 + [ u(s+ a)i ]2); � > 0;is a normalizer of the gradient, and Proj[�; �] is a projection whose i�th component is de�nedfor i 6= 2n, by Proj[p; x] ji = ( xi if kpk < M or pT x � 0;xi � pT xkpk2 pi otherwise;and for i = 2n byProj[p; x] j2n = 8>>>>>>>><>>>>>>>>: x2n if (kpk < M or pT x � 0)and (p2n > bmin or x2n � 0);x2n � pTxkpk2 p2n ; if (kpk � M and pT x > 0)and (p2n > bmin or x2n � 0);0 ; otherwise:There are several features worth commenting upon. First, the augmented error ea(t)consists of the tracking error y(t) � ym(t), as well as the familiar \swapping" term1(s+a)n�m [�T (t)�̂(t)]�[ 1(s+a)n�m�T (t)]�̂(t), which would be zero if �̂(t) were a constant. Second,note that n(t) is slightly di�erent from the usual k (t)k2 since it additionally contains thelower order �ltered terms y(t)(s+a)i and u(t)(s+a)i for 1 � i � n�m� 1, which are absent in (t).Turning to the \projection" mechanism, it has two features. Without any projection, agradient scheme would simply consist of,_̂�(t) = � (t)ea(t)n(t) :However, to keep the estimate inside a sphere of radius M centered at the origin, when theestimate is about to leave the sphere, one would project the drift term so that it evolvestangentially to the sphere. Our projection Proj[p; x] also involves an additional feature toensure that b̂m(t) = 2n�th component of �̂(t) is larger than or equal to bmin.The following easily verifed consequences of projection are important.7

(P0) �̂2n(t) � bmin; k�̂(t)k �M .(P1) k _̂�(t)k � j� (t)ea(t)n(t) j(P2) If k�k � M and �2n = bm � bmin, i.e. � lies in the region to which we con�ne theparameter estimates, then~�T (t) _̂�(t) � ~�T (t)(� (t)ea(t)n(t) ) where ~�(t) := �̂(t)� �.Finally, since the existence of a solution to the above parameter estimator may not beassured due to the discontinuous nature of the projection, we can replace the above projectionby the \smooth" projection due to Pomet and Praly [15], for which existence is assured. Thekey properties (P1) and (P2) continue to hold for the \smooth" projection, and all the resultsof this paper rigorously hold for this \smooth" projection.In what follows, we will throughout suppose that the nominal plant described by � satis�esthe assumptions of (P2).5 Bounding Signals by mIn this section we will show that all the signals in the system can be bounded in terms ofm(t) given in (2).3For brevity of notation in the proof, we de�ne �1(s) = (s+a)n�1; �2(s) = (s+a)nr ; �(s) =�1(s)�2(s); �(t) = 1�2(s) [�T (t)�̂(t)] � [ 1�2(s)�T (t)]�̂(t); 0(t) = [ y(t)�(s); : : : ; y(t)(s+a) ; u(t)�(s); : : : ; u(t)(s+a) ]T ;vf(t) = B(s)F (s)�(s) v(t), and e1(t) = y(t)� ym(t).Lemma 5.1. ju(t)j � Ku 0k 0(t)k+Kuvjv0f(t)j+ kuu + c0exp[�2d0t];8t� 0: (8)3Here and throughout, the values of useful constants used in the bounds are speci�ed in Table C in theAppendix. Certain constants whose exact value is unimportant and which do not depend on Kv; �m; kvwill be denoted generically by C. Any constant whose exact value is unimportant but which depends onKv; �m; kv, and such that its value decreases as Kv; �m; kv decrease, will be denoted generically by c. Anypositive constant which depends only on initial conditions of some �lter, and whose exact value is unimportantwill be denoted by c0. Finally, all constants throughout are positive, unless otherwise noted.8

Proof. Let �I := ( y�(s) ; : : : ; y(s+a) ; y; u�(s); : : : ; u(s+a)) be a sub-vector of �, and �̂I the cor-responding sub-vector of �̂. Then, the control law (6) gives u = (r0 � �TI �̂TI )=b̂m. From(4), we then obtain jy(t)j � Mk 0k + jv0f j + c0exp[�2d0t]: Hence k�Ik � jyj + k 0k �(1 +M)k 0k+ jv0f j+ c0exp[�2d0t]. Since b̂m � bmin, and k�̂Ik � k�̂k �M , we thus concludethat juj � 1bmin (kr +M(1 +M)k 0k+M jv0f j) + c0Mexp[�2d0t]: 2Theorem 5.1.(i) jvf(t)j � Kvfm(t) + kvf + c0exp[�(d0 + d2=2)t] + kvf0exp[�pt];8t� 0;(ii) jv0f(t)j � K 0vfm(t) + kvf + c0exp[�(d0 + d2=2)t] + kvf0exp[�pt];8t� 0;(iii) k 0(t)k � K 0mm(t) + c0exp[�2d0t];8t� 0;(iv) jea(t)j � Kemm(t) + kvf + c0exp[�(d0+ d2=2)t] + kvf0exp[�pt] + c0exp[�2d0t];8t� 0;.(v) jy(t)j � Kymm(t) + kvf + c0exp[�(d0 + d2=2)t] + kvf0exp[�pt] + c0exp[�2d0t];8t � 0;(vi) ju(t)j � Kumm(t)+kum+c0exp[�(d0+d2=2)t]+Kuvkvf0exp[�pt]+c0exp[�2d0t];8t� 0;The proof of this Theorem is based on the following Lemma.Lemma 5.2. LetH(s) be a strictly proper, stable transfer function, whose poles fpjg satisfyRe(pj) � �(d0 + d2) < 0 for all j.(i) If win(t) is the input to a system with transfer function H(s), which satis�es the boundjwin(t)j � kuju(t)j+ kyjy(t)j+ kmm(t) + k0 + k00exp[�p0t];8t � 0, for 0 < p0 � d0, then theoutput wout(t) is bounded by,jwout(t)j � k1kx(0)kexp[�(d0+ d2=2)t] + k3m(t) + k4 + k5exp[�p0t], where x(0) is the initialstate corresponding to a minimal state representation of H(s), and k1; k3; k4 and k5 arerelated positive constants speci�ed below.(ii) Let H 0(s) = kp +H(s). If win(t) is the input to a system with transfer function H 0(s),which satis�es the bound jwin(t)j � kmm(t) + k0 + k00exp[�p0t], 8t � 0, for 0 < p0 � d0,then the output wout(t) is bounded by jw0(t)j � k1kx(0)kexp[�(d0+ d2=2)t] + k6m(t)+ k7+9

k8exp[�p0t], where x(0) is the initial state corresponding to a minimal state representationof H 0(s), and k1; k6; k7 and k8 are related positive constants speci�ed below.4Proof. See [20]. 2Proof of Theorem 5.1. The results (i) and (ii) follow from Lemma 5.2 and (A4) by notingthat vf = A(s)F (s)�(s) v and v0f = vf + �mB(s)F (s)�(s) �m(s)u. The result (iii) follows by applyingLemma 5.2 to each component of 0, noting that each component is either of the form y(s+a)kor u(s+a)k . Since ea = � T ~� + v0f + �2d0 the bound for ea in (iv) follows from the bounds fork k � k 0k and v0f . The proof of (v) is similar to (iv) since y = T� + v0f + �2d0. Finally,(vi) follows from Lemma 5.1 and (ii), (iii) above. 26 Bounding by a Switched SystemLet us introduce the \switched" system,_z(t) = I(t)[�d0z(t)+d1(ju(t)j+ jea(t)j+1)]+(1� I(t))[�g2z(t)+K2]; z(0) � m(0)(1 + 2�Ku 0(a�2d0))(9)where I(t) = 1 if (�d0z(t) + d1(ju(t)j+ jea(t)j+ 1)) � (�g2z(t) +K2);= 0 otherwise. (10)In Theorem 6.1(ii) below we will show that m itself is bounded in terms of z, thus showingfrom Theorem 5.1 that all other signals can also be bounded in terms of z.Theorem 6.1.(i) _m � Kmm+ km + c0exp[�(d0 + d2=2)t] + c0exp[�pt] + c0exp[�2d0t]; (11)(ii) m(t) � Kmzz(t) + kmz; 8t � 0; (12)(iii) _z � Kzz + kz + c0exp[�(d0 + d2=2)t] + c0exp[�pt] + c0exp[�2d0t]: (13)4k3; k4; k6 and k7 are independent of k00, which will mean that initial conditions have no in uence on themagnitude of the bounds or the size of allowable unmodeled dynamics.10

Proof. (i) This follows from _m = �d0m+ d1(juj+ jyj+ 1), by making use of the boundsfor juj and jyj in terms of m given in Theorem 5.1(v),(vi).(ii) By the de�nition of the augmented error ea, we have y = (ea + ym) � �, which impliesjyj � jeaj+ jymj+ j�j: By the Swapping Lemma (Morse [6]; Goodwin & Mayne [13]), �(t) =[hTexp(Q1t)] � (H(t) _~�(t)); where Q1 is an (nr � nr) stable matrix such that det(sI �Q1) =�2(s), HT (t) = ( (t); s (t); : : : ; sn�m�1 (t)), hT is a constant row vector of dimension nr,and \�" denotes convolution. Now,kHT _~�k = kHTProj(�̂; � ea� + k 0k2 )k � �Cjeaj; (14)where the last inequality follows from Property (P1) of the parameter estimator, and sincekHk � Ck 0k. Since �2(s) = (s + a)nr , we have khT exp(Q1t)k � Cexp(�at=2). Thereforej�(t)j � R t0 �Cexp[�a(t� � )=2]jea(� )jd� , and so,Z t0 j�jexp[�d0(t� � )]d� � C� Z t0 exp[�d0(t� � )] Z �0 exp[�a(� � t0)=2]jea(t0)jdt0d�= C� Z t0 exp[�d0t]exp[at0=2] Z tt0 exp[�(a=2� d0)� ]jea(t0)jd�dt0� 2C�a� 2d0 Z t0 exp[�d0(t� t0)]jea(t0)jdt0:This impliesZ t0 jyjexp[�d0(t� � )]d� � Z t0 exp[�d0(t� t0)](1 + 2C�(a� 2d0))jea(t0)jdt0 + kymd0Now, add R t0(ju(� )j+ 1)exp[�d0(t� � )]d� to both sides and multiply by d1. Finally addingexp(�d0t)m(0) to both sides yields the desired result.(iii) The result follows from _z(t) � �(d0 + g2)z(t) + d1(juj+ jeaj+ 1) +K2 and the boundson jeaj and juj in Theorem 5.1(iv, vi) by using the bound for m in (ii). 2From Theorem 6.1 it follows that z can grow at most exponentially fast, and thereforedoes not have a �nite escape time. Since z bounds all signals, it follows that no signal in thesystem has a �nite escape time.In what follows, we choose a large time Tl, and restrict attention to t � Tl, for which thefollowing bounds hold.(i) jv(t)j � Kvm(t) + kv + k(Tl) � KvKmzz(t) +Kvkmz + kv + k(Tl): (15)11

(ii) jv0f(t)j � K 0vfKmzz(t) +K 0vfkmz + kvf + k(Tl): (16)(iii) k 0(t)k � K 0mKmzz(t) +K 0mkmz: (17)(iv) jy(t)j �Mk 0(t)k+K 0vfKmzz(t) +K 0vfkmz + kvf + k(Tl): (18)(iv)0 jy(t)j � Kyzz(t) + kyz + k(Tl): (19)(v) ju(t)j � Ku 0k 0(t)k+K 0vfKuvKmzz(t) + ku + k(Tl): (20)(v)0 ju(t)j � Kuzz(t) + kuz + k(Tl): (21)(vi) _z(t) � Kzz(t) + kz + k(Tl): (22)(vii) �m(t) := �m(s)q(s) u(t) satis�es j�m(t)j � cz(t) + c+ k(Tl); (23)if q(s � d0 � d2) is a Hurwitz polynomial. (The result (vii) follows from Theorem 5.1(vi),Lemma 5.2(ii) and Theorem 6.1(ii).) In all the above inequalities and in what follows, k(Tl)denotes a generic constant which decreases with increasing Tl at a rate faster than exp[�pTl].7 Comparing 0 and zIn this section we compare 0 and z.Theorem 7.1. (i) If I(t) = 1, then n(t)z(t)2 � Knz .(ii) Consider T 0 > 0. Let t1 � Tl + T 0 be any instant such that I(t1) = 1. If z(t) � L2 ;8t 2(t1 � T 0; t1] where L is a large positive constant (speci�ed in Table C), then there exists aKvmax > 0 and a �max > 0 such that for all Kv 2 [0;Kvmax] and for all �m 2 [0; �max], wehave: n(t)z2(t) > �(T 0); 8t 2 (t1 � T 0; t1] (24)Proof. (i) Let us �x t such that I(t) = 1. By (10), we have juj + jeaj � d0�g2d1 z + K2�d1d1 .Using the upper bound (20) on u gives, Ku 0k 0k + jeaj � ( �d0 � �g2)z + �K2 =: RHS: Wenote that we can take �d0 � 0 and �K2 � 0, by the choices of K2; Tl;Kvmax; and �max given inTable C. If k 0k > RHS2Ku 0 , then the result clearly holds, so we consider k 0k � RHS2Ku 0 . Thenclearly jeaj � RHS2 . Therefore using the upper bound (16) for v0f , we obtain by de�nition of12

the augmented error,~Mk k � j~�T j � jeaj � jv0f j � c0exp[�2d0Tl];� j[( �d0 � �g2)z + �K2]=2j � (K 0vfKmzz(t) +K 0vfkmz + kvf + k(Tl) + c0exp[�pTl])Note that ( �d0� �g2) � 4K 0vfKmz and �K2 � 2(K 0vfkmz + kvf + k(Tl)+ kexp[�pTl]), again fromthe choices of K2; Tl;Kvmax; and �max given in Table C. This implies k 0k � k k � [ ( �d0��g2)z4 ~M ].Hence we obtain, n(t)z(t)2 � Knz .(ii) First, we will bound the growth rate of n(t)z2(t). We do not require z to be large for thispart of the proof. Note that ddt( n(t)z2(t)) = _nz2 � 2 nz2 _zz � _nz2 + 2g2 nz2 . Also, _ 0 = �0 � a 0;which implies _n = 2 0T (�0 � a 0) � (1 � 2a)k 0k2 + k�0k2, where, �0T := (s + a) 0T =( y(s+a)2n�m�2 ; : : : ; y(s+a) ; y; u(s+a)2n�m�2 ; : : : ; u(s+a) ; u). Now, k�0k2 � k 0k2 + y2+ u2. Using this,we get _n � 2(1 � a)k 0k2 + y2 + u2. Next, using the upper bounds for u and y from (20),(18), and recalling that n(t) = � + k 0(t)k2, we get ddt( n(t)z2(t)) � Ka( n(t)z2(t)) +Kb + Kcz2(t).Next, using the fact that z(t) � L2 ;8t 2 (t1 � T 0; t1], we get ddt( n(t)z2(t)) � Ka( n(t)z2(t)) +Kd whereKd := Kb + 4KcL2 . This gives,nz2 (t1) � exp[Ka(t1 � t0)][ n(t0)z2(t0) + Z t1t0 exp[Ka(t0 � � )]Kdd� ]� exp[KaT 0][ n(t0)z2(t0) + KdKa (1 � exp[�KaT 0])]; 8t0 2 [t1 � T 0; t1]:Thus, n(t0)z2(t0) � exp[�KaT 0][ nz2 (t1) + KdKa ] � KdKa � exp[�KaT 0][Knz + KdKa ] � KdKa ; 8t0 2 [t1 �T 0; t1]. The fact that there exist Kvmax > 0; �max > 0 such that �(T 0) > 0 for all Kv 2[0;Kvmax]; �m 2 [0; �max] follows from the fact that KdKa can be made as small as we wish bymaking L su�ciently large, and then choosing appropriately small Kvmax and �max. 28 A Nonminimal System RepresentationRecalling y = �T +v0f + �2d0 ; (t) = 1�2(s)�(t) , �T (t)�̂(t) = r0(t), and vm = BF� �mu we notethat, y = 1�2(s)(r0 � ~�T�) + vf + �mvm + �2d0 (25)13

The system (1) can be represented as,_xp = Apxp + bpu+ �mb0p�m ; y = hTp xp + v + �2d0 (26)where (Ap; bp; hTp ) is a minimal state representation of the nominal plant transfer function,�m := �m(s)q(s) u, with q(s) chosen as some polynomial of degree nr�1 such that q(s�2d0�d2)is Hurwitz, thus giving rise to the �2d0 term in (26)). In (26) above, hTp (sI �Ap)�1bp = B(s)A(s)and hTp (sI � Ap)�1b0p = B(s)q(s)A(s) . Note that �m(s)q(s) is a proper, stable transfer function. Thecontrol law (6) is equivalent to �T�+ ~�T� = r0: Hence,bmu = ��Tu �u � �ny � �Ty �y � ~�T�+ r0: (27)De�ning �y = ( y(s+a)n�1 ; : : : ; y(s+a))T and �u like-wise, we note that �(t) =(�Ty (t); y(t); �Tu (t); u(t))T , with_�y(t) = Q�y(t) + qy(t); and _�u(t) = Q�u(t) + qu(t); (28)where Q is stable matrix such that det(sI � Q) = (s + a)n�1, and (Q; q) is a controllablepair.De�ne XTc := [xTp ; �Ty ; �Tu ]. Then, using (26), (28), and (27), we get _Xc = AcXc +bc(r0 � ~�T�) + bcvv + �mbc��m; y = hTc Xc + v + �2d0 , where, Ac is a stable matrix (as we willprove shortly), bTc := [bTp =bm; 0; qT=bm]; bTcv := [��nbTp =bm; qT ;��nqT=bm]; bTc� := [b0Tp ; 0; 0] andhTc := [hTp ; 0; 0]; see [3].For ~� = 0; v � 0 and �m = 0, we have, _Xc = AcXc + bcr0; y = hTc Xc + �2d0 : This impliesy = hTc (sI �Ac)�1bcr0 + �2d0. However, we already know (from (25)) that y = 1�2(s)r0 + �2d0,in such a case. This means that, (after cancellations),hTc (sI �Ac)�1bc = 1�2(s) (29)It can be easily veri�ed (e.g. see [17], pg.136) that we have only stable pole-zero cancellationsbecause the nominal plant is minimum-phase and Q is a stable matrix. This proves that(Ac; hTc ) is detectable. Since the overall transfer function is stable (by (29)), we concludethat Ac is a stable matrix. 14

Thus we can write the following non-minimal representation of 1�2(s) :_Xm = AcXm + bcr0 ; ym = hTc Xmwhere XTm := [xTm; �Tym; �Tum]. Ac being stable and r0 being bounded implies that kXm(t)k �KXm;8t � 0, for some positive constant KXm. De�ne the state error e as e := Xc � Xm.This gives, _e = Ace� bc(~�T�) + bcvv + �mbc��m ; e1 = hTc e+ v (30)where we recall that e1 = y � ym is the tracking error.9 Robust Ultimate BoundednessDe�ne W , which bounds all signals, byW = keeTPe+ 12z2 (31)where P = P T > 0 satis�es PAc +ATc P = �I. Such a P exists since Ac is a stable matrix.Our main result on robust ultimate boundedness of the overall system is given by thefollowing Theorem. It states that eventually W (t) (which bounds all signals) enters a com-pact set, the size of which is independent of the initial conditions. Further, the size of theallowable unmodeled dynamics for which this is guaranteed, is independent of the initialconditions. It should be noted, however, that the time Tlarge that it takes for W (t) to enterthis compact set can depend on the initial conditions.Choose constants 0 < < 1, �0 > 0 small, and �z > 0. Let T; �; Tl;Kvmax; and �max beas in Table C.Theorem 9.1 (Robust Ultimate Boundedness Theorem).There exist a Tlarge � Tl, and positive Kvmax; �max, such that for all Kv 2 [0;Kvmax], andfor all �m 2 [0; �max], W (t) � 8K2wzL2exp[4(Kz + �z)T ]; 8t � Tlarge (32)Furthermore, Kvmax; �max, T , and L are independent of the initial conditions.15

Proof. The idea of the proof, based on the following Lemmas, is to show that wheneverW (t) � KwzL2 throughout an interval of length 2T , then at the end of the interval its valueis smaller than at the beginning of the interval.Lemma 9.1. W (t) � Kwzz2(t) + kWz ;8t � Tl: (33)Proof. Since W = eTPe + 12z2; e = Xc � Xm; kXmk � KXm;Xc = [xTp ; �Ty ; �Tu ]T ; k�yk �cz; k�uk � cz, it su�ces to show that kxpk � cz + c + c0exp[�pt].By (26), de�ning u0 := 1+�m�m(s)q(s) u, y0 = y � v � �2d0(t), and b00p such that hTp (sI �Ap)�1b00p = B(s)q(s)A(s) , with the (�2d0) term in the above equation possibly di�erent from the(�2d0) term in (26), we get, _xp = Apxp + b00pu0; y0 = hTp xp: Now, from Theorem 5.1(v),jy0(t)j � cm(t)+ c+ c0exp[�pt];8t� 0. Noting that B0(s) := B(s)q(s) is such that B0(s�p)is Hurwitz, and applying Lemma A.1, we get kxp(t)k � cm(t) + c + c0exp[�pt];8t � 0:Finally applying Theorem 5.2(ii), we get the desired bound on kxpk. 2Lemma 9.2. Consider an interval [a; b]. ThenW (b)W (a) � exp[Z ba g(� )d� ](1 + �4exp[�(b� a)]�W (a) ) (34)where g(t) := �� + �3 j~�T (t)�(t)jz(t) + �5 j~�T (t) (t)jz(t) , and �3; �4 and �5 are speci�ed in [20].Proof. Recalling from (9) that _z � �(d0 + g2)z + d1(juj+ jeaj+ 1) +K2, we have,_W = ke( _eTPe+ eTP _e) + _zz= �kekek2 � (d0 + g2)z2 +2ke(�(~�T�)bTc Pe+ vbTcvPe+ �m�mbTc�Pe) + d1(juj+ jeaj+ 1)z +K2zusing (30). Then, proceeding as in Ioannou & Tsakalis [12]((4.16)-(4.18)), (with the exceptionbeing that now, we also need to take care of a linear term in kek) we get:_W (t) � ��W (t) + �3 j~�T (t)�(t)jz(t) W (t) + �5 j~�T (t) (t)jz(t) W (t) + �4=: g(t)W (t) + �4; which impliesW (b)W (a) � exp[Z ba g(� )d� ](1 + �4exp[�(b� a)]�W (a) ) since �g(� ) � �.16

Details of the proof are provided in [20]. 2Lemma 9.3. Consider a time interval [a; b] such that (i) b � a � Tl, (ii)W (t) � KwzL2;8t 2[a; b]. Then,(i) 12z2(t) �W (t) � 4Kwzz2(t);8t 2 [a; b].(ii) If I(t) = 0;8t 2 [a; b], then, W (b) � 8Kwz(exp[�g2(b� a)] + 2K2g2L )2W (a).(iii) If (b � a) � 1 and for each t 2 [a; b] such that I(t) = 0 there exists a t0 2 [0; T ] suchthat I(t+ t0) = 1, W (b) � k � exp[��4(b� a)](1 + �4Kwz�L2 exp[�(b� a)])W (a).(iv) W (b) � 8Kwzexp[2(Kz + �z)(b� a)]W (a).Proof. The result (i) follows by Lemma 9.1 sinceW � KwzL2 implies z � L=2. The result(ii) follows from (i) because of _z � �g2z +K2, and since W � KwzL2 implies z � L=2. Toprove (iii), note from Lemmas 9.2, B.3, B.4 and by our choices of �; �max;Kvmax and L, thatZ ba g(t)dt � ��(b� a) + �3�3=�2 + �3[�2�4(T )=�2 + ��5(T )=p�+ �6(T )��+4an�mk(�;L)](b� a) + �5�7=�2 + �5[�2�8(T )=�2 +p�](b� a)� ��4 (b� a) + (�3�3 + �5�7)�2which gives us the desired result after de�ning k = exp[ (�3�3+�5�7)�2 ]. Finally, (iv) follows fromthe bounded growth-rate of z(t) shown in (22), and (i). 2Lemma 9.4 (Contraction Property).If t0 � Tl, and W (t) � KwzL2 8t 2 [t0; t0 + 2T ], then W (t0 + 2T ) � W (t0).Proof. There are four possibilities:Case 1 Suppose I(t) = 0;8t 2 [t0; t0 + 2T ]. Then from Lemma 9.3(ii), we have W (t0 + 2T ) �8Kwz(exp[�2g2T ] + 2K2g2L )2W (t0) � W (t0) , where the last inequality follows by thede�nitions of T and L given in Table C.Let t1 := minft 2 [0; 2T ] : I(t0 + t) = 1g and t2 := maxft 2 [0; 2T ] : I(t0 + t) = 1g.17

Case 2 Suppose 0 � t1 � t2 < T . First, by Lemma 9.3(ii), W (t0+2T ) � 8Kwz(exp[�g2(2T �t2)] + 2K2g2L )2W (t0 + t2).(2A) Suppose t2 < 1. Then, the above inequality implies W (t0 + 2T ) �8Kwz(exp[�g2(2T � 1)] + 2K2g2L )2W (t0 + t2). Using Lemma 9.3(iv), we haveW (t2 + t0) � 8Kwzexp[2(Kz + �z)]W (t0), which implies that W (t0 + 2T ) �64K2wzexp[2(Kz + �z)](exp[�g2(2T � 1)] + 2K2g2L )2W (t0), so that W (t0 + 2T ) � W (t0), the last inequality following from the de�nitions of T and L.(2B) Suppose t2 � 1. In this case, we have by Lemma 9.3(iii) and the de�nitionsof T and L, W (t0 + t2) � exp[��t2=4](k + �0)W (t0), so that W (t0 + 2T ) �8exp[��=4](k + �0)Kwz(exp[�g2T ] + 2K2g2L )2W (t0) � W (t0).Since the rest of the proof is similar, we abbreviate it.Case 3 Suppose 0 � t1 � T � t2 < 2T . We apply the de�nitions of T and L to the followingcases just as in cases 1 and 2 above and in each case get W (t0 + 2T ) � W (t0).(3A) Suppose I(t0+T ) = 1. Then we separately consider the case where I(t0+t) = 1;for some t 2 [t2; t2 + T ], and where I(t0 + t) = 0;8t 2 [t2; t2 + T ].(3B) Suppose I(t0 + T ) = 0. De�ne t3 := maxft 2 [0; T ] : I(t0 + t) = 1g.(1) (t3 < 1)(a) (t2 � T < 1) (i) (I(t0 + t) = 1 ; for some t 2 [t2; t2 + T ])(ii) (I(t0 + t) = 0 ;8t 2 [t2; t2 + T ])(b) (t2 � T � 1) (i) (I(t0 + t) = 1 ; for some t 2 [t2; t2 + T ])(ii) (I(t0 + t) = 0 ;8t 2 [t2; t2 + T ]) :(2) (t3 � 1)(a) (t2 � T < 1) (i) (I(t0 + t) = 1 ; for some t 2 [t2; t2 + T ])(ii) (I(t0 + t) = 0 ;8t 2 [t2; t2 + T ])(b) (t2 � T � 1) (i) (I(t0 + t) = 1 ; for some t 2 [t2; t2 + T ])(ii) (I(t0 + t) = 0 ;8t 2 [t2; t2 + T ]) :18

Here one needs to consider the four cases (t3 < T=2)=(t3 � T=2) and (t2 � T< T=2)=(t2 � T � T=2). (Because of symmetry, these reduce to just two).Case 4 (T � t1 � t2 < 2T ). Again, we apply the de�nitions of T and L as in cases 1 and 2 toget W (t0 + 2T ) � W (t0) in each case.(a) (I(t0+ t) = 0 ;8t 2 [t2; t2 + T ]) (i)(t2 � T < 1) (ii)(t2� T � 1)(b) (I(t0 + t) = 1 ; for some t 2 [t2; t2 + T ]) . 2Proof of Theorem 9.1. Lemma 9.4 and the fact that z has a bounded growth rate givesus the desired result since W (t + 2T ) � 8Kwzexp[4(Kz + �z)T ]W (t) whenever W (� ) �KwzL2;8� 2 [t; t + 2T ]. Both Kwz and L are independent of initial conditions. However,Tlarge can depend on initial conditions. This concludes the proof. 2Theorem 9.2. (i) k k; k 0k 2 L1, (ii) v; vf; v0f 2 L1, (iii) y 2 L1, (iv) u 2 L1, (v)ea 2 L1, (vi) � 2 L1, and (vii) _~� 2 L1.Proof. The result (i) follows from Theorem 5.1(iii). The result (ii) follows from assumption(A4) and Theorem 5.1(i),(ii). The result (iii) follows since y = T�+v0f+�2d0, and k�k �M .Lemma 5.1 implies that (iv) is true, and since ea = � T ~�+v0f+�2d0, and k~�k � ~M , (v) holdstrue. The result (vi) follows since � = hTexp[Q1t] � (HT _~�), and kHT _~�k � C�ea. Finally,result (vii) follows from property (P1) of the parameter estimator. 210 Performance for a Nominal SystemWe now show that if the system has no unmodeled dynamics and disturbances, then exactasymptotic tracking is achieved.Theorem 10.1. If Kv = 0; kv = 0, and �m = 0, then jy(t)� ym(t)j ! 0 as t!1:19

Proof. It su�ces to show ea(t) ! 0 and �(t) ! 0 as t ! 1, since e1(t) = ea(t) � �(t).Note that since Kv = 0; kv = 0; �m = 0 we have v0f(t) � 0. So, by Lemma A.2, e2an �� _V� + cexp[�2d0t]. This implies, since k~�k � ~M , thatZ 10 e2a(� )n(� ) d� � ~M2� + k2d0 : (35)Boundedness of n(�) then implies ea 2 L2. Recall that ea 2 L1. ea = �~�T + �2d0 impliesj _eaj � k _~�kk k + k~�kk _ k + c0exp[�2d0t], which implies _ea 2 L1. Therefore, by Barbalat'sLemma [Popov [1] p.211, or Sastry & Bodson [17] p.19], ea(t) ! 0 as t ! 1. Now, recallthat �(t) = hTexp[Q1t] � (H(t) _~�(t)), and kH _~�k � �Cjeaj, which implies,j�(t)j � C� Z t0 exp[�a(t� � )=2]jea(� )jd�:Using the just established fact that ea ! 0, it follows that � ! 0. 211 Robust PerformanceWe now show that the performance of the adaptive tracker, as measured by the mean squaretracking error, is robust in that it is a quadratic (hence also continuous) in the magnitudeof the unmodeled dynamics and bounded disturbances.Theorem 11.1. lim supT!1 1T2 Z T20 (y(t)� ym(t))2dt � c(K2v + k2v + �2m): (36)where c is a generic constant which can only decrease (or remain constant) as Kv, kv, kv0and �m decrease.Proof. From Lemma A.2, we have e2a�v02fn � � _V� + c0exp[�2d0t]; which im-plies 1T2 R T2Tl e2a�v02fn � kexp[�2d0Tl]T2 ; 8T2 � Tl. Hence, lim supT2!1 1T2 R T2Tl e2a�v02fn �0, i.e. lim supT2!1 1T2 R T2Tl e2an � lim supT2!1 1T2 R T2Tl v02fn : Noting that z(t) �minfz(0);K2=g2g =: zmin, and de�ning zmax := supt�Tl z(t), by (16), we get,v02fz2 � 2(K 0vfKmz)2 +2 (K0vfkmz+kvf+k(Tl))2z2 ; which then gives 1T2 R T2Tl v02fn � [2(K 0vfKmz)2 +20

2 (K0vfkmz+kvf+k(Tl))2z2min ] z2max� ; 8T2 � Tl. This implies lim supT2!1 1T2 R T2Tl e2an � c(K2v + k2v + �2m +k2(Tl)), i.e. lim supT2!1 1T2 R T2Tl e2a � c(K2v + k2v + �2m + k2(Tl)); where we recall that k(Tl)depends only on initial conditions and decays exponentially at a rate faster than exp[�2d0Tl]as Tl increases. Finally, using the fact that ea is bounded and the fact that the expressionabove is true for all Tl > 0, we get lim supT2!1 1T2 R T20 e2a � c(K2v +k2v +�2m): The proportion-ality constant c is basically zmaxnmax and is therefore O(z2max). From the Robust UltimateBoundedness Theorem, we know that it decreases as the unmodeled dynamics and boundeddisturbance decrease. This means that the right-hand side in the expression above indeedgoes to zero as the unmodeled dynamics and bounded disturbances go to zero, thereby givingus robust performance. Now consider the swapping term. Recall from the proof of Theorem6.1 that,Z t0 �2(t0)dt0 � (�C)2 R t0(R t00 exp[�a(t0� � )=2]jea(� )jd� )2dt0 � 8a2 (�C)2 Z t0 e2a(� )d� :This implies that lim supT2!1 1T2 R T20 �2(t0)dt0 � 8a2 (�C)2(K2v + k2v + �2m)c. Finally, recallingthat e1(t) = ea(t)� �(t) yields the desired result. 212 A Simulation ExampleWe now present a simulation example to illustrate the results. Additional examples canbe found in [20]. This example is the same as the one considered in [12], except that wedo not assume knowledge of the (nominal) plant gain, and that we also add a boundeddisturbance to the output. The actual plant is unstable and has a fast, nonminimum phasezero. This plant is modeled as a nominal plant which is minimum-phase and unstable, witha multiplicative uncertainty.Example 1.The true system is given by y(t) = (1� �s)s(s� 1) u(t) + w(t) (37)where 0 < � < 1 and w(t) is a bounded disturbance. The reference model to be matched isym(t) = 1(s+ 1)(s + 2)r(t) (38)21

where r(t) = 10sin(0:5t). We consider a nominal model of the form y(t) = k(s+a)(s+b)u(t),which is parameterized using �1(s) = (s+1); �2(s) = (s+1)2 (so that �(s) = (s+1)3), whichresults in F (s) = (s+4), G(s) = (7s+1), and � = (�6; 7; 3; 1)T . The simulation results with�̂(0) = (�4; 3; 6; 4)T , � = 0:02, w(t) = unit square wave with period 10, parameter estimatorconstants � = 4:0; � = 1 and bmin = 0:5, and constants d0 = 0:7; d1 = 1:0;m(0) = 2:0for the overbounding signal m(t) are given in Fig. 1. Despite the presence of unmodeleddynamics (and a particularly nasty-looking �ltered bounded disturbance), the plant outputapproximately tracks the given reference model output over the time interval considered.

22

13 Concluding RemarksIn this paper, we have obtained boundedness and performance for continuous-time plantsof arbitrary relative degree, with a somewhat wider class of unmodeled dynamics thanin [12], but without any extra modi�cations except projection. Unlike [14], we allow non-di�erentiable bounded disturbances, and non-di�erentiable reference inputs. We also allowsome time-varying and nonlinear uncertainties. The nominal plant is however restricted tobe minimum-phase.We have shown that eventually all the signals enter a closed, compact set, the size ofwhich is independent of initial conditions. Also, the upper-bounds on the size of allowableunmodeled dynamics are independent of initial conditions.Our results thus show that the projection mechanism alone is su�cient to guaranteerobust boundedness and robust performance at least with respect to small unmodeled dy-namics and bounded disturbance. It is important to study the dependence of the bounds onthe parameters of the nominal plant, the constants de�ning the unmodeled dynamics, ini-tial conditions, etc. Also, it is important to reevaluate the various robustness modi�cationswhich have earlier been proposed, to examine the amount of robustness they provide, theperformance guaranteed in the presence of unmodeled dynamics and disturbances, and tothus determine whether they actually provide some improvements with respect to employingjust the projection mechanism.Acknowledgements.We thank Petros Ioannou, Jean-Baptiste Pomet, and Laurent Praly for helpful discus-sions.References[1] V. M. Popov. Hyperstability of Control Systems, Springer-Verlag, Berlin, 1973.[2] C. A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties, AcademicPress, New York, 1975.[3] K. S. Narendra and L. S. Valavani. Stable adaptive controller design - Direct control.IEEE Trans. Aut. Contr., vol. 23, pp. 570-583, 1978.23

[4] A. Feuer and A. S. Morse. Adaptive control of SISO linear systems. IEEE Trans. Aut.Contr., vol. 23, pp. 557-569, 1978.[5] B. Egardt. Stability of adaptive controllers. Lecture Notes in Control and Info. Sciences,vol. 20, Springer-Verlag, Berlin, 1979.[6] A. S. Morse. Global stability of parameter-adaptive control systems. IEEE Trans. Aut.Contr., vol. 25, pp. 433-439, 1980.[7] G. Kreisselmeier and K. S. Narendra. Stable MRAC in the presence of bounded distur-bances. IEEE Trans. Aut. Contr., vol. 27, pp. 1169-1175, 1982.[8] L. Praly. Robustness of model reference adaptive control. K.S.Narendra, Ed., Proc. IIIYale Workshop on Adaptive Systems, pp. 224-226, 1983.[9] L. Praly. Robust model reference adaptive controllers, Part I: Stability Analysis. Proc.23rd IEEE CDC, Dec. 1984.[10] L. Praly. Global stability of a direct adaptive control scheme which is robust w.r.t. agraph topology. Adaptive and Learning Systems: Theory and Applications, K.S. Naren-dra, ed., Plenum Press, New York, 1986.[11] G. Kreisselmeier and B. D.O. Anderson. Robust model reference adaptive control. IEEETrans. Aut. Contr., vol. 31, pp. 127-133, 1986.[12] P. A. Ioannou and K. S. Tsakalis. A robust direct adaptive controller. IEEE Trans. Aut.Contr., vol. 31, pp. 1033-1043, 1986.[13] G. C. Goodwin and D. Q. Mayne. A parameter estimation perspective of continuousMRAC. Automatica, vol. 23, pp.57-70, 1987.[14] P. A. Ioannou and J. Sun. Theory and design of robust direct and indirect adaptive-control schemes. Int. J. Control, vol. 47, pp. 775-813, 1988.[15] J. -B. Pomet and L. Praly. Adaptive nonlinear regulation: equation error from theLyapunov equation. Proc. 28th IEEE C.D.C., Tampa, Florida, pp. 1008-1013, 1989.[16] L. Praly, S. -F. Lin and P. R. Kumar. A robust adaptive minimum variance controller.SIAM J. Control and Optimizn., Vol.27, No.2, pp. 235-266, 1989.[17] S. Sastry and M. Bodson. Adaptive Control : Stability, Convergence, and Robustness,Prentice-Hall, Englewood Cli�s, N.J., 1989.[18] G. Tao and P. A. Ioannou. Robust stability and performance improvement of discrete-time multivariable adaptive control systems. Int. J. Control, vol. 50, pp. 1835-1855,1989.[19] B. E. Ydstie. Stability of discrete MRAC - revisited. Systems & Control Letters, vol. 13,pp. 429-438, 1989. 24

[20] S. M. Naik, P. R. Kumar, and B. E. Ydstie. Foundations of Adaptive Control : The 1990Grainger Lectures, Lecture Notes in Control and Information Sciences, P. V. Kokotovic,Ed., Springer-Verlag, New York, 1991.[21] S. M. Naik and P. R. Kumar. A robust adaptive controller for continuous-time systems.Proc. 1991 American Control Conf., Boston, MA, June 1991.Appendix ALemma A.1.Consider the system: _x = Awx + bwwin; wout = hTwx with zero initial conditions.(Aw; bw; hTw) is a minimal representation of H0(s) = B0(s)A(s) , where H0(s) is strictly properand B0(s� p0) is Hurwitz.If jwoutj � Kom+ ko + c0exp[�pt], then, kxk � Kxm+ kx+ c0exp[�pt] for some positiveconstants Kx and kx.Proof. Without loss of generality, supposeH0(s) = B0(s)Qki=1(s2 + ai1s+ ai2)Qn�2kj=1 (s+ aj)Since H0(s) is minimal, the corresponding states are the states corresponding towins2+ai1s+ai2 ; i = 1; : : : ; k and wins+aj ; j = 1; : : : ; n� 2k. Now,1s+ alwin(t) = Qki=1(s2 + ai1s+ ai2)Qn�2kj=1;j 6=l(s+ aj)B0(s) wout(t)Using Lemma 5.2(ii) (since B 0(s � p0) is Hurwitz), we get j 1s+alwin(t)j � cm(t) + c +c0exp[�pt]; l = 1; : : : ; n� 2k. De�ne wl(t) = 1s2+al1s+al2win(t) =: Hl(s)win(t). Then, sincewl(t) = Qki=1;i 6=l(s2 + ai1s + ai2)Qn�2kj=1 (s+ aj)B0(s) wout(t) =: Hlo(s)wout(t);using Lemma 5.2(i) (since B 0(s � p0) is Hurwitz), we get jwl(t)j � cm(t) + c +c0exp[�pt]; l = 1; : : : ; k. Further, since Hlo(s) is strictly proper, Lemma 5.2(ii) givesj _wl(t)j = jsHlo(s)wout(t)j � cm(t) + c + c0exp[�pt]. Since wl(t) and _wl(t) are the statescorresponding to Hl(s)win(t), we are done. 2Lemma A.2. For the parameter estimator with projection, de�ne V (t) := k~�(t)k2. Then,we have _V (t) � ��(~�T (t) (t))2n(t) + �v02f (t)n(t) + � c0 ~Mqn(t)exp[�2d0t] (39)Furthermore, (39) also holds when the term (~�T (t) (t)) on the right-hand side is replacedby ea(t). 25

Proof. For notational simplicity, let c1(t) := �̂(t)k�̂(t)k, and c2(t) := � (t)ea(t)n(t) . First, considerthe parameter estimator when the projection is not used, i.e. _̂�(t) = c2(t). Then, recallingthat ea(t) = �~�T (t) (t) + v0f (t) + �2d0(t), we obtain_V (t) = 2~�(t)T � (t)ea(t)n(t)= �n(t) [�2(~�T (t) (t))2 + 2(~�T (t) (t))v0f(t) + 2(~�T (t) (t))�2d0(t)]� �n(t) [�(~�T (t) (t))2+ v02f (t)] + � c0 ~Mqn(t)exp[�2d0t]:Now, consider the parameter estimator with projection. Using property (P2) of the projec-tion, we get, _V = 2~�T _~� = 2~�TProj(�̂; c2) � 2~�T c2and so the bound for the estimator without projection still holds. 2Appendix BIn this section, we will assume that (i) t � Tl, and (ii) W (�) � KwzL2, which implies thatz(�) � L=2.Lemma B.1.(i) k kz ; k�ykz ; k�ukz ; k�kz � c.(ii) k (i)kz � ai, i = 1; : : : ; n�m.(iii) j ddt(y � v0f)j � k0yz + k(Tl), for some k0y > 0, and k(Tl) is a positive constant whichdecreases exponentially with increasing Tl.(iv) j ddt(u�r0=b̂m+ �̂nv0f=b̂m)j � k0uz+kv�+k(Tl), for some k0u > 0, where k(Tl) is a positiveconstant which decreases exponentially with increasing Tl and kv� is a positive constantwhich is a weighted linear combination of Kv; kv and �m.Proof. The result (i) is immediate from Theorems 5.1(iii),(v),(vi), and 6.1(ii), while (ii)follows since T = ( y(s+a)2n�m�1 ; : : : ; y(s+a)n�m ; u(s+a)2n�m�1 ; : : : ; u(s+a)n�m ). For (iii) note thaty�v0f = T�+�2d0, so that by (ii) above j ddt(y�v0f)j � a1Mz+k(Tl). For (iv) recall that thecontrol law is u� 1̂bm r0 + �̂nb̂mv0f = � 1̂bm [�̂Tu �u + �̂Ty �y + �̂n(y � v0f)] =: RHE=b̂m. This impliesddt(u� 1̂bm r0 + �̂nb̂mv0f) = � _b̂mb̂2m (RHE) � 1̂bm ( _̂�Tu�u + _̂�Ty �y +�̂Tu _�u + �̂Ty _�y + _̂�n(y � v0f ) + �̂n ddt(y � v0f ))26

Now, k _̂bm�uk � j _̂bmjk�uk � k _̂�kk 0k � �k 0k2jeajn � �jeaj � cz + kv� + k(Tl). Similaranalysis using k�̂k � M and bm � bmin > 0, gives j _b̂mb̂2m (RHE)j � cz + kv� + k(Tl). Also,j _̂�Tu�uj � cz + kv� + k(Tl), etc. Finally, using these results, the fact that �̂ is bounded, and(i), (ii), (iii), etc., we get the desired result. 2For future use, de�neknz(T ) = 1Knz ; if I(t) = 1;= 1=�(T ) ; if I(t) = 0, and I(t+ t0) = 1 for some t0 2 (0; T ]:Lemma B.2. For instants t such that I(t+ t0) = 1 for some t0 2 [0; T ],j ddt( ~�T (i)z )(t)j � ki(T ); i = 0; : : : ; n�m� 1. (40)Proof. Note that j ddt( ~�T (i)z )j � k _~�kk (i)kz + k~�kk (i+1)kz + k~�kk (i)kz j _zjz . Now, the parameter-update law yields k _~�k � �k k(j~�T j+jv0f j)n z2z2 and, z2(t)n(t) � knz(T ): This implies that k _~�k � k(T ).Finally, using (16), (22) and Lemma B.1(ii), we get the desired result. 2Lemma B.3. For instants t such that I(t+ t0) = 1 for some t0 2 [0; T ],j ddt( ~�T (n�m) � (1 � �2nb̂m )r0 + (�n � �2n �̂nb̂m )v0fz )j � kn�m(T ) (41)Proof. Since � = (s + a)(n�m) , we have (n�m) = � �Pn�m�1i=0 ci (i), where ci are theappropriate constants which depend on a. Using the control law (6), we get,~�T� = �̂T�� �T� = r0 � �T�= r0 � (�Tu �u + �Ty �y) � �n(y � v0f)� �2n(u� 1̂bm r0 + �̂nb̂mv0f)� �nv0f � �2nb̂m r0 + �2n �̂nb̂mv0fwhich implies�T ~� � (1 � �2nb̂m )r0 + (�n � �2n �̂nb̂m )v0f = �(�Tu �u + �Ty �y)� �n(y � v0f)��2n(u� 1̂bm r0 + �̂nb̂mv0f)Applying Lemmas B.1, B.2 to the above equation yields the desired result, using (22). 227

Let T 0 � T . Assume that I(t+ t0) = 1 for some t0 2 [T 0; T ]. Then, from Lemma A.2, wehave: 1T 0 Z t+T 0t (~�T )2z2 d�� sup nz2 [V (t)� V (t+ T 0)�T 0 + 1T 0 sup z2n Z t+T 0t (v0fz )2d� + �c0 ~M2�T 0d0 exp[�2d0t]]� 1T 0 (KnnM2� ) +Knn(K 0vfKmz + K 0vfkmz + kvf + k(Tl)L=2 )2knz(T ) + �c0 ~M2�T 0d0 exp[�2d0Tl]where the supremum above is taken over the interval [t; t + T 0]. De�ning � := K 0vfKmz +K0vfkmz+kvf+k(Tl)L=2 , the above inequality becomes,1T 0 Z t+T 0t (~�T )2z2 d� � 1T 0 ((KnnM2� ) + �c0 ~M2�d0 exp[�2d0Tl]) + �2Knnknz(T )Subsequently we will apply Lemma B.4 to this inequality. The important point to note isthat � can be made as small (but positive) as we please by making K 0vf small enough andL large enough, and K 0vf can in turn be made as small as desired by making Kv and �mappropriately small.Remark. The constraint 1 � T 0 in Lemma B.4 stems from the fact that these depend onTheorem C1 of [12], which needs T 0 to be greater than or equal to one. These and all otherdetails are provided in [20].Lemma B.4.(i) If 1 � T 0 � T , and 1T 0 Z t+T 0t ( j~�T jz )2 � c1T 0 + �2c2(T );then, Z t+T 0t j~�T�jz � �3�2 + [�2�4(T )�2 + ��5(T )p� + �6(T )�� + 4an�mk(�;L)]T 0where � 2 (0; 1], � = 2�(n�m+1), and k(�;L) = ( j�2njbmin+1)krL=2 + (j�nj+ j�2njbminM)�.(ii) If 1 � T 0 � T , and 1T 0 Z t+T 0t ( j~�T jz )2 � c1T + �2c2(T );then, Z t+T 0t j~�T jz � �7�2 + [�2�8(T )�2 +p�]T 0where � 2 (0; 1]. 28

Proof. The result (i) is an extension of the proof of Theorem C1 of [12], which uses Lem-mas B.1, B.2 and B.3. Details of the proof are provided in [20]. The result (ii) is a specialcase of the proof of (i). 2Table CKmz = 1 + 2C�a�2d0 kmz = d1kymd0Ku 0 = M(1+M)bmin Kuv = Mbminkuu = krbmin Kvf = Kvckvf = kvc kvf0 = kv0cK 0vf = Kvf + �mc Kym =MK 0m +K 0vfKum = Ku K 0m +KuvK 0vf kum = Kuvkvf + kuuku = kuu +Kuv(K 0vfkmz + kvf ) Km = d1(Kum +Kym)� d0km = d1(1 + kvf + kum Kz = d1Kmz(Kum + ~MK 0m +K 0vf )kz = d1f[( ~MK 0m +K 0vf ) +Kum]kmz �(d0 + g2)+kvf + kum + 1g k00z = kz + k(Tl)�d0 = d0d1 �K 0vfKuvKmz �g2 = g2d1�K2 = K2d1 � 1� k � k(Tl) Knz = ( �d0��g2)24 minf 1K2u 0 ; 14 ~M2gKa = [2(1� a+ g2) + 3(M2 +K2u 0 )] Kb = 3K 02vf (1 +K2uv)K2mzKc = maxf3[(K 0vfkmz + kvf + k(Tl))2 Kd = Kb + 4KcL2+(ku + k(Tl))2]� �Ka + 2g2�; 0g�(T ) = exp[�KaT ][Knz + KdKa ]� KdKa Kwz = 12 + 2�max(P )(C + 2g4)2Kyz = (K 0vf +MK 0m)Kmz Kuz = (K 0vfKuv +Ku 0K 0m)Kmzkyz =MK 0mkmz +K 0vfkmz + kvf kuz = ku +Ku 0K 0mkmzKnn = K2 0m(Kmz + 2kmzL )2 + 4�L2 � = minf 12kPk; d0 + g2g�1 = d1C ke = 2( 4d0+g2�21 + d0+g24 )T = max8<:4; 1 + 1g2 log0@2vuut64K2Wz exp[2(Kz + �z)](k + �0) exp[��=4] 1A ;1 + 1g2 log0@2vuut1024K3Wz exp[2(Kz + �z)](k + �0) exp[��=4] 1A ;12g2 log 2s8KWz ! ; 1g2 log0@2vuut1024K3Wz exp[2(Kz + �z)] 1A ;2� log k + �0 ! ; 1g2 log0@2vuut256K2Wz (k + �0) exp[��=4] 1A ;12 241 + 1g2 log0@2vuut64K2Wz exp[2(Kz + �z)] 1A35 ; 8� log 32KWz(k + �0)p ! ;29

1g2 log0@2s8KWz(k + �0) exp[��=4] 1A ; 2g2 log0@2vuut8KWz(k + �0)p 1A ;4� log 256K2Wz (k + �0)2 exp[2(Kz + �z)] !� 1; 4� log 32KWz(k + �0) ! ;1 + 1g2 log0@2vuut256K3Wz exp[4(Kz + �z)](k + �0) exp[��=4] 1A ;1 + 1g2 log0@2 4vuut4096K4Wz exp[4(Kz + �z)] 1A9=; ;� = min8<: 1=�s �8�3�6(T ); �8�5!29=; ;L = max8<: 2Kz (kz + k(T`)); 2K22Kz � g2 ; 2k00z�z ; 4K2g2 vuut64K2Wz exp[2(Kz + �z)] ;4K2g2 s8KWz(k + �0) exp[��=4] ;s�4k exp[2�T ]�KWz�0 ; 4K2g2 vuut8KWz(k + �0)p ;2K2g2 ; 4K2g2 vuut256K3Wz exp[4(Kz + �z)](k + �0) exp[��=4] ;4K2g2 s8KWz ; 4K2g2 vuut64K2Wz exp[2(Kz + �z)](k + �0) exp[��=4] ;4K2g2 vuut4096K4Wz exp[4(Kz + �z)] ;s 4kWz3KWz ;4K2g2 vuut1024K3Wz exp[2(Kz + �z)](k + �0) exp[��=4] 9=; :Increase L (if necessary) and choose Tl, K2 large enough and Kvmax; �max small enough sothat,( �d0 � �g2) � 4K 0vfKmz ; �K2 � 2(K 0vfkmz + kvf + k(Tl));�(T ) � k; for some positive scalar k, k(�max; L) � �32�3an�m ;�1 � qd0+g28ke ; �2 � d0+g28 ; and�max � minf �p�8�3�5(T ); �q �8�3�4(T ); �q �8�5�8(T )g; where,� := K 0vfKmz + K0vfkmz+kvf+k(Tl)L=2 ; k(�;L) := kr(1+ j�2njbmin )L=2 + (j�nj+ j�2n jbminM)�;�1 := Kv 2Kmz + �m 3c; and �2 := (KvC +K 0vf )d1Kmz :Please refer to [20] for details. 30