Robustness of Discrete-Time Direct Adaptive Controllers - Centre ...

1182 IEEE TRANSACTIONS ON AUTOMATIC COh'TROL. VOL. AC-30, NO. 12, DECEMBER 198514Lproved for invertible systems [I31 by showing that A8, E Oe2.Similar results were obtained in [25], [26].The introduction of the CI posteriori error representation [6],[ 1 I] allows a clear-cut interpretation of the stability proofs, eitherLyapunov or Popov based, available in the literature. Due to thestructure of the integral PAA it is easy to_spow that in the matchedcase e; as given in (3.2) is equal to TO, i$-d, the a posteriorierror. Since the operator H,:e: + is passive (for aconstant gain matrix), even for unbounded +,,-direct applicationof the passivity theorem leads tcthe stability of 8 '41-d. The proofis completed by showing that 8:&d + 0 impfies e, + 0 withbounded 4,. A similar procedure will be required below when wewill seek to prove stability of the adaptive scheme from thestability of the normalized signals.Remark 3.1: It can also be shown that when d > 1 aninterlaced version of (3.1) avoids the necessity of using theaugmented error in (3.2) since for that schemestudy the feedback interconnection is the conic sector stabilitytheorem [17] (see also [2]). It is required then to choose a PAAsuch that sector conditions may be established for the relationHI:e, --f $,.It will be shown below that to obtain +,-independent propertiesfor the PAA (see Remark 2.3) normalization of e, and 4, arecompulsory. In the following (:) will be used to denotenormalized variables and corresponding operators and aredefined as:2 p,-l'2+l-d, g, ~p,-"2e,; $ , = p;"'$, (3.0a)Hi- G * P, -l/2Hi,,1/2 .]; i= 1, 2. (3.0b)The normalization factor p, is introduced in Section V.To gain some insight into the problem of the selection of thePAA we will consider first the approaches and motivations of thematched case, that is when no ROM or BOD are present. A classof PAA for which suitable 110 properties have been established islater presented and its properties stated and proved.A. The Matched CaseMost adaptive schemes reported in the literature use an integralPAA of the form8,=8,-1+F,4,-de; (3.1)where F, is a time-varying matrix (the matrix gain) and e; is anestimate of the prediction error. The increasing complexity of thetreated cases required increasing information fed through e; intothe PAA. Therefore, the choice of e; may be thought of asreflecting the evolution of the adaptive control theory. It wasinitially taken equal to the tracking error to solve the unitary delaycase. Later it was shown that using this same error, a physicallyrealizable globally stab!e solution was still possible for = 2, byproper replacement of 8, by the multiplier operator PL(8,). I. Thislast modification was required to ensure the positive real conditionof the error model. The ingenious inclusion of the augmentederror model allowed proof of convergence of the tracking error bytakingB. PAA Sector ConditionsGiven our objective of uniform asymptotic stability we disregardproportional components in the PAA. In addition, gaindecreasing PAA are discarded to preserve the alertness of theadaptive scheme. Extrapolating from current usage we considerintegral interlaced PAA of the formwhere 5 takes one of the following forms.1) Constant gain (CG) PAA: 5 is a scalar5 2 f>O. (3.4a)2) Regularized least squares (RLS) PAA: 5 is a time-varyingmatrix5 2 F, (3 Ab)where (see [24] for further details)and X, < XI, X are strictly positive scalars.The eigenvalues of F, are all contained in the chosen interval[b, All.Equations (3.3) and (3.4) define an operator RI:P, + 6, (seeFig. 2). Besides this operator we will con_sider for th_e RLS/PAA,its exponentially weighted counterpart HY:Cp + $; where thesuperscript a denotesx; P a'X, : a>o.The I/O properties of the two operators are summarized in thefollowing lemma. Similar results were obtained earlier in [7],[ 141, [ 151, [24]. Notice that A;. = A, when a = 1.Lemma 3.1 (I/O Properties of the PAA):1) CG/PAA: If 5 is given by (3.4a), then+(cR YI-~T-I'#'I-~)/(~ +d':-dFrdl-d). (3.2)However, this new form of e; posed the new stability problem ofensuring boundedness of the auxiliary signal, which was laterfor all 6CG such that' This section's discussion, although restricted to discrete-time systems. isfurther simplified b] choosing the following structure for the operator:PL:Pr(!,) 2 qded- (see [13]) so that the operator retains the basic conceptsof contmuous and hybrid schemes.2) RLS/PAA: If 3 is given by (3.4b). (3.4c), thenis outside CONE ( - 1, v'l - cRLS)Authorized licensed use limited to: ECOLE DES MINES PARIS. Downloaded on November 28, 2009 at 12:57 from IEEE Xplore. Restrictions apply.