A robust adaptive control architecture for disturbance...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2012; 26:1024–1055Published online 24 February 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2281

A robust adaptive control architecture for disturbance rejection anduncertainty suppression with L1 transient and steady-state

performance guarantees

Tansel Yucelen and Wassim M. Haddad*,†

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA

SUMMARY

In this paper, a new adaptive control architecture for linear and nonlinear uncertain dynamical systemsis developed to address the problem of high-gain adaptive control. Specifically, the proposed frameworkinvolves a new and novel controller architecture involving a modification term in the update law that min-imizes an error criterion involving the distance between the weighted regressor vector and the weightedsystem error states. This modification term allows for fast adaptation without hindering system robustness.In particular, we show that the governing tracking closed-loop system error equation approximates a Hurwitzlinear time-invariant dynamical system with L1 input–output signals. This key feature of our frameworkallows for robust stability analysis of the proposed adaptive control law using L1 system theory. We fur-ther show that by properly choosing the design parameters in the modification term, we can guarantee adesired bandwidth of the adaptive controller, guaranteed transient closed-loop performance, and an a prioricharacterization of the size of the ultimate bound of the closed-loop system trajectories. Several illustrativenumerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2012John Wiley & Sons, Ltd.

Received 3 October 2011; Revised 20 January 2012; Accepted 27 January 2012

KEY WORDS: adaptive control; uncertain dynamical systems; disturbance rejection; uncertainty suppres-sion; stabilization; command following; guaranteed transient and steady-state performance;verification and validation

1. INTRODUCTION

One of the fundamental problems in feedback control design is the ability of the control system toguarantee robust stability and robust performance with respect to system uncertainties in the designmodel. To this end, adaptive control along with robust control theory have been developed to addressthe problem of system uncertainty in control-system design. The fundamental differences betweenadaptive control design and robust control design can be traced to the modeling and treatment ofsystem uncertainties as well as the controller architecture structures.

In particular, adaptive control [1–3] is based on constant linearly parameterized system uncer-tainty models of a known structure, but unknown variation, whereas robust control [4, 5] ispredicated on structured and/or unstructured linear or nonlinear (possibly time-varying) operatoruncertainty models consisting of bounded variation. Hence, for systems with constant real para-metric uncertainties with large unknown variations, adaptive control is clearly appropriate, whereasfor systems with time-varying parametric uncertainties and nonparametric uncertainties with normbounded variations, robust control may be more suitable.

*Correspondence to: Wassim M. Haddad, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta,GA 30332-0150, USA.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

ROBUST ADAPTIVE CONTROL 1025

In contrast to fixed-gain robust controllers, which are predicated on a mathematical modelof the system uncertainty, and which maintain specified constants within the feedback con-trol law to sustain robust stability and performance over the range of system uncertainty,adaptive controllers directly or indirectly adjust feedback gains to maintain closed-loop stabil-ity and improve performance in the face of system uncertainties. Specifically, indirect adap-tive controllers utilize parameter update laws to identify unknown system parameters andadjust feedback gains to account for system variation, whereas direct adaptive controllersdirectly adjust the controller gains in response to plant variation. In either case, the over-all process of parameter identification and controller adjustment constitutes a nonlinear con-trol law architecture, which makes validation and verification of guaranteed transient andsteady-state performance, as well as robustness margins of adaptive controllers extremelychallenging [6].

Although adaptive control has been used in numerous applications to achieve system perfor-mance without excessive reliance on system models, the necessity of high-gain feedback forachieving fast adaptation can be a serious limitation of adaptive controllers [7]. Specifically, incertain applications, fast adaptation is required to achieve stringent tracking performance speci-fications in the face of large system uncertainties and abrupt changes in system dynamics. This,for example, is the case for high performance aircraft systems that can be subjected to systemfaults or structural damage, which can result in major changes in aerodynamic system param-eters. In such situations, high-gain adaptive control is necessary to rapidly reduce and main-tain system tracking errors. However, fast adaptation using high-gain feedback can result inhigh-frequency oscillations, which can excite unmodeled system dynamics resulting in systeminstability [7, 8]. Hence, there exists a critical trade-off between system stability and controladaptation rate.

Virtually, all adaptive control methods developed in the literature have averted the problem ofhigh-gain control. Notable exceptions include [9–13]. Specifically, the authors in [9–11] use alow-pass filter that effectively subverts high frequency oscillations that can occur because of fastadaptation, whereas using a predictor model to reconstruct the reference system model. In partic-ular, the authors in [9–11] develop a robust adaptive control architecture that provides sufficientconditions for stability and performance in terms of L1-norms of the underlying system trans-fer functions despite fast adaptation, leading to uniform bounds on the L1-norms of the systeminput–output signals. In [12], an indirect adaptive control architecture is developed using a least-squares parameter estimation scheme to adjust the nominal controller parameters, thereby reducingsystem modeling errors and effectively utilizing a direct adaptive control law with a slower learn-ing rate. More recently, the authors in [13] present a model reference adaptive control architecturefor fast adaptation predicated on an optimal control problem. In particular, a fast adaptation algo-rithm is developed using the minimization of the system tracking error to derive a direct adaptivecontrol law.

In this paper, a new adaptive control architecture for linear and nonlinear uncertain dynamicalsystems is developed to address the problem of high-gain adaptive control. Specifically, the pro-posed framework involves a new and novel controller architecture involving a modification termin the update law that minimizes an error criterion involving the distance between the weightedregressor vector and the weighted system error states. This modification term allows for fast adap-tation without hindering system robustness. In particular, we show that the governing trackingclosed-loop system error equation approximates a Hurwitz linear time-invariant dynamical sys-tem with L1 input–output signals. This key feature of our framework allows for robust stabilityanalysis of the proposed adaptive control law using L1 system theory. Specifically, in the faceof fast adaptation, uniform transient and steady-state system performance bounds are derived interms of L1-norms of the closed-loop system error dynamics. We further show that by prop-erly choosing the design parameters in the modification term, we can adjust the bandwidth ofthe adaptive controller, the transient and steady-state closed-loop performance, and the size ofthe ultimate bound of the closed-loop system trajectories, independently of the system adapta-tion rate. Several illustrative numerical examples are provided to demonstrate the efficacy of theproposed approach.

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2012; 26:1024–1055DOI: 10.1002/acs

1026 T. YUCELEN AND W. M. HADDAD

2. NOTATION, DEFINITIONS, AND MATHEMATICAL PRELIMINARIES

The notation used in this paper is fairly standard. Specifically, R denotes the set of real numbers,ZC denotes the set of nonnegative integers, Rn denotes the set of n � 1 real column vectors,Rn�m denotes the set of n � m real matrices, .�/T denotes transpose, .�/�1 denotes inverse, .�/C

denotes the Moore–Penrose generalized inverse, k�k2 denotes the Euclidian norm, and k�kF denotesthe Frobenius matrix norm. Furthermore, we write �min.M/ (resp., �max.M/) for the minimum(resp., maximum) eigenvalue of the Hermitian matrix M , �min.M/ (resp., �max.M/) for the min-imum (resp., maximum) singular value of the Hermitian matrix M , spec.�/ for the spectrum of asquare matrix, tr.�/ for the trace operator, vec.�/ for the column stacking operator, and .�/0 for theFréchet derivative.

Next, we introduce several definitions and some key results that are necessary for developing themain results of this paper.

Definition 2.1Let A 2 Rn�m and define AL , .ATA/CAT. Then, PA , AAL D A.ATA/CAT is a projectionmatrix and P?A D I �PA is a complementary projection matrix.

Note that PA is symmetric and nonnegative define. Nonnegative-definiteness follows from notingthat �TPA� D �TA.ATA/CAT� D Q�T.ATA/C Q� > 0, where � 2 Rn and Q� , AT�, and by using(xxxi) of Proposition 6.1.6 of [14]. Furthermore, note that PAAD A and P?A AD 0.

Definition 2.2

Let � W Rn! R be a continuously differentiable convex function given by �.�/, ."�C1/�T��2max

"��2max

,

where �max 2R is a projection norm bound imposed on � 2Rn and "� > 0 is a projection tolerancebound. Then, the projection operator Proj WRn �Rn!Rn is defined by

Proj.� , y/,

8<:

y, if �.�/ < 0,y, if �.�/> 0 and �0.�/y 6 0,

y � �0T.�/�0.�/y

�0.�/�0T .�/�.�/, if �.�/> 0 and �0.�/y > 0,

(1)

where y 2Rn.

It follows from Definition 2.2 that

.� � ��/T.Proj.� , y/� y/6 0, �� 2Rn, (2)

holds [15]. The definition of the projection operator can be generalized to matrices asProjm.‚,Y / D .Proj.col1.‚/, col1.Y //, : : : , Proj.colm.‚/, colm.Y ///, where ‚ 2 Rn�m, Y 2Rn�m, and coli .�/ denotes i th column operator. In this case, for a given‚� 2Rn�m, it follows from(2) that

tr�.‚�‚�/T.Projm.‚,Y /� Y /

�D

mXiD1

�coli .‚�‚

�/T.Proj.coli .‚/, coli .Y //� coli .Y //�6 0,

(3)holds. Throughout the paper, we assume that the projection norm bound imposed on each columnof ‚ 2Rn�m is �max.

The following lemma is a statement of Young’s inequality [14].

Lemma 2.1Let x 2Rn and y 2Rn, and let � > 0. Then,

xTy 6 �xTxC 1

4�yTy. (4)



Lemma 2.2 ([14])Let A 2Rn�n and B 2Rn�n be such that A and ACB are nonsingular. Then,

.ACB/�1 D

kXiD0

A�1.�BA�1/i CA�1.�BA�1/kC1.I CBA�1/�1, (5)

for all k 2 ZC.

Lemma 2.3Let X 2 Rn�n and Y 2 Rn�n be symmetric matrices such that X > 0 and Y < 0. Then, XY isHurwitz.

ProofLet � 2 spec.XY / be such that .XY /�D ��, where � 2 Cn and �¤ 0. Now, noting that X

12Y�D

�X�12 � or, equivalently, X

12YX

12

�X�

12 ��D �

�X�

12 ��

, it follows that X12YX

12 Q� D � Q�, where

.�/12 denotes the (unique) positive-definite square root and Q�,X�1�. Hence, because X

12YX

12 < 0

and spec.XY /D spec�X12YX

12

�, it follows that XY is Hurwitz. �

Lemma 2.4Let A 2 Rn�n be Hurwitz and let P 2 Rn�n be the unique, positive-definite solution to theLyapunov equation

0D ATP CPACR, (6)

where R 2 Rn�n is a given positive-definite matrix. Then, for a given symmetric, nonnegativedefinite K 2Rn�n, A�KP is Hurwitz.

ProofRewriting the Lyapunov equation (6) as

� 2PKP D .A�KP/TP CP.A�KP/CR (7)

and noting that PKP > 0, it follows from (7) that

0> .A�KP/TP CP.A�KP/CR, (8)

which implies that A�KP is Hurwitz. �

Next, consider the linear dynamical system given by

Px.t/D Ax.t/CBw.t/, x.0/D x0, t > 0, (9)

´.t/D Cx.t/, (10)

where x.t/ 2 Rn, w.t/ 2 Rm, ´.t/ 2 Rl , t > 0, A 2 Rn�n is Hurwitz, B 2 Rn�m, and C 2 Rl�n.Here, w.�/ is an input signal belonging to the class L1 of bounded Lebesgue measurable functionson Œ0,1/, and ´.�/ is an output signal belonging to the class L1. For the statement of the nextdefinition, jjj�jjjp,q denotes a signal norm with p as temporal norm and q spatial norm.

Definition 2.3Consider the linear dynamical system (9) and (10) with w.�/ 2 L1 and ´.�/ 2 L1. Then, theL1-system norm of (9) and (10) is given by the equi-induced signal norm

jjjGjjj.1,2/,.1,2/ , supw.�/2L1jjj´jjj1,2

jjjwjjj1,2, (11)



where G W L1! L1 denotes the convolution operator of (9) and (10) given by

´.t/D GŒw�.t/D .G �w/.t/,Z 10

G.t � �/w.�/d� , (12)

where G.t/, CeAtB , t > 0.

For ˛ > 0, define the shifted impulse response function G˛ W R ! Rl�m by G˛.t/ ,Ce.AC

˛2 In/tB , t > 0, and let G˛ denote its convolution operator given by

´.t/D G˛Œw�.t/D .G˛ �w/.t/,Z 10

G˛.t � �/w.�/d� . (13)

Next, we provide a computable upper bound for the L1-system norm jjjGjjj.1,2/,.1,2/.

Theorem 2.1 ([16])Let ˛ > 0 be such that AC ˛

2I is Hurwitz, and let Q˛ 2 Rn�n be the unique, nonnegative definite

solution to the Lyapunov equation

0D AQ˛ CQ˛ATC ˛Q˛ CBB

T. (14)

Then, G W L1! L1 and

jjjGjjj.1,2/,.1,2/ 61p˛jjjGjjj˛.1,2/,.2,2/ D

1p˛�1=2max .CQ˛C

T/. (15)

Remark 2.1Note that (15) implies jjj´jjj1,2 6 1p

˛�1=2max

�CQ˛C

T�jjjwjjj1,2, which gives an L1-norm bound on

the system output ´.�/.

Remark 2.2Note that (14) has a nonnegative-definite solution, and .AC ˛

2I , B/ is stabilizable if and only if

AC ˛2I is Hurwitz, which implies ˛ < �2 spabs.A/, where spabs.A/ denotes the spectral abscissa

ofA. To provide the tightest upper bound for the L1-norm of the system (9) and (10), we can replace(15) by

jjjGjjj.1,2/,.1,2/ 6 inf0<˛<�2 spabs.A/

1p˛�1=2max

�CQ˛C

T�

, (16)

where Q˛ satisfies (14).

3. ADAPTIVE CONTROL FOR DISTURBANCE REJECTION

In this section, we consider a linear dynamical system with an unknown time-varying bounded dis-turbance to illustrate the key ideas of the proposed adaptive control architecture. Because for thisproblem, the system loop transfer function (broken at the control input) can be equivalently writtenas a linear time-invariant dynamical system, we resort to classical control theory tools to analyzethe closed-loop system and compare the proposed architecture with a standard adaptive controllerarchitecture for disturbance rejection.

3.1. Problem formulation

Consider the linear dynamical system given by

Px.t/D Ax.t/CBu.t/CBd.t/, x.0/D x0, t > 0, (17)

where x.t/ 2 Rn, t > 0, is the state vector, u.t/ 2 Rm, t > 0, is the control input, d.t/ 2 Rm,t > 0, is an unknown disturbance satisfying kd.t/k2 6 dmax, t > 0, and k Pd.t/k2 6 Pdmax, t > 0, and



A 2Rn�n and B 2Rn�m are known matrices such that the pair .A,B/ is controllable. Furthermore,we assume that the full state is available for feedback, and the control input u.�/ is restricted tothe class of admissible controls consisting of measurable functions such that u.t/ 2 Rm, t > 0. Inaddition, consider the reference system given by

Pxm.t/D Amxm.t/CBmr.t/, xm.0/D x0, t > 0, (18)

where xm.t/ 2 Rn, t > 0, is the reference state vector, r.t/ 2 Rr , t > 0, is a bounded piecewisecontinuous reference input, Am 2Rn�n is Hurwitz, and Bm 2Rn�r .

Next, consider the feedback control law given by

u.t/D un.t/C uad.t/, t > 0, (19)

where the nominal control law un.t/, t > 0, is given by

un.t/D�K1x.t/CK2r.t/, t > 0, (20)

and the adaptive control law uad.t/, t > 0, is given by

uad.t/D� Od.t/, t > 0, (21)

where K1 2 Rm�m and K2 2 Rm�r are nominal control gain matrices, and Od.t/ 2 Rm, t > 0, is anestimate of d.t/, t > 0, satisfying

POd.t/D ProjhOd.t/, BTPe.t/� BT

�AmP

�1CM��1

B Od.t/i

, Od.0/D Od0, t > 0,

(22)where 2 Rm�m is a positive-definite adaptation gain matrix, e.t/ , x.t/ � xm.t/, t > 0, is thesystem error state, P 2Rn�n is a positive-definite solution of the Lyapunov equation

0D ATmP CPAmCR, (23)

where R 2 Rn�n is a positive-definite matrix, > 0 is a scalar adaptation gain, M 2 Rn�n is apositive-definite matrix chosen such that det

�AmP

�1CM�¤ 0 and BT

�AmP

�1CM��1

B

is positive definite, and Odmax 2 R is a norm bound imposed on the update weight Od.t/, t > 0. Notethat because Am is Hurwitz, it follows from converse Lyapunov theory [17, 18] that there exists aunique positive-definite P 2Rn�n satisfying (23) for a given positive-definite matrix R 2Rn�n.

For the nominal control law (20), we have the following assumption involving standard matchingconditions for model reference adaptive control.

Assumption 3.1There exist gain matrices K1 2Rm�n and K2 2Rm�r such that Am D A�BK1 and Bm D BK2.

Remark 3.1Note that setting � 0 in (22) gives the model reference (pure) integral controller ([9])

POd.t/D ProjhOd.t/, BTPe.t/

i, Od.0/D Od0, t > 0. (24)

A block diagram showing the proposed adaptive control architecture is given in Figure 1.

3.2. Stability analysis

The linear dynamical system (17) can be written as

Px.t/D Amx.t/CBmr.t/�B Qd.t/, x.0/D x0, t > 0, (25)



Figure 1. Visualization of proposed adaptive control architecture for the disturbance rejection problem.

where Qd.t/ , Od.t/ � d.t/, t > 0, with the system error and weight update error dynamics givenby, respectively,

Pe.t/DAme.t/�B Qd.t/, e.0/D 0, t > 0, (26)

PQd.t/DProjhOd.t/, BTPe.t/� BT

�AmP

�1CM��1

B Od.t/i� Pd.t/, Qd.0/D Qd0, t > 0.

(27)

Note that because the weight update law for Od.t/, t > 0 is predicated on the projection operator andd.�/ is norm bounded, there exists a norm bound Qdmax such that k Qd.t/k2 6 Qdmax, t > 0. The nexttheorem presents Lyapunov-based stability analysis for the proposed adaptive control architectureas well as guaranteed transient and steady-state performance for the disturbance rejection problem.

Theorem 3.1Consider the linear dynamical system given by (17) with reference system given by (18), and assumethat Assumption 3.1 holds. Furthermore, let the adaptive control law be given by (21) with weightupdate law given by (22). Then, the closed-loop error signals given by (26) and (27) are uniformlybounded for all .e.0/, Qd.0// 2 D˛ , where D˛ is a compact positively invariant set, with ultimatebound ke.t/k2 < ", t > T , where

" >h�max.P /#

2C �max.�1/ Qd2max

i 12

, (28)

# ,��1min.R/

�1kSdkFd

2maxC

1

�2k�1kF Pd

2max

� 12

, (29)

Sd , BT�AmP

�1CM��1

B , and �1 and �2 are arbitrary positive constants satisfying.2 � �1/Sd � �2

�1 > 0. In addition, for t 2 Œ0,T /, the system error and weight update errordynamics, respectively, satisfy the bounds



ke.t/k2 6 k Qd.0/k2�k�1kF

�min.P /

� 12

, (30)

k Qd.t/k2 6 k Qd.0/k2�k�1kF

�min.�1/

� 12

. (31)

ProofConsider the Lyapunov-like function candidate

V.e, Qd/D eTPeC Qd�1 Qd , (32)

where P > 0 satisfies (23). Note that (32) satisfies O .k�k2/ 6 V.�/ 6 O.k�k2/, where � ,ŒeT, QdT�T, O .k�k2/ D O.k�k2/ D k�k22 with k�k22 , eTPe C Qd�1 Qd . Furthermore, note thatO .�/ and O.�/ are class K1 functions. Differentiating (32) along the closed-loop system trajectoriesof (26) and (27) yields

PV .e.t/, Qd.t//D 2eT.t/PhAme.t/�B Qd.t/

iC 2 QdT.t/Proj

hOd.t/, BTPe.t/� Sd Od.t/

i

� 2 QdT�1 Pd.t/

D eT.t/�AT

mP CPAm

�e.t/� 2eT.t/PB Qd.t/C 2 QdT.t/

hBTPe.t/� Sd Od

i

C 2 QdT.t/�Proj

hOd.t/, BTPe.t/� Sd Od.t/

i�hBTPe.t/� Sd Od

i�

� 2 QdT.t/�1 Pd.t/. (33)

Now, using the property of projection operator [15],hOd.t/� d.t/

iT �Proj

hOd.t/, ´.t/

i� ´.t/

�6

0, t > 0, where ´.t/, BTPe.t/� Sd Od.t/, t > 0, and hence, (33) yields

PV .e.t/, Qd.t//6� eT.t/Re.t/� 2eT.t/PB Qd.t/C 2 QdT.t/hBTPe.t/� Sd Od

i� 2 QdT.t/�1 Pd.t/

D� eT.t/Re.t/� 2 QdT.t/Sd Od.t/� 2 QdT.t/�1 Pd.t/

D� eT.t/Re.t/� 2 QdT.t/Sd Qd.t/� 2 QdT.t/Sdd.t/� 2 Qd

T.t/�1 Pd.t/, t > 0.(34)

Furthermore, using Young’s inequality in the last two terms of (34) gives

� 2 QdT.t/Sdd.t/6 j � 2 QdT.t/S12

d S12

d d.t/j6 �1 QdT.t/Sd Qd.t/C

�1dT.t/Sdd.t/, t > 0,

(35)

� 2 QdT.t/�1 Pd.t/6 j � 2 QdT.t/�12�

12 Pd.t/j6 �2 QdT.t/�1 Qd.t/C

1

�2PdT.t/�1 Pd.t/, t > 0.

(36)

Next, using (35) and (36) in (34) and noting .2� �1/Sd � �2�1 > 0, it follows that

PV .e.t/, Qd.t//6� eT.t/Re.t/C

�1dT.t/Sdd.t/C

1

�2PdT.t/�1 Pd.t/

6� �min.R/ke.t/k22C

�1kSdkF d

2maxC

1

�2k�1kF Pd

2max

, t > 0. (37)

Let # be given by (29) and recall that k Qd.t/kF 6 Qdmax, t > 0. Now, for kek2 > # , it follows thatPV .e.t/, Qd.t//6 0 for all .e.t/, Qd.t// 2 De nDr and t > 0, where

De , f.e, Qd/ 2Rn �Rs�mg W x 2Rng, (38)

Dr , f.e, Qd/ 2Rn �Rs�mg W kek2 6 #g. (39)



Finally, define D˛ , f.e, Qd/ 2 Rn �Rs�mg W V.e, QW / 6 ˛g, where ˛ is the maximum value suchthat D˛ � De, and define Dˇ , f.e, Qd/ 2 Rn � Rs�mg W V.e, QW / 6 ˇg, ˇ > O.�/ D �2 D

�max.P /#2C �max.

�1/ Qd2max.To show ultimate boundedness of the closed-loop systems (26) and (27), note that Dˇ � D˛ .

Now, because PV .e.t/, Qd.t// 6 0, t > 0, for all .e.t/, Qd.t// 2 De nDr and Dr � D˛ , it follows thatD˛ is positively invariant. Hence, if .e.0/, Qd.0// 2 D˛ , then it follows from Corollary 4.4 of [17]that the solution .e.t/, Qd.t//, t > 0, to (26) and (27) is ultimately bounded with respect to .e, Qd/with ultimate bound given by O�1.ˇ/D

pˇ, which yields (28).

Finally, over the time interval t 2 Œ0,T /, PV .e.t/, Qd.t// 6 0, because .e.t/, Qd.t// 2 De nDr. Thisimplies that

V.e.t/, Qd.t//6 V.e.0/, Qd.0//, t 2 Œ0,T /. (40)

Using the inequalities �min.P /kek22 6 V.e, Qd/ and V.e.0/, Qd.0// D QdT.0/�1 Qd.0/ 6 k�1kF

�k Qd.0/k22 in (40) gives (30). Identically, using the inequalities �min.�1/k Qdk22 6 V.e, Qd/ and

V.e.0/, Qd.0//6 k�1kFk Qd.0/k22 in (40) gives (31). This completes the proof. �

Remark 3.2Theorem 3.1 guarantees transient performance for the closed-loop system. Specifically, for t 2Œ0,T / the closed-loop error signals, (26) and (27) are bounded from above as given by (30) and(31). This, along with uniform ultimate boundedness of the closed-loop error signals (26) and (27),implies that e.�/ 2 L1 and Qd.�/ 2 L1, and hence, x.�/ 2 L1 and u.�/ 2 L1. Furthermore, notethat e.t/, t 2 Œ0,T /, can be made sufficiently small (satisfying # in (29)) by letting �min./!1.

Remark 3.3Let M D mI in (22). Because

�AmP

�1CmI��1D�AmP

�1C �1mI��1

, it follows from

Lemma 2.2, with A , �1mI and B , AmP�1, that

�AmP

�1C �1mI��1D 0 as ! 0 or

m ! 1. In this case, we obtain the standard adaptive controller given by (24). If, in addition,�min./!1, then, it follows from (29) that # D 0. In this case, we recover asymptotic stability ofthe system dynamics. However, ! 0 or m!1 (resp., �min.M/!1) are not desirable designchoices because they might lead to a high bandwidth adaptive controller that can result in high-frequency oscillations in the closed-loop system response and excite unmodeled dynamics resultingin system instability. This is discussed next in the succeeding section.

3.3. Transient and steady-state performance guarantees

In this section, we show that the governing tracking closed-loop system error equation approxi-mates a Hurwitz linear time-invariant dynamical system with L1 input–output signals. This allowsus to derive uniform transient and steady-state system performance bounds in terms of L1-normsof the closed-loop system error dynamics. The following lemma is key in developing the results ofthis section.

Lemma 3.1Consider the weight update law given by (22). If M D�AmP

�1CmI , m> 0, then

B Od.t/D �1mPBPe.t/� �1mBLT

h�1POd.t/� .t/

i, t > 0, (41)

where PB , B.BTB/CBT, BL , .BTB/CBT, and .t/ 2Rm, t > 0, is a residual error signal.

ProofWriting (22) as

�1POd.t/D Proj

hOd.t/, BTPe.t/� BT

�AmP

�1CM��1

B Od.t/i

, t > 0, (42)



it follows that

0D BTPe.t/� BT�AmP

�1CM��1

B Od.t/� �1POd.t/C .t/, t > 0, (43)

where .t/, t > 0, is a residual error because ProjhOd.t/,y.t/

iis not necessarily equal to y.t/ for

all t > 0. Letting M D�AmP�1CmI , it follows from (43) that

0D BTPe.t/� m�1BTB Od.t/� �1POd.t/C .t/, t > 0. (44)

Pre-multiplying (44) by BLTyields

0DBLTBTPe.t/� m�1BLT

BTB Od.t/�BLT�1POd.t/CBLT

.t/

DPBPe.t/� m�1B Od.t/�BLT

�1POd.t/CBLT

.t/, t > 0, (45)

which can be equivalently written as

0D �1mPBPe.t/�B Od.t/� �1mBLT

�1POd.t/C �1mBLT

.t/, t > 0. (46)

Hence, (41) holds. �

Remark 3.4The residual error .t/ 2 Rm, t > 0, in (41) satisfies k .t/k2 6 max, t > 0 because all thesignals in (41) are bounded by Theorem 3.1. Furthermore, .t/, t > 0, is identically zero whenOdi .t/ 2 .� Odmax, Odmax/, for all i D 1, 2, : : : ,m, where Odmax is the projection norm bound of Od.t/,t > 0. Hence, by choosing a conservative projection norm bound Odmax, we can guarantee that .t/D 0 for all t > 0.

Proposition 3.1Consider the system error dynamics given by (26) and letM D�AmP

�1CmI ,m> 0. Then, thesystem error dynamics (26) can be characterized by the linear, time-invariant system Ge,d given by

Pe.t/D QAe.t/C QBw.t/, e.0/D e0, t > 0, (47)

´.t/D QCe.t/, (48)

where QA , Am � �1mPBP is Hurwitz, QB , B , QC , I , and w.t/ , d.t/ C

�1m.BTB/Ch�1POd.t/� .t/

i, and w.�/ 2 L1.

ProofIt follows from Lemma 3.1 that (26) can be rewritten as

Pe.t/D Ame.t/�B Qd.t/

D Ame.t/�B Od.t/CBd.t/

D Ame.t/�h�1mPBPe.t/�

�1mBLTh�1POd.t/� .t/

iiCBd.t/

D�Am �

�1mPBP�e.t/CB

hd.t/C �1m.BTB/C

h�1POd.t/� .t/

ii

D QAe.t/C QBw.t/, e.0/D e0, t > 0, (49)

which gives (47). Defining the system output signal as ´.t/ , e.t/ gives (48). Next, lettingK D �1mPB > 0, it follows from Lemma 2.4 that QA D Am �

�1mPBP is Hurwitz. Finally,

because the signals in the right-hand side of (22) belong to L1, POd.�/ 2 L1. Hence, it follows fromPOd.�/ 2 L1, .�/ 2 L1, and d.�/ 2 L1, that w.�/ 2 L1. �



Note that (47) and (48) characterizes the closed-loop linear time-invariant dynamical system.Because w.�/ 2 L1 and QA is Hurwitz, it follows that ´.�/ 2 L1. Hence, the following theoremis immediate.

Theorem 3.2Consider the linear, time-invariant system Ge,d given by (47) and (48). Let ˛ > 0 be such thatQAC ˛

2I is Hurwitz and letQ˛ 2Rn�n be the unique, nonnegative definite solution to the Lyapunov

equation

0D QAQ˛ CQ˛QATC ˛Q˛ C QB QB

T. (50)

Then, Ge,d W L1! L1 and

jjjGe,d jjj.1,2/,.1,2/ 61p˛�1=2max .Q˛/, (51)

and hence, jjjejjj1,2 6 1p˛�1=2max .Q˛/jjjwjjj1,2.

ProofThis is a restatement of Theorem 2.1 for Ge,d given by (47) and (48). �

For verification and validation of adaptive control architectures, it is of practical importance tocharacterize the behavior of the closed-loop dynamical system from disturbance d.t/, t > 0, to errorsignals e.t/, t > 0. To achieve such a characterization, we need the following lemma.

Lemma 3.2Let D �� , where � > 0 and � > 0. Then, �1 POd.t/D ��1�1

�

POd.t/D 0, t > 0, as �!1.

ProofConsider the weight update law given by (22). First, assume that Odi .t/ 2 .� Odmax, Odmax/, t > 0, for alli D 1, 2, : : : ,m, where Odmax is the projection norm bound of Od.t/, t > 0. In this case, (22) becomes

POd.t/D hBTPe.t/� BT

�AmP

�1CM��1

B Od.t/iD �Ad Od.t/C �Bde.t/, t > 0, (52)

where Ad , ��BT�AmP

�1CM��1

B is Hurwitz (by Lemma 2.3 with X D � > 0 and

Y D�BT�AmP

�1CM��1

B < 0) and Bd , �BTP . Differentiating (52) with respect to timeyields

ROd.t/D �AdPOd.t/C �Bd Pe.t/, t > 0. (53)

Note that Pe.�/ 2 L1 because the signals in the right-hand side of (26) belong to L1. Define

dd .t/, POd.t/, t > 0, and ed .t/, Pe.t/, t > 0. Then, it follows from (53) that

dd .s/D ŒsI � �Ad ��1 �Bded .s/D

�s��1I �Ad

��1Bded .s/D

�s�I �Ad

��1Bded .s/, (54)

where s denotes the Laplace variable and s� , s��1. It now follows from (54) that dd .s/ is bounded

as �!1, which implies that POd.t/, t > 0, is bounded as �!1. Hence, lim�!1 ��1�1

�

POd.t/D 0.

Next, assume that there exists at least one Odi .t/, i D 1, 2, : : : ,m, that is equal to � Odmax or Odmax

over the time interval t D t� to t D t� C�t . In this case, (52) does not hold. However, becausethe projection operator is continuous and involves a smooth transformation from the original vector

field to an inward or tangent vector field [15], POd.t/ remains bounded for all t 2 Œt�, t�C�t�. Hence,

lim�!1 ��1�1

�

POd.t/D 0 holds when one or more Odi .t/ rides the projection norm boundary over afinite time interval. This completes the proof. �



The following proposition is now immediate.

Proposition 3.2Consider the system error dynamics given by (26) and let M D �AmP

�1 C mI , m > 0, and D �� , � > 0, � > 0. If � ! 1, then the system error dynamics (26) can be characterized

by the linear, time-invariant system Ge,d given by (47) and (48), where QA , Am � �1mPBP is

Hurwitz, QB , B , QC , I , w.t/, d.t/� �1m.BTB/C .t/, t > 0, and w.�/ 2 L1. If, in addition,Odi .t/ 2 .� Odmax, Odmax/, for all i D 1, 2, : : : ,m, where Odmax is the projection norm bound of Od.t/,t > 0, then w.t/D d.t/.

ProofThe proof is a direct consequence of Proposition 3.1 and Lemma 3.2. �

Remark 3.5By choosing a conservative projection norm bound Odmax (Remark 3.4) and letting �min./!1, theclosed-loop linear time-invariant dynamical system given by (47) and (48) characterizes a mappingfrom system disturbance d.t/, t > 0, to system error e.t/, t > 0. Therefore, the properties of theclosed-loop system, such as system bandwidth, transient system performance, and L1-system normof Ge,d (Theorem 3.2), can be analyzed from (47) and (48).

As noted in Remark 3.3, ! 0 or m ! 1 are not always desirable design choices when�min./!1. This is because in this case Ge,d can have infinitely fast decaying poles in the left-half plane (due to QAD Am �

�1mPBP ), and hence, Ge,d can behave similar to a high-bandwidthfilter. This controller design can be very sensitive to variations in w.�/ causing high-frequency oscil-lations in the closed-loop system response that can excite unmodeled dynamics resulting in systeminstability. Although the proposed adaptive control architecture collapses to the standard adaptivecontrol architecture as ! 0 or m ! 1, this analysis implies that standard adaptive controllersare not robust, which is a well known fact. Therefore, the design parameters and m shouldbe selected judiciously by analyzing the characteristics of the closed-loop linear time-invariantdynamical system given by (47) and (48), especially when �min./!1.

Finally, because, for motivational purposes, the results of this section address a linear dynamicalsystem with a time-varying disturbance without system uncertainty, we can resort to classical con-trol theory tools to analyze the closed-loop system. However, in the following section, we generalizethese results to uncertain nonlinear dynamical systems, and hence, we can no longer resort to toolsfrom classical control theory to analyze closed-loop system performance.

3.4. Motivational numerical example

In this section, we present a numerical example to demonstrate the utility and efficacy of the pro-posed robust adaptive control architecture for disturbance rejection. Specifically, consider the lineardynamical system representing a disturbed aircraft rolling dynamics model given by�Px1.t/Px2.t/

�D

�0 1

0 0

� �x1.t/

x2.t/

�C

�0

1

�Œu.t/C d.t/� ,

�x1.0/

x2.0/

�D

�0

0

�, t > 0, (55)

where x1 represents the roll angle in radians, x2 represents the roll rate in radians per second, andd.t/ D .1.65 sin.0.25t/C 3.34 sin.0.05t/C �.t// �

180, where �.t/, t > 0, represents a standard

white noise signal. For our simulations, we choose K1 D Œ0.25, 0.80� and K2 D 0.25, and a refer-ence system corresponding to a natural frequency !n D 0.50 rad/s and a damping ratio � D 0.80.Note that with the above gain matrices, Assumption 3.1 holds. Furthermore, we set R D I2, D 1,M D �AmP

�1 C 4I2, D 25, and Odmax D 2 (a conservative choice for the projection normbound), and design an adaptive controller using Theorem 3.1. It follows from Theorem 3.2 thatjjjGe,d jjj.1,2/,.1,2/ 6 0.28 for ˛ D 3.60. Here, our aim is to track a given filtered square-wave rollangle reference command r.t/, t > 0.



Figure 2 shows the closed-loop system performance of the disturbed aircraft rolling dynamicsmodel without adaptation (uad.t/� 0, t > 0). Figure 3 shows the closed-loop system performanceof a standard adaptive controller given by (24) with an adaptation gain of D 25 and with pro-jection norm bound Odmax D 2. Figure 4 shows the closed-loop system performance of the proposed

0 10 20 30 40 50−20

−15

−10

−5

0

5

10

15

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50−8

−6

−4

−2

0

2

4

6

8

Time [s]

un(t) [deg]

−d(t) ×180/πu

ad(t) [deg]

Figure 2. Closed-loop system response of the disturbed aircraft rolling dynamics model with uad.t/ � 0,t > 0.

0 10 20 30 40 50−10

−5

0

5

10

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50−8

−6

−4

−2

0

2

4

6

8

Time [s]

un(t) [deg]

−d(t) ×180/πu

ad(t) [deg]

Figure 3. Closed-loop system response of the disturbed aircraft rolling dynamics model with the standardadaptive controller ((24) with D 25).



0 10 20 30 40 50−10

−5

0

5

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50−8

−6

−4

−2

0

2

4

6

8

Time [s]

un(t) [deg]

−d(t) ×180/πu

ad(t) [deg]

Figure 4. Closed-loop system response of the disturbed aircraft rolling dynamics model with the proposedadaptive controller

�(22) with D 25, D 1, and M D AmP

�1C 4I2�.

−1.5 −1 −0.5 0

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Nyquist Diagram

Imag

inar

y A

xis

Γ = 1 I

Γ = 25 I

Γ = 1000 I

Real Axis

Figure 5. Nyquist plots of the disturbed aircraft rolling dynamics model with the proposed adaptivecontroller (22) for different adaptation gain settings.

adaptive controller with the same adaptation gain. Note that the tracking performance of both con-trollers depicted in Figures 3 and 4 are identical; however, the proposed adaptive controller achievesthe same objective by using a considerably smoother control signal.

Figure 5 shows the Nyquist plots of the loop transfer function with the proposed adaptive con-troller for different adaptation gains. Note that as the adaptation gain is increased, the system phasemargin increases. This is consistent with Proposition 3.2. Figure 6 shows the Nyquist plots of theloop transfer function with the standard adaptive controller for different adaptation gains. As is well



−1.5 −1 −0.5 0−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05Nyquist Diagram

Imag

inar

y A

xis

Γ = 1000 I

Γ = 25 I

Γ = 1 I

Real Axis

Figure 6. Nyquist plots of the disturbed aircraft rolling dynamics model with the standard adaptive controller(24) for different adaptation gain settings.

0 10 20 30 40 50−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time [s]

e1(t)

predicted e1(t)

0 10 20 30 40 50

−0.2

−0.1

0

0.1

0.2

Time [s]

e2(t)

predicted e2(t)

Figure 7. System error dynamics of the disturbed aircraft rolling dynamics model with the proposed adaptivecontroller (22) and predicted error dynamics from Ge,d (Proposition 3.2).

known in standard adaptive control, the system phase margin diminishes as the adaptation gain isincreased. For example, for D 25, it can be seen from Figure 5 that the system phase marginwith the proposed adaptive controller is 40ı, whereas for the same adaptation gain (as seen fromFigure 6), the system phase margin with the standard adaptive controller is 1ı. The proposedadaptive controller achieves almost the same performance as the standard adaptive controller fordisturbance rejection with far better phase margin, which confirms its robustness margins. Finally,Figure 7 presents the system error dynamics of the disturbed aircraft rolling dynamics model with



the proposed adaptive controller and the system error dynamics predicted through Ge,d . This alsoverifies the assertions of Proposition 3.2.

4. ADAPTIVE CONTROL FOR UNCERTAINTY SUPPRESSION

In this section, we extend the results of the previous section to nonlinear uncertain dynamical sys-tems. We first consider system uncertainty that is perfectly parameterized for all x 2 Rn. Then, weconsider uncertainty characterizations approximated over a compact subset Dx of Rn for the casewhen the system uncertainty cannot be perfectly parameterized and/or the system includes boundedexogenous disturbances.

4.1. Problem formulation

Consider the nonlinear uncertain dynamical system given by

Px.t/D Ax.t/CB�.x.t//CBu.t/, x.0/D x0, t > 0, (56)

where x.t/ 2 Rn, t > 0, is the state vector, u.t/ 2 Rm, t > 0, is the control input, A 2 Rn�n andB 2 Rn�m are known matrices such that the pair .A,B/ is controllable, and � W Rn ! Rm is amatched system uncertainty. Furthermore, we assume that the full state is available for feedback,and the control input u.�/ is restricted to the class of admissible controls consisting of measurablefunctions such that u.t/ 2Rm, t > 0. In addition, consider the reference system given by (18).

Assumption 4.1The matched uncertainty in (56) is perfectly parameterized as

�.x/DW Tˇ.x/, x 2Rn, (57)

where W 2 Rs�m is an unknown constant weighting matrix satisfying kW kF 6 w, w 2 Ris a known positive constant, and ˇ W Rn ! Rs is a basis function of the form ˇ.x/ DŒˇ1.x/, ˇ2.x/, : : : , ˇs.x/� satisfying kˇ.x/k2 6 lˇ0 C lˇ1kxk2, where lˇ0 2 R and lˇ1 2 Rare positive constants.

Remark 4.1In general, application of standard adaptive controllers [1–3] to nonlinear uncertain dynamical sys-tems does not assume the knowledge of an upper bound w of the unknown constant weightingmatrix W . That is, standard adaptive controllers can conceptually deal with uncertainties having(unrealistically) large magnitudes of W for the case where the uncertainty is perfectly param-eterized. However, these controllers do not have any robustness properties, especially when theadaptation gain is chosen to be large to satisfy given performance specifications [6, 19]. This isdue to the fact that high-gain adaptive controllers involve high-bandwidths that can result in high-frequency oscillations in the closed-loop system response and excite unmodeled dynamics resultingin system instability [8]. In contrast, the design of robust adaptive controllers with large adaptationgains requires the knowledge of a conservative upper bound w of the unknown constant weight W(see, for example, [9]).

Remark 4.2In Assumption 4.1, we assume that kˇ.x/k2 6 lˇ0Clˇ1kxk2 for all x 2Rn, which only addresses aspecial class of uncertainties involving a linear growth bound. Later in this section, this assumptionis relaxed by addressing uncertainty parameterizations approximated over a compact set Dx of Rn,similar to what is often done in the neural network literature [20–22]. In this case, we address alarger class of system uncertainties.

Next, consider the feedback control law given by (19), where the nominal control law un.t/,t > 0, is given by (20), and the adaptive control law uad.t/, t > 0, is given by

uad.t/D� OWT.t/ˇ.x.t//, t > 0, (58)



where K1 2 Rm�m and K2 2 Rm�r are nominal control gains satisfying Assumption 3.1, andOW .t/ 2Rs�m, t > 0, is an estimate of W satisfying

POW.t/D Projm

hOW .t/, ˇ.x.t//eT.t/PB C ‰

�ˇ.x.t//, OW .t/

�i, OW .0/D OW0, t > 0, (59)

where 2 Rs�s is a positive-definite adaptation gain matrix, e.t/ , x.t/ � xm.t/, t > 0,is the system error state, P 2 Rn�n is a positive-definite solution of the Lyapunov equationgiven by (23), where R 2 Rn�n is a positive-definite matrix, > 0 is a scalar adaptation gain,‰ WRs �Rs�m!Rs�m is given by

‰�ˇ.x.t//, OW .t/

�, �ˇ.x.t//ˇT.x.t// OW .t/BT

�AmP

�1CM��T

B , t > 0, (60)

where M 2 Rn�n is a positive-definite matrix chosen such that det�AmP

�1CM�¤ 0 and

BT�AmP

�1CM��1

B is positive-definite, and Owmax 2R is a norm bound imposed on the update

weight OW .t/, t > 0.

Remark 4.3For system uncertainty suppression, a standard adaptive controller has the form given by ([9])

POW.t/D Projm

hOW .t/, ˇ.x.t//eT.t/PB

i, OW .0/D OW0, t > 0. (61)

To improve the robustness properties of the standard adaptive controller given by (58) and (61),a � -modification term [23], e-modification term [24], Q-modification term [25], and/or adaptiveloop recovery modification term [26] can be included in the weight update law in (61). Note thatfor the disturbance rejection problem, the weight update law given by (22) can be viewed as astandard adaptive controller with a � -modification term that has a nonstandard positive-definitemodification gain matrix of the form BT

�AmP

�1CM��1

B , which leads to properties given byTheorems 3.1 and 3.2, Lemma 3.1, and Propositions 3.1 and 3.2. For system uncertainty suppres-sion, however, the modification term given by (60) no longer has the form of a � -modification termand can be viewed as a variable and weighted damping term because it depends on OW .t/, t > 0.

The proposed weight update law given by (59) can be viewed as a generalization of the opti-mal control modification approach to the adaptive controller proposed in [13]. Specifically, ifM D AmP

�1 � P�1ATm, then, (59) specializes to the weight update law given in [13]. How-

ever, in this case, M need not be positive definite. As we see in the next sections, by properlychoosing M , we can guarantee a desired bandwidth of the adaptive controller, guaranteed transientclosed-loop performance, and we can guarantee the size of the ultimate bound of the closed-loopsystem trajectories. A block diagram showing the proposed adaptive control architecture is givenin Figure 8.

4.2. Stability analysis

The nonlinear uncertain dynamical system (56) can be written as

Px.t/D Amx.t/CBmr.t/�B QWT.t/ˇ.x.t//, x.0/D x0, t > 0, (62)

where QW .t/ , OW .t/ �W , t > 0, with the system error and weight update error dynamics givenby, respectively,

Pe.t/DAme.t/�B QWT.t/ˇ.x.t//, e.0/D 0, t > 0, (63)

PQW.t/DProjm

hOW .t/, ˇ.x.t//eT.t/PB C ‰

�ˇ.x.t//, OW .t/

�i, QW .0/D QW0, t > 0. (64)

We note here that because the weight update law for OW .t/, t > 0 is predicated on the projec-tion operator, and W is a constant matrix, there exists a norm bound Qwmax such that k QW .t/kF 6Qwmax, t > 0.



Figure 8. Visualization of proposed adaptive control architecture for uncertainty suppression.

Theorem 4.1Consider the nonlinear uncertain dynamical system given by (56) with reference system given by(18), and assume that Assumptions 3.1 and 4.1 hold. Furthermore, let the adaptive control law begiven by (58) with weight update law given by (59). Then, the closed-loop error signals given by(63) and (64) are uniformly bounded for all .e.0/, QW .0// 2 D˛ , where D˛ is a compact positivelyinvariant set, with ultimate bound ke.t/k2 < ", t > T , where

" >��max.P /#

2C �max.�1/ Qw2max

� 12 , (65)

# ,�c224c21Cc3

c1

� 12

Cc2

2c1, (66)

c1 , �min.R/ �12kS

12 k2Fw

2l2ˇ1

> 0, c2 , kS12 k2Fw

2 Nlˇ0lˇ1, c3 , 12kS

12 k2Fw

2l2ˇ0

, S ,BT

�AmP

�1CM��T

B , Nlˇ0 , lˇ0Clˇ1x�m, and kxm.t/k6 x�m, t > 0. In addition, for t 2 Œ0,T /,the system error and weight update error dynamics satisfy

ke.t/k2 6 k QW .0/kF�k�1kF

�min.P /

� 12

, (67)

k QW .t/kF 6 k QW .0/kF�k�1kF

�min.�1/

� 12

. (68)

ProofConsider the Lyapunov-like function candidate

V.e, QW /D eTPeC tr QW T�1 QW , (69)

where P > 0 satisfies (23). Note that (69) satisfies O .k�k2/ 6 V.�/ 6 O.k�k2/, where � ,ŒeT, .vec QW /T�T, O .k�k2/ D O.k�k2/ D k�k22 with k�k22 , eTPe C tr QW T�1 QW . Furthermore,note that O .�/ and O.�/ are class K1 functions. Differentiating (69) along the closed-loop systemtrajectories of (63) and (64) yields



PV .e.t/, QW .t//D 2eT.t/P�Ame.t/�B QW

T.t/ˇ.x.t//�C 2tr QW .t/Projm

hOW .t/,

ˇ.x.t//eT.t/PB � ˇ.x.t//ˇT.x.t// OW .t/Si

D eT.t/�AT

mP CPAm

�e.t/� 2eT.t/PB QW T.t/ˇ.x.t//

C 2tr QW T.t/hˇ.x.t//eT.t/PB � ˇ.x.t//ˇT.x.t// OW .t/S

i

C 2tr QW T.t/�Projm

hOW .t/,Z.t/

i�Z.t/

�, (70)

where Z.t/ , ˇ.x.t//eT.t/PB � ˇ.x.t//ˇT.x.t// OW .t/S , t > 0. Now, using the property of

projection operator [15], trhOW .t/�W

iT �Projm

hOW .t/, Z.t/

i�Z.t/

�6 0, t > 0, and hence,

(70) yields

PV .e.t/, QW .t//6 eT.t/�AT

mP CPAm

�e.t/� 2eT.t/PB QW T.t/ˇ.x.t//

C 2tr QW T.t/hˇ.x.t//eT.t/PB � ˇ.x.t//ˇT.x.t// OW .t/S

i

D� eT.t/Re.t/� 2tr QW T.t/ˇ.x.t//ˇT.x.t// OW .t/S

D� eT.t/Re.t/� 2tr QW T.t/ˇ.x.t//ˇT.x.t// QW .t/S

� 2tr QW T.t/ˇ.x.t//ˇT.x.t//W.t/S

D� eT.t/Re.t/� 2ˇT.x.t// QW .t/S12S

12 QW T.t/ˇ.x.t//

� 2ˇT.x.t//WS12S

12 QW T.t/ˇ.x.t//

6� �min.R/ke.t/k22 � 2kS

12 QW T.t/ˇ.x.t//k22

C 2kS12 QW T.t/ˇ.x.t//k2kS

12W Tˇ.x.t//k2, t > 0. (71)

Furthermore, using Young’s inequality in the last term of (71) gives

2kS12 QW T.t/ˇ.x.t//k2kS

12W Tˇ.x.t//k2 6�kS

12W Tˇ.x.t//k22

C

�kS

12 QW T.t/ˇ.x.t//k22, t > 0, � > 0. (72)

Next, setting � D 12

, (71) yields

PV .e.t/, QW .t//6� �min.R/ke.t/k22C

2kS

12W Tˇ.x.t//k22, t > 0. (73)

Because kˇ.x/k2 6 lˇ0 C lˇ1kxk2 6 lˇ0 C lˇ1kek2 C lˇ1kxmk2 6 Nlˇ0 C lˇ1kek2 and kˇ.x/k22 6Nl2ˇ0C 2 Nlˇ0lˇ1kek2C l

2ˇ1kek22, it follows that

2kS

12W Tˇ.x/k22 6

2kS

12 k2Fw

2�Nl2ˇ0C 2

Nlˇ0lˇ1kek2C l2ˇ1kek

22

�. (74)

Substituting (74) into (73) yields

PV .e.t/, QW .t//6� c1ke.t/k22C c2ke.t/k2C c3

D�

�pc1ke.t/k2 �

c2

2pc1

�2Cc224c1C c3, t > 0. (75)

Let # be given by (66) and recall that k QW .t/kF 6 Qwmax, t > 0. Now, for kek2 > # , it follows thatPV .e.t/, QW .t//6 0 for all .e.t/, QW .t// 2 De nDr and t > 0, where

De , f.e, QW / 2Rn �Rs�mg W x 2Rng, (76)

Dr , f.e, QW / 2Rn �Rs�mg W kek2 6 #g. (77)



Finally, define D˛ , f.e, QW / 2 Rn � Rs�mg W V.e, QW / 6 ˛g, where ˛ is the maximumvalue such that D˛ � De, and define Dˇ , f.e, QW / 2 Rn � Rs�mg W V.e, QW / 6 ˇg, whereˇ > O.�/D �2 D �max.P /#

2C �max.�1/ Qw2max.

To show ultimate boundedness of the closed-loop system (63) and (64), note that Dˇ � D˛because the approximation in (57) holds in Rn. Now, because PV .e.t/, QW .t// 6 0, t > 0, forall .e.t/, QW .t// 2 De n Dr and Dr � D˛ , it follows that D˛ is positively invariant. Hence, if.e.0/, QW .0// 2 D˛ , then it follows from Corollary 4.4 of [17] that the solution .e.t/, QW .t//,t > 0, to (63) and (64) is ultimately bounded with respect to .e, QW / with ultimate bound givenby O�1.ˇ/D

pˇ, which yields (65).

Finally, over the interval t 2 Œ0,T /, PV .e.t/, QW .t// 6 0, because .e.t/, QW .t// 2 De n Dr. Thisimplies that

V.e.t/, QW .t//6 V.e.0/, QW .0//, t 2 Œ0,T /. (78)

Using the inequalities �min.P /kek22 6 V.e, QW / and V.e.0/, QW .0// D tr QW T.0/�1 QW .0/ 6

k�1kF k QW .0/k2F in (78) gives (67). Similarly, using the inequalities �min.

�1/k QW k2F 6 V.e, QW /and V.e.0/, QW .0//6 k�1kFk QW .0/k2F in (78) gives (68). This completes the proof. �

Remark 4.4In Theorem 4.1, we assume that �min.R/ �

12kS

12 k2Fw

2l2ˇ1> 0. Because a conservative upper

bound w of the unknown constant weight W is known, this can be easily satisfied by judi-ciously choosing the design parameters. As noted in Remark 4.1, similar conditions are requiredto be satisfied in the design process for guaranteeing a robust adaptive controller design (see, forexample, [9]).

Remark 4.5Theorem 4.1 shows that over a transient finite-time T , the closed-loop error signals (63) and (64)are bounded from above by (67) and (68), respectively. This implies along with uniform ultimateboundedness of the closed-loop error signals (63) and (64) that e.�/ 2 L1 and vec QW .�/ 2 L1, andhence, x.�/ 2 L1 and u.�/ 2 L1. Furthermore, note that e.t/, t 2 Œ0,T /, can be made sufficientlysmall (satisfying N# in (66)) by letting �min./!1.

Remark 4.6Let M D mI in (60). Because

�AmP

�1CmI��1D�AmP

�1C �1mI��1

, it follows from

Lemma 2.2, with A , �1mI and B , AmP�1 that

�AmP

�1C �1mI��1D 0 as ! 0 or

m ! 1. That is, ‰�ˇ.x.t//, OW .t/

�D 0 as ! 0 or m ! 1. In this case, we obtain the

standard adaptive controller given by (61). Furthermore, it follows from (66) that # D 0, that is,we recover asymptotic stability of the system error dynamics. However, ! 0 or m!1 (resp.,�min.M/ ! 1) are not desirable design choices as they might lead to a high bandwidth adap-tive controller that can result in high-frequency oscillations in the closed-loop system response andexcite unmodeled dynamics resulting in system instability. This is discussed later.

Next, we relax Assumption 4.1 to the case where kˇ.x/k2 6 lˇ0C lˇ1kxk2 does not hold for allx 2Rn, and the system uncertainty �.x/ cannot be perfectly parameterized.

Assumption 4.2The matched uncertainty in (56) is linearly parameterized as

�.x/DW Tˇ.x/C �.x/, x 2Dx , (79)

where W 2 Rs�m is an unknown constant weighting matrix satisfying kW kF 6 w, w 2R is a known positive constant, ˇ W Dx ! Rs is a basis function of the form ˇ.x/ DŒˇ1.x/, ˇ2.x/, : : : , ˇs.x/� satisfying kˇ.x/k2 6 lˇ0Clˇ1kxk2, x 2Dx , where lˇ0 2R and lˇ1 2Rare positive constants, � W Dx ! Rm is the system modeling error satisfying k�.x/k2 6 �max,�max > 0, and Dx is a compact subset of Rn.



Remark 4.7Note that Assumption 4.2 does not assume boundedness of the basis function ˇ.x/, x 2 Dx , asis standard in the neuroadaptive control literature [20, 27–29]. Furthermore, in this case, a basisfunction constructed using neural networks is not the only choice for ˇ.x/ because the boundednessassumption on ˇ.x/, x 2Dx , that is, kˇ.x/k2 6 lˇ0, x 2Dx , is relaxed to kˇ.x/k2 6 lˇ0Clˇ1kxk2,x 2Dx .

Using the uncertainty parametrization given by (79), the system error dynamics are now given by

Pe.t/D Ame.t/�B QWT.t/ˇ.x.t//CB�.x.t//, e.0/D 0, t > 0, (80)

where QW .t/, t > 0, is given by (64).

Theorem 4.2Consider the nonlinear uncertain dynamical system given by (56) with reference system given by(18), and assume that Assumptions 3.1 and 4.2 hold. Furthermore, let the adaptive control law begiven by (58) with weight update law given by (59). Then, the closed-loop error signals given by(80) and (64) are uniformly bounded for all .e.0/, QW .0// 2 D˛ , where D˛ is a compact positivelyinvariant set, with ultimate bound ke.t/k2 < N", t > T , where

N" >��max.P / N#

2C �max.�1/ Qw2max

� 12 , (81)

N# ,�Nc224c21Cc3

c1

� 12

CNc2

2c1, (82)

c1 , �min.R/ �12kS

12 k2Fw

2l2ˇ1

> 0, Nc2 , kS12 k2Fw

2 Nlˇ0lˇ1 C 2kPBkF�max, c3 ,12kS

12 k2Fw

2l2ˇ0

, S , BT�AmP

�1CM��T

B , Nlˇ0 , lˇ0 C lˇ1x�m, and kxm.t/k 6 x�m,

t > 0. In addition, the system error and weight update error dynamics satisfy the bounds (67) and(68), respectively.

ProofConsider the Lyapunov-like function candidate given by (69). Differentiating (69) along theclosed-loop trajectories of (80) and (64), and after considerable algebraic manipulation, we obtain

PV .e.t/, QW .t//6� c1ke.t/k22C Nc2ke.t/k2C c3

D�

�pc1ke.t/k2 �

Nc2

2pc1

�2CNc224c1C c3, t > 0. (83)

Now, for kek2 > N# , it follows that PV .e.t/, QW .t// 6 0 for all .e.t/, QW .t// 2 De n Dr and t > 0,where

De , f.e, QW / 2Rn �Rs�mg W x 2Dx �Rng, (84)

Dr , f.e, QW / 2Rn �Rs�mg W kek2 6 N#g. (85)

Next, define D˛ , f.e, QW / 2 Rn � Rs�mg W V.e, QW / 6 ˛g, where ˛ is the maximum valuesuch that D˛ � De, and define Dˇ , f.e, QW / 2 Rn � Rs�mg W V.e, QW / 6 ˇg, whereˇ > O.�/D �2 D �max.P / N#

2C �max.�1/ Qw2max.

To show ultimate boundedness of the closed-loop system (80) and (64), assume‡ that Dˇ � D˛ .Now, as PV .e.t/, QW .t// 6 0, t > 0, for all .e.t/, QW .t// 2 De nDr and Dr � D˛ , it follows that D˛is positively invariant. Hence, if .e.0/, QW .0// 2 D˛ , then it follows from Corollary 4.4 of [17] that

‡This assumption ensures that in the error space De, there exists at least one Lyapunov level set Dˇ �D˛ . Equivalently,imposing bounds on the adaptation gains ensures Dˇ �D˛ [30]. In the case where the approximation holds in Rn, thisassumption is automatically satisfied. See Remark 4.8 for further details.



the solution .e.t/, QW .t//, t > 0, to (80) and (64) is ultimately bounded with respect to .e, QW / withultimate bound given by O�1.ˇ/D

pˇ, which yields (81). Finally, the bounds (67) and (68) on the

error dynamics follow as in the proof of Theorem 4.2. �

Remark 4.8As is common in the neural network literature, in the proof of Theorem 4.2, we assume that for agiven arbitrarily large compact set Dx � Rn, there exists an approximator for the unknown nonlin-ear map �.�/ up to a desired accuracy. This assumption ensures that in the error space De definedby (84), there exists at least one Lyapunov level set such that the set inclusions invoked in the proofof Theorem 4.2 are satisfied. In the case where�.�/ is continuous on Rn, it follows from the Stone–Weierstrass theorem [31] that �.�/ can be approximated over an arbitrarily large compact set Dx inthe sense of (79).

4.3. Transient and steady-state performance guarantees

In this section, we show that the governing tracking closed-loop system error equation approximatesa Hurwitz linear time-invariant dynamical system with L1 input–output signals. This allows us toderive uniform transient and steady-state performance bounds in terms of L1-norms of the closed-loop system error dynamics that are independent of the system adaptation rate. The following lemmais key in developing the results of this section.

Lemma 4.1Consider the weight update law given by (59). If M D�AmP

�1CmI , m> 0, then

B OW T.t/ˇ.x.t//D �1mPBPe.t/� �1mBLT

hˇL.x.t//

h�1

POW.t/� .t/iiT

, t > 0, (86)

where PB , B.BTB/CBT, BL , .BTB/CBT, ˇL.x.t// ,�ˇT.x.t//ˇ.x.t//

�Cˇ.x.t//T, and

.t/ 2Rs�m, t > 0, is a residual error signal.

ProofWriting (59) as

�1POW.t/DProjm

hOW .t/, ˇ.x.t//eT.t/PB � ˇ.x.t//ˇT.x.t// OW .t/BT

�AmP

�1

CM��T Bi

, t > 0, (87)

it follows that

0Dˇ.x.t//eT.t/PB � ˇ.x.t//ˇT.x.t// OW .t/BTŒAmP�1CM��TB � �1

POW.t/

C .t/, t > 0, (88)

where .t/, t > 0, is a residual error, as ProjmŒ OW .t/,X.t/� is not necessarily equal to X.t/ for allt > 0. Equivalently, (88) can be written as

0DBTPe.t/ˇT.x.t//� BTŒAmP�1CM��1B OW T.t/ˇ.x.t//ˇT.x.t//�

POW T.t/�1

C T.t/, t > 0. (89)

Post-multiplying (89) by ˇ.x/�ˇT.x/ˇ.x/

�Cand setting M D�AmP

�1CmI yields

0DBTPe.t/� m�1BTB OW T.t/ˇ.x.t//�POW T.T /�1ˇ.x.t//ˇs.x.t//

C T.t/ˇ.x.t//ˇs.x.t//, t > 0, (90)



where ˇs.x/,�ˇT.x/ˇ.x/

�C. Next, pre-multiplying (90) by BLT

yields

0DBLTBTPe.t/� m�1BLT

BTB OW T.t/ˇ.x.t//�BLT POW T.T /�1ˇ.x.t//ˇs.x.t//

CBLT T.t/ˇ.x.t//ˇs.x.t//

DPBPe.t/� m�1B OW T.t/ˇ.x.t//�BLT

h�1

POW.t/� .t/iTˇ.x.t//ˇs.x.t//, t > 0,

(91)

which can be equivalently written as

0D �1mPBPe.t/�B OWT.t/ˇ.x.t//� �1mBLT

hˇL.x.t//

h�1

POW.t/� .t/iiT

, t > 0.

(92)Hence, (86) holds. �

Remark 4.9The residual error .t/ 2 Rs�m, t > 0 in (86) satisfies k .t/kF 6 max, t > 0, as all the signalsin (86) are bounded by Theorem 4.1 (resp., Theorem 4.2). Furthermore, .t/, t > 0, is identicallyzero when OWij .t/ 2 .� Owmax, Owmax/, for all i D 1, 2, : : : , s and j D 1, 2, : : : ,m, where Owmax is theprojection bound of OW .t/, t > 0. Hence, by choosing a conservative projection norm bound Owmax, .t/D 0 for all t > 0.

Proposition 4.1Consider the system error dynamics given by (63) (resp., (80)) and let M D �AmP

�1 C mI ,m> 0. Furthermore, assume Assumption 4.1 (resp., Assumption 4.2) holds. Then, the system errordynamics (63) (resp., (80)) can be characterized by the linear, time-invariant system Ge,w given by

Pe.t/D QAe.t/C QBw.t/, e.0/D e0, t > 0, (93)

´.t/D QCe.t/, (94)

where QA , Am � �1mPBP is Hurwitz, QB , B , QC , I , w.t/ , �1m.BTB/C

�ˇL.x.t//�h

�1POW.t/� .t/

iiTCW Tˇ.x.t//

resp.,w.t/, �1m.BTB/C

hˇL.x.t//

h�1

POW.t/� .t/iiT

CW Tˇ.x.t//C �.x.t//

, and w.�/ 2 L1.

ProofFirst, assume that Assumption 4.1 holds. It follows from Lemma 4.1 that (63) can be rewritten as

Pe.t/D Ame.t/�B QWT.t/ˇ.x.t//

D Ame.t/�B OWT.t/ˇ.x.t//CBW Tˇ.x.t//

D Ame.t/� �1mPBPe.t/C

�1mBLThˇL.x.t//

h�1

POW.t/� .t/iiTCBW Tˇ.x.t//

D�Am �

�1mPBP�e.t/CB

��1m.BTB/C

hˇL.x.t//

h�1

POW.t/� .t/iiT

CW Tˇ.x.t//

�

D QAe.t/C QBw.t/, e.0/D e0, t > 0, (95)

which gives (93). Defining the system output signal as ´.t/ , e.t/ gives (94). Next, lettingK D �1mPB > 0, it follows from Lemma 2.4 that QA D Am �

�1mPBP is Hurwitz. Finally,

as the signals in the right-hand side of (59) belong to L1, vec POW.�/ 2 L1. Hence, it follows from



vec POW.�/ 2 L1, .�/ 2 L1, and ˇ.x.�// 2 L1 that w.�/ 2 L1. The proof for the case whereAssumption 4.2 holds in place of Assumption 4.1 is identical. �

Note that (93) and (94) characterizes the closed-loop linear time-invariant dynamical system.Because w.�/ 2 L1 and QA is Hurwitz, it follows that ´.�/ 2 L1. Hence, the following theoremis immediate.

Theorem 4.3Consider the linear, time-invariant system Ge,w given by (93) and (94). Let ˛ > 0 be such thatQAC ˛

2I is Hurwitz and letQ˛ 2Rn�n be the unique, nonnegative definite solution to the Lyapunov

equation (50). Then, Ge,w W L1! L1 and (51) holds, and hence, jjjejjj1,2 6 1p˛�1=2max .Q˛/jjjwjjj1,2.

ProofThis is a restatement of Theorem 2.1 for Ge,w given by (93) and (94). �

The following proposition discusses the special case when �min./ ! 1 and OWij .t/ 2.� Owmax, Owmax/ for all i D 1, 2, : : : , s and j D 1, 2, : : : ,m, where Owmax is the projection boundof OW .t/, t > 0. In this case, �.x.t//, t > 0, in (57) (resp., in (79)) is directly related to ´.t/, t > 0,in (94), through Ge,w .

Proposition 4.2Consider the system error dynamics given by (63), and let M D �AmP

�1 C mI , m > 0, and D �� , � > 0, � > 0. Furthermore, assume Assumption 4.1 (resp., Assumption 4.2) holds

and POW.t/, t > 0 is bounded as � ! 1. If � ! 1, then, the system error dynamics (63) (resp.(80)) can be characterized by the linear, time-invariant system Ge,w given by (93) and (94), whereQA , Am �

�1mPBP is Hurwitz, QB , B , QC , I ,w.t/ , �1m.BTB/C��ˇL.x.t// .t/

�TC

W Tˇ.x.t//�

resp.,w.t/, �1m.BTB/C��ˇL.x.t// .t/

�TCW Tˇ.x.t//C�.x.t//

�, andw.�/ 2

L1. If, in addition, OWij .t/ 2 .� Owmax, Owmax/, for all i D 1, 2, : : : , s and j D 1, 2, : : : ,m, where Owmax

is the projection bound of OW .t/, t > 0, then w.t/ , �.x.t//, where �.x.t//, t > 0, is given by(57) (resp., (79)), and w.�/ 2 L1.

ProofThe proof is a direct consequence of Proposition 4.1. �

Remark 4.10Proposition 4.2 characterizes a closed-loop linear time-invariant dynamical system from the systeminput �.x.t//, t > 0, given by (57) (resp., (79)), to the system output e.t/, t > 0. Note that for bothresults, we assume the time derivative of OW .t/, t > 0, remains bounded as � !1. This assump-tion is not restrictive because the modification term given by (60) can be viewed as a variable andweighted damping term because of its sign definiteness and dependence on OW .t/, t > 0, and hence,it limits the frequency content of the weight update law given by (59) resulting in a bounded timederivative of OW .t/, t > 0. This observation holds even in the face of infinitely large adaptation gains.

Remark 4.11By choosing a conservative projection norm bound Owmax (see Remark 4.9) and letting �min./!1,the closed-loop linear time-invariant dynamical system given by (93) and (94) characterizes a map-ping when �.x.t//, t > 0, given by (57) (resp., (79)) is viewed as a system input and e.t/, t > 0,is viewed as a system output. Hence, closed-loop system properties such as system bandwidth,transient system performance, and L1 system norm of Ge,w (Theorem 4.3) can be analyzed using(93) and (94).



As noted in Remark 4.6, ! 0 or m ! 1 (resp., �min.M/ ! 1) are not always desirabledesign choices. This is because in this case, Ge,w has infinitely fast decaying poles in the left-halfplane, and hence, Ge,w behaves similar to a high-bandwidth filter, which can be very sensitive tovariations in w.t/ causing high-frequency oscillations in the system response. Hence, by varyingthe parameters and M , we can guarantee a prescribed closed-loop system bandwidth.

4.4. Connections to gradient minimization

Finally, we show that the proposed modification term (60) in the adaptive update law (59) can bederived from a minimization problem involving an error criterion capturing the distance betweenthe weighted regressor vector and the weighted error states. Specifically, consider the cost functiongiven by

J .e, OW /D kPe �B OW Tˇ.x/k22 (96)

and note that the gradient of (96) with respect to OW is given by

@J .e.t/, OW .t//@ OW .t/

D�ˇ.x.t//eT.t/PB C ˇ.x.t//ˇT.x.t// OW .t/BTB , t > 0. (97)

Now, constructing the weight update law to be the negative gradient of (96), as in [25, 32–35],we obtain

POW.t/D�@J .e.t/, OW .t//

@ OW .t/, OW .0/D OW0, t > 0, (98)

where > 0. Hence, choosing sufficiently large it follows that (96) is minimized. If

M D�AmP�1C I , (99)

then (59) specializes to (98). If M in (99) is positive definite, then (59), with M given by (99), canbe viewed as a gradient optimization based weight update law that minimizes (96).

As noted in Remarks 3.3 and 4.6, the controller architectures given by (21) and (22) for distur-bance rejection and (58) and (59) for uncertainty suppression involve a modification term in theadaptive law with design parameters that can be easily tuned to guarantee a desired bandwidth, tran-sient and steady-state performance, as well as an a priori characterization of the size of the ultimatebound of the closed-loop system. In contrast, this architecture is significantly different from the L1adaptive control architecture of [9, 10], which requires additional states to be propagated and storedowing to the inclusion of a predictor system in the design architecture. In addition, the L1 adaptivecontroller of [9, 10] can sometimes be difficult to tune and analyze because the controller archi-tecture is set up so that a predictor system rather than the actual system is being tracked. Finally,the selection of the design parameters of the low-pass filter required in designing the L1 adaptivecontroller in [9, 10] can effect system stability as well as the allowable system uncertainty. Theaforementioned factors add additional design complexity to the L1 adaptive controller of [9, 10] ascompared with the proposed approach.

5. ILLUSTRATIVE NUMERICAL EXAMPLES

In this section, we present two numerical examples to demonstrate the utility and efficacy of theproposed robust adaptive control architecture.

5.1. Linear uncertainty suppression

First, we present a numerical example to illustrate the proposed adaptive controller design based onTheorem 4.1. Specifically, consider the linear dynamical system representing an uncertain aircraftrolling dynamics model given by



�Px1.t/Px2.t/

�D

�0 1

0 0

� �x1.t/

x2.t/

�C

�0

1

��.x.t//C

�0

1

�u.t/,

�x1.0/

x2.0/

�D

�0

0

�, t > 0,

(100)where x1 represents the roll angle in radians, x2 represents the roll rate in radians per second, and�.x/ D Œ0.2314, 0.7848�T. For our simulation, we choose K1 D Œ0.25, 0.80� and K2 D 0.25,and a reference system corresponding to a natural frequency !n D 0.50 rad/s and a damping ratio� D 0.80. Note that Assumptions 3.1 and 4.1 hold. We set ˇ.x/ D x, R D I2, D 1, M D 4I2, D 100000I2, and Owmax D 2 (a conservative choice for projection norm bound). It follows fromTheorem 4.3 that jjjGe,w jjj.1,2/,.1,2/ 6 0.35 for ˛ D 2.85. Here, our aim is to track a given filteredsquare-wave roll angle reference command r.t/, t > 0.

Figure 9 shows that the nominal closed-loop system is very lightly damped without adaptation(uad.t/ � 0). Figures 10 and 11 show the closed-loop system performance of a standard adap-tive controller given by (61) with an adaptation gain of D 100I2 and the same projection normbound for the two different reference command profiles having a maximum amplitude of 10ı and50ı, respectively. It can be seen from these two figures that the system response is oscillatory, andthe controller performance significantly changes when we change the reference command profile.Figures 12 and 13 show the closed-loop system performance of the proposed adaptive controllergiven by (59) for the two different reference command profiles having a maximum amplitude of 10ı

and 50ı, respectively. Note that the tracking performance of the proposed controller is clearly supe-rior as compared with the standard adaptive controller. Furthermore, the proposed controller perfor-mance does not change when we change the reference command profile. In contrast, the standardadaptive controller performance changes significantly, leading to high frequency oscillations.

This system performance is expected because the proposed robust adaptive controller does nothave a high bandwidth in the face of high-gain control. This implies that the proposed adaptive con-troller possesses robustness margins against input time-delays because its bandwidth is limited. Tosee this, we inserted a time-delay to the control input and determined the achieved time-delay mar-gin of the uncertain aircraft rolling dynamics model with both the standard and proposed adaptivecontrollers. Figure 14 shows that the proposed adaptive controller’s achieved time-delay margin isbounded away from zero and goes to 0.12 s as we increase the system adaptation gain. In contrast,the standard adaptive controller’s achieved time-delay margin approaches zero as we increase the

0 50 100 150 200 250−600

−400

−200

0

200

400

600

Time [s]

Time [s]

r(t) [deg]

x (t) [deg]

x (t) [deg/s]

0 50 100 150 200 250

−150

−100

−50

0

50

100

150

u (t) [deg]

u (t) [deg]

Figure 9. Closed-loop system response of the uncertain aircraft rolling dynamics model with uad.t/ � 0,t > 0.



0 10 20 30 40 50−10

−5

0

5

10

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50−6

−4

−2

0

2

4

Time [s]

un(t) [deg]

uad

(t) [deg]

Figure 10. Closed-loop system response of the uncertain aircraft rolling dynamics model with the standardadaptive controller for a reference command having a maximum amplitude of 10 ((61) with D 100I2).

0 10 20 30 40 50

−60

−40

−20

0

20

40

60

80

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [5 × deg/s]

0 10 20 30 40 50

−20

−10

0

10

20

Time [s]

un(t) [deg]

uad

(t) [deg]

Figure 11. Closed-loop system response of the uncertain aircraft rolling dynamics model with the standardadaptive controller for a reference command having a maximum amplitude of 50 ((61) with D 100I2).

system adaptation gain. In future research, we will investigate the guaranteed robustness propertiesof the proposed adaptive controller against input time-delays.

5.2. Nonlinear uncertainty suppression

Next, we present a numerical example to illustrate the proposed adaptive controller design based onTheorem 4.2. Specifically, consider the nonlinear dynamical system representing a controlled wing



0 10 20 30 40 50−10

−5

0

5

10

Time [s]

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50

−2

−1

0

1

2 un(t) [deg]

uad

(t) [deg]

Figure 12. Closed-loop system response of the uncertain aircraft rolling dynamics model with the proposedadaptive controller for a reference command having a maximum amplitude of 10 ((59) with D 100000I2,

D 1, and M D 4I2).

0 10 20 30 40 50−50

0

50

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50

−10

−5

0

5

10

Time [s]

un(t) [deg]

uad

(t) [deg]

Figure 13. Closed-loop system response of the uncertain aircraft rolling dynamics model with the proposedadaptive controller for a reference command having a maximum amplitude of 50 ((59) with D 100000I2,

D 1, and M D 4I2).

rock aircraft dynamics model given by (100) with �.x/D ˛1x1C ˛2x2C ˛3jx1jx2C ˛4jx2jx2C˛5x

31 , where ˛i , i D 1, : : : , 5, are unknown parameters that are derived from the aircraft aerody-

namic coefficients. For our simulation, we set ˛1 D 0.2314, ˛2 D 0.7848, ˛3 D �0.0624, ˛4 D0.0095, and ˛5 D 0.0215 [36]. We use the same reference system as given in the previous example.



100 101 102 103 104 105 106

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Ach

ieve

d T

ime

Del

ay M

argi

n

Standard Adaptive ControllerProposed Adaptive Controller

Figure 14. Achieved time-delay margin of the uncertain aircraft rolling dynamics model with the standardand proposed adaptive controllers for a reference command having a maximum amplitude of 10.

0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 1 2 3 4 5 6 7 8 9

−30

−25

−20

−15

−10

−5

0

Time [s]

un(t) [deg]

uad

(t) [deg]

Figure 15. Closed-loop system response of wing rock aircraft dynamics model with uad.t/� 0, t > 0.

In addition, we use sigmoidal type basis functions of the form ˇ.x/ Dh1, 1

1Ce�x1, 11Ce�x2

iTso

that Assumption 4.2 is satisfied. Furthermore, we set R D I2, D 1, M D 4I2, D 100000I2,and Owmax D 2. Once again, it follows from Theorem 4.3 that jjjGe,w jjj.1,2/,.1,2/ 6 0.35 for ˛ D 2.85.Here, our aim is to track a given filtered square-wave roll angle reference command r.t/, t > 0.

Figure 15 shows that the nominal closed-loop system is unstable without adaptation (uad.t/� 0,t > 0). Figure 16 shows the closed-loop system performance of a standard adaptive controller given



0 10 20 30 40 50−10

−5

0

5

10

Time [s]

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50

−150

−100

−50

0

50

100

150 un(t) [deg]

uad

(t) [deg]

Figure 16. Closed-loop system response of wing rock aircraft dynamics model with the standard adaptivecontroller ((61) with D 100I2).

0 10 20 30 40 50−10

−5

0

5

10

Time [s]

Time [s]

r(t) [deg]x

1(t) [deg]

x2(t) [deg/s]

0 10 20 30 40 50

−2

−1

0

1

2 un(t) [deg]

uad

(t) [deg]

Figure 17. Closed-loop system response of wing rock aircraft dynamics model with the proposed adaptivecontroller ((59) with D 100000I2, D 1, and M D 4I2).

by (61) with an adaptation gain of D 100I2 and the same projection norm bound. The closed-loopsystem performance presented in this figure is clearly not acceptable. Figure 17 shows the closed-loop system performance of the proposed adaptive controller given by (59). Once again, the trackingperformance of the proposed controller is clearly superior as compared with the standard adaptivecontroller. Note that the controller performance depicted in this figure is identical to the controllerperformance depicted in Figure 12, even though we use a different set of basis functions and weapply the proposed controller to a system having nonlinear uncertainty.



6. CONCLUSION

It is well known that standard model reference adaptive control methods employ high-gain feed-back to achieve fast adaptation in order to rapidly reduce system tracking errors in the face of largesystem uncertainties. High-gain feedback, however, leads to increased controller effort and reducedrelative stability margins, and can excite unmodeled dynamics and drive the system to instability. Inthis paper, we presented a new robust adaptive control architecture that allows for fast adaptation ofnonlinear uncertain dynamical systems while guaranteeing transient and steady-state performancebounds. The proposed architecture allows for a trade-off between tracking performance and stabil-ity robustness providing a systematic framework for verification and validation of adaptive controlsystems. Future research will include analyzing relative stability gain and time delay margins andextensions to output feedback control for minimum and non-minimum phase uncertain systems.

ACKNOWLEDGEMENT

This research was supported in part by the Air Force Office of Scientific Research under Grant FA9550-09-1-0429.

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