L1 Adaptive Output-Feedback Controller forNon-Strictly-Positive-Real Reference Systems:

Missile Longitudinal Autopilot Design

Chengyu Cao∗

University of Connecticut, Storrs, Connecticut 06269

and

Naira Hovakimyan†

University of Illinois at Urbana–Champaign, Urbana, Illinois 61801

DOI: 10.2514/1.40877

This paper presents an extension of the L1 adaptive output-feedback controller to systems of unknown relative

degree in the presence of time-varying uncertainties without restricting the rate of their variation. As compared with

earlier results in this direction, a new piecewise continuous adaptive law is introduced, along with the low-pass-

filtered control signal that allows for achieving arbitrarily close tracking of the input and the output signals of the

reference system, the transfer function of which is not required to be strictly positive real. Stability of this reference

system is proved using a small-gain-type argument. The performance bounds between the closed-loop reference

system and the closed-loop L1 adaptive system can be rendered arbitrarily small by reducing the step size of

integration. Missile longitudinal autopilot design is used as an example to illustrate the theoretical results.

I. Introduction

M ODERN fighter aircraft and munitions need to operate inhighly nonlinear and uncertain flight regimes, for which

accurate aerodynamic modeling is not possible or is overlyexpensive. The control design challenge for these vehicles is toensure safe operation in the presence of such uncertainties, withoutsacrificing the maneuverability of the vehicle. Both classical andmodern control design methods have been extensively investigatedfor performance improvement of such vehicles in the presenceof uncertainties. Their performance is well known to depend uponaccurate knowledge of the system dynamics and is limited bythe presence of unmodeled high-frequency effects. The same isparticularly true for adaptive methods that lose their robustnessin the presence of fast adaptation [1–4]. The recently developed L1

adaptive controller [5–7] opened new opportunities for addressingthe control challenge for highly uncertain vehicles in the presenceof component failures, aerodynamic uncertainties, and disturban-ces. It is important at this point to emphasize the significance ofoutput-feedback architectures, which have the potential to relax thematching assumption, typically appearing in state-feedback adaptivecontrol architectures. One of the fundamental limitations of adaptiveoutput-feedback approaches is the requirement for the reference-system dynamics to have a strictly-positive-real (SPR) transferfunction between the input and the regulated output [8]. When theSPR condition does not hold, then the obtained results lead toultimate boundedness, which are hard to quantify rigorously and/orto reduce in desirable way [9,10]. On the other hand, the SPRrequirement for the reference system restricts the class of reference

behaviors that can be achieved by adaptation for the given unknownplant [6,11].

This paper extends the results of [6] to an output-feedbackframework by considering reference systems that do not verify theSPR condition for their input–output transfer function. The keydifference from the results in [6] is the new piecewise continuousadaptive law. The adaptive control is defined as the output of a low-pass filter, resulting in a continuous signal despite the discontinuityof the adaptive law. Similar to [6], theL1-norms of both input/outputerror signals between the closed-loop adaptive system and thereference system can be rendered arbitrarily small by reducing thestep size of integration.

We note that adaptive algorithms achieving arbitrarily improvedtransient performance for a system’s output were reported in [12–26]. As compared with those results, [5,6] presented the opportunityto also regulate the performance bound for a system’s input signal byrendering it arbitrarily close to the corresponding signal of a boundedlinear time-invariant reference system. Unlike conventional adaptivecontrollers, the L1 adaptive controllers adapt fast, leading to desiredtransient and asymptotic tracking with a guaranteed, bounded awayfrom zero, time-delay margin [27]. The results from [5,6] have beenintensively applied in flight tests [11,28–31] and various mid- tohigh-fidelity simulation environments [32–35]. Insights into theperformance of L1 adaptive controller can be obtained from theanalysis of a simple linear system in [36], in which sensitivity andcosensitivity transfer functions are analyzed for disturbance rejec-tion and noise tolerance in the presence of a large adaptation rate.Further, the results in [37] provide systematic design guidelines forselection of the underlying filter to achieve the desired performancebound while retaining a guaranteed time-delay margin.

This paper presents the adaptive output-feedback counterpartof the results in [5,6], without enforcing the SPR condition on theinput/output transfer function of the desired reference system,which typically appears in conventional adaptive output-feedbackschemes. The paper also uses the longitudinal dynamics of themissile from [38] to demonstrate the usefulness of the methodology.Different applications of this methodology can be found in [39,40].

The paper is organized as follows. Section II gives the problemformulation. In Sec. III, the closed-loop reference system is intro-duced. In Sec. IV, some preliminary results are developed towardthe definition of the L1 adaptive controller. In Sec. V, the novel L1

adaptive control architecture is presented. Stability and uniformperformance bounds are presented in Sec. VI. In Sec. VII, simulation

Presented as Paper 7288 at the AIAA Guidance, Navigation, and ControlConference, Honolulu, HI, 18–21 August 2008; received 8 September 2008;revision received 24 November 2008; accepted for publication 22 December2008. Copyright © 2009 by Chengyu Cao and Naira Hovakimyan. Publishedby the American Institute of Aeronautics and Astronautics, Inc., withpermission. Copies of this paper may be made for personal or internal use, oncondition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0731-5090/09 $10.00 in correspondence with the CCC.

∗Assistant Professor, Department of Mechanical Engineering; ccao@engr.uconn.edu. Member AIAA.

†Professor, Department of Mechanical Science and Engineering;nhovakim@illinois.edu. Associate Fellow AIAA.

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 32, No. 3, May–June 2009

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results are presented, and Sec. VIII concludes the paper. The small-gain theorem and some basic definitions from linear systems theoryused throughout the paper are given in the Appendix. Unlessotherwise mentioned, k � k will be used for the 2-norm of the vector.

II. Problem Formulation

Consider the following single-input/single-output (SISO) system:

y�s� � A�s��u�s� � d�s��; y�0� � 0 (1)

where u�t� 2 R is the input; y�t� 2 R is the system output; A�s� is astrictly proper unknown transfer function of unknown relative degreenr, for which only a known lower bound 1< dr � nr is available;d�s� is the Laplace transform of the time-varying uncertainties anddisturbances d�t� � f�t; y�t�� and f is an unknown map, subject tothe following assumptions.

Assumption 1. There exist constants L > 0 and L0 > 0 such that

jf�t; y1� � f�t; y2�j � Ljy1 � y2j; jf�t; y�j � Ljyj � L0

hold uniformly in t � 0, where the numbers L and L0 can bearbitrarily large.

Assumption 2. There exist constants L1 > 0, L2 > 0, and L3 > 0such that

j _d�t�j � L1j _y�t�j � L2jy�t�j � L3

for all t � 0, where the numbers L1, L2, and L3 can be arbitrarilylarge.

Let r�t� be a given bounded continuous reference input signal. Thecontrol objective is to design an adaptive output-feedback controlleru�t� such that the system output y�t� tracks the reference input r�t�following a desired reference model: that is,

y�s� M�s�r�s�

whereM�s� is a minimum-phase stable transfer function of relativedegree dr. We rewrite the system in Eq. (1) as

y�s� �M�s��u�s� � ��s��; y�0� � 0 (2)

��s� � ��A�s� �M�s��u�s� � A�s�d�s��=M�s� (3)

Let �Am; bm; cm� be the minimal realization of M�s�; that is, it iscontrollable and observable andAm is Hurwitz. The system in Eq. (2)can be rewritten as

_x�t� � Amx�t� � bm�u�t� � ��t��y�t� � c>mx�t�; x�0� � x0 � 0 (4)

III. Closed-Loop Reference System

Consider the following closed-loop reference system:

yref�s� �M�s��uref�s� � �ref�s�� (5)

�ref�s� ��A�s� �M�s��uref�s� � A�s�dref�s�

M�s� (6)

uref�s� � C�s��r�s� � �ref�s�� (7)

where dref�t� � f�t; yref�t��, and C�s� is a strictly proper system ofrelative order dr, with its dc gain C�0� � 1. Further, the selection ofC�s� andM�s� must ensure that

H�s� � A�s�M�s�=�C�s�A�s� � �1 � C�s��M�s�� (8)

is stable and

kG�s�kL1L < 1 (9)

where G�s� �H�s��1 � C�s��. This in turn restricts the class ofsystems A�s� in Eq. (1), for which the proposed approach in thispaper can achieve stabilization. However, as discussed later inRemark 3, the class of such systems is not empty. Letting

A�s� � An�s�Ad�s�

; C�s� � Cn�s�Cd�s�

; M�s� �Mn�s�Md�s�

(10)

where the numerators and the denominators are all polynomials of s.It follows from Eq. (8) that

H�s� � Cd�s�Mn�s�An�s�Hd�s�

(11)

where

Hd�s� � Cn�s�An�s�Md�s� �Mn�s�Ad�s��Cd�s� � Cn�s�� (12)

A strictly proper C�s� implies that the order of Cd�s� � Cn�s� andCd�s� is the same. Because the order of Ad�s� is higher thanthat of An�s�, the transfer function H�s� is strictly proper.

Lemma 1. If C�s� and M�s� verify the condition in Eq. (9), theclosed-loop reference system in Eqs. (5–7) is stable.

Proof. It follows from Eqs. (6) and (7) that

uref�s� �C�s�M�s�r�s� � C�s�A�s�dref�s�C�s�A�s� � �1 � C�s��M�s� (13)

From Eqs. (5) and (6), one can derive

yref�s� �H�s��C�s�r�s� � �1 � C�s��dref�s�� (14)

Because H�s� is strictly proper and stable,

G�s� �H�s��1 � C�s��

is also strictly proper and stable; therefore,

kyrefkL1 � kH�s�C�s�kL1krkL1 � kG�s�kL1

�LkyrefkL1 � L0�

Using the condition in Eq. (9), one can write

kyrefkL1 � �; ��kH�s�C�s�kL1

krkL1 � kG�s�kL1L0

1 � kG�s�kL1L

<1

(15)

Hence, kyrefkL1 is bounded, and the proof is complete. □

IV. Preliminaries for the Main Result

Let

H0�s� � A�s�=�C�s�A�s� � �1 � C�s��M�s�� (16)

H1�s����A�s��M�s��C�s��=�C�s�A�s���1�C�s��M�s�� (17)

H2�s� �H�s�C�s�=M�s� (18)

H3�s� � ��M�s�C�s��=��C�s�A�s� � �1 � C�s��M�s��� (19)

Using the expressions from Eqs. (10) and (12),

H0�s� � Cd�s�An�s�Md�s�=Hd�s�

and H1�s� can be rewritten as

H1�s� � �Cn�s�An�s�Md�s� � Cn�s�Ad�s�Mn�s��=Hd�s� (20)

Because the degree of Cd�s� � Cn�s� is larger than Cn�s� by dr,the degree of

Mn�s�Ad�s��Cd�s� � Cn�s��

is larger than

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Cn�s�Ad�s�Mn�s�

by dr. Because the degree of Ad�s� is larger than An�s� by � drand the degree ofMn�s� is larger thanMd�s� by dr, the degree of

Mn�s�Ad�s��Cd�s� � Cn�s��

is larger than that of

Cn�s�An�s�Md�s�

Therefore, H1�s� is strictly proper with relative degree dr. Wenote from Eqs. (11) and (20) that H1�s� has the same denominatoras H�s�, and it therefore follows from Eq. (9) that H1�s� is stable.Using similar arguments, it can be verified that H0�s� is proper andstable. Similarly, H2�s� is strictly proper and stable. Let

�� kH1�s�kL1krkL1 � kH0�s�kL1

�L�� L0�

� ��

�kH1�s�=M�s�kL1

� LkH0�s�kL1

kH2�s�kL1

1 � kG�s�kL1L

�(21)

where �� > 0 is an arbitrary constant. Because both H1�s� andM�s� are stable and strictly proper with relative degree dr andM�s�is minimum-phase, H1�s�=M�s� is stable and proper. Hence,kH1�s�=M�s�kL1

exists. Therefore, � is bounded. Further, becauseAm is Hurwitz, there existsP� P> > 0, which satisfies the algebraicLyapunov equation:

A>mP� PAm ��Q; Q > 0

From the properties of P, it follows that there exits nonsingular����Pp

such that

P� �����Pp�>

����Pp

Given the vector c>m�����Pp��1, let D be a �n � 1� n matrix that

contains the null space of c>m�����Pp��1:

D�c>m�����Pp��1�> � 0 (22)

and further let

�� c>mD

����Pp

� �

Lemma 2. For any

�� yz

� �2 Rn

where y 2 R and z 2 Rn�1, there exist p1 > 0 and positive definiteP2 2 R�n�1��n�1� such that

�>���1�>P��1�� p1y2 � z>P2z

Proof. Using P� �����Pp�>

����Pp

, one can write

�>���1�>P��1�� �>�����Pp

��1�>�����Pp

��1��

We note that

������Pp��1 � c>m�

����Pp��1

D

� �

Let

q1 � �c>m�����Pp��1��c>m�

����Pp��1�>; Q2 �DD>

From Eq. (22), we have

�������Pp��1����

����Pp��1�> � q1 0

0 Q2

� �

Nonsingularity of � and����Pp

implies that

�������Pp��1����

����Pp��1�>

is nonsingular, and therefore Q2 is also nonsingular. Hence,� ����Pp

��1�>� ����

Pp

��1�����

����Pp��1��

������Pp��1�>

��1����

����Pp��1��>� ����

Pp

��1��

q�11 0

0 Q�12

" #

Denoting p1 � q�11 and P2 �Q�12 completes the proof. □

Let T be any positive constant, which can be associated with thesampling rate of the available CPU, and let 11 2 Rn be the basisvector with the first element equal to 1 and all other elements equal to0. Let ��T� 2 Rn�1 be a vector, which consists of 2 to n elements of1>1 exp��Am��1T�, and let

��T� �ZT

0

j1>1 exp��Am��1�T � ����bmjd� (23)

Further, let

&�T� � k��T�k�������������������

�

max�P2�

r� ��T��

�� max���>P��1��

2�k��>Pbmkmin���>Q��1�

�2

(24)

Letting

1 >1 exp��Am��1t� � �1�t�>2 �t��

where 1�t� 2 R and 2�t� 2 Rn�1 contain the first and 2 to nelements of the row vector 1>1 exp��Am��1t�, respectively, weintroduce the following functions:

�1�T� � maxt2�0;T�j1�t�j; �2�T� � max

t2�0;T�k2�t�k (25)

Further, let ��T� be the n n matrix:

��T� �ZT

0

exp��Am��1�T � ����d� (26)

�3�T� � maxt2�0;T�

3�t�; �4�T� � maxt2�0;T�

4�t� (27)

where

3�t� �Zt

0

j1>1 exp��Am��1�t

� ������1�T� exp��Am��1T�11jd�;

4�t� �Zt

0

j1>1 exp��Am��1�t � ����bmjd�

Finally, let

�0�T� � �1�T�&�T� � �2�T��������������������

�

max�P2�

r� �3�T�&�T� � �4�T��

(28)

Lemma 3. The following limiting relationship is true:

limT!0

�0�T� � 0

Proof. Note that because �1�T�, �3�T��, and � are bounded, it issufficient to prove that

limT!0

&�T� � 0 (29)

limT!0

�2�T� � 0 (30)

limT!0

�4�T� � 0 (31)

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Because

limT!0

1>1 exp��Am��1T� � 1>1

then

limT!0

��T� � 0n�1

which implies

limT!0k��T�k � 0

Further, it follows from the definition of ��T� in Eq. (23) that

limT!0

��T� � 0

Because � and � are bounded,

limT!0

&�T� � 0

which proves Eq. (29). Because 2�t� is continuous, it followsfrom Eq. (25) that

limT!0

�2�T� � limt!0k2�t�k

Because

limt!0

1>1 exp��Am��1t� � 1>1

we have

limt!0k2�t�k � 0

which proves Eq. (30). Similarly,

limT!0

�4�T� � limt!0k4�t�k � 0

which proves Eq. (31). The boundedness of �, �3�T�, and� implies

limT!0

��1�T�&�T� � �2�T�

��������������������

max�P2�

r� �3�T�&�T�

� �4�T���� 0

which completes the proof. □

V. L1 Adaptive Output-Feedback Controller

We consider the following state predictor (or passive identifier):

_x�t� � Amx�t� � bmu�t� � ��t�y�t� � c>mx�t�; x�0� � x0 � 0 (32)

where ��t� 2 Rn is the vector of adaptive parameters. Note thatalthough ��t� 2 R in Eq. (4) (i.e., the unknown disturbance) ismatched, the uncertainty estimation in Eq. (32), ��t� 2 Rn, isunmatched. This is the key step of the solution and the subsequentanalysis.

Letting ~y�t� � y�t� � y�t�, the update law for ��t� is given by

��t� � ��iT�; t 2 �iT; �i� 1�T���iT� � ���1�T���iT�; i� 0; 1; 2; . . .

(33)

where ��T� is defined in Eq. (26) and

��iT� � exp��Am��1T�11 ~y�iT�; i� 0; 1; 2; 3; . . . (34)

The control signal is defined as follows:

u�s� � C�s�r�s� � C�s�c>m�sI � Am��1bm

c>m�sI � Am��1��s� (35)

whereC�s�wasfirst introduced in Eq. (7). TheL1 adaptive controllerconsists of Eqs. (32), (33), and (35), subject to the condition inEq. (9).

We will now proceed with the computation of error bounds. Let~x�t� � x�t� � x�t�. The error dynamics between Eqs. (4) and (32) are

_~x�t� � Am ~x�t� � ��t� � bm��t�~y�t� � c>m ~x�t�; ~x�0� � 0 (36)

Lemma 4. Let e�t� � y�t� � yref�t�. Then

ketkL1 �kH2�s�kL1

1 � kG�s�kL1Lk ~ytkL1 (37)

Proof. Let

~��s� � C�s�c>m�sI� Am��1bm

c>m�sI� Am��1��s� � C�s���s� (38)

It follows from Eq. (35) that

u�s� � C�s�r�s� � C�s���s� � ~��s� (39)

and the system in Eq. (2) consequently takes the form

y�s� �M�s��C�s�r�s� � �1 � C�s����s� � ~��s�� (40)

Substituting u�s� from Eq. (39) into Eq. (3) gives

��s� � ��A�s� �M�s���C�s�r�s�� C�s���s� � ~��s�� � A�s�d�s��=M�s�

and hence,

��s� � �A�s� �M�s���C�s�r�s� � ~��s�� � A�s�d�s�M�s� � C�s��A�s� �M�s�� (41)

Using the definitions of H0�s� and H1�s� in Eqs. (16) and (17), wecan write

��s� �H1�s�r�s� �H1�s�C�s� ~��s� �H0�s�d�s� (42)

Substitution into Eq. (40) leads to

y�s� �M�s��C�s� �H1�s��1 � C�s����r�s� � ~��s�

C�s�

��H0�s�M�s��1 � C�s��d�s�

Recalling the definition of H�s� from Eq. (8), one can verify that

M�s��C�s� �H1�s��1 � C�s��� �H�s�C�s�

H�s� �H0�s�M�s�

which implies

y�s� �H�s��C�s�r�s� � ~��s�� �H�s��1 � C�s��d�s�

Using the expression for yref�s� fromEq. (14) and letting de�s� be theLaplace transform of

de�t� � f�t; y�t�� � f�t; yref�t��

one can derive

e�s� �H�s���1 � C�s��de�s� � ~��s��

Lemma 5 in the Appendix and Assumption 1 give the followingupper bound:

ketkL1 � LkH�s��1 � C�s��kL1ketkL1 � kr1tkL1 (43)

where r1�t� is the signal, with its Laplace transform being

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r1�s� �H�s� ~��s�

Using the expression for ~��s� from Eq. (38), along with the ex-pression for y�s� from Eq. (2), and taking into consideration that

y�s� �M�s�u�s� � c>m�sI� Am��1��s�

we have

~y�s� � c>m�sI� Am��1��s� �M�s���s�

�M�s�C�s�

C�s�M�s� c

>m�sI � Am��1��s� �

M�s�C�s� C�s���s�

�M�s�C�s� ~��s� (44)

This implies that r1�s� can be rewritten as

r1�s� �C�s�H�s�M�s�

M�s�C�s� ~��s� �H2�s� ~y�s�

and hence

kr1tkL1 � kH2�s�kL1k ~ytkL1

Substituting this back into Eq. (43) completes the proof. □

VI. Analysis of L1 Adaptive Controller

In this section, we analyze the stability and the performance oftheL1 adaptive controller. Using the definitions fromEq. (10),H3�s�in Eq. (19) can be rewritten as

H3�s� ��Cn�s�Ad�s�Mn�s�

Hd�s�(45)

where Hd�s� is defined in Eq. (12). Because

deg�Cd�s� � Cn�s�� � degCn�s� � dr

it can be checked straightforwardly thatH3�s� is strictly proper. Wenote from Eqs. (11) and (45) that H3�s� has the same denominatoras H�s�, and therefore it follows from Eq. (9) that H3�s� is stable.Because H3�s� is strictly proper and stable with relative degree dr,H3�s�=M�s� is stable and proper and, therefore its L1 norm is finite.Consider the state transformation

~���~x

It follows from Eq. (36) that

_~��t� ��Am��1 ~��t� ����t� ��bm��t�; ~��0� � 0 (46)

~y�t� � ~�1�t� (47)

where ~�1�t� is the first element of ~��t�.Theorem 1. Given the system in Eq. (1) and the L1 adaptive con-

troller in Eqs. (32), (33), and (35), subject to Eq. (9), if we choose Tto ensure

�0�T�< �� (48)

where �� is an arbitrary positive constant introduced in Eq. (21), then

k ~ykL1 < �� (49)

ky � yrefkL1 � �1; ku � urefkL1 � �2 (50)

with

�1 � kH2�s�kL1��=�1 � kG�s�kL1

L��2 � LkH2�s�kL1

�1 � kH3�s�=M�s�kL1��

Proof. First, we prove the bound in Eq. (49) by a contradictionargument. Because ~y�0� � 0 and ~y�t� is continuous, then assumingthe opposite implies that there exists t0 such that

k ~y�t�k< ��; 8 0 � t < t0 (51)

k ~y�t0�k � �� (52)

which leads to

k ~yt0 kL1 � �� (53)

Because y�t� � yref�t� � e�t�, the upper bound in Eq. (15) can beused to arrive at

kyt0 kL1 � kyreft0 kL1 � ket0 kL1 � �� kC�s�H�s�=M�s�kL1

��=�1 � kG�s�kL1L� (54)

It follows from Eqs. (42) and (44) that

��s� �H1�s�r�s� �H1�s� ~y�s�=M�s� �H0�s�d�s�

and hence Eq. (53) implies that

k�t0 kL1 � kH1�s�kL1krkL1 � kH1�s�=M�s�kL1

��

� kH0�s�kL1�Lkyt0 kL1 � L0�

which, along with Eq. (54), leads to

k�t0 kL1 � � (55)

It follows from Eq. (46) that

~��iT � t� � exp��Am��1t� ~��iT�

�ZiT�t

iT

exp��Am��1�iT � t� ������iT�d�

�ZiT�t

iT

exp��Am��1�iT � t � ����bm����d�

� exp��Am��1t� ~��iT� �Zt

0

exp��Am��1�t � ������iT�d�

�Zt

0

exp��Am��1�t � ����bm��iT � ��d� (56)

Because

~��iT� � ~y�iT�0

� �� 0

~z�iT�

� �

it follows from Eq. (56) that

~��iT � t� � �iT � t� � ��iT � t� (57)

where

�iT � t� � exp��Am��1t�~y�iT�0

� ��

Zt

0

exp��Am��1�t � ������iT�d�(58)

��iT � t� � exp��Am��1t�0

~z�iT�

" #

�Zt

0

exp��Am��1�t � ����bm��iT � ��d� (59)

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In what follows, we prove that for all iT � t0, one has

j ~y�iT�j � &�T� (60)

~z >�iT�P2 ~z�iT� � � (61)

where &�T� and � are defined in Eq. (24). Because ~��0� � 0, it isstraightforward that j ~y�0�j � &�T� and ~z>�0�P2 ~z�0� � �. For any�j� 1�T � t0, we will prove that if

j ~y�jT�j � &�T� (62)

~z >�jT�P2 ~z�jT� � � (63)

then Eqs. (62) and (63) hold for j� 1 too. Hence, Eqs. (60) and (61)hold for all iT � t0.

Assume that Eqs. (62) and (63) hold for j and, in addition,

�j� 1�T � t0 (64)

It follows from Eq. (57) that

~���j� 1�T� � ��j� 1�T� � ���j� 1�T� (65)

where

��j� 1�T� � exp��Am��1T�~y�jT�0

" #

�ZT

0

exp��Am��1�T � ������jT�d� (66)

���j� 1�T� � exp��Am��1T�0

~z�jT�

" #

�ZT

0

exp��Am��1�T � ����bm��jT � ��d� (67)

Substituting the adaptive law from Eq. (33) in Eq. (66), we have

��j� 1�T� � 0 (68)

It follows from Eq. (59) that ��t� is the solution of the followingdynamics:

_��t� ��Am��1��t� ��bm��t� (69)

��jT� � 0

~z�jT�

� �; t 2 �jT; �j� 1�T� (70)

Consider the following function

V�t� � �>�t���>P��1��t�

over t 2 �jT; �j� 1�T�. Because � is nonsingular and P is positivedefinite, ��>P��1 is positive definite, and hence V�t� is a posi-tive definite function. It follows from Lemma 2 and the relationshipin Eq. (70) that

V���jT�� � ~z>�jT�P2 ~z�jT�

which further, along with the upper bound in Eq. (63), leads to thefollowing:

V���jT�� � � (71)

It follows from Eq. (69) that over t 2 �jT; �j� 1�T�,

_V�t���>�t���>P��1�Am��1��t���>�t���>A>m�>��>P>��1��t��2�>�t���>P��1�bm��t����>�t���>Q��1��t��2�>�t���>Pbm��t�

Using the upper bound from Eq. (55), one can derive over t 2�jT; �j� 1�T�

_V�t� � �min���>Q��1�k��t�k2 � 2k��t�kk��>Pbmk� (72)

Note that for all t 2 �jT; �j� 1�T�, if

V�t� � � (73)

we have

k��t�k �����������������������������������

�

max���>P��1�

r� 2�k��>Pbmkmin���>Q��1�

and the upper bound in Eq. (72) yields

_V�t� � 0 (74)

It follows from Eqs. (71), (73), and (74) that

V�t� � �; 8 t 2 �jT; �j� 1�T�

and therefore

V��j� 1�T� � �>��j� 1�T����>P��1����j� 1�T� � � (75)

Because

~���j� 1�T� � ��j� 1�T� � ���j� 1�T� (76)

the equality in Eq. (68) and the upper bound in Eq. (75) lead to thefollowing inequality:

~� >��j� 1�T����>P��1� ~���j� 1�T� � �

Using the result of Lemma 2, one can derive that

~z>��j� 1�T�P2 ~z��j� 1�T�

� ~�>��j� 1�T����>P��1� ~���j� 1�T� � �

which implies that the upper bound in Eq. (63) holds for j� 1.It follows from Eqs. (47), (68), and (76), that

~y��j� 1�T� � 1>1 ���j� 1�T�

and the definition of ���j� 1�T� in Eq. (67) leads to the followingexpression:

~y��j� 1�T� � 1>1 exp��Am��1T�0

~z�jT�

� �

� 1>1

ZT

0

exp��Am��1�T � ����bm��jT � ��d� (77)

The upper bounds in Eqs. (55) and (63) allow for the following upperbound:

j ~y��j� 1�T�j � k��T�kk~z�jT�k

�ZT

0

j1>1 exp��Am��1�T � ����bmjj��jT � ��jd�

� k��T�k�������������������

�

max�P2�

r� ��T��� &�T� (78)

where ��T� and ��T� are defined in Eq. (23), and &�T� is defined inEq. (24). This confirms the upper bound in Eq. (62) for j� 1. Hence,Eqs. (60) and (61) hold for all iT � t0.

For all iT � t � t0, where 0 � t � T, using the expression fromEq. (56), we can write that

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~y�iT � t� � 1>1 exp��Am��1t� ~��iT�

� 1>1

Zt

0

exp��Am��1�t� ������iT�d�

� 1>1

Zt

0

exp��Am��1�t � ����bm��iT � ��d�

The upper bound in Eq. (55) and definitions of 1�t�, 2�t�, 3�t�, and4�t� allow for the following upper bound:

j ~y�iT � t�j � j1�t�jj ~y�iT�j � k2�t�kk~z�iT�k� 3�t�j ~y�iT�j � 4�t��

Taking into consideration Eqs. (62) and (63) and recalling thedefinitions of �1�T�, �2�T�, �3�T�, and �4�T� in Eqs. (25–27) for all0 � t � T and for any nonnegative integer i subject to iT � t � t0,we have

j ~y�iT � t�j � �1�T�&�T� � �2�T��������������������

�

max�P2�

r� �3�T�&�T� � �4�T��

Because the right-hand side coincides with the definition of �0�T� inEq. (28), then for all t 2 �0; t0�, we have

j ~y�t�j � �0�T�

which, along with the assumption on T introduced in Eq. (48), yields

k ~yt0 kL1 < ��

This clearly contradicts the statement in Eq. (53). Therefore,

k ~ykL1 < ��

which proves Eq. (49). Further, it follows from Lemma 4 that

ketkL1 �kH2�s�kL1

1 � kG�s�kL1L��

which holds uniformly for all t � 0 and therefore leads to the firstupper bound in Eq. (50).

It follows from Eqs. (39) and (41) that

u�s� �M�s�C�s�r�s� �M�s� ~��s� � C�s�A�s�d�s�C�s�A�s� � �1 � C�s��M�s�

To prove the second bound in Eq. (50), we use the expression ofuref�s� from Eq. (13) to derive

u�s� � uref�s� � �H2�s�de�s� �H3�s�C�s� ~��s�

� �H2�s�de�s� �H3�s�M�s�

M�s�C�s� ~��s� (79)

It follows from Eqs. (44) and (79) that

ku � urefkL1 � LkH2�s�kL1ky � yrefkL1

� kH3�s�=M�s�kL1k ~ykL1

which, along with Eqs. (49) and (50), leads to the second bound inEq. (50). The proof is complete. □

Thus, the tracking error between y�t� and yref�t�, as well be-tween u�t� and uref�t�, is uniformly bounded by a constant propor-tional to T. This implies that during the transient one can achievearbitrary improvement of tracking performance by uniformlyreducing T.

Remark 1. Note that the parameter T is the fixed time step in thedefinition of the adaptive law. The adaptive parameters in ��t� 2 Rn

take constant values during �iT; �i� 1�T� for every i� 0; 1; . . ..Reducing T imposes hardware (CPU) requirements, and Theorem 1further implies that the performance limitations are consistent with

the hardware limitations. This in turn is consistent with the earlierresults in [5,6], in which improvement of the transient performancewas achieved by increasing the adaptation rate in the continuous-time adaptive laws.

Remark 2. We note that the following ideal control signal

uideal�t� � r�t� � �ref�t�

is the one that leads to desired system response

yideal�s� �M�s�r�s�

by canceling the uncertainties exactly. Thus, the reference system inEqs. (5–7) has a different response from the ideal one. It only cancelsthe uncertainties within the bandwidth ofC�s�, which can be selectedto be compatible with the control channel specifications. This isexactly what one can hope to achieve with any feedback in thepresence of uncertainties.

Remark 3. We note that stability of H�s� is equivalent tostabilization of A�s� by

C�s�M�s��1 � C�s�� (80)

Indeed, consider the closed-loop system, composed of the systemA�s� and negative feedback of Eq. (80). The closed-loop transferfunction is

A�s���

1� A�s� C�s�M�s��1 � C�s��

�(81)

Incorporating Eq. (10), one can verify that the denominator of thesystem in Eq. (81) is exactly Hd�s�. Hence, stability of H�s� isequivalent to the stability of the closed-loop system in Eq. (81). Thisimplies that the class of systems A�s� that can be stabilized by theL1

adaptive output-feedback controller Eqs. (32), (33), and (35) is notempty.

Remark 4. Finally, we note that although the feedback in Eq. (80)may stabilize the system in Eq. (1) for some classes of unknownnonlinearities, it will not ensure uniform transient performance in thepresence of unknown A�s�. On the contrary, the L1 adaptivecontroller ensures uniform transient performance for both of thesystem’s signals, independent of the unknown nonlinearity andindependent of A�s�.

VII. Simulations

A. Numerical Example

Consider the system in Eq. (1) with

A�s� � �s� 1�=�s3 � s2 � 2s� 8�

We note that A�s� has poles in the right-half complex plane. Weconsider the L1 adaptive output-feedback controller defined viaEqs. (32), (33), and (35), where

M�s� � 1

s2 � 1:4s� 1

C�s� � 100

s2 � 14s� 100; T � 10�4

First, we consider the step response for d�t� � 0. The simulationresults of L1 adaptive controller are shown in Figs. 1a and 1b. Next,we consider

d�t� � f�t; y�t�� � sin�0:1t�y2�t� � sin�0:4t�

and apply the same controller without retuning. The control signaland the system response are plotted in Figs. 2a and 2b. Further, weconsider a time-varying reference input r�t� � 0:5 sin�0:3t� and notethat without any retuning of the controller, the system response andthe control signal behave as expected (Figs. 3a and 3b).

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0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time t0 5 10 15 20

−2

0

2

4

6

8

time t

a) y(t) (solid) and r(t) (dashed) b) Time history of u(t)

Fig. 1 Performance for r�t� � 1 and d�t� � 0.

0 5 10 15 20−2

0

2

4

6

8

time t0 5 10 15 20

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time t

a) y(t) (solid) and r(t) (dashed) b) Time history of u(t)

Fig. 2 Performance for r�t� � 1 and d�t� � sin�0:1t�y2�t� � sin�0:4t�.

0 10 20 30 40−0.5

0

0.5

Time t0 10 20 30 40

−5

0

5

time t

a) y(t) (solid) and r(t) (dashed) b) Time history of u(t)Fig. 3 Performance for r�t� � 0:5 sin�0:3t� and d�t� � sin�0:1t�y2�t� � sin�0:4t�.

0 0.5 1 1.5−4

−3

−2

−1

0

1

2

Time, [s]

Con

trol

sig

nal,

[rad

]

K=10%K=25%K=50%K=100%

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, [s]

Ang

le o

f atta

ck, [

deg]

K=10%K=25%K=50%K=100%

a) System response b) Control historyFig. 4 Simulation results for missile longitudinal autopilot design: Scaled response for scaled (unmatched) uncertainties.

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B. Missile Longitudinal Autopilot Design

We consider the missile dynamics from [38], which are given bythe following state-space representation:

_x�t� �

ZaV

1 ZdV

0

Ma 0 Md 0

0 0 0 1

0 0 �!2a �2�a!a

266664

377775x�t� ��Ax�t�

�

0

0

0

!2a

266664

377775�u�t� � k>mx�t��

y�t� � � 1 0 0 0 �x�t�

where x�t� � �Az q �a _�a �> is the system state, in which Az(ft=s2) is the vertical acceleration, q (rad=s) is the pitch rate, �a (rad)

is the fin deflection, _�a (rad=s) is the fin deflection rate, km is thevector of matched (in the sense of state-feedback) parametricuncertainties, and�A contains unmatched uncertainties. For detailedanalysis of the missile dynamics, the reader is referred to [38].

In simulations, the following numerical values have been used forthe missile dynamics [38]:

ZaV��1:3046

�1

s

�;

ZdV��0:2142

�1

s

�

Ma � 47:7109

�1

s2

�; Md ��104:8346

�1

s2

�

�A�

�0:4996K 0 �0:4996K 0

0:4996K 0 �0:4996K 0

0 0 0 0

0 0 0 0

266664

377775

km � 0:3

0

0

�1�2 �a

!a

2666664

3777775

Here, the matrix �A is selected to enforce the worst direction forthe change of aerodynamic parameters, withK being the uncertaintyscaling factor for unmatched uncertainties. In this example, K � 0corresponds to nonperturbed aerodynamics, and the value K � 1corresponds to uncertainties that would turn the system into anoscillator in the presence of an linear quadratic regulator state-feedback controller only.

For simulation of the L1 adaptive output-feedback controller, thefollowingM�s� and C�s� have been selected, along with a samplingrate T � 0:0001 s:

M�s� � 1

1=!s2 � 2�!s� 1; C�s� � 1

1=!cs2 � 2�c!cs� 1

where !� 8 rad=s, �� 0:9, !c � 120 rad=s, and �c � 1:85.Figures 4a and 4b show the scaled response for scaled uncertainties,dependent upon the changes inK, achieved via scaled control efforts.

VIII. Conclusions

We presented the L1 adaptive output-feedback controller forreference systems that do not verify the SPR condition for theirinput–output transfer function. The new piecewise constant adaptivelaw, along with the low-pass-filtered control signal, ensures uniform

performance bounds for the system’s input/output signals simul-taneously. The performance bounds can be systematically improvedby reducing the integration time step.

Appendix: Facts from Linear Systems Theory

Definition 1 [41]. For a signal ��t�, where t � 0 and � 2 Rn, itsL1and truncated L1 norms are

k�kL1 � maxi�1;...;n

�sup��0j�i���j

�

k�tkL1 � maxi�1;...;n

�sup0���tj�i���j

�

where �i is the ith component of �.Definition 2 [41]. The L1 gain of a stable proper SISO system is

defined:

jjH�s�jjL1�Z 10

jh�t�jdt

where h�t� is the impulse response of H�s�.Lemma5 [41]. For a stable propermulti-input/multi-output system

H�s� with input r�t� 2 Rm and output x�t� 2 Rn, we have

kxtkL1 � kH�s�kL1krtkL1 8 t � 0

Acknowledgments

This research is supported by the U.S. Air Force Office ofScientific Research under contract FA9550-08-1-0135 and byNASA under contracts NNX08AB97A and NNX08AC81A. Theauthors are thankful toKevinWise for sharing hismissile code and toEvgeny Kharisov for integrating L1 adaptive output-feedbackcontroller into that code.

References

[1] Khalil, H. K., and Esfandiari, F., “Semiglobal Stabilization of a Classof Nonlinear Systems Using Output Feedback,” IEEE Transactions on

Automatic Control, Vol. 38, No. 9, 1993, pp. 1412–1415.doi:10.1109/9.237658

[2] Krstic, M., Kanellakopoulos, I., and Kokotovic, P., Nonlinear and

Adaptive Control Design, Wiley, New York, 1995.[3] Marino, R., and Tomei, P., “Global Adaptive Output Feedback Control

of Nonlinear Systems, Part 1: Linear Parameterization,” IEEE

Transactions on Automatic Control, Vol. 38, No. 1, 1993, pp. 17–32.doi:10.1109/9.186309

[4] Marino, R., and Tomei, P., “Global Adaptive Output Feedback Controlof Nonlinear Systems, Part 2: Nonlinear Parameterization,” IEEE

Transactions on Automatic Control, Vol. 38, No. 1, 1993, pp. 33–48.doi:10.1109/9.186310

[5] Cao, C., and Hovakimyan, N., “Design and Analysis of a Novel L1

Adaptive Control Architecture with Guaranteed Transient Perform-ance,” IEEE Transactions on Automatic Control, Vol. 53, No. 2,Mar. 2008, pp. 586–591.doi:10.1109/TAC.2007.914282

[6] Cao, C., and Hovakimyan, N., “L1 Adaptive Output FeedbackController for Systems of Unknown Dimension,” IEEE Transactions

on Automatic Control, Vol. 53, No. 3, Apr. 2008, pp. 815–821.doi:10.1109/TAC.2008.919550

[7] Cao, C., and Hovakimyan, N., “L1 Adaptive Controller for Systemswith Unknown Time-Varying Parameters and Disturbances in thePresence of Nonzero Trajectory Initialization Error,” International

Journal of Control, Vol. 81, No. 7, 2008, pp. 1147–1161.doi:10.1080/00207170701670939

[8] Calise, A. J., Hovakimyan, N., and Idan, M., “Adaptive OutputFeedback Control of Nonlinear Systems Using Neural Networks,”Automatica, Vol. 37, No. 8, 2001, pp. 1201–1211.doi:10.1016/S0005-1098(01)00070-X

[9] Hovakimyan, N., Nardi, F., and Calise, A., “A Novel Error ObserverBased Adaptive Output Feedback Approach for Control of UncertainSystems,” IEEE Transactions on Automatic Control, Vol. 47, No. 8,

CAO AND HOVAKIMYAN 725

Dow

nloa

ded

by B

EIH

AN

G U

NIV

ER

SIT

Y o

n D

ecem

ber

20, 2

016

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.408

77

2002, pp. 1310–1314.doi:10.1109/TAC.2002.800766

[10] Hovakimyan, N., Nardi, F., Calise, A., and Kim, N., “Adaptive OutputFeedback Control of Uncertain Systems Using Single Hidden LayerNeural Networks”, IEEE Transactions on Neural Networks, Vol. 13,No. 6, 2002, pp. 1420–1431.doi:10.1109/TNN.2002.804289

[11] Cao, C., Hovakimyan, N., Kaminer, I., Patel, V., andDobrokhodov, V.,“Stabilization of Cascaded Systems via L1 Adaptive Controller withApplication to a UAV Path Following Problem and Flight TestResults,” American Control Conference, Inst. of Electrical andElectronics Engineers, Piscataway, NJ, 2007, pp. 1787–1792.

[12] Tsakalis, K. S., and Ioannou, P. A., “Adaptive Control of LinearTime-Varying Plants,” Automatica, Vol. 23, No. 4, 1987, pp. 459–468.doi:10.1016/0005-1098(87)90075-6

[13] Middleton, R. H., and Goodwin, G. C., “Adaptive Control of Time-Varying Linear Systems,” IEEE Transactions on Automatic Control,Vol. 33, No. 2, 1988, pp. 150–155.doi:10.1109/9.382

[14] Bartolini, G., Ferrara, A., and Stotsky, A. A., “Robustness andPerformance of an Indirect Adaptive Control Scheme in the Presence ofBounded Disturbances,” IEEE Transactions on Automatic Control,Vol. 44, No. 4, Apr. 1999, pp. 789–793.doi:10.1109/9.754819

[15] Sun, J., “A Modified Model Reference Adaptive Control Scheme forImproved Transient Performance,” IEEE Transactions on Automatic

Control, Vol. 38, No. 8, July 1993, pp. 1255–1259.doi:10.1109/9.233162

[16] Miller, D. E., and Davison, E. J., “Adaptive ControlWhich Provides anArbitrarily Good Transient and Steady-State Response,” IEEE

Transactions on Automatic Control, Vol. 36, No. 1, Jan. 1991,pp. 68–81.doi:10.1109/9.62269

[17] Costa, R., “Improving Transient Behavior of Model-ReferenceAdaptive Control.” Proceedings of the American Control Conference,Inst. of Electrical and Electronics Engineers, Piscataway, NJ, 1999,pp. 576–580.

[18] Ydstie, B. E., “Transient Performance and Robustness of DirectAdaptive Control,” IEEE Transactions on Automatic Control, Vol. 37,No. 8, Aug. 1992, pp. 1091–1105.doi:10.1109/9.151091

[19] Krstic, M., Kokotovic, P. V., and Kanellakopoulos, I., “TransientPerformance Improvement with a New Class of Adaptive Controllers,”Systems and Control Letters, Vol. 21, No. 6, 1993, pp. 451–461.doi:10.1016/0167-6911(93)90050-G

[20] Ortega, R., “Morse’s New Adaptive Controller: ParameterConvergence and Transient Performance,” IEEE Transactions on

Automatic Control, Vol. 38, No. 8, Aug. 1993, pp. 1191–1202.doi:10.1109/9.233152

[21] Zang, Z., and Bitmead, R., “Transient Bounds for Adaptive ControlSystems,” Proceedings of the 30th IEEE Conference on Decision and

Control, Inst. of Electrical and Electronics Engineers, Piscataway, NJ,Dec. 1990, pp. 2724–2729

[22] Zang, Y., and Ioannou, P. A., “Adaptive Control of Linear Time-Varying Systems,” Proceedings of the 35th IEEE Conference on

Decision and Control, Inst. of Electrical and Electronics Engineers,Piscataway, NJ, 1996, pp. 837–842.

[23] Datta, A., and Ioannou, P., “Performance Analysis and Improvement inModel Reference Adaptive Control,” IEEE Transactions on Automatic

Control, Vol. 39, No. 12, pp. 2370–2387.doi:10.1109/9.362856, Dec. 1994.

[24] Narendra, K. S., and Balakrishnan, J., “Improving Transient Responseof Adaptive Control Systems Using Multiple Models and Switching,”IEEE Transactions on Automatic Control, Vol. 39, No. 9, Sept. 1994,pp. 1861–1866.doi:10.1109/9.317113

[25] Marino, R., and Tomei, P., “An Adaptive Output Feedback Control fora Class of Nonlinear Systems with Time-Varying Parameters,” IEEE

Transactions on Automatic Control, Vol. 44, No. 11, 1999, pp. 2190–2194.doi:10.1109/9.802943

[26] Arteaga, A. M., and Tang, Y., “Adaptive Control of Robots with anImproved Transient Performance,” IEEE Transactions on Automatic

Control, Vol. 47, No. 7, July 2002, pp. 1198–1202.doi:10.1109/TAC.2002.800672

[27] Cao, C., and Hovakimyan, N., “Stability Margins of L1 AdaptiveController: Part 2,” American Control Conference, Inst. of Electricaland Electronics Engineers, Piscataway, NJ, 2007, pp. 3931–3936.

[28] Beard, R.W., Knoebel, N., Cao, C., Hovakimyan, N., andMatthews, J.,“An L1 Adaptive Pitch Controller for Miniature Air Vehicles,” AIAAGuidance, Navigation and Control Conference, AIAA Paper 2006-6777, 2006.

[29] Kaminer, I., Yakimenko, O., Dobrokhodov, V., Pascoal, A.,Hovakimyan, N., Cao, C., Young, A., and Patel, V. V., “CoordinatedPath Following for Time-Critical Missions of Multiple UAVs via L1

Adaptive Output Feedback Controllers,” AIAA Guidance, Navigationand Control Conference, Hilton Head Island, SC, AIAA Paper 2007-6409, Aug. 2007.

[30] Dobrokhodov, V., Kitsiois, I., Kaminer, I., Jones, K., Xargay, E.,Hovakimyan, N., Cao, C., Lizarraga, M., and Gregory, I., “FlightValidation of Metrics Driven L1 Adaptive Control,”. AIAA Guidance,Navigation, and Control Conference, Honolulu, HI, AIAA Paper 2008-6987, 2008.

[31] Aguiar, P., Pascoal, A., Kaminer, I., Dobrokhodov, V., Xargay, E.,Hovakimyan, N., Cao, C., and Ghabchello, R., “Time-CoordinatedPath Following ofMultiple UAVs over Time-Varying Networks UsingL1 Adaptation,”AIAAGuidance, Navigation and Control Conference,Honolulu, HI, AIAA Paper 2008-7131, 2008.

[32] Wise, K., Lavretsky, E., Hovakimyan, N., Cao, C., and Wang, J.,“Verifiable Adaptive Control: UCAV and Aerial Refueling,”AIAAGuidance, Navigation, and Control Conference and Exhibit, AIAAPaper 2008-6658, 2008.

[33] Lei, Y., Cao, C., Cliff, E., Hovakimyan, N., Kurdila, A., Bolender, M,and Doman, D., “Design of an L1 Adaptive Controller for an Air-Breathing Hypersonic Vehicle Model with Unmodelled Dynamics,”AIAA Guidance, Navigation and Control Conference, Hilton HeadIsland, SC, AIAA Paper 2007-6527, 2007.

[34] Gregory, I., Patel, V., Cao, C., and Hovakimyan, N., “L1

Adaptive Control Laws for Flexible Wind Tunnel Model of High-Aspect Ratio Flying Wing,”AIAA Guidance, Navigation, and ControlConference, Hilton Head Island, SC, AIAA Paper 2007-6525,2007.

[35] Cotting, M. C., Cao, C., Hovakimyan, N., Kraus, R., and Durham, W.,“Simulator Testing of Longitudinal Flying Qualities with L1 AdaptiveController,” AIAA Guidance, Navigation and Control Conference,Honolulu, HI, AIAA Paper 2008-6551, 2008.

[36] Cao, C., Patel, V. V., Reddy, K., Hovakimyan, N., and Lavretsky, E.,“Are the Phase and Time-Delay Margins Always Adversely AffectedbyHigh-Gain?,”AIAAGuidance, Navigation andControl Conference,AIAA Paper 2006-6347, 2006.

[37] Li, D., Hovakimyan, N., Cao, C., and Wise, K., “Filter Design forFeedback-Loop Trade-Off of L1 Adaptive Controller. A Linear MatrixInequality Approach,” AIAA Guidance, Navigation and ControlConference, AIAA Paper 2008-6280, 2008.

[38] Wise, K., “Robust Stability Analysis of Adaptive Missile Autopilots,”AIAA Guidance, Navigation, and Control Conference and Exhibit,AIAA Paper 2008-6999, 2008.

[39] Wang, J., Cao, C., Hovakimyan, N., Hindman, R., and Ridgely, B. “L1

Adaptive Controller for a Missile Longitudinal Autopilot Design,”.AIAA Guidance, Navigation, and Control Conference and Exhibit,AIAA Paper 2008-6282, 2008.

[40] Kharisov, E., Gregory, I., Cao, C., and Hovakimyan, N., “L1 AdaptiveControl Law for Flexible Space Launch Vehicle and Proposed Plan forFlight Test Validation,” AIAA Guidance, Navigation, and ControlConference and Exhibit, AIAA Paper 2008-7128, 2008.

[41] Khalil, H. K., Nonlinear Systems, Prentice–Hall, Englewood Cliffs,NJ, 2002.

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