L1 Adaptive Output Feedback Controller for Non Strictly ... · L1 Adaptive Output Feedback...

L1 Adaptive Output Feedback Controller for Non StrictlyPositive Real Reference Systems

with Applications to Aerospace Examples∗

Chengyu Cao†

University of Connecticut, Storrs, CT 06269

Naira Hovakimyan‡

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

This paper presents an extension of theL1 adaptive output feedback controller to systems of unknownrelative degree in the presence of time-varying uncertainties without restricting the rate of their variation. Ascompared to earlier results in this direction, a new piece-wise continuous adaptive law is introduced alongwith the low-pass filtered control signal that allows for achieving arbitrarily close tracking of the input and theoutput signals of the reference system, the transfer function of which is not required to be strictly positive real(SPR). Stability of this reference system is proved using small-gain type argument. The performance boundsbetween the closed-loop reference system and the closed-loop L1 adaptive system can be rendered arbitrarilysmall by reducing the step size of integration. Simulationsverify the theoretical findings.

I. Introduction

This paper extends the results of Ref.1 by considering reference systems that do not verify the SPR condition fortheir input-output transfer function. Similar to Ref.,1 theL∞-norms of both input/output error signals between theclosed-loop adaptive system and the reference system can berendered arbitrarily small by reducing the step-size ofintegration. The key difference from the results in Ref.1 is the new piece-wise continuous adaptive law. The adaptivecontrol is defined as output of a low-pass filter, resulting ina continuous signal despite the discontinuity of the adaptivelaw.

We notice that adaptive algorithms achieving arbitrarily improved transient performance for system’s output werereported in Refs.2–16 References1, 17 presented the opportunity to regulate also the performancebound for system’sinput signal, by rendering it arbitrarily close to the corresponding signal of a bounded reference LTI system. Thispaper presents the adaptive output feedback counterpart ofthe results in Refs.,1, 17without enforcing the SPR conditionon the input/output transfer function of the desired reference system, which typically appears in conventional adaptiveoutput feedback schemes. Application of the methodology presented in this paper can be found in Refs.18, 19

The paper is organized as follows. Section II gives the problem formulation. In Section III, the closed-loopreference system is introduced. In Section IV, some preliminary results are developed towards the definition of theL1 adaptive controller. In Section V, the novelL1 adaptive control architecture is presented. Stability anduniformperformance bounds are presented in Section VI. In Section VII, simulation results are presented and two aerospaceapplications are discussed, while Section VIII concludes the paper. The small-gain theorem and some basic definitionsfrom linear systems theory used throughout the paper are given in Appendix. Unless otherwise mentioned,|| · || willbe used for the2-norm of the vector.

∗Research is supported by AFOSR under Contract No. FA9550-05-1-0157 and NASA under contracts NNX08AB97A and NNX08AC81A.†Assistant Professor, Department of Mechanical Engineering, AIAA Member, [email protected]‡Professor, Department of Mechanical Science and Engineering, and AIAA Associate Fellow, [email protected]

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American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation and Control Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-7288

Copyright © 2008 by C. Cao and Naira Hovakimyan. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

II. Problem Formulation

Consider the following single–input single–output (SISO)system:

y(s) = A(s)(u(s) + d(s)) , y(0) = 0 , (1)

whereu(t) ∈ R is the input,y(t) ∈ R is the system output,A(s) is a strictly proper unknown transfer function ofunknown relative degreenr, for which only a known lower bound1 < dr ≤ nr is available,d(s) is the Laplacetransform of the time-varying uncertainties and disturbancesd(t) = f(t, y(t)), whilef is an unknown map, subject tothe following assumptions.

Assumption 1 There exist constantsL > 0 andL0 > 0 such that the following inequalities hold uniformly int ≥ 0:

|f(t, y1) − f(t, y2)| ≤ L|y1 − y2| , |f(t, y)| ≤ L|y|+ L0.

Assumption 2 There exist constantsL1 > 0, L2 > 0 andL3 > 0 such that for allt ≥ 0:

|d(t)| ≤ L1|y(t)| + L2|y(t)| + L3 , (2)

where the numbersL, L0, L1, L2, L3 can be arbitrarily large. Letr(t) be a given bounded continuous reference inputsignal. The control objective is to design an adaptive output feedback controlleru(t) such that the system outputy(t) tracks the reference inputr(t) following a desired reference model, i.e.y(s) ≈ M(s)r(s), whereM(s) is aminimum-phase stable transfer function of relative degreedr > 1. We rewrite the system in (1) as:

y(s) = M(s) (u(s) + σ(s)) , (3)

σ(s) =(

(A(s) − M(s))u(s) + A(s)d(s))

/M(s) . (4)

Let (Am, bm, cm) be the minimal realization ofM(s), i.e. it is controllable and observable andAm is Hurwitz. Thesystem in (3) can be rewritten as:

x(t) = Amx(t) + bm(u(t) + σ(t)) , (5)

y(t) = c⊤mx(t) , x(0) = x0 = 0.

III. Closed-loop Reference System

Consider the following closed-loop reference system:

yref (s) = M(s)(uref (s) + σref (s)) (6)

σref (s) =(A(s) − M(s))uref (s) + A(s)dref (s)

M(s)(7)

uref(s) = C(s)(r(s) − σref (s)) , (8)

wheredref (t) = f(t, yref(t)), andC(s) is a strictly proper system of relative orderdr, with its DC gainC(0) = 1.We notice that there is no algebraic loop involved in the definition of σ(s), u(s) andσref (s), uref (s).

For the proof of stability of the reference system in (6)-(8), the selection ofC(s) andM(s) must ensure that

H(s) = A(s)M(s)/(

C(s)A(s) + (1 − C(s))M(s))

(9)

is stable and‖G(s)‖L1

L < 1 , (10)

whereG(s) = H(s)(1 − C(s)). This in its turn restricts the class of systemsA(s) in (1), for which the proposedapproach in this paper can achieve stabilization. However,as discussed later in Remark 3, the class of such systems isnot empty. Letting

A(s) =An(s)

Ad(s), C(s) =

Cn(s)

Cd(s), M(s) =

Mn(s)

Md(s), (11)

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it follows from (9) that

H(s) =Cd(s)Mn(s)An(s)

Hd(s), (12)

whereHd(s) = Cn(s)An(s)Md(s) + Mn(s)Ad(s)(Cd(s) − Cn(s)). (13)

A strictly properC(s) implies that the order ofCd(s) − Cn(s) andCd(s) is the same. Since the order ofAd(s) ishigher than that ofAn(s), the transfer functionH(s) is strictly proper.

The next Lemma establishes the stability of the closed-loopsystem in (6)-(8).

Lemma 1 If C(s) andM(s) verify the condition in (10), the closed-loop reference system in (6)-(8) is stable.

Proof. It follows from (7)-(8) thaturef (s) = C(s)r(s) − C(s)((A(s) − M(s))uref (s) + A(s)dref (s))/M(s), andhence

uref(s) =C(s)M(s)r(s) − C(s)A(s)dref (s)

C(s)A(s) + (1 − C(s))M(s). (14)

From (6)-(7) one can deriveyref (s) = A(s)(uref (s) + dref (s)), which upon substitution of (14), leads to

yref (s) = H(s)(

C(s)r(s) + (1 − C(s))dref (s))

. (15)

SinceH(s) is strictly proper and stable,G(s) is also strictly proper and stable and therefore‖yref‖L∞≤ ‖H(s)C(s)‖L1

‖r‖L∞+

‖G(s)‖L1(L‖yref‖L∞

+ L0) . Using the condition in (10), one can write

‖yref‖L∞≤ ρ , (16)

where

ρ =‖H(s)C(s)‖L1

‖r‖L∞+ ‖G(s)‖L1

L0

1 − ‖G(s)‖L1L

< ∞ . (17)

Hence,‖yref‖L∞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))

(18)

H1(s) =(A(s) − M(s))C(s)

C(s)A(s) + (1 − C(s))M(s)(19)

H2(s) = − M(s)C(s)

(C(s)A(s) + (1 − C(s))M(s))(20)

H3(s) = H(s)C(s)/M(s) . (21)

Using the expressions from (11) and (13),H0(s) andH1(s) can be rewritten as:

H0(s) = Cd(s)An(s)Md(s)/Hd(s),

H1(s) =(

Cn(s)An(s)Md(s) − Cn(s)Ad(s)Mn(s))

/Hd(s). (22)

Since the degree ofCd(s) − Cn(s) is larger ofCn(s) by dr, the degree ofMn(s)Ad(s)(Cd(s) − Cn(s)) is larger ofCn(s)Ad(s)Mn(s) by dr. Since the degree ofAd(s) is larger ofAn(s) by≥ dr, while the degree ofMn(s) is largerof Md(s) by dr, the degree ofMn(s)Ad(s)(Cd(s) − Cn(s)) is larger than that ofCn(s)An(s)Md(s). Therefore,H1(s) is strictly proper with relative degreedr. We notice from (12) and (22) thatH1(s) has the same denominator asH(s), and therefore it follows from (10) thatH1(s) is stable. Using similar arguments, it can be verified thatH0(s) isproper and stable. Similarly,H3(s) is strictly proper and stable.

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Let

∆ = ‖H1(s)‖L1‖r‖L∞

+ ‖H0(s)‖L1(L ρ + L0) + γ

(

‖H1(s)/M(s)‖L1+ L‖H0(s)‖L1

‖H3(s)‖L1

1 − ‖G(s)‖L1L

)

,

whereγ > 0 is an arbitrary constant. Since bothH1(s) andM(s) are stable and strictly proper with relative degreedr, andM(s) is minimum-phase,H1(s)/M(s) is stable and proper. Hence,‖H1(s)/M(s)‖L1

exists. Therefore∆ isbounded.

Further, sinceAm is Hurwitz, there existsP = P⊤ > 0 that satisfies the algebraic Lyapunov equationA⊤mP +

PAm = −Q, Q > 0 . From the properties ofP it follows that there exits non-singular√

P such thatP = (√

P )⊤√

P .Given the vectorc⊤m(

√P )−1, let D be a(n − 1) × n matrix that contains the null space ofc⊤m(

√P )−1, i.e.

D(c⊤m(√

P )−1)⊤ = 0 , (23)

and further letΛ =

[

c⊤mD√

P

]

.

Lemma 2 For any ξ =

[

y

z

]

∈ Rn, wherey ∈ R and z ∈ R

n−1, there existp1 > 0 and positive definite

P2 ∈ R(n−1)×(n−1) such thatξ⊤(Λ−1)⊤PΛ−1ξ = p1y

2 + z⊤P2z .

Proof. UsingP = (√

P )⊤√

P , one can write

ξ⊤(Λ−1)⊤PΛ−1ξ = ξ⊤(√

PΛ−1)⊤(√

PΛ−1)ξ . (24)

We notice that

Λ(√

P )−1 =

[

c⊤m(√

P )−1

D

]

.

Lettingq1 = (c⊤m(√

P )−1)(c⊤m(√

P )−1)⊤, Q2 = DD⊤ , it follows from (23) that

(Λ(√

P )−1)(Λ(√

P )−1)⊤ =

[

q1 0

0 Q2

]

. (25)

Non-singularity ofΛ and√

P implies that(Λ(√

P )−1)(Λ(√

P )−1)⊤ is non-singular, and thereforeQ2 is also non-singular. Hence,

(√

PΛ−1)⊤(√

PΛ−1) = (Λ(√

P )−1)(Λ(√

P )−1)⊤)−1 = (Λ(√

P )−1)−⊤(√

PΛ−1) =

[

q−11 0

0 Q−12

]

. (26)

Lettingp1 = q−11 andP2 = Q−1

2 , it follows from (24) that

ξ⊤(√

PΛ−1)⊤(√

PΛ−1)ξ = p1y2 + z⊤P2z , (27)

which completes the proof. �

Let T be any positive constant and11 ∈ Rn be the basis vector with first element1 and all other elements zero.

Let φ(T ) ∈ Rn−1 be a vector which consists of2 to n elements of1⊤

1 eΛAmΛ−1T and let

κ(T ) =

∫ T

0

|1⊤1 eΛAmΛ−1(T−τ)Λbm|dτ . (28)

Further, let

ς(T ) = ‖φ(T )‖√

α

λmax(P2)+ κ(T )∆ , (29)

α = λmax(Λ−⊤PΛ−1)

(

2∆‖Λ−⊤Pbm‖λmin(Λ−⊤QΛ−1)

)2

.

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Letting11e

ΛAmΛ−1t = [η1(t) η⊤2 (t)] , (30)

whereη1(t) ∈ R andη2(t) ∈ Rn−1 contain the first and2 to n elements of the row vector11e

ΛAmΛ−1t, respectively,we introduce the following definitions

β1(T ) = maxt∈[0, T ]

|η1(t)| , (31)

β2(T ) = maxt∈[0, T ]

‖η2(t)‖ . (32)

Further, letΦ(T ) be then × n matrix

Φ(T ) =

∫ T

0

eΛAmΛ−1(T−τ)Λdτ , (33)

and let

β3(T ) = maxt∈[0, T ]

η3(t) , (34)

β4(T ) = maxt∈[0, T ]

η4(t) , (35)

where

η3(t) =

∫ t

0

|1⊤1 eΛAmΛ−1(t−τ)ΛΦ−1(T )eΛAmΛ−1T

11|dτ ,

η4(t) =

∫ t

0

|1⊤1 eΛAmΛ−1(t−τ)Λbm|dτ .

Finally, let

γ0(T ) = β1(T )ς(T ) + β2(T )

√

α

λmax(P2)+ β3(T )ς(T ) + β4(T )∆ . (36)

Lemma 3limT→0

γ0(T ) = 0 . (37)

Proof. Notice that sinceβ1(T ), β3(T ), α and∆ are bounded, it is sufficient to prove that

limT→0

ς(T ) = 0 , (38)

limT→0

β2(T ) = 0 , (39)

limT→0

β4(T ) = 0 . (40)

Since limT→0

1⊤1 eΛAmΛ−1T = 1

⊤1 , then lim

T→0φ(T ) = 0n−1 , which implies lim

T→0‖φ(T )‖ = 0 . Further, it follows from

the definition ofκ(T ) in (28) that limT→0

κ(T ) = 0 . Sinceα and∆ are bounded, thenlimT→0

ς(T ) = 0, which proves (38).

Sinceη2(t) is continuous, it follows from (32) thatlimT→0

β2(T ) = limt→0

‖η2(t)‖ . Hence, the upper bound in (39) follows

from limt→0

1⊤1 eΛAmΛ−1t = 1

⊤1 . Thus,lim

t→0‖η2(t)‖ = 0, which proves (39). Similarlylim

T→0β4(T ) = lim

t→0‖η4(t)‖ = 0 ,

which proves (40). Boundedness ofα, β3(T ) and∆ implies thatlimT→0

(

β1(T )ς(T )+β2(T )

√

α

λmax(P2)+β3(T )ς(T )+

β4(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) + σ(t) (41)

y(t) = c⊤mx(t) , x(0) = x0 = 0 ,

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whereσ(t) ∈ Rn is the vector of adaptive parameters. Notice that whileσ(t) ∈ R in (5), i.e. the unknown disturbance

is matched, the uncertainty estimation in (41) isσ(t) ∈ Rn, i.e. the estimation of it is unmatched. This is the key step

of the solution and the subsequent analysis.Taking the Laplace transform of (41), gives

y(s) = M(s)u(s) + c⊤m(sI − Am)−1σ(s) . (42)

Letting y(t) = y(t) − y(t), the update law forσ(t) is given by

σ(t) = σ(iT ), t ∈ [iT, (i + 1)T ) (43)

σ(iT ) = −Φ−1(T )µ(iT ) , i = 0, 1, 2, · · · , (44)

whereΦ(T ) is defined in (33), and

µ(iT ) = eΛAmΛ−1T11y(iT ) , i = 0, 1, 2, 3, · · · . (45)

The control signal is the output of the low-pass filter:

u(s) = C(s)r(s) − C(s)

c⊤m(sI − Am)−1bm

c⊤m(sI − Am)−1σ(s) . (46)

The completeL1 adaptive controller consists of (41), (44) and (46), subject to theL1-gain upper bound in (10).Let x(t) = x(t) − x(t). The error dynamics between (5) and (41) are

˙x(t) = Amx(t) + σ(t) − bmσ(t) (47)

y(t) = c⊤mx(t), x(0) = 0 .

Lemma 4 Let e(t) = y(t) − yref (t). Then

‖et‖L∞≤

‖H3(s)‖L1

1 − ‖G(s)‖L1L‖yt‖L∞

. (48)

Proof. Let

σ(s) =C(s)

c⊤m(sI − Am)−1bm

c⊤m(sI − Am)−1σ(s) − C(s)σ(s) . (49)

It follows from (46) that

u(s) = C(s)r(s) − C(s)σ(s) − σ(s) , (50)

and the system in (3) consequently takes the form:

y(s) = M(s)(

C(s)r(s) + (1 − C(s))σ(s) − σ(s))

. (51)

Substitutingu(s) from (50) into (4), we have

σ(s) =(

(

A(s) − M(s))(

C(s)r(s) − C(s)σ(s) − σ(s))

+ A(s)d(s))

/M(s) , (52)

and hence

σ(s) =

(

A(s) − M(s))(

C(s)r(s) − σ(s))

+ A(s)d(s)

M(s) + C(s)(A(s) − M(s)). (53)

Using the definitions ofH0(s) andH1(s) in (18) and (19), we can rewriteσ(s) into the following form:

σ(s) = H1(s)r(s) −H1(s)

C(s)σ(s) + H0(s)d(s) . (54)

Substitution into (51) leads toy(s) = M(s)(

C(s)+H1(s)(1−C(s))) (

r(s) − σ(s)C(s)

)

+H0(s)M(s)(

1−C(s))

d(s) .

Recalling the definition ofH(s) from (9), one can verify thatM(s)(C(s) + H1(s)(1 − C(s))) = H(s)C(s),

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H(s) = H0(s)M(s), which consequently impliesy(s) = H(s)(

C(s)r(s) − σ(s))

+ H(s)(1 − C(s))d(s) . Us-

ing the expression foryref (s) from (15), one can derivee(s) = H(s) ((1 − C(s))de(s) − σ(s)), wherede(s) isintroduced to denote the Laplace transform ofde(t) = f(t, y(t)) − f(t, yref (t)). Lemma 5 and Assumption 1 givethe following upper bound:

‖et‖L∞≤ L‖H(s)(1 − C(s))‖L1

‖et‖L∞+ ‖r1t

‖L∞, (55)

wherer1(t) is the signal with its Laplace transform being

r1(s) = H(s)σ(s). (56)

Using the expression forσ(s) from (49) along with the expressions fory(s) from (3) andy(s) from (42), one canderive

y(s) = c⊤m(sI − Am)−1σ(s) − M(s)σ(s)

=M(s)

C(s)

C(s)

M(s)c⊤m(sI − Am)−1σ(s) − M(s)

C(s)C(s)σ(s) =

M(s)

C(s)σ(s) . (57)

This implies thatr1(s) can be rewritten asr1(s) = C(s)H(s)M(s)

M(s)C(s) σ(s) = H3(s)y(s), and hence‖r1t

‖L∞≤

‖H3(s)‖L1‖yt‖L∞

. Substituting this back into (55) completes the proof.

VI. Analysis of L1 Adaptive Controller

In this section, we analyze the stability and the performance of theL1 adaptive controller. Using the definitionsfrom (11),H2(s) in (20) can be rewritten as

H2(s) =−Cn(s)Ad(s)Mn(s)

Hd(s), (58)

whereHd(s) is defined in (13). Sincedeg(Cd(s)−Cn(s))−degCn(s) = dr, it can be checked straightforwardly thatH2(s) is strictly proper. We notice from (12) and (58) thatH2(s) has the same denominator asH(s), and therefore itfollows from (10) thatH2(s) is stable. SinceH2(s) is strictly proper and stable with relative degreedr, H2(s)/M(s)is stable and proper and, therefore itsL1 gain is finite.

Consider the state transformation:ξ = Λx . (59)


˙ξ(t) = ΛAmΛ−1ξ(t) + Λσ(t) − Λbmσ(t) (60)

y(t) = ξ1(t) , (61)

whereξ1(t) is the first element ofξ(t).

Theorem 1 Given the system in (1) and theL1 adaptive controller in (41), (44), (46) subject to (10), if we chooseTto ensure

γ0(T ) < γ , (62)

whereγ is an arbitrary positive constant introduced in (23), then

‖y‖L∞< γ (63)

‖y − yref‖L∞≤ γ1 (64)

‖u − uref‖L∞≤ γ2, (65)

with γ1 = ‖H3(s)‖L1γ/(1 − ‖G(s)‖L1

L), γ2 = L‖H3(s)‖L1γ1 + ‖H2(s)/M(s)‖L1

γ .

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Proof. First we prove the bound in (63) by a contradiction argument.Sincey(0) = 0 andy(t) is continuous, thenassuming the opposite implies that there existst′ such that

‖y(t)‖ < γ, ∀ 0 ≤ t < t′ , (66)

‖y(t′)‖ = γ , (67)

which leads to‖yt′‖L∞

= γ . (68)

Sincey(t) = yref (t) + e(t), the upper bound in (16) can be used to arrive at

‖yt′‖L∞≤ ‖yref

t′‖L∞

+ ‖et′‖L∞≤ ρ + ‖C(s)H(s)/M(s)‖L1

γ/(1 − ‖G(s)‖L1L) .

It follows from (54) and (57) thatσ(s) = H1(s)r(s) − H1(s)y(s)/M(s) + H0(s)d(s), and hence (68) implies that‖σt′‖L∞

≤ ‖H1(s)‖L1‖r‖L∞

+ ‖H1(s)/M(s)‖L1γ + ‖H0(s)‖L1

(L‖yt′‖L∞+ L0), which along with (69) leads to

‖σt′‖L∞≤ ∆ . (69)


ξ(iT + t) = eΛAmΛ−1tξ(iT ) +∫ iT+t

iT

eΛAmΛ−1(iT+t−τ)Λσ(iT )dτ −∫ iT+t

iT

eΛAmΛ−1(iT+t−τ)Λbmσ(τ)dτ (70)

= eΛAmΛ−1tξ(iT ) +

∫ t

0

eΛAmΛ−1(t−τ)Λσ(iT )dτ −∫ t

0

eΛAmΛ−1(t−τ)Λbmσ(iT + τ)dτ .

Sinceξ(iT ) =

[

y(iT )

0

]

+

[

0

z(iT )

]

, it follows from (70) that

ξ(iT + t) = χ(iT + t) + ζ(iT + t) , (71)

where

χ(iT + t) = eΛAmΛ−1t

[

y(iT )

0

]

+

∫ t

0

eΛAmΛ−1(t−τ)Λσ(iT )dτ , (72)

ζ(iT + t) = eΛAmΛ−1t

[

0

z(iT )

]

−∫ t

0

eΛAmΛ−1(t−τ)Λbmσ(iT + τ)dτ . (73)

In what follows, we prove that for alliT ≤ t′ one has

|y(iT )| ≤ ς(T ) , (74)

z⊤(iT )P2z(iT ) ≤ α , (75)

whereς(T ) andα are defined in (29). Sinceξ(0) = 0, it is straightforward that|y(0)| ≤ ς(T ) , z⊤(0)P2z(0) ≤ α .For any(j + 1)T ≤ t′, we will prove that if

|y(jT )| ≤ ς(T ) , (76)

z⊤(jT )P2z(jT ) ≤ α , (77)

then (76)-(77) hold forj + 1 too. Hence, (74)-(75) hold for alliT ≤ t′.Assume (76)-(77) hold forj, and in addition,

(j + 1)T ≤ t′ . (78)


ξ((j + 1)T ) = χ((j + 1)T ) + ζ((j + 1)T ) , (79)

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where

χ((j + 1)T ) = eΛAmΛ−1T

[

y(jT )

0

]

+

∫ T

0

eΛAmΛ−1(T−τ)Λσ(jT )dτ , (80)

ζ((j + 1)T ) = eΛAmΛ−1T

[

0

z(jT )

]

−∫ T

0

eΛAmΛ−1(T−τ)Λbmσ(jT + τ)dτ . (81)

Substituting the adaptive law from (44) in (80), we have

χ((j + 1)T ) = 0 . (82)

It follows from (73) thatζ(t) is the solution of the following dynamics:

ζ(t) = ΛAmΛ−1ζ(t) − Λbmσ(t) , (83)

ζ(jT ) =

[

0

z(jT )

]

, t ∈ [jT, (j + 1)T ]. (84)

Consider the following function:

V (t) = ζ⊤(t)Λ−⊤PΛ−1ζ(t) , (85)

overt ∈ [iT, (i + 1)T ]. SinceΛ is non-singular andP is positive definite,Λ−⊤PΛ−1 is positive definite and, hence,V (t) is a positive definite function. It follows from (83) that over t ∈ [jT, (j + 1)T ]

V (t) = ζ⊤(t)Λ−⊤PΛ−1ΛAmΛ−1ζ(t) + ζ⊤(t)Λ−⊤A⊤mΛ⊤Λ−⊤P⊤Λ−1ζ(t) − 2ζ⊤(t)Λ−⊤PΛ−1Λbmσ(t)

= ζ⊤(t)Λ−⊤(PAm + A⊤mP⊤)Λ−1ζ(t) − 2ζ⊤(t)Λ−⊤Pbmσ(t) (86)

= −ζ⊤(t)Λ−⊤QΛ−1ζ(t) − 2ζ⊤(t)Λ−⊤Pbmσ(t) .

Using the upper bound from (69) we can compute the following upper bound overt ∈ [iT, (i + 1)T ]

V (t) ≤ −λmin(Λ−⊤QΛ−1)‖ζ(t)‖2 + 2‖ζ(t)‖‖Λ−⊤Pbm‖∆ . (87)

Notice that for allt ∈ [jT, (j + 1)T ], ifV (t) ≥ α , (88)

we have‖ζ(t)‖ ≥√

α

λmax(Λ−⊤PΛ−1)=

2∆‖Λ−⊤Pbm‖λmin(Λ−⊤QΛ−1)

, and hence the upper bound in (87) yields

V (t) ≤ 0 . (89)

It follows from Lemma 2 and the relationship in (84) thatV (ζ(jT )) = z⊤(jT )P2z(jT ) , which further along withthe upper bound in (77) leads to the following

V (ζ(jT )) ≤ α . (90)

It follows from (88)-(89) and (90) that

V (t) ≤ α , ∀ t ∈ [jT, (j + 1)T ] , (91)

and thereforeV ((j + 1)T ) = ζ⊤((j + 1)T )(Λ−⊤PΛ−1)ζ((j + 1)T ) ≤ α . (92)

Sinceξ((j + 1)T ) = χ((j + 1)T ) + ζ((j + 1)T ) , (93)

then the equality in (82) and the upper bound in (92) lead to the following inequalityξ⊤((j +1)T )(Λ−⊤PΛ−1)ξ((j +1)T ) ≤ α . Using the result of Lemma 2 one can derive thatz⊤((i+1)T )P2z((i+1)T ) ≤ ξ⊤((i+1)T )(Λ−⊤PΛ−1)ξ((i+1)T ) ≤ α, which implies that the upper bound in (77) holds forj + 1.

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It follows from (61), (82) and (93) thaty((j + 1)T ) = 1⊤1 ζ((j + 1)T ) , and the definition ofζ((j + 1)T ) in (81)

leads to the following expression:

y((j + 1)T ) = 1⊤1 eΛAmΛ−1T

[

0

z(jT )

]

− 1⊤1

∫ T

0

eΛAmΛ−1(T−τ)Λbmσ(jT + τ)dτ . (94)

The upper bounds in (69) and (77) allow for the following upper bound:

|y((j + 1)T )| ≤ ‖φ(T )‖ ‖z(iT )‖ +

∫ T

0

|1⊤1 eΛAmΛ−1(T−τ)Λbm| |σ(jT + τ)|dτ

≤ ‖φ(T )‖√

α

λmax(P2)+ κ(T )∆ = ς(T ) , (95)

whereφ(T ) andκ(T ) are defined in (28), andς(T ) is defined in (29). This confirms the upper bound in (76) forj +1.Hence, (74)-(75) hold for alliT ≤ t′.

For all iT + t ≤ t′, where0 ≤ t ≤ T , using the expression from (70) we can write that

y(iT + t) = 11eΛAmΛ−1tξ(iT ) + 1

⊤1

∫ t

0

eΛAmΛ−1(t−τ)Λσ(iT )dτ − 1⊤1

∫ t

0

eΛAmΛ−1(t−τ)Λbmσ(iT + τ)dτ .

The upper bound in (69) and definitions ofη1(t), η2(t), η3(t) andη4(t) allow for the following representation

|y(iT + t)| ≤ |η1(t)| |y(iT )| + ‖η2(t)‖ ‖z(iT )‖ + η3(t)|y(iT )| + η4(t)∆ . (96)

Taking into consideration (76)-(77) and recalling the definitions of β1(T ), β2(T ), β3(T ), β4(T ) in (31)-(35), for all0 ≤ t ≤ T and for any non-negative integeri subject toiT + t ≤ t′, we have

|y(iT + t)| ≤ β1(T )ς(T ) + β2(T )

√

α

λmax(P2)+ β3(T )ς(T ) + β4(T )∆ (97)

Since the right hand side coincides with the definition ofγ0(T ) in (36), then for allt ∈ [0, t′]

|y(t)| ≤ γ0(T ) , (98)

which along with our assumption onT introduced in (62) yields

‖yt′‖L∞< γ . (99)

This clearly contradicts the statement in (68). Therefore,‖y‖L∞< γ, which proves (63). Further, it follows from (48)

that‖et‖L∞≤ ‖H3(s)‖

L1

1−‖G(s)‖L1L

γ, which holds uniformly for allt ≥ 0, and therefore leads to the upper bound in (64).It follows from (50) and (53) that

u(s) =M(s)C(s)r(s) − M(s)σ(s) − C(s)A(s)d(s)

C(s)A(s) + (1 − C(s))M(s).

To prove the bound in (65), we use the expression ofuref (s) from (14) to derive

u(s) − uref (s) = −H3(s)r2(s) +H2(s)

C(s)σ(s) = −H3(s)r2(s) +

H2(s)

M(s)

M(s)

C(s)σ(s) , (100)

wherer2(t) = f(t, y(t)) − f(t, yref (t)) . It follows from (57) and (100) that‖u − uref‖L∞≤ L‖H3(s)‖L1

‖y −yref‖L∞

+ ‖H2(s)/M(s)‖L1‖y‖L∞

, which along with (63)-(64) leads to (65). The proof is complete. �

The main result can be summarized as follows.

Theorem 2 Given the system in (1) and theL1 adaptive controller in (41), (44), (46) subject to (10), we have:

limT→0

(y(t) − yref (t)) = 0 , ∀ t ≥ 0, (101)

limT→0

(u(t) − uref (t)) = 0 , ∀ t ≥ 0 . (102)

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The proof follows from Theorem 1 directly. Thus, the tracking error betweeny(t) andyref (t), as well betweenu(t)anduref (t), is uniformly bounded by a constant proportional toT . This implies that during the transient one canachieve arbitrary improvement of tracking performance forboth signals simultaneously by uniformly reducingT .

Remark 1 Notice that the parameterT is the fixed time-step in the definition of the adaptive law. The adaptiveparameters inσ(t) ∈ R

n take constant values during[iT, (i + 1)T ) for everyi = 0, 1, · · · . ReducingT imposeshardware requirements and implies that the performance limitations are consistent with the hardware limitations.This in turn is consistent with the results in Refs.,1, 17 where improvement of the transient performance was achievedby increasing the adaptation rate in the continuous adaptive laws.

Remark 2 We notice that the followingidealcontrol signaluideal(t) = r(t)−σref (t) is the one that leads to desiredsystem responseyideal(s) = M(s)r(s) by cancelling the uncertainties exactly. Thus, the reference system in (6)-(8)has a different response as compared to this one. It only cancels the uncertainties within the bandwidth ofC(s), whichcan be selected comparable to the control channel bandwidth. This is exactly what one can hope to achieve with anyfeedback in the presence of uncertainties.

Remark 3 We notice that stability ofH(s) is equivalent to stabilization ofA(s) by

C(s)

M(s)(1 − C(s)). (103)

Indeed, consider the closed-loop system, comprised of the systemA(s) and negative feedback of (103). The closed-loop transfer function is:

A(s)

1 + A(s) C(s)M(s)(1−C(s))

. (104)

Incorporating (11), one can verify that the denominator of the system in (104) is exactlyHd(s). Hence, stability ofH(s) is equivalent to the stability of the closed-loop system in (104). This implies that the class of systemsA(s) thatcan be stabilized by theL1 adaptive output feedback controller (41), (44) and (46) is not empty.

Remark 4 Finally, we notice that while the feedback in (103) may stabilize the system in (1) for some classes ofunknown nonlinearities, it will not ensure uniform transient performance in the presence of unknownA(s). On thecontrary, theL1 adaptive controller ensures uniform transient performance for system’s both signals, independent ofthe unknown nonlinearity and independent ofA(s).

VII. Simulations

Numerical Example.As an illustrative example, consider the system in (1) withA(s) = (s + 1)/(s3 − s2 − 2s + 8) . We notice that

A(s) has poles in the right half complex plane. We considerL1 adaptive controller defined via (41), (44) and (46),where

M(s) =1

s2 + 1.4s + 1, C(s) =

100

s2 + 14s + 100, T = 10−4 .

First, we consider the step response whend(t) = 0. The simulation results ofL1 adaptive controller are givenin Figs 1(a)-1(b). Next, we considerd(t) = f(t, y(t)) = sin(0.1t)y2(t) + sin(0.4t), and apply the same controllerwithout re-tuning. The control signal and the system response are plotted in Figs. 2(a)-2(b). Further, we consider atime-varying reference inputr(t) = 0.5 sin(0.3t) and notice that without any re-tuning of the controller the systemresponse and the control signal behave as expected, Figs. 3(a)-3(b).

Missile Longitudinal Autopilot Design.

We consider the missile dynamics from Ref.20, which is given by the following state-space representation:

x(t) =

Za 1 Zd 0

Ma 0 Md 0

0 0 0 1

0 0 −ω2a −2ζaωa

x(t) + ∆Ax(t) +

0

0

0

ω2a

(

u(t) + k⊤mx(t)

)

y(t) =[

1 0 0 0]

x(t) ,

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0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time t

(a) y(t) (solid) andr(t) (dashed)

0 5 10 15 20−2

0

2

4

6

8

time t

(b) Time-history ofu(t)

Figure 1. Performance forr(t) = 1 and d(t) = 0.

0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time t


0 5 10 15 20−2

0

2

4

6

8

time t


Figure 2. Performance forr(t) = 1 and d(t) = sin(0.1t)y2(t) + sin(0.4t).

0 10 20 30 40−0.5

0

0.5

Time t


0 10 20 30 40−5

0

5

time t


Figure 3. Performance forr(t) = 0.5 sin(0.3t) and d(t) = sin(0.1t)y2(t) + sin(0.4t).

wherex(t) = [ Az q δa δa ]⊤ is the system state, in whichAz is the vertical acceleration,q is the pitch rate,

δa is the fin deflection, andδa is the fin deflection rate,km is the vector of matched parametric uncertainties, and∆Acontains both unmatched and additional matched uncertainties. In simulations, the following numerical values havebeen used for the missile dynamics20:

Za = −1.3046, Zd = −0.2142, Ma = 47.7109, Md = −104.8346,

∆A =

−K 0 −K 0

K 0 −K 0

0 0 0 0

a1 a2 a3 a4

, km = 0.3

0

0

−1

−2 ζa

ωa

,

whereK is the uncertainty scaling factor for unmatched uncertainties, anda1, a2, a3, a4 can assume arbitrary values,modeling additional matched uncertainties. For simulation of theL1 adaptive output feedback controller, the following

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(a) System response (b) Control history

Figure 4. Simulation results for missile longitudinal autopilot design

M(s) andC(s) have been selected, along with a sampling rateT = 0.00001 s:

M(s) =1

1/ωs2 + 2ζωs + 1, C(s) =

1

1/ωcs2 + 2ζcωcs + 1.

whereω = 10 rads , ζ = 0.8, ωc = 150 rad

s , andζc = 0.8. Figure 9 shows scaled response for scaled uncertainties,with K denoting the scaling factor for unmatched uncertainties. We notice that the system response does not changefrom variation ofa1, a2, a3, a4.

Flexible Wing Application

We now apply theL1 adaptive output feedback control architecture to a highly flexible semi-span wind tunnelmodel, considered earlier in the work of authors in Ref.21 The wind-tunnel model is instrumented with accelerometersalong the spar, strain gauges at the root and mid-spar, a rategyro at the wing tip, a gust sensor vane in front of thewing, and a rate gyro and accelerometers at the tunnel attachment point. The accelerometers, strain gauges and rategyro allow the control system to sense the bending modes and the structural stresses. The test objectives are, first, tocontrol the first and second bending modes of the wing, while executing altitude hold and controlling pitch momentat the pivot point. The second control objective is GLA/flutter suppression, particularly around the frequency of thefirst bending mode. In the following figures, we compare the performance ofL1 adaptive output feedback controllerto a baseline linear design that was provided with the model.Control signals are generated by the leading and thetrailing edge control surfaces. In Figs. 5(a)-5(b), we notice that theL1 adaptive output feedback controller leads to thesame performance as the baseline controller in the absence of turbulence. In the presence of turbulence, Figs. 6(a)-7(b)demonstrate that theL1 adaptive output feedback controller improves its performance in the presence of turbulence.We further modify the originalA matrix of the system by adding0.1 to its every element. Figs. 8(a)-9(b) demonstratethat theL1 adaptive controller outperforms the baseline controller for this class of uncertainty as well.

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0 2 4 6 80

0.02

0.04

0.06

0.08

0.1Output y(t)

time t

Ver

tical

dis

plac

emen

t, fe

et

Attidue outputCommand

(a) Tracking performance

0 2 4 6 8−0.15

−0.1

−0.05

0

0.05Control Signal u(t)

time t

Sur

face

def

lect

ion,

deg

LETE1TE2TE3TE4

(b) Control history

Figure 5. Simulation results of baseline andL1 adaptive controller in the absence of turbulence

0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12Output y(t)

time t

Ver

tical

dis

plac

emen

t, fe

et



0 2 4 6 8−0.15

−0.1

−0.05

0

0.05


time t

Sur

face

def

lect

ion,

deg

LETE1TE2TE3TE4

(b) Control history

Figure 6. Simulation results of baseline controller in the presence of turbulence

0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12Output y(t)

time t

Ver

tical

dis

plac

emen

t, fe

et



0 2 4 6 8−0.4

−0.3

−0.2

−0.1

0

0.1

0.2


time t

Sur

face

def

lect

ion,

deg

LETE1TE2TE3TE4

(b) Control history

Figure 7. Simulation results ofL1 adaptive controller in the presence of turbulence

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0 2 4 6 80

0.2

0.4

0.6

0.8

1Output y(t)

time t

Ver

tical

dis

plac

emen

t, fe

et



0 2 4 6 8−1.5

−1

−0.5

0

0.5

1

1.5

2Control Signal u(t)

time t

Sur

face

def

lect

ion,

deg

LETE1TE2TE3TE4

(b) Control history

Figure 8. Simulation results of baseline controller in the presence of additive uncertainty in theA matrix

0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12Output y(t)

time t

Ver

tical

dis

plac

emen

t, fe

et



0 2 4 6 8−0.3

−0.2

−0.1

0

0.1

0.2

0.3


time t

Sur

face

def

lect

ion,

deg

LETE1TE2TE3TE4

(b) Control history

Figure 9. Simulation results ofL1 adaptive controller in the presence of additive uncertainty in the A matrix

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VIII. Conclusions

We presented theL1 adaptive output feedback controller for reference systemsthat do not verify the SPR conditionfor their input-output transfer function. The new piece-wise constant adaptive law along with low-pass filtered controlsignal ensures uniform performance bounds for system’s both input/output signals simultaneously. The performancebounds can be systematically improved by reducing the integration time-step.

References1C. Cao and N. Hovakimyan.L1 adaptive output feedback controller for systems of unknowndimension.IEEE Transactions on Automatic

Control, 53:815–821, April 2008.2K.S. Tsakalis and P.A. Ioannou. Adaptive control of linear time-varying plants.Automatica, 23:459–468, 1987.3R. H. Middleton and G. C. Goodwin. Adaptive control of time-varying linear systems.IEEE Trans. Autom. Contr., 33(1):150–155, 1988.4G. Bartolini, A. Ferrara, and A. A. Stotsky. Robustness and performance of an indirect adaptive control scheme in the presence of bounded

disturbances.IEEE Trans. Autom. Contr., 44(4):789–793, April 1999.5J. Sun. A modified model reference adaptive control scheme for improved transient performance.IEEE Trans. Autom. Contr., 38(7):1255–

1259, July 1993.6D.E. Miller and E.J. Davison. Adaptive control which provides an arbitrarily good transient and steady-state response. IEEE Trans. Autom.

Contr., 36(1):68–81, January 1991.7R. Costa. Improving transient behavior of model-referenceadaptive control.Proc. of American Control Conference, pages 576–580, 1999.8B.E. Ydstie. Transient performance and robustness of direct adaptive control.IEEE Trans. Autom. Contr., 37(8):1091–1105, August 1992.9M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos. Transient performance improvement with a new class of adaptive controllers. Systems

& Control Letters, 21:451–461, 1993.10R. Ortega. Morse’s new adaptive controller: Parameter convergence and transient performance.IEEE Trans. Autom. Contr., 38(8):1191–

1202, August 1993.11Z. Zang and R. Bitmead. Transient bounds for adaptive control systems.Proc. of30th IEEE Conference on Decision and Control, pages

2724–2729, December 1990.12Y. Zang and P. A. Ioannou. Adaptive control of linear time-varying systems.In Proc. of IEEE 35th Conf. Decision Contr., pages 837–842,

1996.13A. Datta and P. Ioannou. Performance analysis and improvement in model reference adaptive control.IEEE Trans. Autom. Contr.,

39(12):2370–2387, December 1994.14K. S. Narendra and J. Balakrishnan. Improving transient response of adaptive control systems using multiple models andswitching. IEEE

Trans. Autom. Contr., 39(9):1861–1866, September 1994.15R. Marino and P. Tomei. An adaptive output feedback control for a class of nonlinear systems with time-varying parameters. IEEE Trans.

Autom. Contr., 44(11):2190–2194, 1999.16A. M. Arteaga and Y. Tang. Adaptive control of robots with an improved transient performance.IEEE Trans. Autom. Contr., 47(7):1198–

1202, July 2002.17C. Cao and N. Hovakimyan. Design and analysis of a novelL1 adaptive control architecture with guaranteed transient performance.IEEE

Transactions on Automatic Control, 53:586–591, March 2008.18J. Wang, C. Cao, N. Hovakimyan, R. hindman, and B. Ridgely. ”L1 Adaptive Controller for a Missile Longitudinal Autopilot Design”.

AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA-2008-6282, 2008.19E. Kharisov, I. Gregory, C. Cao, and N. Hovakimyan. ”L1 Adaptive Control Law for Flexible Space Launch Vehicle and Proposed Plan for

Flight Test Validation”.AIAA Guidance, Navigation, and Control Conference and Exhibit, 2008.20K. Wise. ”Robust Stability Analysis of Adaptive Missile Autopilots”. AIAA Guidance, Navigation, and Control Conference and Exhibit,

2008.21I. Gregory, V. Patel, C. Cao, and N. Hovakimyan.L1 adaptive control laws for flexible wind tunnel model of high-aspect ratio flying wing.

In Proc. of AIAA Guidance, Navigation & Control Conf., Hilton Head Island, SC, 2007. AIAA 2007-6525.22H. K. Khalil. Nonlinear Systems. Prentice Hall, Englewood Cliffs, NJ, 2002.

IX. Appendix

Definition 1 22 For a signal ξ(t), t ≥ 0, ξ ∈ Rn, its truncatedL∞ and L∞ norms are‖ξt‖L∞

= maxi=1,..,n

( sup0≤τ≤t

|ξi(τ )|),

‖ξ‖L∞= max

i=1,..,n(supτ≥0

|ξi(τ )|), whereξi is theith component ofξ.

Definition 2 22 TheL1 gain of a stable proper SISO system is defined||H(s)||L1=

∫ ∞

0|h(t)|dt, whereh(t) is the impulse

response ofH(s).

Lemma 5 22 For a stable proper multi-input multi-output (MIMO) systemH(s) with inputr(t) ∈ Rm and outputx(t) ∈ R

n, wehave‖xt‖L∞

≤ ‖H(s)‖L1‖rt‖L∞

,∀ t ≥ 0.

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