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Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2012, Article ID 502149, 16 pagesdoi:10.1155/2012/502149

Research Article

Adaptive Control Allocation in the Presence of Actuator Failures

Yu Liu and Luis G. Crespo

National Institute of Aerospace, Hampton, VA 23666, USA

Correspondence should be addressed to Yu Liu, [email protected]

Received 3 November 2011; Revised 17 February 2012; Accepted 2 March 2012

Academic Editor: Yunjun Xu

Copyright © 2012 Y. Liu and L. G. Crespo. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper proposes a control allocation framework where a feedback adaptive signal is designed for a group of redundant actuatorsand then it is adaptively allocated among all group members. In the adaptive control allocation structure, cooperative actuatorsare grouped and treated as an equivalent control effector. A state feedback adaptive control signal is designed for the equivalenteffector and adaptively allocated to the member actuators. Two adaptive control allocation algorithms, guaranteeing closed-loopstability and asymptotic state tracking when partial and total loss of control effectiveness occur, are developed. Proper groupingof the actuators reduces the controller complexity without reducing their efficacy. The implementation and effectiveness of thestrategies proposed is demonstrated in detail using several examples.

1. Introduction

Actuator redundancy is highly desirable for fault-tolerantcontrol. This redundancy yields multiple ways to implementthe forces and moments prescribed by the controller. How-ever, this freedom creates the need for properly allocatingthe control inputs among all individual actuators. Whilemultiple actuator configurations do generate the desiredforces and moments, some of them may yield unintendedoutcomes, for example, the effect of some control surfacedeflections counteracts the effect of other ones. Redundancymanagement is the need for properly allocating the controlinputs among functionally redundant actuators when someof them may not be fully functional.

The purpose of control allocation is to distribute thecontrol signals to the available actuators such that the desiredmoments and forces are efficiently generated. Traditionalcontrol allocation methods include explicit ganging [1],daisy chaining [2], pseudinverse [3, 4], and error andcontrol minimization [5–10]. Explicit ganging performscontrol allocation by finding a fixed relation between thedesired control moments and forces and the designedcontrol signals. Multiple actuators (e.g., two aileron surfaces)can be combined to generate the desired effects. Daisychaining allocates inputs in a prioritized fashion. It utilizes

the actuators in sequence to generate certain effect. If anactuator is unable to generate such an effect, say due toactuator saturation, the next actuator in the sequence willbe used. The pseudoinverse approach, which accounts forinput saturation and failure, performs control allocationby solving a linear optimization problem in real time.Error and control minimization is another common controlallocation approach. This approach minimizes the errorbetween the desired and generated control moments subjectto control constraints. Several approaches can accommodatefor actuator failure and saturation compensation, providedthat the actuator failure or saturation has been identified.Thus one obvious drawback of these approaches is thatthey require a fault detection system. These systems, whichare usually complex, require the characterization of severalfailure modes a priori. Furthermore, they may requiresolving optimization problems in real time; thus they cansubstantially increase the software and hardware demands ofthe flight control system.

Adaptive control, on the other hand, does not requireknowing which controllers have failed nor the class orseverity of the failure. This is due to its ability to changecontrol parameters according to the existing flight con-dition. Due to parameter adaptation, it is also able toaccommodate for parametric uncertainties in the vehicle

2 Journal of Control Science and Engineering

dynamics. Substantial developments in adaptive control foractuator failures have been made in the last decade [11,12]. Adaptive control’s ability to seamlessly compensate foractuator failures requires for the system’s built-in actuationredundancy to be sufficient. This is usually described as arank condition on the input matrix B [11, 13]. To take advan-tage of all redundancy available, one common approachin multivariable adaptive control is to generate a controlsignal for each control surface. Such an approach endows thecontroller the maximum degree of freedom to compensatefor failure. When certain control surfaces are stuck or havereduced control effectiveness, the remaining control surfaceswill adaptively cooperate until a new combination of inputsfor the remaining control surfaces is found. This will occurautomatically without knowing which surfaces have failed,or when such failures occur. Although adaptive controlensures closed-loop stability and tracking performance,it does not differentiate between control generation andcontrol allocation, and the resulting actuation scheme maybe unacceptable. For instance, separately designed controlsignals for multiple control surface segments may cancel eachother’s effects, for example, it has been observed that thesteady-state deflection of both rudder segments for a directadaptive control had opposite angles, resulting in a wingsleveled flight with increased drag.

The lack of control allocation in the current directadaptive control framework motivates this research effort.In this study, we aim at separating control generation fromcontrol allocation. In the adaptive control allocation frame-work, a key step is to combine redundant control surfacessimilar to explicit ganging. For each group of combinedactuators, we design an adaptive control signal, which is thenallocated among group members by an adaptive gain. If nofailure occurs, a nonadaptive control allocation scheme setin advance is enforced. In the presence of uncertainty ofactuator failure, the adaptive flight controller modifies theallocation of input accordingly.

The structure of the adaptive control allocation frame-work is illustrated in Figure 1. This is a simple aircraft controlexample with the elevator controlling the pitch motion.The aircraft longitudinal state, denoted as x(t), should trackthe state of a reference system for a given reference inputr(t). The elevator consists of four segments, namely, leftoutboard, left inboard, right outboard, and right inboardsegments. For pitch control, they can be grouped togetherand considered as an equivalent elevator by adding the fourcolumns of the input matrix. A “virtual” elevator signalv0 is generated by the adaptive controller for the desiredpitching maneuver. This elevator signal is then allocated bythe adaptive allocation gains αi(t), i = 1, . . . , 4. The resultingelevator signals v0i(t) = αi(t)v0(t), i = 1, . . . , 4 will be fedto the four elevator segments. The allocation gains can beupdated on-line based on the knowledge of the nominalplant and v0 to mitigate the uncertainties of actuator failures.Conversely, in a fixed allocation scheme αi are constant, forexample, αi = 1/4, i = 1, . . . , 4.

One advantage of this adaptive control allocation struc-ture is the ease in solving problems such as the counteractingactuation. To remedy this problem, the signs of the adaptive

allocation gains αi can be made the same, say, via theprojection algorithm [14] so that the allocated control signalscannot go in opposite directions. Another advantage of thisstructure is the reduction of the controller complexity. Forinstance, let us consider a state feedback adaptive controldesign for an n-state system with m controls. The numberof controller parameters to be updated is m × n. If theproblem is solved by having all actuators in one group,there are m + n adaptive parameters. For example, if wetry to stabilize a 5-state system with 3 controls, the totaladaptive parameters using a direct model reference adaptivecontroller would be 15. With a grouping of control inputsfor which the system remains controllable (i.e., the systemis controllable for the control input matrix associated withthe grouping), the total number of adaptive parametersrequired is 8 (1 × 5 gain vector and 3 α’s). The largerthe family of redundant actuators, the bigger the benefit.Another advantage of this approach is that the virtual controlsignal can be designed using a nonadaptive (fixed) statefeedback gain, and failure compensation is achieved by onlyadapting the control allocation gains α’s. This implies thatthe proposed structure could be added to a conventionallydesigned state feedback controllers without having to usefault detection and isolation.

In this paper, we develop the mathematical foundation ofthis adaptive control allocation structure for a single groupof actuators. Two adaptive control allocation algorithmsare presented for both loss of effectiveness and constant-magnitude actuator failures. Technical issues such as designconditions, adaptive law designs, and stability analysis areaddressed. The proposed schemes are shown to guaranteestability and asymptotic state tracking in the presence ofunknown failures. Simulation-based examples are used todemonstrate the strategy proposed.

The paper is organized as follows. Section 2 presentsthe adaptive control allocation algorithm for loss of controleffectiveness. This is followed by Section 3, where a schemefor constant-magnitude actuator failure compensation isdeveloped in Section 3. Finally, a few concluding remarksclose the paper.

2. Adaptive Control Allocation Design forLoss of Effectiveness Failures

2.1. Problem Formulation

2.1.1. Plant. Consider the linear time-invariant (LTI) system

x(t) = Ax(t) + Bu(t), (1)

where x ∈ Rn and u ∈ Rm are the system state and the controlinput. The matricesA ∈ Rn×n and B ∈ Rn×m are constant andknown. The matrix B is the control gain matrix for the groupof actuators.

The control signal u(t) can be expressed as

u(t) = Λv(t), (2)

where v(t) is the allocated control signal (i.e., input to theactuator) and Λ is a piecewise constant uncertain diagonal

Journal of Control Science and Engineering 3

Longitudinal dynamics

LOB

LIB

ROB

RIB elevator

elevator

elevator

elevator

x(t)r(t) v0Controller

Control generation

Control allocation

α1(t)

α2(t)

α3(t)

α4(t)

α1υ0

α2υ0

α3υ0

α4υ0

Figure 1: Aircraft pitch control with adaptive control allocation.

control effectiveness matrix with Λ = diag{λ1, λ2, . . . , λm},and 0 < λi ≤ 1, i = 1, . . . ,m. The ith actuator is fullyfunctional when λi = 1 and has a loss of effectivenessfailure when 0 < λi < 1. For the design in this section, weassume that λi /= 0, that is, no actuator outage occurs. For thereminder of this section we assume that Λ is always positivedefinite.

The control input v(t) = [v1(t), v2(t), . . . , vm(t)]� is givenby

vj(t) = αj(t)v0(t), j = 1, 2, . . . ,m, (3)

where v0(t) is a control signal designed for the group,and αj(t) is the adaptive allocation gain for jth actuator.We assume that a nominal allocation scheme has beenprescribed. This scheme will be enforced under nominaloperating conditions. The nominal allocation gain vector isdenoted as α∗ ∈ Rm. We also define the equivalent controlgain vector as b0 � Bα∗. The vector b0 can be seen as theequivalent control gain matrix for the equivalent controleffector representing the actuator group. We further assumethat the pair (A, b0) is stabilizable.

2.1.2. Reference Model. For the adaptive control, the desiredclosed-loop dynamics is given by

xm(t) = Amxm(t) + Bmr(t), (4)

where Am ∈ Rn×n is a Hurwitz matrix, and Bm ∈ Rn. Thesignal r(t) ∈ R is a bounded piecewise continuous referenceinput, and xm(t) is the desired state. For a given symmetricpositive definite matrix Q ∈ Rn×n, there exists a unique P ∈Rn×n that satisfies

PAm + A�mP = −Q, P = P� > 0. (5)

For the adaptive control design, we need the followingstandard plant model matching condition.

Assumption 1. There exist constant K∗1 ∈ Rn, K∗2 ∈ R suchthat

A + b0K∗�1 = Am, b0K

∗2 = Bm. (6)

2.1.3. Control Objective. The control objective is to designthe virtual control signal v0(t) and adaptive allocation gainsαj , j = 1, . . . ,m, such that all the closed-loop signals arebounded and the system state x(t) tracks the desired statexm(t) asymptotically in the presence of uncertain controleffectiveness Λ.

2.2. Adaptive Control Allocation Design

2.2.1. Nominal Controller. The plant model matching con-dition in Assumption 1 indicates the existence of a nominalcontroller v∗(t) for the system without failures and a nom-inal constant allocation gain vector α∗ such that the closed-loop response is identical to that of the reference model whenthe responses of any unmatched initial conditions vanishexponentially. This signal takes the form

v∗0 (t) = K∗�1 x(t) + K∗2 r(t) � θ∗�ω(t),

v∗(t) = α∗v∗0 (t),(7)

where θ∗ � [K∗�1 ,K∗2 ]� ∈ Rn+1, and ω(t) = [x�(t), r(t)]�.The above state feedback control v∗0 (t) together with aprespecified distribution α∗ ensures the state tracking errore(t) = x(t)− xm(t) approaches zero exponentially.

2.2.2. Adaptive Controller. When a failure occurs, the allo-cation gain will be adaptively adjusted to accommodate forfailure, but α∗ may no longer be effective. For this, we usethe adaptive versions of control signal and allocation gain

v0(t) = K�1 (t)x(t) + K2(t)r(t) = θ�(t)ω(t),

v(t) = α(t)v0(t),(8)

where K1(t) and K2(t) are estimates of K∗1 and K∗2 , andθ(t) = [K�1 (t),K2(t)]� ∈ Rn+1. The updated α(t) =[α1(t), . . . ,αm(t)]� is an estimate of α∗.


2.2.3. Error Dynamics. With the plant and control in (1) and(2), nominal controller in (7), and adaptive controller in (8),the closed-loop dynamics can be expressed as

x(t) = Ax(t) + BΛα(t)v0(t)

= Ax(t) + Bα∗v0(t) + Bα(t)v0(t)

= Ax(t) + b0v0(t) + Bα(t)v0(t)

= Ax(t) + b0v∗0 (t) + b0

˜θ�(t)ω(t) + Bα(t)v0(t)

= (A + b0K∗�1

)

x(t) + b0K∗2 r(t) + b0

˜θ�(t)ω(t)

+ Bα(t)v0(t)

= Amx(t) + Bmr(t) + b0˜θ�(t)ω(t) + Bα(t)v0(t),

(9)

where α(t) � Λα(t)− α∗(t) and ˜θ(t) = θ(t)− θ∗.From the closed-loop dynamics in (9) and reference

model in (4), the error dynamics can be obtained as

e(t) = Ame(t) + b0˜θ�(t)ω(t) + Bα(t)v0(t). (10)

It can be seen that the error dynamics in (10) is suitable foradaptive law design in that the latter half of its right-hand

side is linear in ˜θ(t) and α(t).

2.2.4. Adaptive Laws. Based on the error dynamics in (10),we can design the adaptive laws for the control parameterθ(t) and allocation gain α(t) as

θ(t) = −Γθω(t)e�(t)Pb0, (11)

α(t) = −ΓαB�Pe(t)v0(t), (12)

where Γθ and Γα are symmetric positive definite matrices andP is determined by (5). From the adaptive laws, we can seethat the controller parameters of the virtual control signalfor the actuator group are updated using the information ofthe equivalent control gain vector b0. The allocation gains areupdated with the B matrix since successful allocation of thevirtual control signal to each actuator requires the knowledgeof each column of B.

The properties of the adaptive control allocation schemecan be summarized in the following theorem whose proof ispresented in the appendix.

Theorem 1. For the system in (1), the adaptive controller andallocation scheme in (8), and the adaptive laws in (11) and(12) guarantee that the all the closed-loop signals are boundedand limt→∞[x(t)−xm(t)] = 0 in the presence of uncertain lossof effectiveness actuator failures in (2).

Remark 2. From the definition of α(t) in (9), we can seethat the adaptation of α(t) is essential for compensating theuncertainties in Λ, while K1(t) and K2(t) can be fixed to theirnominal values K∗1 and K∗2 . In this case, with ˜θ(t) = 0 in(10) and (A.1), the closed-loop stability and asymptotic statetracking results still hold.

2.3. Examples

2.3.1. Linear Plants. Two case studies based on linearizedaircraft models are presented next.

Plant. Consider the linearized lateral dynamic model of alarge transport aircraft flying in a steady wings-level cruisecondition with u = 778 ft/s [15]. The aircraft model is

x = Ax + Bu, (13)

x = [vb, pb, rb,φ,ψ]�, u = [δa, δr]

�. (14)

The state includes the lateral velocity vb (ft/s), roll rate pb(rad/s), yaw rate rb (rad/s) (all in body-axis frame), roll angleφ (rad), and yaw angle ψ (rad). The control inputs are aileronδa and rudder δr deflections (deg). The system matrices Aand B are

A =

⎡

⎢

⎢

⎢

⎢

⎢

⎣

−0.129 28.328 −774.92 32.145 0−0.012 −1.4419 0.9409 0 00.004 −0.0409 −0.1757 −0.0001 0

0 1 0.0372 0 00 0 1.0007 0 0

⎤

⎥

⎥

⎥

⎥

⎥

⎦

,

B =

⎡

⎢

⎢

⎢

⎢

⎢

⎣

0.0542 0.46690.0443 0.02000.0025 −0.0382

0 00 0

⎤

⎥

⎥

⎥

⎥

⎥

⎦

.

(15)

Hereafter we will group the aileron and rudder into a singlegroup. Note that these surfaces have related but differentfunctionalities. If adaptive control signals are designedseparately for the two control inputs with the state feedbackstructure, the total updated parameters would be (5+1)×2 =12. They include 5 state feedback parameters and 1 feed-forward parameter (for tracking the reference input) for eachcontrol surface. The adaptive control allocation scheme, onthe other hand, requires 5 + 1 + 2 = 8 parameters since onlyone adaptive control signal is designed and allocated withtwo updated allocation gains.

Nominal Parameters. The nominal allocation gain vector isselected as α∗ = [8.6103,−1]�, thus the equivalent controlgain vector is b0 = Bα∗ = [0, 0.3614, 0.0594, 0, 0]�. Thenominal allocation gain is chosen so that the first element inb0, that is, the control gain for the lateral acceleration is zero.The purpose of this choice is to attain a coordinated turn.

The nominal controller is designed based on the LQRapproach for (A, b0). The resulting gains are

K∗1 = [−0.8963, 22.1655,−73.6645, 28.8488, 4.0825]�,

K∗2 = 1.(16)

Reference Model. For this example, the reference model ischosen as the closed-loop dynamics with the above LQRcontroller, that is,

Am = A + b0K∗�1 , Bm = b0K

∗2 . (17)


The reference input to the reference model is chosen as r(t) =2.1376 for t ≥ 0, which leads to a desired state trajectory withsteady-state values

xm(∞) = [0, 0, 0, 0, 0.5236]�. (18)

This reference command yields a right turn with a change inthe yaw angle of 30 degrees.

Actuator Failure. We will consider the 80% loss in controleffectiveness:

u1(t) = 0.2v1(t), for t ≥ 10 seconds, (19)

where u1(t) is the output of the aileron actuator and v1(t) isthe designed control input to aileron.

Simulation Results. The following cases will be studied.

(i) Case 1: adaptive allocation of the adaptive controldesigned in Section 2.2 ((11) and (12)).

(ii) Case 2: adaptive allocation (as in (12)) of a nonadap-tive control signal with fixed gains (as in (16)).

Case 1. The time history of the system state under failure andthe desired state is shown in Figure 2. It can be seen that theasymptotic state tracking is achieved after failure.

Figure 3 shows the designed control signal v0(t), allo-cated control signals v(t) and actuator outputs u(t). Thedesigned control signal is allocated by the adaptive allocationgains. The actuator outputs are different from the allocatedcontrol signals due to the failure.

The adaptive parameters are shown in Figure 4. Wecan see that K2(t) also plays an important role in thecompensation of actuator failures. The adaptive allocationgains are shown in Figure 5. Both allocations gains adaptimmediately after the failure occurs and settle to a newcombination of steady-state values that, together with K2(t)and K1(t), guarantee state tracking after failure.

Case 2. The system state and desired state are shown inFigure 6. Despite the uncertain actuator failure, asymptoticstate tracking is achieved. The designed control signal,allocated control signals and the actuator outputs are shownin Figure 7. Similar to the previous case, the allocated controlsignals and the actuator outputs are not identical due to theactuator failure. The allocation gains are shown in Figure 8.The allocation gains are updated autonomously for failurecompensation.

In this simulation case, we have achieved similar resultsto Case 1 with a slightly degraded transient response (seeFigures 2 and 6). A possible explanation for it is that the con-troller parameters are fixed in this case and cannot contributeto the failure compensation and trajectory tracking as theydo in Case 1. However, closed-loop stability and asymptoticstate tracking are achieved even though only the allocationgains are adapted. The successful demonstration of failurecompensation in Case 2 implies that the proposed adaptiveallocation unit can be added to a control loop having any

02

velo

city

(ft

/s)

Late

ral

010

012

rate

(de

g/s)

Yaw

rate

(de

g/s)

Rol

l

05

10

angl

e (d

eg)

Rol

l

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

02040

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

angl

e (d

eg)

Yaw

−2

−10

Figure 2: Time history of plant state (solid) and reference modelstate (dashed) (Case 1).

40

012

V

irtu

al c

ontr

ol

in

put

v 0 (

deg)

05

1015

inpu

ts v

(de

g)

Act

uat

or

AileronRudder

0 5 10 15 20 25 30 35

400 5 10 15 20 25 30 35

400 5 10 15 20 25 30 35

05

1015

Time (s)

Time (s)

Time (s)

outp

uts

u (

deg)

Act

uat

or

−1

−10−5

−5

Figure 3: Time history of control signals and actuator outputs(Case 1).

state feedback controller. The added adaptive allocation tothe nonadaptive controller ensures closed-loop stability andasymptotic state tracking despite uncertain actuator failures.

2.3.2. NASA Generic Transport Model. In this section, weapply the control allocation strategy to the NASA GenericTransport Model (GTM). The NASA GTM is a high-fidelitymodel of the NASA AirSTAR UAV testbed [16, 17]. Thepurpose of this example is to show that this adaptive scheme,with the grouping of actuators having different physical


0 5 10 15 20 25 30 35 40

0204060

Time (s)

0 5 10 15 20 25 30 35 40

Time (s)

0.75

0.8

0.85

K1(t

)K

2(t

)

−20−40

Figure 4: Time history of controller parameters (Case 1).

0 5 10 15 20 25 30 35 40

0

2

4

6

8

10

12

Time (s)

Allo

cati

on g

ain

α(t

)

−2

α1(t): gain for aileron

α2(t): gain for rudder

Figure 5: Time history of allocation gains (Case 1).

02

010

012

05

10

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

02040

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

−2

−10

velo

city

(ft

/s)

Late

ral

rate

(de

g/s)

Yaw

rate

(de

g/s)

Rol

l

angl

e (d

eg)

Rol

lan

gle

(deg

)Ya

w

Figure 6: Time history of plant state (solid) and reference modelstate (dashed) (Case 2).

0123

inpu

t v

(de

g)

Vir

tual

con

trol

01020

inpu

ts v

(de

g)A

ctu

ator

AileronRudder

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0

0

5 10 15 20 25 30 35 40

05

101520

Time (s)

Time (s)

Time (s)

outp

uts

u (

deg)

Act

uat

or

−1

−10

−5

Figure 7: Time history of control signals and actuator outputs(Case 2).

0 5 10 15 20 25 30 35 40

0

2

4

6

8

10

Time (s)

Allo

cati

on g

ain

(t

)

α1(t): gain for aileron

α2(t): gain for rudder

α

−2

Figure 8: Time history of allocation gains (Case 2).

functions, can be applied to the nonlinear plant to achieveclosed-loop stability and asymptotic tracking.

LTI Model. For this simulation study, we trim and linearizethe GTM at a wings-level flight for an aerodynamic speed of92.09 knots. The same states and controls of (14) are used.

Flight Conditions. As in the previous example, the aircraftis commanded to turn right from the initial wings-levelhorizontal flight. The turn starts at 10 seconds and at steadystate the heading angle will increase 60 degrees.

Actuator Failure. We let the left aileron lose 90% of itseffectiveness at 12 seconds. The failure magnitude and itsonset time instant are unknown to the controller. The controlobjective is for the aircraft to achieve an accurate turn in thepresence of the failure.


−101

−100

10

−505

10

02040

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

050

100

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

velo

city

v (

ft/s

)La

tera

lra

te r

(de

g/s)

Yaw

rate

p (

deg/

s)R

oll

angl

e (d

eg)

Rol

lan

gle

(deg

)Ya

w

Figure 9: Time history of lateral states (solid) and reference modelstates (dashed).

100

150

200

Forw

ard

velo

city

u (

ft/s

)

68

10

Ver

tica

lve

loci

ty w

(ft

/s)

0

5

Pit

chra

te q

(de

g/s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

010

Pit

ch a

ngl

e

−5

−10θ(d

eg)

Figure 10: Time history of longitudinal states.

Simulation Results. The simulation results are shown inFigures 9–14. Figure 9 shows the relevant states of the ref-erence model and of the plant. The states track the referencetrajectories asymptotically in spite of the disturbance causedby the failure. The yaw angle is shown to reach 60 degreesaccurately. This accurate turning is also shown in Figure 11by the ground track of the aircraft.

The longitudinal states are shown in Figure 10. Thereare fluctuations in the longitudinal states since they are notcontrolled by aileron and rudder.

0 1 2 3 4 5 6

0

1

2

3

4

5

6

West-east displacement (ft)

Sou

th-n

orth

dis

plac

emen

t (f

t)

×104

×104

−1−1

Figure 11: Ground track of the aircraft.

inpu

t v

(de

g)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 500

Vir

tual

con

trol

Time (s)

Time (s)

inpu

ts v

(de

g)A

ctu

ator

AileronRudder

0.5

0

−0.5

−1

−1.5

0.040.02

00.02−−

0.04−0.06−0.08−0.1−0.12

Figure 12: Time history of virtual control signal and actuatorinputs.

Figures 12 and 13 show the time history of the virtualcontrol signal, allocated control signals, and actual aileronand rudder deflections. The discontinuity in Figure 13 at thetop is a consequence of failure.

Figure 14 shows the control allocation gains and con-troller parameters. The parameters adjust autonomouslyafter the failure occurs. Note that the allocation gains start toupdate at the beginning of the turn at t = 10 seconds, beforethe failure occurs. The reason for this phenomenon is thatthe allocation gains are improving the tracking performancebeyond of what the adaptive gains can do alone.

3. Adaptive Control Allocation Design forConstant Failures

3.1. Problem Formulation. When constant failures occur, thecontrol signal can be rewritten as [11]

u(t) = v(t) + σ f (u− v(t)), (20)


defl

ecti

ons

(deg

)

Act

ual

aile

ron

Left aileron

Right aileron

0 5 10 15 20 25 30 35 40 45 50

Time (s)

0 5 10 15 20 25 30 35 40 45 50

Time (s) de

flec

tion

s (d

eg)

Act

ual

ru

dder

Upper rudderLower rudder

1

0.5

0

−0.5

−1

−1.5

0.040.02

0−0.02−0.04−0.06−0.08−0.1

Figure 13: Time history of actuator deflections.

0 10 20 30 40 5012.52

12.54

12.56

12.58

12.6

12.62

12.64

for

aile

ron

Time (s)

Time (s) Time (s)

0 10 20 30 40 500.68

0.7

0.72

0.74

0.76

0.78

0.8

for

rudd

er

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Ada

ptiv

e co

ntr

olle

r pa

ram

eter

sαα

−0.1

Figure 14: Time history of allocation gains and adaptive parameters.

where u = [u1, . . . ,um]� is the failure vector whose elementsare unknown constants, and σ f represents the failure pattern,which is defined as

σ f = diag{σ1, σ2, . . . , σm}, (21)

with σi = 1 if the ith actuator has failed, that is, ui = ui,and σi = 0 otherwise. The failures are assumed to occurinstantaneously, that is, σi are piecewise constant function oftime. An example of such actuator failures is when a controlsurface (such as the rudder or an aileron) is stuck at someunknown fixed angle at an unknown time instant. This type


of failures could be caused by failed hydraulic systems ormechanical linkages.

The plant dynamics can then be rewritten as

x(t) = Ax(t) + B(

I − σ f)

v(t) + Bσ f u. (22)

The constant failure u introduces an uncertain disturbancethat needs to be accommodated for. The control objective forthis adaptive control allocation scheme is to design v(t) toguarantee closed-loop stability and asymptotic state trackingwhen uncertain constant failures occur.

For adaptive control of constant actuator failures, suffi-cient built-in actuation redundancy is required. The redun-dancy condition is described in the following assumption.

Assumption 2. The rank of B matrix satisfies that rank[B] =1, and there is at least one operable actuator in the system.

The rank condition characterizes the redundancy ofactuation which is necessary for a successful constant failurecompensation. This rank condition suggests that the systemremains controllable after a failure, and the effect of theconstant failure can be properly matched and canceled by theallocated control signals through other columns of B. Basedon this condition, there can be up to m− 1 constant actuatorfailures at any given time.

3.1.1. Nominal Controller. Consider the nominal controllerstructure

v∗0 (t) = K∗�1 x(t) + K∗2 r(t) + K∗3 � θ∗�ω(t),

v∗(t) = α∗v∗0 (t),(23)

where θ∗ = [K∗�1 ,K∗2 ,K∗3 ]� and ω = [x�(t), r(t), 1]�.

When there is no failure in the system, K∗1 , K∗2 , and α∗

are chosen as in (6), and K∗3 is set to be zero. In thisway, the controller ensures the match between the referencemodel and the nominal plant without failures. Similar to theprevious section, we also define Bα∗ = b0 which is known inadvance for controller design.

Next we will show that the controller in (23) attainsmodel matching when failures occur. When there are failuresin the system, K∗3 cannot generally be zero, and a new set ofallocation gains, denoted as α∗, may be needed. From (22)and (23) we obtain

x = Ax + B(

I − σ f)

α∗(

K∗�1 x + K∗2 r + K∗3)

+ Bσ f u. (24)

We assume that at most p actuators can fail with p ≤ m − 1so that there is at least one operable actuator left. Define anindex set for failed actuators as F = {i1, . . . , ip} such thatσk = 1 for any k ∈ F . Then B(I − σ f )α∗ can be expressed as

B(

I − σ f)

α∗ =∑

j /∈F

bjα∗j , (25)

where bj is the jth column of B and α∗j is the jth element ofα∗.

Based on the rank condition in Assumption 2, we knowthat the columns of B are linearly dependent, that is, for any

two columns of B, bi and bj , bi = ci jb j where ci j is a constantscalar. Thus we know that the vector b0 = Bα∗, which is thelinear combination of all columns of B, is also parallel to anycolumn in B. So for each column bk in B, k = 1, 2, . . . ,m, wecan find a scalar ck satisfying bk = ckb0. Therefore (25) canbe expressed as

B(

I − σ f)

α∗ =∑

j /∈F

bjα∗j =

∑

j /∈F

cjα∗j b0. (26)

If α∗j , j /∈ F are chosen such that

∑

j /∈F

cjα∗j = 1, (27)

then we may obtain

B(

I − σ f)

α∗ = b0. (28)

One possible choice for α∗j , j /∈ F is

α∗j =1

cj(

m− p) . (29)

Equation (28) indicates that, under constant failures, theplant model condition in (6) can still be satisfied. Note thatfor the system without failures

Bα∗ =m∑

j=1

α∗j b j =m∑

j=1

cjα∗j b0 = b0 (30)

which implies that

m∑

j=1

c jα∗j = 1. (31)

Comparing (27) and (31), we can see that α∗ is generallydifferent from α∗.

For Bσ f u, we can also get

Bσ f u =∑

k∈F

bkuk =∑

k∈F

ckukb0 � d∗b0. (32)

With (28) and (32), (24) can be expressed as

x(t) = Ax(t) + b0K∗�1 x(t) + b0K

∗2 r(t) + b0K

∗3 + d∗b0. (33)

By choosing K∗3 = −d∗, we have b0K∗3 = −Bσ f u, and (33)

can be reduced to

x(t) = Ax(t) + b0K∗�1 x(t) + b0K

∗2 r(t)

= (A + b0K∗�1

)

x(t) + b0K∗2 r(t).

(34)

Therefore when failures are present, a nominal controllercan always be found, with K∗1 and K∗2 specified in (6), α∗

characterized in (27), andK∗3 = −d∗ ensures that the closed-loop system is stable under constant failures, and the stateconverges to the desired state xm in (4) exponentially.


3.2. Adaptive Control Allocation Design

3.2.1. Adaptive Controller. Due to the uncertain nature ofthe failures, the controller parameters must be adjusted.Consider the adaptive control allocation scheme

v0(t) = K�1 (t)x(t) + K2(t)r(t) + K3(t) � θ�

(t)ω(t),v(t) = α(t)v0(t),

(35)

where K1(t), K2(t), and K3(t) are the estimates of K∗1 , K∗2 ,and K∗3 in (23). The signal ω(t) = [x�(t), r(t), 1]�. With theadaptive controller in (35) and the plant dynamics in (22),the closed-loop system is

x(t) = Ax(t) + B(

I − σ f)

α(t)

× [K�1 (t)x(t) + K2(t)r(t) + K3(t)]

+ Bσ f u.(36)

3.2.2. Error Dynamics. For this adaptive control schemedesign, we define α(t) � α(t)− α∗, and (36) becomes

x(t) = Ax(t) + B(

I − σ f)

α(t)v0(t) + Bσ f u

= Ax(t) + B(

I − σ f)

α(t)v0(t) + B(

I − σ f)

α∗v0(t)

+ Bσ f u

= Ax(t) + B(

I − σ f)

α(t)v0(t) + B(

I − σ f)

α∗v∗0 (t)

+ B(

I − σ f)

α∗v0(t) + Bσ f u,

(37)

where v0(t) = (θ(t)− θ∗)�ω(t) � ˜θ�(t)ω(t).With (28), B(I − σ f )α∗v∗0 (t) in (37) becomes

B(

I − σ f)

α∗v∗0 (t) = b0v∗0 (t) = b0K

∗�1 x(t) + b0K

∗2 r(t)

+ b0K∗3 .

(38)

With (38), (37), and the plant model matching condition in(6), we have

x(t) = Amx(t) + Bmr(t) + B(

I − σ f)

α(t)v0(t) + b0˜θ�(t)ω(t).

(39)

From the closed-loop dynamics in (39) and the referencemodel in (4), we obtain the error dynamics

e(t) = Ame(t) + B(

I − σ f)

α(t)v0(t) + b0˜θ�(t)ω(t). (40)

3.2.3. Adaptive Laws. From the error dynamics in (40), thefollowing adaptive laws are derived:

θ(t) = −Γθω(t)e�(t)Pb0, (41)

α j(t) = −γje�(t)Pbjv0(t), j = 1, 2, . . . ,m, (42)

where bj is the jth column of B, γj > 0 and Γθ = Γ�θ > 0 areadaptive gains.

The following theorem summarizes the properties of theadaptive control allocation scheme and the proof is providedin the Appendix:

−100

1020

TAS

(ft/

s)

−2−1

01

AO

A (

deg)

−10−5

05

(deg

/s)

Pit

ch r

ate

0 5 10 15 20 25 30−5

0

5

Time (s)

Time (s)

Time (s)

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

(deg

)P

itch

an

gle

Figure 15: Time history of plant state (solid) and reference modelstate (dashed).

12345

elev

ator

(de

g)D

esig

ned

05

10

eleLOBeleLIB

eleROB

eleRIB

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

05

10

Time (s)

outp

uts

(de

g)

Ele

vato

rin

puts

(de

g)

Ele

vato

r

−5

−5

Figure 16: Time history of control signals and actuator outputs.

Theorem 3. The adaptive control allocation scheme in (35)with the adaptive laws in (41) and (42) applied to the plantin (22) in the presence of constant failures guarantees that allclosed-loop signals are bounded and limt→∞(x(t)−xm(t)) = 0.

3.3. Example

3.3.1. Linear Plant. Consider the longitudinal LTI model ofthe NASA GTM given by

x = Ax + Bv,

x = [VT ,αa, q, θ]�,

v = [δelob, δelib, δerob, δerib]�,

(43)

where the state includes the true airspeed VT (ft/s), angle ofattack αa (rad), pitch rate q (rad/s), and pitch angle θ (rad).The control inputs are the deflections of the four elevator


150155160165170175

Forw

ard

0102030

Pit

ch r

ate

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

010

Pit

ch a

ngl

eve

loci

ty (

ft/s

)(d

eg/s

)(d

eg)

Time (s)

−10−20

−10−20−30

Figure 17: Time history of longitudinal states (solid) and reference model states (dashed).

05

Rol

l rat

e

012

Yaw

rat

e

05

10

Rol

l an

gle

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

024

Yaw

an

gle

(deg

/s)

(deg

/s)

(deg

)(d

eg)

−5−10

−5−10

−2

−2−1

−4

Figure 18: Time history of lateral states (solid) and reference model states (dashed).

segments: left outboard elevator δelob, left inboard elevatorδelib, right outboard elevator δerob, and right inboard elevatorδerib (deg). The system matrices A and B are

A =

⎡

⎢

⎢

⎢

⎣

−0.0450 −8.9632 0.0349 −32.1740−0.0035 −2.7429 0.9514 0−0.0056 −42.6233 −3.5616 0

0 0 1 0

⎤

⎥

⎥

⎥

⎦

,

B =

⎡

⎢

⎢

⎢

⎣

−0.0110 −0.0110 −0.0110 −0.0110−0.0012 −0.0012 −0.0012 −0.0012−0.1962 −0.1962 −0.1962 −0.1962

0 0 0 0

⎤

⎥

⎥

⎥

⎦

.

(44)

Obviously, rank[B] = 1. For the simulation study, we willinclude the four elevator surfaces into one group, for whichan elevator control signal will be designed and allocated.

Nominal Parameters. The nominal allocation gain α∗ ischosen as α∗ = [0.25, 0.25, 0.25, 0.25]� and b0 = Bα∗ =[−0.011,−0.0012,−0.1962, 0]� so the deflection of the four

segments will be the same if no failure occurs. The nominalcontroller is designed using the LQR approach for (A, b0).The resulting gains are

K∗1 = [0.1494, 25.0761,−3.2841,−27.1293]�, K∗2 = 1.(45)

Reference Model. Similar to the simulation study inSection 2, the reference model is chosen as the closed-loopdynamics of the LQR controller, that is,

Am = A + b0K∗�1 , Bm = b0K

∗2 . (46)

The reference input to the reference model is chosen as r(t) =4.1841 for t ≥ 0, which leads to a reference trajectory withsteady-state values

xm(∞) = [10,−0.0139, 0,−0.0110]�, (47)

whose physical meaning is that the aircraft speed is increasedby 10 ft/s; its angle of attack is reduced by 0.0139 rad; and itspitch angle is reduced by 0.0110 rad.


00.5

1

01

23

500

1000

1500

2000

Alt

itu

de (

ft)

1 0 0.5 1

0

0.5

1

1.5

2

2.5

3

0 50 100 150400

600

800

1000

1200

1400

1600

1800

2000

Time (s)

Alt

itu

de (

ft)

West-east displacement (ft)

West–east displacement (ft)×104×104

×105

×105 −−1− 1−

0.5−0.5

−0.5

Sou

th-n

orth

dis

plac

emen

t (f

t)

South–north displacement (ft)

Figure 19: Flight trajectory, ground track, and altitude of the aircraft.

−1012

elev

ator

(de

g)D

esig

ned

−5

−5

05

10

inpu

ts (

deg)

Ele

vato

rs

LOBLIB

ROBRIB

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

05

1015

defl

ecti

ons

(deg

)E

leva

tors

Figure 20: Time history of lumped and allocated elevators andelevator deflections.

Actuator Failure. The left outboard elevator is stuck at −5degrees after 1 second, that is,

u1(t) = −5 deg, for t ≥ t f second, (48)

where t f = 1 second. The signal u1(t) is the output of the leftoutboard elevator. For t > t f the elevator does not respondto the elevator input v1(t).

Simulation Results. The results are shown in Figures 15 and16. From Figure 15, we can see that the plant state tracksthe reference state after failure. Figure 16 shows the designedelevator signal, elevator inputs, and elevator outputs. Thefailure occurs at t f = 1 second and the other elevators canbe seen to accommodate the failure with the adaptation.

3.3.2. NASA Generic Transport Model. Here we apply theadaptive control scheme to the nonlinear NASA GTM.

LTI Model for Controller Design. For adaptive control design,we consider a LTI model for a wings levelled flight having a3 deg angle of attack at trim. Similar to the linear simulationstudy, we study the compensation for constant elevatorfailures and will consider multiple elevator failures in whichboth the lateral and longitudinal are active. The state andcontrol vectors are given by

x = [u, q, θ, p, r,φ,ψ]

,

u = [δelob, δelib, δerob, δerib, δtl, δtr, δal, δar, δru, δrl],(49)

where δelob, δelib, δerob, and δerib are the four elevatorsegments, δtl and δtr are the two engine throttles, δal andδar are left and right aileron segments, and δru and δrl areupper and lower rudder segments. The actuators will begrouped into elevator, engine, aileron, and rudder groups. Avirtual control signal is designed for each group and allocatedto its members adaptively. The design is based on that inSection 3.2 and is similar to that in the linear simulationstudy. We include lateral states and lateral actuators to


1

0.5

0

0.5

Des

ign

ed a

lero

n (

deg)

0 50 100 150

Time (s)

1.5−−

−

(a)

2

0

1

2

Aile

ron

defl

ecti

ons

(deg

)

Left aileronRight aileron

0 50 100 150

Time (s)

1−−

(b)

0

0.2

Des

ign

ed r

udd

er (

deg)

0 50 100 150

Time (s)

0.6

0.4

0.2

−

−−−

0.8

(c)

0.6

0.4

0.2

0

0.2

Ru

dder

def

ecti

ons

(deg

)Upper rudder

Lower rudder

0 50 100 150

Time (s)

−

−−−

0.8

(d)

0 50 100 150

0

5

10

Des

ign

ed t

hro

ttle

(%

)

Time (s)

5−

(e)

0 50 100 150

Time (s)

0

5

10

15

thro

ttle

s (%

)

Allo

cate

d en

gin

e

Left engine

Right engine

5−

(f)

Figure 21: Time history of the inputs and outputs of ailerons, rudders, and engines.

regulate the disturbance of the elevator failures propagatedto the lateral dynamics.

Flight Conditions. A set of commands that aim to make theaircraft climb at 4-degree pitch angle is applied at 5 seconds.The nominal parameters and reference inputs are obtainedas in the linear simulation example.

Actuator Failures. The right outboard elevator is locked at10 degrees at 10 seconds and the right inboard elevator at 5degrees at 20 seconds. The control objective is to maintainthe climbing flight in the presence of these failures.

The simulation results are shown in Figures 17–23.Figure 17 shows the time history of the longitudinal statesand reference model states. It can be seen that the tracking ofthe desired longitudinal attitude is accurately achieved.

The lateral states are shown in Figure 18. Some distur-bances can be observed after the occurrence of asymmetric

elevator failures. These disturbances are regulated by theailerons and asymptotic tracking of the lateral states can bealso achieved.

Figure 19 shows the flight trajectory, ground track, andaltitude of the aircraft. From the ground track plot, we cansee the effect of asymmetric failure on the lateral dynamics.The deviation is corrected by the lateral actuators. Figure 20shows the lumped elevator signal, allocated elevator signals,and the actual elevator deflections. The elevator failuresappear as constant (step signals) on the lower plot, andthe designed and allocated elevator signals are shown toaccommodate for the failures after their occurrence. Theinput signals and outputs of other actuators are shown inFigure 21. Note the actuation of the aileron and rudderrequired to compensate for the asymmetric failure. The timehistory of the control allocation gains and some selectedadaptive parameters are shown in Figures 22 and 23. It canbe seen that the allocation gains and controller parametersare updated autonomously to ensure the desired flight.


0 50 100 1507

7.5

8

8.5

for

elev

ator

s

Time (s)

LOBLIB

ROBRIB

α

(a)

0 50 100 1500.99

1

1.01

1.02

1.03

1.04

for

aile

ron

s

Time (s)

Left aileronRight aileron

α

(b)

0 50 100 1500.995

1

1.005

1.01

1.015

1.02

1.025

for

rudd

ers

Time (s)

Upper rudderLower rudder

α

(c)

0 50 100 1500.9

1

1.1

1.2

1.3

1.4

for

engi

nes

Time (s)

Left engineRight engine

α

(d)

Figure 22: Time history of control allocation gains.

para

met

ers

of K

Sele

cted

para

met

ers

of K

Sele

cted

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

para

met

ers

of K

Sele

cted

Time (s)

Time (s)

Time (s)

0.10

12

3

−0.1−0.2−0.3−0.4

0.10.080.060.040.02

0.030.020.01

0−0.01

Figure 23: Time history of selected adaptive controller parameters.

4. Conclusions

A novel adaptive control allocation framework is proposedherein. The adaptive allocation scheme includes an adaptivecontrol signal and a control allocation unit with adaptivelyupdated allocation gains. Two adaptive control allocation

algorithms have been proposed for the compensation ofuncertain failures. The proposed algorithms have beenshown to guarantee closed-loop stability and asymptoticstate tracking. It has also been shown that the proposedadaptive control allocation framework reduces the controllercomplexity with proper grouping of the actuators. In this


framework, the control signal to be adaptively allocatedcan be actuated by nonadaptive controllers. The simulationresults demonstrate the performance of the proposed algo-rithms and their applicability to aircraft flight control. Somefuture research topics in this direction include the extensionof the adaptive control allocation framework to systems withmultiple groups of actuators and the strict enforcement ofa pure allocation, that is, exactly enforcing the designedcontrol signal with

∑

αi = 1.

Appendix

Proof of Theorem 1. When a failure occurs, the control effec-tiveness matrix Λ has a discontinuity. It would lead to a finitejump in α(t) as well, which could result in a a finite jumpin the error dynamics. Let us assume that there are l failuresoccurring in the system, and the actuator failures occur attime instants tk, with tk < tk+1, k = 1, 2, . . . ,N . For theclosed-loop stability and state tracking analysis, we choosethe following Lyapunov-like function:

V(t) = 12e�(t)Pe(t) +

12α�(t)Γ−1

α Λ−1α(t) +12˜θ�(t)Γ−1

θ˜θ(t),

(A.1)

for each time interval (tk, tk+1), k = 0, 1, . . . ,N with t0 = 0and tN+1 = ∞. V(t) is thus discontinuous with finite jumpsat tk, k = 1, . . . ,N . Taking the time derivative of V(t) andsubstituting the adaptive laws in (11) and (12) into the resultfor each (tk, tk+1), we obtain

V(t) = e�(t)Pe(t) + α�(t)Γ−1α α(t) + ˜θ�(t)Γ−1

θ θ(t)

= e�(t)P[

Ame(t) + b0˜θ�(t)ω(t) + Bα(t)v0(t)

]

− α�(t)B�Pe(t)v0(t)− ˜θ�(t)ω(t)e�(t)Pb0

= −12e�Qe ≤ 0.

(A.2)

Thus we can conclude that for each (tk, tk+1), k = 0, 1, . . . ,N ,V(t) is bounded. Since V(t) only has finite jumps at tk,k = 1, . . . ,N , we can conclude that V(t) is bounded fort ∈ [0,∞), and e(t) ∈ L∞, α(t) ∈ L∞, ˜θ(t) ∈ L∞, α(t) ∈ L∞,θ(t) ∈ L∞, x(t) ∈ L∞, and ω(t) ∈ L∞. Integrating both sidesof (A.2), we can obtain

V(

t+k)

−V(

t−k+1

)

= 12

∫ tk+1

tk e�(τ)Q(τ)e(τ)dτ. (A.3)

For N + 1 intervals: [0, t1), (t1, t2), . . ., (tN−1, tN ), and (tN ,∞),(A.3) holds. Summing both sides of (A.3) for k = 0, 1, . . . ,N ,

we obtain

12

∫∞

0e�(τ)Q(τ)e(τ)dτ = V(0)−V(t−1

)

+V(

t+1)

−V(t−2)

+V(

t+2)−· · ·−V

(

t−k)

+V(

t+k)

−· · ·−V(t−N)

+V(

t+N)

−V(∞)

= V(0) +N∑

i=1

[

V(

t+i)−V(t−i

)]

−V(∞) <∞,(A.4)

because the jumps V(t+i ) − V(t−i ) are finite and the numberN of jumps is also finite. Thus we have e(t) ∈ L2. We can alsoconclude from (10) that e(t) ∈ L∞. So from e(t) ∈ L2 ∩ L∞,and e(t) ∈ L∞, we have lim0 → ∞e(t) = 0.

Proof of Theorem 3. In Section 3.1, we have assumed thatthere are at most p ≤ m − 1 constant actuator failures anddefined an index set for failed actuators as F = {i1, . . . , ip}such that σk = 1 for all k ∈ F . Here we further assume thatthe failures occur at instants tk, with tk < tk+1, k = 1, 2, . . . ,Nwith 1 ≤ N ≤ p. The number of failure instants maybe smaller than the total number of failures since multiplefailures may happen at the same time. For the stability proof,we choose the following Lyapunov-like function

V(t) = 12e�(t)Pe(t) +

12

∑

i /∈F

γ−1i α2

i (t) +12˜θ�(t)Γ−1

θ˜θ(t),

(A.5)

for each time interval (tk, tk+1), k = 0, 1, . . . ,N , with t0 =0 and tN+1 = ∞. The time derivative in each time interval(tk, tk+1) is

V(t) = e�(t)Pe(t) +∑

i /∈F

γ−1i αi(t)αi(t) + ˜θ�(t)Γ−1

θ θ(t)

= e(t)�PAme(t) + e�(t)PB(

I − σ f)

α(t)v0(t)

+ e�(t)Pb0˜θ�(t)ω(t)

− e�(t)Pv0(t)∑

i /∈F

αi(t)bi − ˜θ�(t)ω(t)e�(t)Pb0

= e�(t)PAme(t) = −12e�(t)Qe(t) ≤ 0

(A.6)

with the fact that B(I − σ f )α(t) = ∑

i /∈F αi(t)bi. Followinga similar approach to the stability analysis in Section 2, wecan conclude that for any t ∈ [0,∞), V(t) ∈ L∞, e(t) ∈L∞, x(t) ∈ L∞, ω(t) ∈ L∞, θ(t) ∈ L∞, v0(t) ∈ L∞, e(t) ∈L2. Since V(t) only includes αi(t) with i /∈ F , we can onlyconclude the boundedness of αi(t) and αi(t) for i /∈ F . Toshow the boundedness of αj(t), j ∈ F , we note that for anyj ∈ F

αj(t) = αj(0)−∫ t

0γje

�(τ)Pv0(τ)bj dτ, (A.7)


based on the adaptive law in (42). Note that all the columnsof B are linearly dependent based on the rank condition inAssumption 2. So we can have

bj = c∗j bk, ∀ j ∈ F , (A.8)

where bk is the column of B that corresponds to an arbitraryhealthy actuator, that is, k /∈ F , and c∗j is a nonzero constant.Equation (A.7) can thus be expressed as

αj(t)= αj(0)−∫ t

0γje

�(t)Pv0(t)c∗j bk dt

= αj(0)− γjγkc∗j

∫ t

0γke

�(t)Pv0(t)bk dt

= αj(0) +γjγkc∗j

∫ t

0

(−γke�(t)Pv0(t)bk)

dt

= αj(0) +γjγkc∗j

∫ t

0αk(t)dt

= αj(0) +γjγkc∗j [αk(t)− αk(0)], ∀ j ∈ F .

(A.9)

Since αk(t), k /∈ F has been proved to be bounded, we haveαj(t) ∈ L∞, for j ∈ F .

We can further obtain that e(t) ∈ L∞ from (40). Withe(t) ∈ L∞ ∩ L2 and e(t) ∈ L∞, we can have limt→∞e(t) =0.

Acknowledgments

This work was supported by the NRA NNX08AC62A of theIRAC project of NASA. The authors would like to thank Drs.Suresh M. Joshi and Sean P. Kenny at the NASA LangleyResearch Center for their valuable comments.

References

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