Engineering NotesDirect Model Reference Adaptive

Control with Actuator Failures

and Sensor Bias

Suresh M. Joshi∗

NASA Langley Research Center, Hampton, Virginia 23681

and

Parag Patre†

Siemens Corporation, Corporate Technology, Princeton,

New Jersey 08540

DOI: 10.2514/1.61446

I. Introduction

ACTUATOR and sensor faults have been implicated in severalaircraft loss-of-control accidents and incidents. Direct model

reference adaptive control (MRAC) methods have been suggested asa promising approach for maintaining stability and controllability inthe presence of uncertainties and actuator failures without requiringexplicit fault detection, identification, and controller reconfiguration(e.g., [1,2]). (An extensive amount of literature is available onadaptive control in the presence of actuator failures, but could not belisted in this brief Note due to space limitations; however, a detailedreference list is available in [2]). In addition to actuator faults, sensorfaults may also compromise safety [3–5]. A common type of sensorfault is unknown sensor bias, which can develop during operation inone or more sensors, such as rate gyros, accelerometers, altimeter,etc. If used directly in an MRAC law, such offsets in sensormeasurements can have detrimental effects on closed-loop stability,which can no longer be theoretically guaranteed. Accommodation ofsensor faults in an MRAC setting has been addressed (e.g., [6–8]).However, adaptive control of systemswith simultaneous actuator andsensor faults has not been adequately addressed. Toward that goal,MRAC control laws using state feedback for state tracking weredeveloped in [9] for the case with simultaneous sensor bias andactuator failures, and signal boundedness as well as bounded orasymptotic tracking were obtained. This Note elaborates on theresults of [9] and presents numerical examples to illustrate themethods.

II. Problem Formulation

Consider a linear time-invariant plant, subject to actuator failuresand sensor biases, described by

_x�t� � Ax�t� � Bu�t� y�t� � x�t� � β (1)

where A ∈ Rn×n and B ∈ Rn×m are the system and input matrices(assumed to be unknown and uncertain), x�t� ∈ Rn is the systemstate, and u�t� ∈ Rm is the control input. The state measurement isy�t� ∈ Rn includes an unknown constant bias β ∈ Rn, which may bepresent at the outset ormay develop or change during operation. As iscustomary in theMRAC literature, sensor noise and process noise arenot included in the analysis to facilitate analytical proofs of trackingstability and signal boundedness, although the simulation examplesto be presented include noise. In addition to sensor bias, some of theactuators u�t� ∈ Rm (e.g., control surfaces or engines in aircraft flightcontrol) may fail during operation. Actuator failures are modeled as

uj�t� � �uj; t ≥ tj; j ∈ J p (2)

where the failure patternJ p � fj1; j2; : : : ; jpg ⊆ f1; 2; : : : ; mg, thefailure value �uj (assumed to be constant), and the failure time ofoccurrence tj are all unknown. Let v�t� � �v1; v2; : : : ; vm�⊤ ∈ Rmdenote the applied (commanded) control input signal. In the presenceof actuator failures, the actual input vector u�t� to the system can bedescribed as

u�t� � v�t� � σ� �u − v�t�� � �I − σ�v�t� � σ �u (3)

where

�u � � �u1; �u2; : : : ; �um�⊤ σ � diagfσ1; σ2; : : : σmg (4)

where σ is a diagonal matrix (“failure pattern”matrix) whose entriesare piecewise constant signals that take on the values of zero or one,σi � 1 if the ith actuator fails, and σi � 0 otherwise.The actuator failures are uncertain in value ( �uj), pattern (J p), and

time of occurrence (tj). The objective is to design an adaptivefeedback control law using the available measurement y�t� withunknown bias β, such that closed-loop signal boundedness is ensuredand the system state x�t� tracks the state of a reference modeldescribed by

_xm�t� � Amxm�t� � Bmr�t� (5)

where xm ∈ Rn is the reference model state, Am ∈ Rn×n, is Hurwitz,Bm ∈ Rn×mr and r�t� ∈ Rmr (1 ≤ mr ≤ m) is a bounded referenceinput used in system operation. The reference model �Am; Bm� isusually based on the nominal plant parameters and is designed tocapture the desired closed-loop response to the reference input (e.g.,pilot input in the case of aircraft). For example, the reference modelmay be designed using optimal and robust control methods, such aslinear quadratic regulator (LQR), H2, or H∞ methods.This Note considers the single-reference input case [i.e., r is a

scalar (mr � 1) and Bm ∈ Rn]. The actuators are assumed to besimilar (e.g., segments of the same control surface), that is, thecolumns (bi) of the B matrix can differ only by unknown scalarmultipliers. It is also assumed that bi are parallel to the referencemodel input matrix Bm ∈ Rn, that is,

bi � Bm∕αi; i � 1; : : : ; m (6)

for some unknown (finite and nonzero) αi values whose signs areassumed known. The multiplier αi represents the uncertainty in theeffectiveness of the ith actuator due to modeling uncertainties and/orreduced effectiveness (e.g., caused by partial loss of an aircraftcontrol surface or ice accumulation on a control surface). Theobjective is to design an adaptive control law that will ensureclosed-loop signal boundedness and asymptotic state tracking [i.e.,limt→∞�x�t� − xm�t�� � 0] despite system uncertainties, actuator

Received12December 2012; revision received21February 2013; acceptedfor publication 21 February 2013; published online 29 July 2013. Thismaterial is declared a work of the U.S. Government and is not subject tocopyright protection in theUnitedStates.Copies of this papermaybemade forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 1533-3884/13 and $10.00 in correspondencewith the CCC.

*Senior Scientist for Control Theory,Mail Stop 308; suresh.m.joshi@nasa.gov. Fellow AIAA.

†Mechatronics and Control Scientist; formerly with NASA LangleyResearch Center; parag.patre@gmail.com.

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failures, and sensor bias faults. The adaptive controller shouldsynthesize the control signal v�t� capable of compensating foractuator failures and sensor bias faults automatically.It is assumed that the following matching conditions hold: There

exist gains K1 ∈ Rn×m, and k2, k3 ∈ Rm, such that

Am � A� B�I − σ�K⊤1 ; Bm � B�I − σ�k2;

Bσ �u � −B�I − σ��K⊤1 β� k3� (7)

The first two matching conditions are typical MRAC matchingconditions (e.g., as in [1], which addressed actuator failures withoutsensor bias), whereas the third condition contains a modificationbecause of sensor bias. At least one actuatormust be functional for thematching conditions to be satisfied. The reference model �Am;Bm�represents the desired closed-loop characteristics, and it is usuallydesigned using an appropriate state feedback for the nominal plant�A;B�. (Thus, the matching conditions are always satisfied undernominal conditions). For the adaptive control scheme, only Am andBm need to be known. Because Am is a Hurwitz matrix, there existpositive definite matrices P � P⊤, Q � Q⊤ ∈ Rn×n, such that thefollowing Lyapunov inequality holds:

A⊤mP� PAm ≤ −Q (8)

III. MRAC Using Adaptive Sensor Bias Estimation

The sensor measurements available for feedback have unknownbiases as inEq. (1). Let β�t� denote an estimate of the unknown sensorbias β. Using β, define the “corrected” state �x�t� ∈ Rn as

�x � y − β � x� β − β � x� ~β (9)

where ~β � β − β. Design an adaptive control law as

v � K⊤1 y� k2r� k3 (10)

where K1�t� ∈ Rn×m and k2�t�, k3�t� ∈ Rm are the adaptive gains.Therefore, the closed-loop corrected-state equation is

_�x � Ax� B�I − σ��K⊤1 y� k2r� k3� � _~β� Bσ �u

� �A� B�I − σ�K⊤1 �x� B�I − σ�� ~K⊤

1 y� ~k2r� ~k3�

� B�I − σ�k2r� B�I − σ�K⊤1 β� B�I − σ�k3 � _~β� Bσ �u

(11)

where ~K1 � K1 − K1, ~k2 � k2 − k2, and ~k3 � k3 − k3. UsingEq. (7) in Eq. (11), we get

_�x � Am �x� Bmr� B�I − σ�� ~K⊤1 y� ~k2r� ~k3� − Am ~β� _~β (12)

Define a measurable auxiliary error signal e�t� ∈ Rn as

e � �x − xm � x − xm � ~β � e� ~β (13)

where e � x − xm denotes the state tracking error. DifferentiatingEq. (13) with respect to time and using Eqs. (5) and (12), the closed-loop auxiliary error system can be expressed as

_e � _�x − _xm � Ame� B�I − σ�� ~K⊤1 y� ~k2r� ~k3� − Am ~β� _~β

� Ame� BmXmj∈=J p

�1∕αj�� ~K⊤1jy� ~k2jr� ~k3j� − Am ~β� _~β (14)

where the subscript j denotes the jth column of ~K1 and the jthelement of ~k2, ~k3 (similar notation is used for K1, K1, k2, k2, k3, k3).The following theorem gives adaptive gain update and bias

estimation laws that guarantee closed-loop signal boundedness, aswell as the bounded tracking error.Theorem 1: For the system given by Eqs. (1), (3), and (5), the

adaptive controller (10), the gain adaptation laws

_K1j � −sgn�αj�Γ1jB

⊤mPey

_k2j � −sgn�αj�γ2jB⊤

mPer

_k3j � −sgn�αj�γ3jB⊤

mPe (15)

for j � 1; 2; : : : ; m, where Γ1j ∈ Rn×n is a constant symmetricpositive definite matrix, γ2j, γ3j, are constant positive scalars, P wasdefined in Eq. (8); and the bias estimation law

_β � −ηP−1A⊤

mPe (16)

where η ∈ R is a tunable positive constant gain, guarantee that e→ 0and all the closed-loop signals including e�t�, the adaptive gains, andbias estimate are bounded.Proof: Define

V � e⊤Pe�Xmj∈=J p

1

jαjj� ~K⊤

1jΓ−11j

~K1j � ~k22jγ−12j � ~k23jγ

−13j � �

1

η~β⊤P ~β

(17)

Differentiating Eq. (17) with respect to time and using Eqs. (8) and(14), and the gain update laws in Eq. (15), the following expression isobtained upon simplification:

_V ≤ −e⊤Qe − 2e⊤PAm ~β − 2e⊤P_β −

2

η~β⊤P

_β (18)

Using the bias estimation law of Eq. (16) in Eq. (18), we get

_V ≤ −e⊤Qe − 2e⊤PAm ~β� 2ηe⊤A⊤mPe� 2 ~β⊤A⊤

mPe

≤ −e⊤Qe − ηe⊤Qe � −�1� η�e⊤Qe

Therefore, _V ≤ 0 [i.e., V�t� is bounded for all t], and e�t�, β�t�, y�t�,K1, k2, k3 are all bounded and e�t� ∈ L2. FromEqs. (14) and (16) andclosed-loop signal boundedness, we have

_~β, _e�t� ∈ L∞, therefore,using Barbalat’s lemma [1], limt→∞ e�t� � 0. That is, all signals andestimates are bounded, and limt→∞� �x − xm� � 0.□The adaptive control law in Theorem 1 guarantees stability (signal

boundedness) and bounded tracking error. However, although e→ 0as t→ ∞, it cannot be concluded that e�t� → 0 unless ~β�t� → 0. Ifpersistent excitation is present, ~β�t� would approach zero as t→ ∞,in which case, e�t� → 0 as t→ ∞.In an effort to achieve asymptotic tracking without the need for

persistent excitation, the next section addresses the case when aseparate asymptotically stable bias estimator is available.

IV. MRAC Using Asymptotic Bias Estimator

Suppose an external sensor bias estimator can be constructed (e.g.,using methods such as those found in [10]), such that the estimationerror asymptotically approaches zero, and that the estimation errordynamics is of the form

_~β � Aβ~β (19)

where β�t� is an estimate of β and ~β�t� � β − β�t� is the estimationerror. Aβ ∈ Rn×n is a known Hurwitz matrix, which implies that

limt→∞ ~β�t� � 0. Defining the adaptive control law as in Eq. (10) andproceeding as in Sec. III, we obtain Eq. (14). The adaptive controllaw (10) along with the gain adaptation laws (15) guarantees signalboundedness and asymptotic tracking.

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Theorem 2: For the system given byEqs. (1), (3), (5), and (10)witha bias estimator that satisfies Eq. (19), and the gain adaptation laws(15) guarantee that all the closed-loop signals including adaptivegains are bounded and the tracking error e�t� → 0 as t→ ∞.Proof: Define

V � e⊤Pe�Xmj∈=J p

1

jαjj� ~K⊤

1jΓ−11j

~K1j � ~k22jγ−12j � ~k23jγ

−13j � � ~β⊤Pβ

~β

(20)

where Pβ � P⊤β ∈ Rn×n is a positive definite solution of the

Lyapunov inequality

A⊤βPβ � PβAβ ≤ −Qβ (21)

for someQβ � Q⊤β > 0 ∈ Rn×n. DifferentiatingEq. (20)with respect

to time and using Eqs. (8), (14), (15), (19), and (21), the followingexpression is obtained upon simplification:

_V ≤ −e⊤Qe − 2e⊤P�Am − Aβ� ~β − ~β⊤Qβ~β ≤ −z⊤ �Qz

where z � � e⊤ ~β⊤ �⊤ and �Q ∈ R2n×2n is defined as

�Q ��

Q P�Am − Aβ��Am − Aβ�⊤P Qβ

�

Because Q is positive definite, the matrix �Q is positive definite if(using Schur complement)

Qβ − �Am − Aβ�⊤PQ−1P�Am − Aβ� > 0 (22)

Qβ can be chosen such that Eq. (22) is satisfied; therefore, _V ≤ 0, i.e.,V�t� is bounded for all t, and e�t�, β�t�, y�t�, K1, k2, k3 are allbounded and e�t�, ~β�t� ∈ L2. Using arguments similar to those in theproof of Theorem 1, and noting from Eq. (19) that ~β�t� → 0 ast→ ∞, it can be concluded that all signals including the adaptivegains are bounded, and limt→∞ e�t� � 0, that is, x�t� → xm�t�.□

V. Special Case: Bias in Rate Measurements Only

This section considers a special casewherein sensor bias is presentfor some or all of the rate measurements and it is assumed that bias-free measurements of the corresponding position variables areavailable. For example, in aircraft, rate gyros, whichmeasure angularvelocities, can be prone to constant or slowly varying biases, whereasbias-free angle (attitude) measurements can be available by usingappropriately processed GPS data. Although, in theory, rates can beobtained by differentiating the position signals, this is undesirable inpractice because of sensor noise.In this case, the output is given by

y ��yξy2

���ξx2

���β0

�(23)

where ξ ∈ Rn1 denotes the rate variables whose measurements (yξ)include a constant unknownbias β ∈ Rn1 , and x2 ∈ Rn−n1 denotes theremaining state variables. In addition, suppose bias-free measure-ments of the position vector ρ corresponding to ξ are available,where

_ρ � ξ � yξ − β (24)

For example, for a linear longitudinal aircraft model with a bias errorin the pitch rate gyro, ξ � q, the pitch rate, and ρ � θ, the pitch angle.(As stated previously, differentiation of ρ to get β is not allowedbecause of sensor noise.) In general, ρmay include some componentsof the state vector x.

A. Model-Independent Observer

Using the bias-free position measurements, a non-model-basedobserver can be designed to estimate the velocity sensor bias, asshown next.Augmenting Eq. (24) with the equation _β � 0, the following

system is obtained:

�_ρ_β

���0 −I0 0

��ρβ

���I0

�yξ; yρ � � I 0 �

�ρβ

�(25)

Observability of the preceding system can be readily verified, and anobserver gain L � �L⊤

1L⊤2 �⊤ can be designed to yield an

asymptotically stable state estimator:

� _ρ_β

���0 −I0 0

��ρβ

���I0

�yξ �

�L1

L2

��ρ − ρ� (26)

The estimation error equation is

�_~ρ_~β

���−L1 −I−L2 0

��~ρ~β

�≔ Af

�~ρ~β

�(27)

where ~ρ � ρ − ρ and Af is a Hurwitz matrix. Therefore, given anysymmetric positive definite matrix Qf ∈ R2n1×2n1 , there exists asymmetric positive definite matrix Pf ∈ R2n1×2n1 , such that

A⊤f Pf � PfAf ≤ −Qf (28)

B. Asymptotic State Tracking

Note that the system state variables are reordered as �ξT; xT2 �T [seeEq. (23)]; therefore, the state variables of the reference model inEq. (5) are also similarly reordered for consistency. The bias estimateβ can be used to define the corrected state vector

�x � y −�β0

�� x�

�β0

�−�β0

�� x�

�~β0

�(29)

As in Sec. III, the auxiliary tracking error (which can be measured) isdefined as

e � �x − xm � x − xm ��~β0

�� e�

�~β0

�(30)

Proceeding as in Sec. III, a slightly modified version of Eq. (14) isobtained:

_e�Ame�BmXmj∈=J p

�1∕αj�� ~K⊤1jy� ~k2jr� ~k3j�−AmI ~β�I _~β (31)

where I � �In10n1×�n−n1��T (Il and 0l×k denote the identitymatrix andthe zero matrix of the subscript dimensions). The following resultgives an adaptive control law that guarantees closed-loop signalboundedness and asymptotic tracking for systems with rate sensorbias and actuator failures.Theorem 3: For the system given by Eqs. (1), (3), (5), (10), (23),

and (26) and the gain adaptation laws (15), all closed-loop signalsincluding the adaptive gains are bounded and the tracking errore�t� → 0 as t→ ∞.Proof: Define

V � e⊤Pe�Xmj∈=J p

1

jαjj� ~K⊤

1jΓ−11j

~K1j � ~k22jγ−12j � ~k23jγ

−13j � � ~z⊤Pf ~z

(32)

where ~z � � ~ρ⊤ ~β⊤�⊤. Differentiating Eq. (32) with respect to timeand using Eq. (8), (27), (28), and (31), the following expression

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is obtained upon simplification after using the gain updatelaws (15):

_V ≤ −e⊤Qe − 2e⊤P� IL2 AmI � ~z − ~z⊤Qf ~z � _V ≤ −χ⊤ �Qχ

where χ � � e⊤ ~z⊤ �⊤ ∈ Rn�2n1 and �Q ∈ R�n�2n1�×�n�2n1� is definedas

�Q ��

Q P� IL2 AmI �� IL2 AmI �⊤P Qf

�(33)

Because Q is positive definite, the matrix �Q is positive definite if(using Schur complement)

Qf − � IL2 AmI �⊤PQ−1P� IL2 AmI � > 0 (34)

Qf can be chosen such that Eq. (34) is satisfied; therefore, _V ≤ 0, thatis, V�t� is bounded for all t. Using arguments similar to those in theproof of Theorem 1, and noting from Eq. (27) that ~β�t� → 0 ast→ ∞, it can be concluded that all signals including the adaptivegains are bounded, and limt→∞ e�t� � 0, that is, x�t� → xm�t�.□

VI. Application Example: Large Transport Aircraft

Todemonstrate themethods, simulation studies are performed on afourth-order longitudinal dynamicsmodel of a large transport aircraftin a wings-level cruise condition with known nominal trimconditions. The state variables are as follows: pitch rate q�t� (deg ∕s),true airspeed v�t� (m∕s), angle of attack α�t� (deg), and pitch angleθ�t� (deg) {i.e., x�t� � � q v α θ �T}. The actuators are twoidentical elevators (i.e., two pairs of elevators operating symmetri-cally): ue1�t� and ue2�t� (deg), respectively. The system is describedby Eq. (1), where the nominal values of the system and inputmatricesare given in [2]. The sensor outputs are contaminated with additivezero-mean Gaussian white noise in the simulations.From t � 0 to t � 10 s, the available measurement is bias free

(i.e., y � x). At t � 10 s, the measurement develops an unknownconstant bias β � � 2 7 −4 3 �T such that y � x� β fort > 10 s. (β has the same units as x).The control input is u�t� � �ue1 ue2 �T . The aircraft experiences

an unknown elevator failure pattern, which elevator, time of failure,and the failed elevator deflection are unknown), such that the secondelevator gets stuck at its instantaneous location at t � 15.1 s, whereasthe first elevator remains functional throughout.The reference model is chosen as the closed-loop system:

Am � A� BKT1 ; bm � B1 � B2

where K1 is the LQR gain designed to minimize a quadraticperformance function.

The control objective is to track the response (state vector) of areference model to a reference command input, as shown in Fig. 1.1) Case 1 (MRAC using adaptive sensor bias estimation): The

adaptive control law in Eq. (15) is implemented, where the adaptivesensor bias estimate is generated using the bias estimation law inEq. (16). All control and adaptation parameters �Γij; γij� are chosenby trial and error. The plant and reference model states, control input,and bias estimates are shown in Figs. 2–4, respectively. Except forvelocity, the controller and the bias estimator effectively compensatefor the sensor bias and achieve an acceptable tracking performance

0 5 10 15 20 25

−10

−8

−6

−4

−2

0

Ref

eren

ce C

omm

and,

r(t

)

Time, s

Fig. 1 Reference command for all cases.

0 5 10 15 20 25−20

0

20

q (d

eg/s

)

0 5 10 15 20 25−20

0

20

v (m

/s)

0 5 10 15 20 25−10

0

10

α (d

eg)

0 5 10 15 20 25−10

0

10

θ (d

eg)

Time, s

xx

m

Fig. 2 Case 1: Plant and reference model states.

0 5 10 15 20 25−10

0

10

u e1 (

deg)

0 5 10 15 20 25−10

0

10

u e2 (

deg)

Time, s

Elevator gets stuck at t = 15.1 s

Fig. 3 Case 1: Control inputs.

0 5 10 15 20 25−10

−5

0

5

10

15

Bia

s es

timat

e

Time (s)

β

β

β βθ

β β θ βα

β α

β = True values(dashed)β^ = Estimates(solid)

Unknown biasappears at t = 10 s

Fig. 4 Case 1: Adaptive sensor bias estimates.

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despite the occurrence of sensor bias and actuator failure duringoperation.2) Case 2 (MRAC using asymptotic bias estimator): The adaptive

control law in Eq. (15) is implemented, and the sensor bias estimate isgenerated using an arbitrarily designed asymptotic estimator(Sec. IV). As expected, the bias estimates converge to their truevalues. The system states closely follow the reference model states.(The response figures are not included due to space limitation.)3) Case 3 (Special case: bias in rate measurements only): For this

special case, the sensor bias only exists in rate measurements, pitchrate q and airspeed v (i.e., β � � 2 7 0 0 �T). It is assumed thatreliable bias-free measurements of θ and the ground position (e.g.,using differential GPS measurements and GPS position) areavailable. Estimates of sensor bias in pitch rate and airspeed can begenerated using Eq. (26) (for details of estimator construction, please

see [8]). The bias estimates converge to their true values, as shown inFig. 5. The system states asymptotically follow the reference modelstates, as shown in Fig. 6.

VII. Conclusions

Accommodation of simultaneous actuator failures and sensor biasfaults was addressed in a model reference adaptive control (MRAC)setting. A modified MRAC law was presented, which includes biasestimation as well as control gain adaptation. This control law wasshown to provide bounded tracking error and stability (signalboundedness). When an external asymptotically stable sensor biasestimator is available, an MRAC law was developed to accomplishasymptotic state tracking. When biases are present only in the ratemeasurements and bias-free position measurements are available, anMRAC law that incorporates a model-independent bias estimatorwas developed and was shown to provide asymptotic state tracking.Simulation examples were presented to illustrate the methods.Although the results presented are for the single reference input case,they can also be extended to the case with multiple actuator groupsand multiple reference inputs. Future work will also includeextension to the case when the sensor biases are time varying.

References

[1] Tao, G., Chen, S. H., Tang, X. D., and Joshi, S. M., Adaptive Control ofSystems with Actuator Failures, Springer–Verlag, London, 2004,pp. 15–54.

[2] Joshi, S. M., Patre, P., and Tao, G., “Adaptive Control of Systems withActuator Failures Using an Adaptive Reference Model,” Journal of

Guidance, Control, and Dynamics, Vol. 35, No. 3, 2012, pp. 938–949.doi:10.2514/1.54332

[3] Ale, B. J. M., Bellamy, L. J., Cooper, J., Ababei, D., Kurowicka, D.,Morales, O., and Spouge, J., “Analysis of the Crash of TK 1951 UsingCATS,” Reliability Engineering & System Safety, Vol. 95, No. 5,May 2010, pp. 469–477.doi:10.1016/j.ress.2009.11.014

[4] “Qantas Airbus A330 Accident Media Conference,” AustralianTransport Safety Bureau, Australian Transport Safety Bureau MediaRelease Alert, 14 Oct. 2008, http://www.atsb.gov.au/newsroom/2008/release/2008_43.aspx.

[5] “Airbus Gives New Warning on Speed Sensors,” Associated Press,Canadian Broadcasting Corp. News, Canadian Press, 21 Dec. 2010,http://www.cbc.ca/world/story/2010/12/21/airbus-sensor-warning.html.

[6] Guo, J., and Tao, G., “A Multivariable MRAC Scheme with SensorUncertainty Compensation,” Joint 48th IEEE Conference on Decision

and Control and 28th Chinese Control Conference, IEEE Publications,Piscataway, NJ, 2009, pp. 6632–6637.

[7] Burkholder, J., and Tao, G., “Adaptive Detection of SensorUncertainties and Failures,” Journal of Guidance, Control, and

Dynamics. Vol. 34, No. 6, 2011, pp. 1605–1612.doi:10.2514/1.50285

[8] Patre, P., and Joshi, S. M., “Accommodating Sensor Bias in MRAC forState Tracking,” AIAA Paper 2011-6605, 2011.

[9] Joshi, S. M., “Adaptive Control in the Presence of Simultaneous SensorBias and Actuator Failures,” NASA TM-2012-217341, Feb. 2012.

[10] Tereshkov, V. M., “An Intuitive Approach to Inertial Sensor BiasEstimation,” Cornell University Library, 4 Dec. 2012, http://arxiv.org/abs/1212.0892.

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Fig. 5 Case 3: Adaptive sensor bias estimates.

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