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Suresh M. Joshi 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
Abstract

The problem of simultaneously accommodating unknown sensor biases and unknown

actuator failures in uncertain systems is considered in a direct model reference adaptive control (MRAC) setting for state tracking using state feedback. Sensor biases

and actuator faults may be present at the outset or may occur at unknown instants

of time during operation. A modified MRAC law is proposed, which combines sensor bias estimation with control gain adaptation for accommodation of sensor biases

and actuator failures. This control law is shown to provide signal boundedness in

the resulting system. For the case when an external asymptotically stable sensor

bias estimator is available, an MRAC law is developed to accomplish asymptotic

state tracking and signal boundedness. For a special case wherein biases are only

present in the rate measurements and bias-free position measurements are available,

an MRAC law is developed using a model-independent bias estimator, and is shown

to provide asymptotic state tracking with signal boundedness.

1 Introduction

Loss of control due to anomaly and upsets has been known to be a major cause of

aircraft accidents and incidents. Actuator and sensor faults have been implicated in

several aircraft accidents. Some examples of accidents caused by actuator failures

include: hydraulics/multiple actuator failures [1], rudder failure [2], and horizontal

stabilizer failure [3]. Sensor failures have also been implicated in a number of accidents and incidents, for example, radio altimeter fault [4], angle-of-attack sensor

fault [5], and airspeed sensor fault [6].

Direct model reference adaptive control (MRAC) methods offer an approach for

maintaining stability and controllability in the presence of uncertainties and failures,

without requiring explicit fault detection, isolation, and controller reconfiguration.

Direct MRAC methods use controller gains that are adaptively adjusted to achieve

a performance close to that of a reference model while maintaining system stability

and close tracking of the reference model response. Direct MRAC methods for stateor output- tracking using state feedback have been well established. In particular,

direct MRAC schemes that use state feedback for state-tracking (SFST) have the

advantage of simplicity of implementation and can achieve effective state tracking

in the presence of parameter uncertainties [7]. Such schemes have been extended to

the case with actuator failures [8], [9], [10]; to plant-model mismatch due to damage

or icing [11]; and to simultaneous plant-model mismatch and actuator failures [12].

In addition to actuator faults, sensor faults may also compromise safety. A

common type of sensor fault is sensor bias, which can develop during operation in

sensors such as rate gyros, accelerometers, altimeter, etc. One approach to deal with

sensor faults is to use redundant sensor packages. However, common-mode failures

can occur across all the sensors, and each sensor can typically develop a different

unknown bias. If used directly in an MRAC law, such offsets in sensor measurements can have detrimental effects on closed-loop stability, which can no longer be

theoretically guaranteed. Literature addressing accommodation of sensor faults (or

1
simultaneous actuator and sensor faults) in an MRAC setting is relatively limited.

In [13], a modified MRAC law was developed, which combines a bias estimator with

control gain adaptation, to obtain signal boundedness and bounded tracking error.

Further, for the case wherein an asymptotically stable sensor bias estimator is available, an MRAC control law was developed to accomplish asymptotic tracking and

signal boundedness.

In this paper, the results of [13] are extended to the case with simultaneous actuator failures and sensor bias faults. Such faults (of unknown magnitude) may occur

in unknown sensors and/or actuators at unknown instants of time. The problem

formulation is presented in Section 2, and an MRAC law that incorporates adaptive sensor bias estimation to ensure signal boundedness is developed in Section 3.

Section 4 considers the case when an external asymptotically stable sensor bias estimator is available, and Section 5 addresses a special case when sensor bias exists

only in the rate measurements. Section 6 contains the concluding remarks.

2 Problem Formulation

Consider a linear time-invariant plant, subject to actuator failures and sensor biases,

described by:

x ̇ (t) = Ax (t) + Bu (t) (1)

y (t) = x(t) + β

where A ∈ Rn×n is the system matrix assumed to be unknown, B ∈ Rn×m is the

input matrix, x (t) ∈ Rn is the system state, and u (t) ∈ Rm is the control input.

y (t) ∈ Rn is the available state measurement with an unknown constant bias β ∈ Rn.

In practice, additive sensor noise and process noise are also present in the system.

However, as is customary in the MRAC literature, sensor noise and process noise

are not included in the analysis in order to facilitate analytical proofs of tracking

stability and signal boundedness.

In addition, the actuators u(t) ∈ Rm (e.g., control surfaces or engines in aircraft

flight control) may fail during the operation. Actuator failures are modeled as

uj (t)= ̄uj , t ≥ tj , j ∈ Jp (2)

Jp = {j1, j2,...,jp}⊆{1, 2,...,m}

where the failure pattern Jp, the failure value  ̄uj (assumed to be constant), and

the failure time of occurrence tj are all unknown. For example, an aircraft control

surface may be locked at some unknown fixed value due to hydraulics failure. Let

v(t)=[v1, v2,...,vm]

∈ Rm be the applied(commanded) control input signal. In

the presence of actuator failures, the actual input vector u(t) to the system can be

described as

u(t) = v(t) + σ( ̄u − v(t)) = (I − σ)v(t) + σu ̄ (3)

2
where

u ̄ = [ ̄u1, u ̄2,..., u ̄m]



σ = diag{σ1, σ2,...,σm}

σi = 1 if the ith actuator fails, i.e., ui =  ̄ui

σi = 0 otherwise. (4)

That is, σ is a diagonal matrix (“failure pattern matrix”) whose entries are piecewise

constant signals that take on the values of zero or one. The components of the applied

input signal v(t)=[v1(t), v2(t),...,vm(t)]which correspond to the failed actuators,

cannot affect the system dynamics. The actuator failures are uncertain in value,

pattern and time of occurrence.

The objective is to design an adaptive feedback control law using the available

measurement y (t) with unknown bias β, such that closed-loop signal boundedness

is ensured and the system state x (t) tracks the state of a reference model described

by

x ̇ m (t) = Amxm (t) + Bmr (t) (5)

where xm ∈ Rn is the reference model state, Am ∈ Rn×n, Bm ∈ Rn×mr , and

r(t) ∈ Rmr (1 ≤ mr ≤ m) is a bounded reference input used in system operation

(e.g., pilot input in the case of aircraft).

This paper considers the single-reference-input case, i.e., r is a scalar (mr = 1)

and Bm ∈ Rn. The actuators are assumed to be similar (e.g., segments of the same

control surface), i.e., the columns (bi) of the B matrix can differ only by an unknown

scalar multiplier. It is also assumed that bi are parallel to the reference model input

matrix Bm ∈ Rn, i.e.,

bi = Bm/αi, i = 1,...,m (6)

for some unknown (finite and non-zero) αi’s whose signs are assumed known. The

objective is to design an adaptive control law that will ensure closed-loop signal

boundedness and asymptotic state tracking (i.e., limt→∞(x(t) − xm(t)) = 0) despite

system uncertainties, actuator failures, and sensor bias faults. The adaptive controller should synthesize the control signal v(t) so as to ensure system stability and

asymptotic tracking regardless of whether (or which) actuators have failed, or the

failure values, in the presence of unknown biases in the sensor outputs. That is, v(t)

should be capable of compensating for the actuator failures and sensor bias faults

automatically.

As in the case of actuator failures with no sensor bias [9], it is assumed that

the system (A, B) and the reference model (Am, bm) satisfy the following SFST

matching conditions (similar to those in [9], modified for actuator failure and sensor

bias accommodation): For every failure pattern, there exist gains K1 ∈ Rn×m, and

k2, k3 ∈ Rm, such that

Am = A + B(I − σ)K

1 ; Bm = B(I − σ)k2; Bσu ̄ = −B(I − σ)(K1 β + k3). (7)

The reference model (Am, Bm) is usually designed to capture the desired closed-loop

response of the plant. For example, the reference model may be designed using

3
optimal and robust control methods such as LQR, H2, or H∞ methods. For the

adaptive control scheme, only Am and Bm need to be known. Because Am is a

Hurwitz matrix, there exist positive definite matrices P = P , Q = Q∈ Rn×n,

such that the following Lyapunov inequality holds:

A

mP + P Am ≤ −Q. (8)

Remark- Another type of actuator failure, reduced effectiveness, can occur due

to reasons such as partial loss of a control surface, or control surface icing in aircraft.

When such failures are present in addition to the failures defined in (2), the actual

control input generated by the non-failed actuators is reduced by a factor between 0

and 1. Reduced actuator effectiveness can be readily incorporated in the unknown

coefficients αi’s in Eq. (6).

3 MRAC Using Adaptive Sensor Bias Estimation

The sensor measurements available for feedback have unknown biases as in Eq. (1).

This section considers a modified adaptive control law that includes sensor bias

estimation similar to the approach considered in [13]. Let βˆ (t) denote an estimate

of the unknown sensor bias β. Using βˆ, define the ‘corrected’ state  ̄x(t) ∈ Rn as

x ̄ = y − β. ˆ (9)

Therefore,

x ̄ = x + β − βˆ = x + β.  ̃ (10)

where β ̃ = β − βˆ. Design an adaptive control law as

v = Kˆ 

1 y + ˆ

k2r + ˆk3 (11)

where Kˆ1 (t) ∈ Rn×m, and ˆ

k2, ˆk3 (t) ∈ Rm are the adaptive gains. Therefore, the

closed-loop corrected-state equation is

x ̄ ̇ = Ax + B(I − σ)



Kˆ 

1 y + ˆ

k2r + ˆk3



+  ̃ ̇β + Bσu ̄

= Ax + B(I − σ) (K

1 y + k2r + k3) + B(I − σ)



K ̃ 

1 y +  ̃

k2r +  ̃k3



+  ̃ ̇β + Bσu ̄

= (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 + Bσu ̄ (12)

where K ̃1 = Kˆ1 − K1,  ̃

k2 = ˆk2 − k2, and  ̃k3 = ˆk3 − k3. The matching conditions of

(7) are assumed to be satisfied. Using (7) and (10) in (12), we get

x ̄ ̇ = Amx ̄ + Bmr + B(I − σ)



K ̃ 

1 y +  ̃

k2r +  ̃k3



− Amβ ̃ +  ̃ ̇

β. (13)

Define a measurable auxiliary error signal ˆe(t) ∈ Rn as

eˆ =  ̄x − xm. (14)

4
Therefore, from (10), we have

eˆ = x − xm + β ̃ = e + β ̃ (15)

where e = x − xm denotes the state tracking error. Differentiating (14) with respect

to time, the closed-loop auxiliary error system can be expressed as

 ̇

eˆ = x ̄ ̇ − x ̇ m. (16)

Substituting (13) and (5) into (16) yields

 ̇

eˆ = Ameˆ+ B(I − σ)



K ̃ 

1 y +  ̃

k2r +  ̃k3



− Amβ ̃ +  ̃ ̇

β

= Ameˆ+ m

j /∈Jp

bj (K 

1jy + k2jr + k3j ) − Amβ ̃ +  ̃ ̇

β

= Ameˆ+ Bm

m

j /∈Jp

(1/αj )(K 

1jy + k2jr + k3j ) − Amβ ̃ +  ̃ ̇

β (17)

where the subscript j denotes the jth column of K ̃1 and the jth element of  ̃

k2,  ̃k3

(similar notation for K1, Kˆ1, k2, ˆk2, k3, ˆk3). The following theorem gives adaptive

gain update and bias estimation laws that guarantee closed-loop signal boundedness

as well as bounded tracking error.

Theorem 1: For the system given by (1), (3), (5); the adaptive controller (11), the

gain adaptation laws

 ̇

Kˆ1j = −sgn(αj )Γ1jB

mPeyˆ

 ̇

ˆ

k2j = −sgn(αj )γ2jB

mPerˆ (18)

 ̇

ˆ

k3j = −sgn(αj )γ3jB

mPeˆ

for j = 1, 2,...,m, where Γ1j ∈ Rn×n is a constant symmetric positive definite

matrix, γ2j , γ3j , are constant positive scalars, and P was defined in (8); and the bias

estimation law  ̇

βˆ = ηP −1A

mPeˆ (19)

where η ∈ R is a tunable positive constant gain, guarantee that all the closed-loop

signals including the adaptive gains and bias estimate are bounded and the tracking

error e (t) is bounded.

Proof: Define

V = ˆePeˆ+ m

j /∈Jp

1

|αj |

(K ̃ 

1jΓ−1

1j K ̃1j +  ̃

k2

2jγ−1

2j +  ̃

k2

3jγ−1

3j ) + 1

η

β ̃Pβ.  ̃ (20)

Differentiating (20) with respect to time, and using (8), (17), and the gain update

laws in (18), the following expression is obtained upon simplification:

V ̇ ≤ −eˆQeˆ− 2ˆeP Amβ ̃ + 2ˆeP  ̇

βˆ +

2

η

β ̃P  ̇

β. ˆ (21)

5
Using the bias estimation law of (19) in (21), we get

V ̇ ≤ −eˆQeˆ− 2ˆeP Amβ ̃ + 2ηeˆA

mPeˆ+ 2β ̃AmPeˆ

≤ −eˆQeˆ− ηeˆQeˆ = − (1 + η) ˆeQe. ˆ

Therefore, V ̇ ≤ 0, i.e., V (t) is bounded for all t, and ˆe(t), βˆ (t), y(t), Kˆ1, ˆ

k2, ˆk3 are

all bounded and ˆe(t) ∈ L2. From (17), (19) and closed-loop signal boundedness, we

have  ̃ ̇β,  ̇eˆ(t) ∈ L∞, therefore, using Barbalat’s lemma [7], limt→∞ eˆ(t) = 0. That

is, all signals and estimates are bounded, and limt→∞( ̄x − xm)=0. 

The adaptive control law in Theorem 1 guarantees stability (signal boundedness)

and bounded tracking error. However, although ˆe → 0 as t → ∞, it cannot be

concluded that e(t) → 0 unless β ̃(t) → 0. If persistent excitation is present, β ̃(t)

would approach 0 as t → ∞, in which case, e (t) → 0 as t → ∞.

In an effort to accomplish asymptotic tracking without the need for persistent

excitation, the next section addresses the case when a separate asymptotically stable

bias estimator is available.

4 MRAC Using Asymptotic Bias Estimator

In the presence of actuator failures and sensor bias as defined in Section 2, suppose

an external bias estimator is available, such that the estimation error dynamics is of

the form

 ̃ ̇

β = Aββ ̃ (22)

where βˆ (t) is an estimate of β, and

β ̃ (t) = β − βˆ (t)

is the estimation error; Aβ ∈ Rn×n is a known Hurwitz (asymptotically stable)

matrix, which implies that limt→∞β ̃ (t) = 0. Defining the adaptive control law as

in (11) and proceeding as in Section 3, we obtain (17). The adaptive control law

(11) along with the gain adaptation laws (18) guarantee signal boundedness and

asymptotic tracking.

Theorem 2: For the system given by (1), (3), (5), (11) with a bias estimator that

satisfies (22), and the gain adaptation laws (18) guarantee that all the closed-loop

signals including adaptive gains are bounded and the tracking error e (t) → 0 as

t → ∞.

Proof: Define

V = ˆePeˆ+ m

j /∈Jp

1

|αj |

(K ̃ 

1jΓ−1

1j K ̃1j +  ̃

k2

2jγ−1

2j +  ̃

k2

3jγ−1

3j ) + β ̃Pββ ̃ (23)

where Pβ = P 

β ∈ Rn×n is a positive definite solution of the Lyapunov inequality

A

βPβ + PβAβ ≤ −Qβ (24)

6
for some Qβ = Q

β > 0 ∈ Rn×n. Differentiating (23) with respect to time and using

(8), (17), (18), (22), (24), the following expression is obtained upon simplification:

V ̇ ≤ −eˆQeˆ− 2ˆeP (Am − Aβ) β ̃ − β ̃Qββ ̃

≤ −zQz ̄

where z ∈ R2n is defined as

z = 

eˆβ ̃

and Q ̄ ∈ R2n×2n is defined as

Q ̄ =

 Q P (Am − Aβ)

(Am − Aβ)

P Qβ

	

.

Since Q is positive definite, the matrix Q ̄ is positive definite iff

Qβ − (Am − Aβ)

P Q−1P (Am − Aβ) > 0. (25)

Qβ can be chosen such that (25) is satisfied; therefore, V ̇ ≤ 0, i.e., V (t) is bounded

for all t, and ˆe(t), βˆ (t), y(t), Kˆ1, ˆ

k2, ˆk3 are all bounded and ˆe(t), β ̃ (t) ∈ L2.

Using arguments similar to those in the proof of Theorem 1, and noting from (22)

that β ̃(t) → 0 as t → ∞, it can be concluded that all signals including the adaptive

gains are bounded, and limt→∞ e(t) = 0, i.e., x(t) → xm(t) 

5 A Special Case- Bias in Rate Measurements Only

This section considers a special case wherein sensor bias is present only in the rate

measurements, but bias-free measurements of the corresponding position variables

are available. For example, rate gyros, which measure angular velocities, can be

prone to constant or slowly-varying biases, while bias-free angle (attitude) measurements can be available by using GPS data. Suppose the state vector x(t) in (1) is

composed of rate variables ξ(t) ∈ Rn1 , and the remaining states x2(t) ∈ R(n−n1)

,

and that bias exists only in the measurements of ξ(t). Then (1) can be written in

the form:

x ̇ = d

dt  ξ

x2

	

= [A]

 ξ

x2

	

+

 B1

B2

	

u (26)

y =

 yξ

y2

	

=

 ξ

x2

	

+

 β

0

	

(27)

where β ∈ Rn1 is a constant unknown bias in the rate measurements. (For example,

for a longitudinal aircraft model in a wings-level cruise condition, having a bias error

in the pitch rate gyro, the state variables can be represented by: ξ = q, x2 = [v, α, θ]

,

where q, v, α, θ denote the pitch rate, airspeed, angle of attack, and pitch angle

respectively). In general, some components of ρ may be included in x2.

The position variable vector ρ corresponding to ξ is given by

ρ ̇ = ξ = yξ − β. (28)

It will be assumed that a bias-free measurement of ρ(t) is available, i.e.,

yρ = ρ. (29)

7
5.1 Model-Independent Observer

Using the bias-free position measurements, a non-model-based observer can be designed to estimate the velocity sensor bias as shown below.

From (28) and (29), and augmenting the equation

β ̇ = 0

the following system is obtained.

 ρ ̇

β ̇

	

=

 0 −I

0 0 	  pβ

	

+

 I

0

	

yξ (30)

yρ = [I 0]  ρ

β

	

. (31)

The observability of the above system can be readily verified, and an observer gain

L = [L

1 L2]

can be designed to yield an asymptotically stable state estimator:


  ̇

ρˆ

 ̇

βˆ



=

 0 −I

0 0 	  ρˆ

βˆ

	

+

 I

0

	

yξ +

 L1

L2

	

(ρ − ρˆ). (32)

The estimation error equation is:


 ρ ̃ ̇

 ̃ ̇

β



=

 −L1 −I

−L2 0

	  ρ ̃

β ̃

	

:= [Af ]

 ρ ̃

β ̃

	

(33)

where  ̃ρ = ρ − ρˆ and Af is a Hurwitz matrix. Therefore, given any symmetric

positive definite matrix Qf ∈ R2n1×2n1 , there exists a symmetric positive definite

matrix Pf ∈ R2n1×2n1 such that

A

fPf + PfAf ≤ −Qf . (34)

5.2 Asymptotic State Tracking

Note that the system state variables are re-ordered as in (26); therefore the state

variables of the reference model in (5) are also similarly re-ordered for consistency.

The bias estimate βˆ can be used to define the “corrected” state vector

x ̄ = y −

 βˆ

0

	

= x +

 β

0

	

−

 βˆ

0

	

= x +

 β ̃

0

	

. (35)

As in Section 3, the auxiliary tracking error (which can be measured) is defined as

eˆ =  ̄x − xm = x − xm +

 β ̃

0

	

= e +

 β ̃

0

	

. (36)

Proceeding as in Section 3, a slightly modified version of (17) is obtained:

 ̇

eˆ = Ame + Bm

m

j /∈Jp

(1/αj )(K 

1jy + k2jr + k3j ) − AmIβ ̃ + I  ̃ ̇

β (37)

8
where I = [In1 0n1×(n−n1)]

(Il and 0l×k denote the identity matrix and the zero matrix of the subscript dimensions). The following result gives an adaptive control law

that guarantees closed-loop signal boundedness and asymptotic tracking for systems

with rate sensor bias and actuator failures.

Theorem 3: For the system given by (26), (5), (11), (32) and the gain adaptation

laws (18), all closed-loop signals including the adaptive gains are bounded and the

tracking error e (t) → 0 as t → ∞.

Proof: Define

V = ˆePeˆ+ m

j /∈Jp

1

|αj |

(K ̃ 

1jΓ−1

1j K ̃1j +  ̃

k2

2jγ−1

2j +  ̃

k2

3jγ−1

3j )+ ̃zPf z ̃ (38)

where  ̃z = [ ̃ρβ ̃]

. Differentiating (38) with respect to time and using (8), (37),

(33), (34), the following expression is obtained upon simplification after using the

gain update laws (18):

V ̇ ≤ −eˆQeˆ− 2ˆeP[IL2 AmI] ̃z − z ̃Qf z ̃

that is,

V ̇ ≤ −χQχ ̄ (39)

where χ ∈ Rn+2n1 is defined as

χ = 

eˆz ̃

and Q ̄ ∈ R(n+2n1)×(n+2n1) is defined as

Q ̄ =

 Q P(IL2 AmI)

(IL2 AmI)P Qf

	

. (40)

Since Q is positive definite, the matrix Q ̄ is positive definite iff

Qf − [IL2 AmI]

P Q−1P[IL2 AmI] > 0. (41)

Qf can be chosen such that (41) is satisfied; therefore, V ̇ ≤ 0, i.e., V (t) is bounded

for all t. Using arguments similar to those in the proof of Theorem 1, and noting

from (33) that β ̃(t) → 0 as t → ∞, it can be concluded that all signals including the

adaptive gains are bounded, and limt→∞ e(t) = 0, i.e., x(t) → xm(t). 

6 Concluding Remarks

This paper addressed accommodation of simultaneous actuator failures and sensor

bias faults in a model reference adaptive control (MRAC) setting. A modified MRAC

law, which combines bias estimation with control gain adaptation, was developed and

shown to provide bounded tracking error and signal boundedness. For the case when

an external asymptotically stable sensor bias estimator is available, an MRAC law

was developed to accomplish asymptotic state tracking and signal boundedness. For

a special case wherein biases are present only in the rate measurements and bias-free

9
position measurements are available, an MRAC law was developed using a modelindependent bias estimator, and was shown to provide asymptotic state tracking

with signal boundedness. This paper focused on analytical results. Application

examples and simulation results will be included in future papers on this topic.

Future work will also address extension to cases with multiple actuator groups and

multiple reference inputs.

References

1. Anon: Aircraft Accident Report- United Airlines Flight 232, McDonnell

Douglas DC-10, Sioux Gateway Airport, Sioux City, Iowa, July 19, 1989,

Report No. NTSB/AAR-90106 (PBSO-910406), National Transportation

Safety Board, November 1,1990.

2. Anon: Aircraft Accident Report- Uncontrolled Descent and Collision with

Terrain USAir Flight 427 Boeing 737-300, N513AU Near Aliquippa, Pennsylvania September 8, 1994, NTSB/AAR-99/01 (PB99-910401), March

24, 1999.

3. Anon:Aircraft Accident Report- Loss of Control and Impact with Pacific Ocean Alaska Airlines Flight 261 McDonnell Douglas MD-83,

N963AS About 2.7 Miles North of Anacapa Island, California January

31, 2000, NTSB/AAR-02/01,PB2002-910402 Notation 7263E, National

Transportation Safety Board, December 30, 2002.

4. Alea, B.J.M., Bellamyb, L.J., Cooper, J., Ababeid, D., Kurowickad,D.,

Moralesd, O., and Spougee, J.: Analysis of the Crash of TK 1951 Using

CATS, Reliability Engineering & System Safety, Volume 95, Issue 5, May

2010, pp. 469-477

5. Anon: Qantas Airbus A330 Accident Media Conference, Press release of

the Australian Transport Safety Bureau, 14 October 2008.

6. Anon: Airbus Gives New Warning on Speed Sensors, (Associated Press article), http://www.cbc.ca/world/story/2010/12/21/airbussensor-warning.html, Canadian Broadcasting Corp. (CBC) News, December 21, 2010.

7. Tao, G.: Adaptive Control Design and Analysis, N.J.: John Wiley & Sons,

2003.

8. Tao, G., Joshi, S. M., and Ma, X.: Adaptive State Feedback and Tracking

Control of Systems with Actuator Failures, IEEE Trans. on Automatic

Control, vol. 46, No. 1, January 2001, pp. 78-95.

9. Tao, G., Chen, S. H., Tang, X. D., and Joshi, S. M. Adaptive Control of

Systems with Actuator Failures, London: Springer-Verlag, 2004.

10
10. Tao, G. and Joshi, S. M.: Direct Adaptive Control of Systems with Actuator Failures: State of the Art and Continuing Challenges, Proc. 2008

AIAA Guidance, Navigation, and Control Conference, Honolulu, HI, Aug

18-21.

11. Joshi, S. M., Tao, G., and Patre, P.: Direct Adaptive Control Using an

Adaptive Reference Model, International Journal of Control, vol. 84, No.

1, January 2011, pp. 180-196.

12. Joshi, S. M., Patre, P., and Tao, G.: Adaptive Control of Systems with

Actuator Failures Using an Adaptive Reference Model, AIAA J. Guidance, Control, and Dynamics (to appear).

13. Patre, P. and Joshi, S. M.: Accommodating Sensor Bias in MRAC for

State Tracking, Proc. AIAA Guidance, Naviagation, and Control Conference, Portland, OR, August 8-10, 2011.

11
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OMB No. 0704-0188

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Technical Memorandum

 4. TITLE AND SUBTITLE

Adaptive Control in the Presence of Simultaneous Sensor Bias and 

Actuator Failures

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Joshi, Suresh M.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

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NASA

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14. ABSTRACT

The problem of simultaneously accommodating unknown sensor biases and unknown actuator failures in uncertain systems is 

considered in a direct model reference adaptive control (MRAC) setting for state tracking using state feedback. Sensor biases 

and actuator faults may be present at the outset or may occur at unknown instants of time during operation. A modified MRAC 

law is proposed, which combines sensor bias estimation with control gain adaptation for accommodation of sensor biases and 

actuator failures. This control law is shown to provide signal boundedness in the resulting system. For the case when an 

external asymptotically stable sensor bias estimator is available, an MRAC law is developed to accomplish asymptotic state 

tracking and signal boundedness. For a special case wherein biases are only present in the rate measurements and bias-free 

position measurements are available, an MRAC law is developed using a model-independent bias estimator, and is shown to 

provide asymptotic state tracking with signal boundedness. 

15. SUBJECT TERMS

Actuator faults; Adaptive control; Fault-tolerant control; Sensor faults

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