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Adaptive failure compensation for aircraft tracking control using engine differential based model

TL;DR: In this paper, an adaptive feedback control scheme for asymptotic state tracking is developed and applied to a transport aircraft model in the presence of two types of failures during operation, rudder failure and aileron failure.
Abstract: An aircraft model that incorporates independently adjustable engine throttles and ailerons is employed to develop an adaptive control scheme in the presence of actuator failures. This model captures the key features of aircraft flight dynamics when in the engine differential mode. Based on this model an adaptive feedback control scheme for asymptotic state tracking is developed and applied to a transport aircraft model in the presence of two types of failures during operation, rudder failure and aileron failure. Simulation results are presented to demonstrate the adaptive failure compensation scheme.

Summary (2 min read)

Introduction

  • Actuator failures, adaptive compensation, aircraft flight control, engine differentials, tracking, also known as Keywords.
  • Effective compensation of control component failures is crucial for aircraft flight safety.
  • In [2], several multivariable adaptive control algorithms for flight control reconfiguration were presented with a failure characterized by a locked left horizontal tail surface.
  • The problem was formulated as a nonlinear disturbance rejection problem in the presence of actuator failures and simulation results using an F-16 aircraft model were discussed.
  • For the design of such control schemes, an aircraft model with independently adjustable engine thrusts is necessary.

II. ENGINE DIFFERENTIAL BASED MODEL

  • As described in [8], a nonlinear aircraft dynamic model in body-axis coordinate system which incorporates engine differentials can be described by the force equations m(u̇+ qw − rv) = u, v and w are the bodyaxis components of the velocity of the center of mass.
  • P, q and r are the body-axis components of the angular velocity of the aircraft.
  • X , Y and Z are the body-axis aerodynamic forces about the center of mass.
  • (II.13) represent the effect of engine thrust differentials, that is, if the left and right engine thrusts are equal, these matrices are zero.
  • This engine differential based model in which the two engine thrusts and the ailerons are taken into account separately captures the essential dynamics of the aircraft in the engine differential mode, and is capable of coping with some actuator failures such as rudder failures or engine failure, which cannot be achieved without using engine differentials.

A. Problem Formulation

  • An example of such actuator failures is when an aircraft control surface (such as the rudder or an aileron) is stuck at some unknown fixed position at an unknown time instant.
  • The control objective is to design an adaptive state feedback control signal to be applied to the actuators in u, to ensure closed-loop signal boundedness, and asymptotic tracking: limt→∞(x(t)−xd(t)) = 0, where xd(t) is a desired state trajectory, in the presence of unknown actuator failures.

B. Adaptive Compensator Designs

  • The failures are assumed to occur instantaneously, i.e., σi are piecewise constant functions of time.
  • For asymptotic tracking, the authors first present a desired nominal design for the system (III.1) without any actuator failures.
  • Assumption 3.1 also requires the knowledge of A and B. (This condition basically implies that the nominal LQ regulator used for generating the desired trajectory is designed to be robust to parameter uncertainties).

IV. APPLICATION TO FLIGHT CONTROL

  • The authors demonstrate application of the adaptive failure compensation technique to a transport aircraft by presenting some simulation results for trajectory tracking in the presence of unknown rudder and aileron failures.
  • The authors shall first describe the aircraft model used in simulation, and then present the simulation results.

A. Aircraft Model for Simulation Study

  • For their simulation study, the authors use a transport aircraft model.
  • The non-zero terms in A(3) and B(3) represent the engine thrust differential effect.
  • The authors consider two types of constant actuator failures: rudder failure and aileron failure.
  • It represents the rudder stuck in its position at instant tf , and cannot be moved.

B. Simulation Results

  • The authors present the simulation results for the asymptotic tracking of xd(t) by x(t) to demonstrate the performance of the system with the adaptive failure compensation scheme described in Section 3.2.
  • The initial value of the state vector is zero, i.e., the airplane is in steady wings-level flight.
  • For their simulation study, the authors examine two cases: (I) system responses with adaptive failure compensation with failure (IV.2) and (II) system responses with adaptive failure compensation with aileron failure (IV.3).
  • These design parameters were chosen by trial and error.
  • This objective cannot be achieved with a fixed controller.

V. CONCLUDING REMARKS

  • A dynamic model of aircraft with independently adjustable engine throttles and ailerons was considered for failure compensation in the presence of rudder or aileron failure.
  • This model captures the key features of aircraft flight dynamics when in the engine differential mode and facilitates the development of an adaptive failure compensation approach to handle actuator failures using functioning actuators that can be of types different from the failed actuators.
  • First, robustness of the adaptive scheme to model errors, and relaxation of the requirement of knowledge of the system matrices, need to be investigated.
  • In addition, the effects of actuator nonlinearities including output and rate saturation, as well as actuator dynamics, need to be addressed.

ACKNOWLEDGMENTS

  • This research was partially supported by NASA Langley Research Center under grant NCC-1-02006.
  • The authors would like to thank the reviewers for their helpful comments.

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Adaptive Failure Compensation for Aircraft Tracking Control Using
Engine Differential Based Model
Yu Liu, Xidong Tang and Gang Tao
Department of Electrical and Computer Engineering
University of Virginia
Charlottesville, VA 22903
Suresh M. Joshi
Mail Stop 308
NASA Langley Research Center
Hampton, VA 23681
AbstractAn aircraft model that incorporates independently
adjustable engine throttles and ailerons is employed to develop
an adaptive control scheme in the presence of actuator failures.
This model captures the key features of aircraft flight dynamics
when in the engine differential mode. Based on this model an
adaptive feedback control scheme for asymptotic state tracking
is developed and applied to a transport aircraft model in the
presence of two types of failures during operation, rudder
failure and aileron failure. Simulation results are presented to
demonstrate the adaptive failure compensation scheme.
Keywords: Actuator failures, adaptive compensation, aircraft
flight control, engine differentials, tracking.
I. INTRODUCTION
Effective compensation of control component failures is
crucial for aircraft flight safety. Considerable research has
focused on the design of control systems that can provide
safe performance when failures occur. In [5], an emergency
flight control system that can utilize engine thrusts to ma-
neuver an aircraft was developed and tested on an MD-
11 airplane. In [7], a propulsion controlled aircraft design
by H-infinity model matching was introduced. In [1], an
indirect adaptive LQ controller was developed for aircraft
control, which is able to implicitly reconfigure the control
law using on-line estimates of the changed aircraft dynamics,
so that the failures in the pitch control channel or the
horizontal stabilizer can be accommodated. In [2], several
multivariable adaptive control algorithms for flight control
reconfiguration were presented with a failure characterized
by a locked left horizontal tail surface. An adaptive controller
was used to compensate this failure. In [13], a direct adaptive
reconfigurable flight control algorithm was presented. An on-
line adaptive neural network was applied to regulate the error
between the plant model and the actual aircraft, and appli-
cation of this control approach to a tailless advanced fighter
aircraft was demonstrated. In [9], an algorithm for aircraft
failure detection and compensation was presented, which
incorporated multiple model adaptive estimation methods.
In this approach failures are detected by a bank of parallel
Kalman filters and a reconfiguration algorithm is used to
redistribute control commands to the non-failed surfaces.
In [3], a new parametrization for the modeling of control
effector failures in flight control applications was proposed,
including lock in place, hard over and loss of effectiveness
patterns. An algorithm based on multiple model adaptive re-
configuration control approach was presented and illustrated
by simulation results of the F/A-18 aircraft during carrier
landing. In [12], an F-16 fighter aircraft subject to asymmet-
ric actuator failure was discussed, including system modeling
and control system design. The problem was formulated as
a nonlinear disturbance rejection problem in the presence
of actuator failures and simulation results using an F-16
aircraft model were discussed. In [6], fault-tolerant control
system design against stuck actuators was investigated using
an iterative learning observer that provides information of the
system state estimates and fault compensation transients. The
performance of the controller design was evaluated using an
F-8 aircraft model.
In this paper, we present a failure compensation scheme
based on an adaptive control approach that can utilize the
remaining (functioning) controls to achieve desired perfor-
mance in the presence of uncertain system failures. To com-
pensate for aircraft failures such as rudder failure or engine
malfunction, asymmetric engine thrusts may be inevitably
needed [10]. For the design of such control schemes, an
aircraft model with independently adjustable engine thrusts
is necessary. In [8], we derived such an aircraft model and
used it to develop an adaptive failure compensation control
scheme using engine differentials for state regulation. In this
paper, we shall use this aircraft model to develop an adaptive
failure compensation control scheme for state tracking, and
apply the scheme to a transport aircraft model. We shall
consider two simulation cases representing realistic scenarios
in which the rudder or an aileron are stuck at unknown
constant values at unknown time instants.
The paper is organized as follows. In Section 2, we
describe an engine differential based aircraft flight dynamic
model. In Section 3, we develop an adaptive compensation
scheme that is able to handle uncertain actuator failures and
guarantee asymptotic state tracking. In Section 4, we apply
this compensation scheme to a transport aircraft model and
present simulation results to illustrate the effectiveness of the
scheme.
II. ENGINE DIFFERENTIAL BASED MODEL
As described in [8], a nonlinear aircraft dynamic model
in body-axis coordinate system which incorporates engine
1

differentials can be described by the force equations
m( ˙u + qw rv) = X mg sin θ + (T
L
+ T
R
) cos ǫ (II.1)
m( ˙v + ru pw) = Y + mg cos θ sin φ (II.2)
m( ˙w+pvqu) = Z+mg cos θ cos φ(T
L
+T
R
) sin ǫ (II.3)
and moment equations
I
x
˙p+I
xz
˙r+(I
z
I
y
)qr+I
xz
qp = L+l(T
L
T
R
) sin ǫ (II.4)
I
y
˙q + (I
x
I
z
)pr + I
xz
(r
2
p
2
) = M (II.5)
I
z
˙r + I
xz
˙p + (I
y
I
x
)qp I
xz
qr = N + l(T
L
T
R
) cos ǫ
(II.6)
where m is the mass of the aircraft. u, v and w are the body-
axis components of the velocity of the center of mass. p, q
and r are the body-axis components of the angular velocity
of the aircraft. X, Y and Z are the body-axis aerodynamic
forces about the center of mass. L, M and N are the body-
axis aerodynamic torques about the center of mass. θ and φ
are the Euler pitch and roll angles of the aircraft. ǫ represents
the angle between thrust and body x-axis. I
i
are the moments
(or products) of inertia in body axes. g is the gravitational
force per unit mass. T
L
and T
R
are the left and right engine
thrusts, and l is the distance between engines and xz plane.
By applying the linearization procedure around the equi-
librium point of interest, we can obtain the linearized aircraft
model with engine differentials. For this purpose, the state
and control vectors of the linearized model are
x = [ u w q θ v r p φ ψ ]
T
(II.7)
U = [ δ
e
δ
t
l
δ
t
r
δ
a
l
δ
a
r
δ
r
]
T
(II.8)
where the notation δ has been dropped from δx and δU
for simplicity of presentation. Thus (u, v, w) represent the
velocity perturbations along each axis and (p, q, r) are the
angular velocity perturbations about each axis. (θ, φ, ψ) are
the pitch, roll and yaw angle perturbations, and δ
e
, δ
a
l
, δ
a
r
,
δ
r
are the deflection perturbations of the elevator, the left
and right ailerons and the rudder. δ
t
l
and δ
t
r
are the left and
right throttle perturbations.
In our study, we consider a steady-state rectilinear wings-
level flight condition as the equilibrium point. For this
steady-state flight condition, the derivatives of all states, the
angular velocity components (p, q, r) and the roll angle φ at
the equilibrium point are all zero, that is,
[ ˙u ˙w ˙q
˙
θ ˙v ˙r ˙p
˙
φ
˙
ψ ]
x
o
, U
o
= 0 (II.9)
p
o
= q
o
= r
o
= φ
o
= ψ
o
= v
o
= 0, (II.10)
where x
o
and U
o
are determined as
x
o
=[ u
o
w
o
0 θ
o
0 0 0 0 0 ]
T
,
U
o
=[ δ
eo
δ
t
lo
δ
t
ro
δ
alo
δ
aro
δ
ro
]
T
. (II.11)
By applying the linearization around this equilibrium point,
we can obtain the linearized aircraft model as
˙x =
A
(1)
4×4
A
(2)
4×5
A
(3)
5×4
A
(4)
5×5
x +
B
(1)
4×3
B
(2)
4×3
B
(3)
5×3
B
(4)
5×3
U, (II.12)
where A
(2)
and B
(2)
are zero matrices, A
(1)
, A
(4)
, B
(1)
and
B
(4)
are of the same forms as in the literature [4], and the
matrices
A
(3)
=
0 0 0 0
¯
T
u
¯
T
w
0 0
¯
T
u
¯
T
w
0 0
0 0 0 0
0 0 0 0
, B
(3)
=
0 0 0
0
¯
T
′′
δ
t
l
¯
T
′′
δ
t
r
0
¯
T
′′′
δ
t
l
¯
T
′′′
δ
t
r
0 0 0
0 0 0
(II.13)
represent the effect of engine thrust differentials, that is, if
the left and right engine thrusts are equal, these matrices are
zero. See [8] for details of this model.
We note that this aircraft model is different from standard
models used in most of the literature [4] that assume equal
engine thrusts and aileron angles. This engine differential
based model in which the two engine thrusts and the ailerons
are taken into account separately captures the essential
dynamics of the aircraft in the engine differential mode,
and is capable of coping with some actuator failures such as
rudder failures or engine failure, which cannot be achieved
without using engine differentials. Therefore it is desirable
to develop an adaptive control scheme for aircraft actuator
failure compensation using engine differentials.
III. ADAPTIVE FAILURE COMPENSATION
In this section, we shall first formulate an actuator failure
compensation problem for linear systems, and then develop
an adaptive failure compensation scheme for closed-loop
stability and asymptotic tracking of the system state variables
in the presence of certain actuator failures.
A. Problem Formulation
Consider the linear time-invariant system
˙x = Ax + Bu, x R
n
, u R
m
, (III.1)
whose actuators u = [u
1
, u
2
, . . . , u
m
]
T
may fail during
system operation. A typical failure model is
u
i
(t) = ¯u
i
, t t
i
, i {1, 2, . . . , m}, (III.2)
where t
i
is the unknown failure time instant and ¯u
i
is the
unknown failure constant [11]. An example of such actuator
failures is when an aircraft control surface (such as the rudder
or an aileron) is stuck at some unknown fixed position at an
unknown time instant.
The control objective is to design an adaptive state feed-
back control signal to be applied to the actuators in u,
to ensure closed-loop signal boundedness, and asymptotic
tracking: lim
t→∞
(x(t)x
d
(t)) = 0, where x
d
(t) is a desired
state trajectory, in the presence of unknown actuator failures.
B. Adaptive Compensator Designs
In the presence of actuator failures, u(t) can be expressed
as
u(t) = v(t) + σ(¯u v(t)), (III.3)
2

where v(t) R
m
is the applied control input vector, ¯u =
[¯u
1
, ¯u
2
, . . . , ¯u
m
]
T
is the failure vector, and σ represents the
failure pattern and is defined as
σ = diag{σ
1
, σ
2
, . . . , σ
m
} (III.4)
with σ
i
= 1 if the ith actuator has failed, that is, u
i
= ¯u
i
,
and σ
i
= 0 otherwise. The failures are assumed to occur
instantaneously, i.e., σ
i
are piecewise constant functions of
time. There are 2
m
possible combinations of actuator states
(each actuator is either normal or failed), and therefore 2
m
1
possible failure patterns that constitute a set denoted by
¯
Σ.
The system (III.1) can then be rewritten as
˙x(t) = Ax(t) + B(I σ)v(t) + Bσ ¯u. (III.5)
For our adaptive control design for actuator failure compen-
sation, the following assumption is needed:
Assumption 3.1: (A, B) is known and stabilizable, and
there exists a non-empty set Σ of “recoverable” failures such
that rank[B(I σ)] = rank[B] σ Σ. Σ is a subset of
¯
Σ.
Remark 3.1: This assumption characterizes the built-in
redundancy needed for failure compensation as well as all the
failure patterns that can be accommodated. This condition is
needed for the existence of a (fixed) failure compensation
controller that can achieve the desired performance when
the system and failure parameters are known. The adaptive
control design for unknown failure parameters is developed
based on the same condition. For instance, when the aircraft
rudder fails during flight, this condition can still be satisfied
so that the aircraft can be controlled by the remaining
actuators and the failure can be accommodated (which is
demonstrated in the simulation in Section 4.2). 2
For asymptotic tracking, we first present a desired nominal
design for the system (III.1) without any actuator failures.
The nonadaptive nominal controller is
u(t) = Kx(t) + κr
d
(t), (III.6)
where K = R
1
B
T
P R
m×n
is an optimal LQ gain
with P satisfying the Riccati equation
A
T
P + P A P BR
1
B
T
P + Q = 0 (III.7)
for some chosen n×n matrix Q = Q
T
> 0 and m×m matrix
R = R
T
> 0, and the reference input r
d
(t) R
m
r
and
κ R
m×m
r
are chosen for some desired system trajectory.
With this nominal controller, the closed-loop system is
˙x(t) = (A+BK)x(t)+Bκr
d
(t), based on which, we define
the desired state trajectory x
d
(t) from the reference system
˙x
d
(t) = (A + BK)x
d
(t) + Bκr
d
(t). (III.8)
Define the tracking error e(t) = x(t) x
d
(t). As the new
adaptive failure compensation scheme for asymptotic state
tracking, the feedback control law is
v(t) =
ˆ
Kx(t) + ˆκr
d
(t) +
ˆ
θ, (III.9)
where
ˆ
K = [
ˆ
K
1
,
ˆ
K
2
, . . . ,
ˆ
K
m
]
T
R
m×n
, ˆκ =
[ˆκ
1
, ˆκ
2
, . . . , ˆκ
m
]
T
R
m×m
r
, and
ˆ
θ = [
ˆ
θ
1
,
ˆ
θ
2
, . . . ,
ˆ
θ
m
]
T
R
m×1
, are the parameters updated from the adaptive laws
˙
ˆ
K
i
= Γ
i
xe
T
P b
i
, i = 1, 2, . . . , m (III.10)
˙
ˆκ
i
= γ
i
r
d
e
T
P b
i
, i = 1, 2, . . . , m (III.11)
˙
ˆ
θ
i
= λ
i
e
T
P b
i
, i = 1, 2, . . . , m, (III.12)
where Γ
i
= Γ
T
i
> 0, γ
i
= γ
T
i
> 0, λ
i
> 0, b
i
is the ith
column of B, i = 1, 2, . . . , m, and P = P
T
> 0 satisfying
(III.7). Γ
i
, γ
i
, and λ
i
denote the design parameters for the
adaptive laws. This adaptive actuator failure compensation
scheme has the following desired properties:
Theorem 3.1: The control law (III.9), updated from
(III.10)–(III.12) and applied to the system (III.1) subject
to the actuator failures (III.2) under Assumption 3.1, en-
sures that all closed-loop system signals are bounded and
lim
t→∞
(x(t) x
d
(t)) = 0, for any failure pattern σ Σ
with uncertain parameters.
Proof: For
¯
Q = Q + P BR
1
B
T
P , using (III.7), we obtain
P (A + BK) + (A + BK)
T
P =
¯
Q < 0. (III.13)
Suppose that at time t there are p < m actuator failures,
that is, u
i
(t) = ¯u
i
, i = i
1
, i
2
, . . . , i
p
, {i
1
, i
2
, . . . , i
p
}
{1, 2, . . . , m}, and that actuator failures happen at time
instants t
k
, with t
k
< t
k+1
, k = 1, 2, . . . , N .
From the condition of Assumption 3.1: rank[B(I σ)] =
rank[B], σ Σ, it follows that for each σ Σ, there exist
constant matrices K
σ
R
m×n
and κ
σ
R
m×m
r
such that
B(I σ)K
σ
= BK, B(I σ)κ
σ
= Bκ. (III.14)
Therefore, for each σ, there are constant K
σ
satisfying
P [A + B(I σ)K
σ
] + [A
T
+ (I σ)K
σ
T
B
T
]P =
¯
Q < 0,
(III.15)
and κ
σ
satisfying B(I σ)κ
σ
r
d
(t) = Bκr
d
(t). In addition
there exists constant θ = [θ
1
, θ
2
, . . . , θ
m
]
T
R
m
, where θ
i
,
i 6= i
1
, i
2
, . . . , i
p
, are solutions of the following equation
X
i6=i
1
,i
2
,...,i
p
b
i
θ
i
=
X
j=i
1
,i
2
,...,i
p
b
j
¯u
j
, (III.16)
and θ
i
= 0, for i = i
1
, i
2
, . . . , i
p
, such that B(I σ)θ =
Bσ¯u.
From (III.5), (III.8), and III.9, we have
˙e(t) = Ax(t) + B(I σ)(
ˆ
Kx(t) + ˆκr
d
(t) +
ˆ
θ) + Bσ ¯u
(A + BK)x
d
(t) Bκr
d
(t)
=Ax(t)+B(I σ)(K
σ
x(t)+κ
σ
r
d
(t)+θ)+Bσ ¯u
(A+BK)x
d
(t)Bκr
d
(t)+B(I σ)[(
ˆ
K K
σ
)x(t)
+(ˆκ κ
σ
)r
d
(t) + (
ˆ
θ θ)] (III.17)
With equations (III.14) and (III.16), the dynamic equation
for tracking error can be simplified as
˙e(t) = (A + BK)e(t) + B(I σ)[(
ˆ
K K
σ
)x(t)
+(ˆκ κ
σ
)r
d
(t) + (
ˆ
θ θ)]. (III.18)
3

With the adaptive laws (III.10)–(III.12), a Lyapunov func-
tion candidate can be chosen as
V = e
T
P e +
X
i6=i
1
,i
2
,...,i
p
(
ˆ
K
i
K
i
)
T
Γ
1
i
(
ˆ
K
i
K
i
)
+
X
i6=i
1
,i
2
,...,i
p
(ˆκ
i
κ
i
)
T
γ
1
i
(ˆκ
i
κ
i
) +
X
i6=i
1
,i
2
,...,i
p
λ
1
i
(
ˆ
θ
i
θ
i
)
2
for each time interval (t
k
, t
k+1
), k = 0, 1, . . . , N, with t
0
=
0 and t
N+1
= , where K
i
is the ith row of K
σ
and κ
i
is
the ith row of κ
σ
. The time-derivative of V in each (t
k
, t
k+1
)
is
˙
V = e
T
[P (A + BK) + (A
T
+ K
T
B
T
)P ]e
+2e
T
P B(I σ)[(
ˆ
K K
σ
)x(t)+(ˆκ κ
σ
)r
d
(t)+(
ˆ
θ θ)]
+2
X
i6=i
1
,i
2
,...,i
p
(
ˆ
K
i
K
i
)
T
Γ
1
i
˙
ˆ
K
i
+ 2
X
i6=i
1
,i
2
,...,i
p
(ˆκ
i
κ
i
)
T
γ
1
i
˙
ˆκ
i
+2
X
i6=i
1
,i
2
,...,i
p
λ
1
i
(
ˆ
θ
i
θ
i
)
˙
ˆ
θ
i
= e
T
[P (A + BK) + (A
T
+ K
T
B
T
)P ]e
+2e
T
P B(I σ)[(
ˆ
K K
σ
)x(t)+(ˆκ κ
σ
)r
d
(t)+(
ˆ
θ θ)]
2
X
i6=i
1
,i
2
,...,i
p
(
ˆ
K
i
K
i
)
T
xe
T
P b
i
2
X
i6=i
1
,i
2
,...,i
p
(ˆκ
i
κ
i
)
T
r
d
e
T
P b
i
2
X
i6=i
1
,i
2
,...,i
p
(
ˆ
θ
i
θ
i
)e
T
P b
i
,
For the considered actuator failure pattern, that is, u
i
(t) =
¯u
i
, σ
i
= 1, i = i
1
, i
2
, . . . , i
p
, using the fact that B(I
σ)(
ˆ
K K
σ
) =
P
i6=i
1
,i
2
,...,i
p
b
i
(
ˆ
K
i
K
i
)
T
and the commu-
tativity property of the matrix trace operator, i.e., T r(XY ) =
T r(Y X), the following equalities hold:
e
T
P B(I σ)(
ˆ
K K
σ
)x(t) =
X
i6=i
1
,i
2
,...,i
p
(
ˆ
K
i
K
i
)
T
xe
T
P b
i
,
e
T
P B(I σ)(ˆκκ
σ
)r
d
(t) =
X
i6=i
1
,i
2
,...,i
p
(ˆκ
i
κ
i
)
T
r
d
e
T
P b
i
,
e
T
P B(I σ)(
ˆ
θθ) =
X
i6=i
1
,i
2
,...,i
p
(
ˆ
θ
i
θ
i
)e
T
P b
i
.
So the time-derivative of V in each (t
k
, t
k+1
) is simplified
as
˙
V = e
T
¯
Qe 0, (III.19)
where (III.13) is used for the last equality. It follows that e
L
2
L
, and
ˆ
K
i
L
and
ˆ
θ
i
L
for i 6= i
1
, i
2
, . . . , i
p
,
where i
1
, i
2
, . . . , i
p
are the indexes of failed actuators.
From (III.10)–(III.12), we have
h
Γ
1
1
˙
ˆ
K
1
, Γ
1
2
˙
ˆ
K
2
, . . . , Γ
1
m
˙
ˆ
K
m
i
= xe
T
P B,
h
γ
1
1
˙
ˆκ
1
, γ
1
2
˙
ˆκ
2
, . . . , γ
1
m
˙
ˆκ
m
i
= r
d
e
T
P B,
h
λ
1
1
˙
ˆ
θ
1
, λ
1
2
˙
ˆ
θ
2
, . . . , λ
1
m
˙
ˆ
θ
m
i
= e
T
P B, (III.20)
which implies that
ˆ
K
i
L
, ˆκ
i
L
and
ˆ
θ
i
L
for
i = i
1
, i
2
, . . . , i
p
, because B can be represented by a linear
combination of b
i
, i 6= i
1
, i
2
, . . . , i
p
.
The function V is not continuous at t
k
, k = 0, 1, . . . , N ,
and only has finite value jumps at those time instants. So
we can conclude that e L
2
L
,
ˆ
K L
, ˆκ L
,
and
ˆ
θ L
. For the nominal design (III.6), A + BK is
asymptotically stable such that x
d
(t) L
with a bounded
reference input r
d
(t). Hence we conclude that x(t) L
,
so does v(t). Furthermore, since ˙x(t) L
and ˙x
d
L
,
given that e(t) L
2
, we also have lim
t→∞
e(t) = 0. 2
The physical meaning of state tracking for the aircraft
dynamic model linearized at an equilibrium point (x
o
, U
o
) is
that the operation of the aircraft follows a desired trajectory
in a neighborhood of the equilibrium point. When we apply
this adaptive failure compensation scheme to aircraft flight
control, we want the aircraft to maintain the desired trajec-
tory that was originally set for the nominal case of no failure,
even if unknown actuator failures occur. Theorem 3.1 gives a
solution to the problem of state tracking. The stabilizability
and rank condition in Assumption 3.1 characterizes the
system redundancy condition needed for actuator failure
compensation. As shown in next section, it is satisfied for
the rudder or aileron failure case and the system state is
able to track the desired trajectory asymptotically, which
implies that the aircraft can maintain the desired performance
under normal as well as failure conditions. Assumption
3.1 also requires the knowledge of A and B. However, it
may be noted that, if the actual system matrix given by
A
p
= A + δA (where δA is the parameter error) is such that
(A
p
+BK)
T
P +P (A
p
+BK) < 0, the asymptotic tracking
and signal boundedness of Theorem 3.1 will still hold. (This
condition basically implies that the nominal LQ regulator
used for generating the desired trajectory is designed to be
robust to parameter uncertainties). Further research is needed
in order to investigate robustness of the adaptive scheme to
model errors, and to relax the requirement of knowledge of
A and B.
IV. APPLICATION TO FLIGHT CONTROL
In this section, we demonstrate application of the adaptive
failure compensation technique to a transport aircraft by
presenting some simulation results for trajectory tracking in
the presence of unknown rudder and aileron failures. We
shall first describe the aircraft model used in simulation, and
then present the simulation results.
A. Aircraft Model for Simulation Study
For our simulation study, we use a transport aircraft model.
The airplane flies at a velocity of 774 ft/sec and an altitude
of 40 kft. The linearized dynamic model is
˙x(t) =
A
(1)
4×4
A
(2)
4×5
A
(3)
5×4
A
(4)
5×5
x(t) +
B
(1)
4×3
B
(2)
4×3
B
(3)
5×3
B
(4)
5×3
U(t)(IV.1)
where A
(2)
and B
(2)
are zero matrices, and A
(1)
, A
(4)
, B
(1)
and B
(4)
are of the same forms as in [8], and
x = [ u w q θ v r p φ ψ ]
T
,
U = [ δ
e
δ
t
l
δ
t
r
δ
a
l
δ
a
r
δ
r
]
T
,
A
(3)
=
0 0 0 0
0.001 0.001 0 0
0.001 0.001 0 0
0 0 0 0
0 0 0 0
, B
(3)
=
0 0 0
0 0.8 0.7
0 0.5 0.6
0 0 0
0 0 0
4

The non-zero terms in A
(3)
and B
(3)
represent the engine
thrust differential effect. The basic units used in this model
are ft, sec and crad (0.01 radian).
We consider two types of constant actuator failures: rudder
failure and aileron failure. The rudder failure is denoted as
U
6
(t) = U
6
(t
f
) t t
f
, (IV.2)
where t
f
is the failure time instant. It represents the rudder
stuck in its position at instant t
f
, and cannot be moved. The
aileron failure we consider is
U
5
(t) = 0 t t
f
, (IV.3)
which indicates that at failure time instant t
f
, the right
aileron angle drops to zero and is stuck from then on. These
failure patterns satisfy Assumption 3.1.
B. Simulation Results
In this subsection, we present the simulation results for
the asymptotic tracking of x
d
(t) by x(t) to demonstrate
the performance of the system with the adaptive failure
compensation scheme described in Section 3.2. For our
simulation, the values of κ and r
d
were chosen as r
d
= 1,
and
κ = [ 0.7344 0.6842 1.5974 0.5817 0.7853 1 ]
T
that is, r
d
= r
0
d
R is a scalar, which leads to the final
values of the desired states:
x
d
= [ 4 1.06 0 1 0 0 0 0 1.8 ]
T
This trajectory represents a steady state flight condition
in which the aircraft is climbing with a 4 ft/sec velocity
perturbation along the x-axis, a -1.06 ft/sec velocity per-
turbation along the z-axis, a pitch angle of 1 crads (0.57
degrees), and a yaw angle of -1.8 crads (-1.0 degree). The
initial value of the state vector is zero, i.e., the airplane
is in steady wings-level flight. The physical meaning of
state tracking is that the aircraft flies from one steady state
flight condition to another steady state flight condition while
closely following a reference state-trajectory. The choice
of the reference trajectory (in particular, κ and r
d
) in this
paper was arbitrary, the main purpose being demonstration
of the adaptive scheme. For the controller design, we choose
Q = I
9
and R = diag{2, 6, 6, 2, 2, 2}.
For our simulation study, we examine two cases: (I)
system responses with adaptive failure compensation with
failure (IV.2) and (II) system responses with adaptive failure
compensation with aileron failure (IV.3).
Case (I). System performances with adaptive
compensation scheme and rudder failure (IV.2). The failure
instant is t
f
= 5 seconds. Γ
i
(i = 1, . . . , 6) are chosen
as [0.01 0.01 0.01 0.06 0.01 0.01 0.01 0.04 0.08].
γ
i
and λ
i
(i = 1, . . . , 6) are chosen as
[0.01 0.05 0.05 0.02 0.02 0.01]. These design
parameters were chosen by trial and error. Some selected
states and control signals are shown in Figures 1 and 2,
which demonstrate how the rudder failure is accommodated
0 20 40 60 80 100 120 140 160 180 200
−0.5
0
0.5
1
1.5
y−axis velocity x
5
=v (solid) and desired x
d5
(dashed) (ft/sec) vs. time (sec)
0 20 40 60 80 100 120 140 160 180 200
−1
−0.5
0
0.5
roll angle x
8
=φ (solid) and desired x
d8
(dashed) (deg) vs. time (sec)
0 20 40 60 80 100 120 140 160 180 200
−1
−0.5
0
yaw angle x
9
=ψ (solid) and desired x
d9
(dashed) (deg) vs. time (sec)
Fig. 1. System states x
5
= v, x
8
= φ, x
9
= ψ (Case I).
0 20 40 60 80 100 120 140 160 180 200
−0.5
0
0.5
1
left engine throttle δ
tl
(solid) and right engine throttle δ
tr
(dashed) (ft/sec
2
) vs. time (sec)
0 20 40 60 80 100 120 140 160 180 200
−2
−1
0
1
left aileron δ
al
(solid) and right aileron δ
ar
(dashed) (deg) vs. time (sec)
0 20 40 60 80 100 120 140 160 180 200
−0.4
−0.2
0
0.2
0.4
rudder angle δ
r
(deg) vs. time (sec)
Fig. 2. Control signals: δ
tl
, δ
tr
, δ
al
, δ
ar
, and δ
r
(Case I).
immediately after its occurrence by the adaptive controller.
(The other states can also converge to the desired trajectory
after rudder failure, but are not shown in the figures due to
the limitation of space). The initial transients (before the
failure occurs) are because of the adaptive system response
when the system is first turned on with some arbitrary initial
values of the adaptive control gains.
Case (II). System performances with adaptive compensa-
tion scheme and aileron failure (IV.3). The failure instant is
t
f
= 20 seconds and the parameter setting is the same as
Case (I). The results are shown in Figures 3 and 4.
0 50 100 150 200 250 300
−0.5
0
0.5
1
1.5
y−axis velocity x
5
=v (solid) and desired x
d5
(dashed) (ft/sec) vs. time (sec)
0 50 100 150 200 250 300
−1
−0.5
0
0.5
roll angle x
8
=φ (solid) and desired x
d8
(dashed) (deg) vs. time (sec)
0 20 40 60 80 100 120 140 160 180 200
−1
−0.5
0
yaw angle x
9
=ψ (solid) and desired x
d9
(dashed) (deg) vs. time (sec)
Fig. 3. System states x
5
= v, x
8
= φ, x
9
= ψ (Case II).
In summary, in this study we simulated some typical
aircraft motions for realistic rudder and aileron failure con-
5

Citations
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Proceedings ArticleDOI
18 Aug 2008
TL;DR: Closing the gap between state-of-the-art methodologies used to certify conventional flight control system software and what will likely to be needed to satisfy FAA airworthiness requirements will require certain advances in simulation methods and comprehensive methods to determine learning algorithm stability and convergence rates.
Abstract: Over the last five decades, extensive research has been performed to design and develop adaptive control systems for aerospace systems and other applications where the capability to change controller behavior at different operating conditions is highly desirable. Although adaptive flight control has been partially implemented through the use of gain-scheduled control, truly adaptive control systems using learning algorithms and on-line system identification methods have not seen commercial deployment. The reason is that the certification process for adaptive flight control software for use in national air space has not yet been decided. The purpose of this paper is to examine the gaps between the state-of-the-art methodologies used to certify conventional (i.e., non-adaptive) flight control system software and what will likely to be needed to satisfy FAA airworthiness requirements. These gaps include the lack of a certification plan or process guide, the need to develop verification and validation tools and methodologies to analyze adaptive controller stability and convergence, as well as the development of metrics to evaluate adaptive controller performance at off-nominal flight conditions. This paper presents the major certification gap areas, a description of the current state of the verification methodologies, and what further research efforts will likely be needed to close the gaps remaining in current certification practices. It is envisioned that closing the gap will require certain advances in simulation methods, comprehensive methods to determine learning algorithm stability and convergence rates, the development of performance metrics for adaptive controllers, the application of formal software assurance methods, the application of on-line software monitoring tools for adaptive controller health assessment, and the development of a certification case for adaptive system safety of flight.

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Cites background from "Adaptive failure compensation for a..."

  • ...Current State of the Art: It is not possible to mention the many on-going efforts by industry and government projects to develop adaptive flight control systems.[10, 13, 15, 16, 35, 50-53] Although most of the industry development programs are proprietary, the Air Force VVIACS (Verification and Validation of Intelligent and Adaptive Control Systems)[54] and NASA IRAC (Intelligent Resilient Adaptive Control)[55] efforts represent multi-year programs with industry partners have been initiated to define methodologies and test procedures for adaptive flight control systems....

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TL;DR: A brief background of the state of the art in adaptive fault tolerant flight control is provided and a comparison study of several novel direct neural network based adaptive control methods are presented.
Abstract: In flight control systems it is crucial to accommodate actuator failures for safe operation. Modern flight control systems are equipped with multiple power sources and actuation systems that offer redundancy in the event of actuator failures. Traditional approaches to flight control system design involve scheduling of fixed gain controllers and control allocation to take full advantage of redundant actuators. Alternative adaptive control approaches have been extensively investigated recently to provide enhanced fault tolerance by augmenting an existing flight control design with adaptive elements. This paper provides a brief background of the state of the art in adaptive fault tolerant flight control and presents a comparison study of several novel direct neural network based adaptive control methods.

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08 Aug 2011
TL;DR: The SAFE-Cue system as discussed by the authors provides force feedback to the pilot via an active control inceptor with corresponding command path gain adjustments, thus ensuring pilot-vehicle system stability and performance in the presence of damage or failures.
Abstract: As a means to enhance aviation safety, numerous adaptive control techniques have been developed to maintain aircraft stability and performance in the presence of failures or damage. The techniques apply a wide array of adaptations from simple gain scheduling to online learning algorithms. While the ready availability of low cost, reduced scale unmanned systems have allowed for many successful flight test demonstrations, applications to piloted aircraft have been more limited. Flight evaluations of various adaptive control applications conducted by NASA and others have shown great promise. In some cases, however, unfavorable pilot-vehicle interactions including pilot-induced oscillations have occurred. Susceptibility to such interactions is more likely when the pilot interacts with a highly nonlinear vehicle that may no longer have predictable response characteristics due to the intervention of the adaptive controller and/or presence of flight control system nonlinearities such as actuator rate and position limits. To address the challenge of adverse pilot interactions with an adaptive control system, the Smart Adaptive Flight Effective Cue or SAFE-Cue system is introduced. This innovative cueing system provides force feedback to the pilot via an active control inceptor with corresponding command path gain adjustments. The SAFE-Cue will alert the pilot that the adaptive control system is active, provide guidance via force feedback cues, and attenuate commands, thus ensuring pilot-vehicle system stability and performance in the presence of damage or failures. Piloted simulation results found that the SAFE-Cue system was effective at suppressing pilot-vehicle system oscillations for failure/damage scenarios, allowing the pilots to focus on the task at hand rather than simply maintaining control.

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Proceedings ArticleDOI
10 Aug 2009
TL;DR: In this article, an adaptive control technique for a damaged large transport aircraft subject to unknown atmospheric disturbances such as wind gust or turbulence is presented, where the damage results in vertical tail loss with no rudder authority, which is replaced with a differential thrust input.
Abstract: The paper presents an adaptive control technique for a damaged large transport aircraft subject to unknown atmospheric disturbances such as wind gust or turbulence It is assumed that the damage results in vertical tail loss with no rudder authority, which is replaced with a differential thrust input The proposed technique uses the adaptive prediction based control design in conjunction with the time scale separation principle, based on the singular perturbation theory The application of later is necessitated by the fact that the engine response to a throttle command is substantially slow that the angular rate dynamics of the aircraft It is shown that this control technique guarantees the stability of the closed-loop system and the tracking of a given reference model The simulation example shows the benefits of the approach

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Proceedings ArticleDOI
18 Aug 2008
TL;DR: In th is p ap er, the p ro b lem of con trol ling sy stems with failures a nd fa ults is in tro du c ed ,an d an o v erview of recen t w ork on d irect ad aptiv e co n trol for comp en sa tion o f un certainactu ator fail u res is pr es en ted.
Abstract: In this paper, the problem of controlling systems with failures and faults is introduced, and an overview of recent work on direct adaptive control for compensation of uncertain actuator failures is presented. Actuator failures may be characterized by some unknown system inputs being stuck at some unknown (fixed or varying) values at unknown time instants, that cannot be influenced by the control signals. The key task of adaptive compensation is to design the control signals in such a manner that the remaining actuators can automatically and seamlessly take over for the failed ones, and achieve desired stability and asymptotic tracking. A certain degree of redundancy is necessary to accomplish failure compensation. The objective of adaptive control design is to effectively use the available actuation redundancy to handle failures without the knowledge of the failure patterns, parameters, and time of occurrence. This is a challenging problem because failures introduce large uncertainties in the dynamic structure of the system, in addition to parametric uncertainties and unknown disturbances. The paper addresses some theoretical issues in adaptive actuator failure compensation: actuator failure modeling, redundant actuation requirements, plant-model matching, error system dynamics, adaptation laws, and stability, tracking, and performance analysis. Adaptive control designs can be shown to effectively handle uncertain actuator failures without explicit failure detection. Some open technical challenges and research problems in this important research area are discussed.

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References
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Journal ArticleDOI
TL;DR: The results demonstrate the ability of the adaptive algorithms to maintain trim after a failure, to restore tracking of the pilot commands despite the loss of actuator effectiveness, and to coordinate the use of the remaining active control surfaces in order to guarantee the decoupling of the rotational axes.
Abstract: The application of multivariable adaptive control techniques to flight control reconfiguration is considered. The objective is to redesign automatically flight control laws to compensate for actuator failures or surface damage. Three adaptive algorithms for multivariable model reference control are compared. The availability of state measurements in this application leads to relatively simple algorithms. The respective advantages and disadvantages of the adaptive algorithms are discussed, considering their complexity and the assumptions that they require. An equation-error based algorithm is found to be preferable. Simulations obtained using a full nonlinear model of a twin-engine jet aircraft are presented. The results demonstrate the ability of the adaptive algorithms to maintain trim after a failure, to restore tracking of the pilot commands despite the loss of actuator effectiveness, and to coordinate the use of the remaining active control surfaces in order to guarantee the decoupling of the rotational axes. A new adaptive algorithm with a variable forgetting feature is also used and is found to yield a useful alternative to covariance resetting as a solution to covariance wind-up in least-squares algorithms.

468 citations


Additional excerpts

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Book
01 Jan 1994
TL;DR: In this paper, the linear-quadratic-regulator (LQR) method of feedback control synthesis is used to coordinate multiple controls, producing graceful maneuvers comparable to those of an expert pilot.
Abstract: Here a leading researcher provides a comprehensive treatment of the design of automatic control logic for spacecraft and aircraft. In this book Arthur Bryson describes the linear-quadratic-regulator (LQR) method of feedback control synthesis, which coordinates multiple controls, producing graceful maneuvers comparable to those of an expert pilot. The first half of the work is about attitude control of rigid and flexible spacecraft using momentum wheels, spin, fixed thrusters, and gimbaled engines. Guidance for nearly circular orbits is discussed. The second half is about aircraft attitude and flight path control. This section discusses autopilot designs for cruise, climb-descent, coordinated turns, and automatic landing. One chapter deals with controlling helicopters near hover, and another offers an introduction to the stabilization of aeroelastic instabilities. Throughout the book there is a strong emphasis on the mathematical modeling necessary for designing a good feedback control system. The appendixes summarize analysis of linear dynamic systems, synthesis of analog and digital feedback control, simulation, and modeling of flexible vehicles.

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Book
Gang Tao1
10 May 2004
TL;DR: Some results on using adaptive control techniques to compensate unknown actuator failures in dynamic control systems are presented.
Abstract: In this talk, we present some results on using adaptive control techniques to compensate unknown actuator failures in dynamic control systems.

275 citations


"Adaptive failure compensation for a..." refers background in this paper

  • ...where ti is the unknown failure time instant and ūi is the unknown constant failure value [11]....

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Journal ArticleDOI
TL;DR: In this paper, multiple model adaptive estimation (MMAE) methods have been incorporated into the design of a flight control system for the variable in-flight stability test aircraft (VISTA) F-16, providing it with the capability to detect and compensate for sensor and control surface/actuator failures.
Abstract: Multiple model adaptive estimation (MMAE) methods have been incorporated into the design of a flight control system for the variable in-flight stability test aircraft (VISTA) F-16, providing it with the capability to detect and compensate for sensor and control surface/actuator failures. The algorithm consists of a ‘front end’ estimator for the control system, composed of a bank of parallel Kalman filters, each matched to a specific hypothesis about the failure status of the system (fully functional or a failure in any one sensor or surface/actuator), and a means of blending the filter outputs through a probability-weighted average. For multiple failures, a hierarchical structure is used to keep the number of online filters to a minimum. To compensate for failed control surfaces or actuators, a ‘back end’ algorithm redistributes control commands (that would normally be sent to surfaces detected as having failed) to the non-failed surfaces, accomplishing the same control action on the aircraft. Failures are demonstrated detectable in less than one second, even at low dynamic pressure (20000ft and 0.4 Mach), with an aircraft output nearly identical to that anticipated from a fully functional aircraft in the same environment. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government work and is in the public domain in the United States.

188 citations


Additional excerpts

  • ...In [ 9 ], an algorithm for aircraft failure detection and compensation was presented, which incorporated multiple model adaptive estimation methods....

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Proceedings ArticleDOI
14 Dec 1994
TL;DR: It is shown that simple algorithms can be obtained if full state feedback is assumed and the objective is to design automatically a flight control law in the presence of actuator failures or surface damage.
Abstract: The application of multivariable adaptive control techniques to flight control reconfiguration is considered. The paper first discusses three adaptation mechanisms for model reference control. It is shown that simple algorithms can be obtained if full state feedback is assumed. The respective advantages and disadvantages of the three algorithms are discussed in general terms, considering their complexity and the assumptions that they require. Next, the application of the adaptive algorithms to reconfigurable flight control is investigated. The objective is to design automatically a flight control law in the presence of actuator failures or surface damage. Design considerations for the adaptive algorithms are discussed in this context. Simulations obtained using a full nonlinear simulation of a twin-engine jet aircraft are included to illustrate the results. >

182 citations


"Adaptive failure compensation for a..." refers methods in this paper

  • ...In [2], several multivariable adaptive control algorithms for flight contr ol reconfiguration were presented with a failure characterize d by a locked left horizontal tail surface....

    [...]

Frequently Asked Questions (14)
Q1. What contributions have the authors mentioned in the paper "Adaptive failure compensation for aircraft tracking control using engine differential based model" ?

In this paper, an aircraft model that incorporates independently adjustable engine throttles and ailerons is employed to develop an adaptive control scheme in the presence of actuator failures. 

Several important and challenging issues need to be addressed in future research. 

In addition, the effects of actuator nonlinearities including output and rate saturation, as well as actuator dynamics, need to be addressed. 

By applying the linearization procedure around the equilibrium point of interest, the authors can obtain the linearized aircraft model with engine differentials. 

As the new adaptive failure compensation scheme for asymptotic state tracking, the feedback control law isv(t) = K̂x(t) + κ̂rd(t) + θ̂, (III.9)where K̂ = [K̂1, K̂2, . . . , K̂m] 

This engine differential based model in which the two engine thrusts and the ailerons are taken into account separately captures the essential dynamics of the aircraft in the engine differential mode, and is capable of coping with some actuator failures such as rudder failures or engine failure, which cannot be achieved without using engine differentials. 

0. 2The physical meaning of state tracking for the aircraft dynamic model linearized at an equilibrium point (xo, Uo) is that the operation of the aircraft follows a desired trajectory in a neighborhood of the equilibrium point. 

(This condition basically implies that the nominal LQ regulator used for generating the desired trajectory is designed to be robust to parameter uncertainties). 

This adaptive actuator failure compensation scheme has the following desired properties:Theorem 3.1: The control law (III.9), updated from (III.10)–(III.12) and applied to the system (III.1) subject to the actuator failures (III.2) under Assumption 3.1, ensures that all closed-loop system signals are bounded and limt→∞(x(t) − xd(t)) = 0, for any failure pattern σ ∈ Σ with uncertain parameters. 

T̄ ′′′δtl−T̄ ′′′δtr 0 0 0 0 0 0 (II.13) represent the effect of engine thrust differentials, that is, if the left and right engine thrusts are equal, these matrices are zero. 

There are 2m possible combinations of actuator states (each actuator is either normal or failed), and therefore 2m−1 possible failure patterns that constitute a set denoted by Σ̄. 

The stabilizability and rank condition in Assumption 3.1 characterizes the system redundancy condition needed for actuator failure compensation. 

T and the commutativity property of the matrix trace operator, i.e., Tr(XY ) = Tr(Y X), the following equalities hold:eTPB(I−σ)(K̂−Kσ)x(t) = ∑i6=i1,i2,...,ip(K̂i−Ki) TxeTPbi,eTPB 

As described in [8], a nonlinear aircraft dynamic model in body-axis coordinate system which incorporates enginedifferentials can be described by the force equationsm(u̇+ qw − rv) = X −mg sin θ + (TL + TR) cos ǫ (II.1)m(v̇ + ru− pw) = Y +mg cos θ sinφ (II.2)m(ẇ+pv−qu) = Z+mg cos θ cosφ−(TL+TR) sin ǫ (II.3)and moment equationsIxṗ+Ixz ṙ+(Iz−Iy)qr+Ixzqp = L+l(TL−TR) sin ǫ (II.4)Iy q̇ + (Ix − Iz)pr + Ixz(r 2 − p2) = M (II.5)Iz ṙ + Ixz ṗ+ (Iy − Ix)qp− Ixzqr = N + l(TL − TR) cos ǫ (II.6) where m is the mass of the aircraft.