Simple Adaptive Control for Aircraft Control Surface Failures
Abderrazak I. Belkharraz, Member, IEEE
LaGuardia Community College
Kenneth Sobel, Senior Member, IEEE
The City College of New York
Keywords: adaptive control, fault tolerant control, aircraft flight control
This work extends the so-called simple adaptive control approach to direct model reference adaptive control
of multi-input multi-output systems to include loss of control effectiveness failures. It is proven that all signals
are bounded for loss of control effectiveness failures during a bounded input disturbance. A state space approach is introduced for computing the feedforward compensator that is required by the stability result. The
adaptive algorithm is applied to a three input model of the linearized lateral dynamics of the F/A-18 aircraft.
Simulation results are obtained with single, double, and triple control effectiveness failures of 88% during
the occurrence of a lateral gust. These results show that the adaptive controller exhibits improved model
following as compared to a fixed gain eigenstructure assignment controller.
An earlier version of this paper was presented at the 2004 Guidance, Navigation, and Control Conference.
This work was partially supported by a grant from The City University of New York Collaborative Incentive Research Program
and by a grant from The City University of New York PSC-CUNY Research Award Program.
Corresponding author’s address: Prof. Kenneth Sobel, Department of Electrical Engineering, The City College of New York,
New York, NY 10031. email: sobel@ccny.cuny.edu
I
Introduction
Aircraft flight control systems are designed with extensive redundancy to ensure a low probability of
failure. During recent years, however, several aircraft have experienced major control system failures.
These have caused an increased interest in fault tolerant flight control systems. The objective of a fault
tolerant flight control system is to control and safely land the aircraft in case of severely damaged or
inoperable control surfaces. One of the approaches to fault tolerant control is active control. An active
fault tolerant control system has to either reconfigure or adapt the controller in response to the failure.
Typical design methods include multiple model, switching, and tuning designs; adaptive designs; and fault
detection and diagnosis designs. Adaptive failure accomodation designs have simpler control structures
and do not rely on knowledge of the actuator failures. Direct adaptive designs use the system response
error to achieve desired performance.
Boskovic and Mehra [1] propose an indirect adaptive scheme for loss of aircraft control effectiveness
failures. However, the authors assume that the system state is available and that the number of control effectors exceeds the number of variables that can be affected by the controls. Bodson and Groszkiewicz [2]
consider adaptive schemes for aircraft control effectiveness failure accomodation. However, the authors
require the system state to be available and the aircraft to be minimum phase and relative degree one. Tao
et. al. [3] use a direct adaptive scheme to accomodate locked in place actuator failures. However, the
authors require the aircraft to be minimum phase.
Sobel et. al. [5] proposed simple adaptive control algorithms for multiple input multiple output systems.
Later, Barkana [6] inserted a feedforward compensator around the plant so that the augmented system
is almost strictly positive real (ASPR). Kaufman et. al. [7] summarize stability results that show all
2
signals in the adaptive system are bounded and that the augmented error is asymptotically vanishing if the
augmented plant is ASPR. Morse and Ossman [4] use simple adaptive control to accomodate failures that
can modelled by a zero column in the linearized aircraft control derivative matrix. However, the authors
choose the outputs to be linear combinations of the states instead of the usual aircraft sensor measurements.
Furthermore, no proof of stability is shown.
The present paper extends the work described in Kaufman et. al. [7] to include loss of control effectiveness
failures. It is proven that all signals are bounded for loss of control effectiveness failures during a bounded
input disturbance. The percentage loss of control effectiveness is unknown and may be arbitrarily close to
a complete loss subject to the satisfaction of the sufficient conditions for stability. A state space approach
is introduced for computing the feedforward compensator which guarantees that the augmented plant is
ASPR by using the MATLAB R LMI and Optimization toolboxes. The adaptive algorithm is applied to a
three input model of the linearized lateral dynamics of the F/A-18 aircraft. Simulation results are obtained
with single, double,and triple control effectiveness failures of 88% during the occurrence of a lateral gust.
The maximum failure is chosen to be 88% because this is the largest failure for which a feedforward
compensator could be designed that ensures that the aircraft plus feedforward is ASPR. The simulation
results show that the adaptive controller exhibits improved model following as compared to a fixed gain
eigenstructure assignment controller.
II
Problem Statement
Let (Ti , Ti+1 ), i = 0, 1, . . . , q0 , with q0 finite and T0 = 0, be the time intervals on which the control surface
failure pattern is fixed. That is, control surfaces only fail at time Ti , i = 1, . . . , q0 .
3
Then, the plant on the interval (Ti , Ti+1 ), i = 0, 1, . . . , q0 is described by



 ẋ p (t) = A p x p (t) + Bip u p (t) + d(t)
Pi :


 y p (t) = C p x p (t)
(1)
where A p ∈ ℜn p ×n p , B p ∈ ℜn p ×m , C p ∈ ℜq×n p , Bip = B p αi , B p ≡ B0p , and d(t) denotes a bounded input
disturbance vector. The value of the bound on the disturbance is unknown and not required by the adaptive
algorithm.
αi =



 diag{αi1 , αi2 , . . . , αim }; i = 1, 2, . . . , q0


 I;
(2)
i=0



 0 < αik < 1, if kth control surface fails, k = 1, . . . , m


 αi = 1,
k
if kth surface does not fail, k = 1, . . . , m
Here the failure times are Ti , i = 1, . . . , q0 ; which control surfaces fail at Ti , i = 1, . . . , q0 is unknown; the
amount of the loss of effectiveness at Ti given by αi , where αik ∈ (0, 1) is unknown. Furthermore, once a
control surface fails it may fail again later with a different amount of loss of effectiveness.
The plant is augmented with a single fixed feedforward compensator of the form
H
−1
(s) =
K −1
f
(3)
1 + τs
A state space realization of H −1 (s) is given by
ṡ p (t) = As s p (t) + Bs u p (t)
(4)
h p (t) = Cs s p (t)
(5)
The augmented plant is then given by



 ẋap (t) = Aap xap (t) + Bap αi u p (t) + D(t)
a
Pi :


 yap (t) = Cap xap (t)
4
(6)
xap
where








 d(t) 
 x p  a  A p 0  a i  B p αi  a

; C p = ( C C ); D(t) = 
; B p α = 
; A p = 
=
p
s








0
Bs
0 As
sp
and where xap (t) ∈ ℜn ; u p (t) ∈ ℜm ; and yap (t) ∈ ℜq .
Then the augmented output to be controlled is
yap (t) = y p (t) + h p (t)
(7)
where h p (t) is the inverse Laplace transform of H −1 (s).
The control objective is to design an adaptive control signal u p (t) such that all signals in the closed loop
system are bounded and the augmented plant output yap (t) tracks the output of a reference model given by
ẋm (t) = Am xm (t) + Bm um (t)
(8)
ym (t) = Cm xm (t)
(9)
where Am ∈ ℜnm ×nm , Bm ∈ ℜnm ×mm , Cm ∈ ℜq×nm .
Remark 1 We remark that the order of the plant may be much greater than the order of the reference
model.
III
The General Tracking Problem
We summarize the general tracking problem for a known plant. These results are shown in Kaufman et al.
[7]. Let the input command um (t) be the output of an unknown command generating system of the form
v̇m (t) = Av vm (t)
(10)
um (t) = Cv vm (t)
(11)
5
Thus, the allowable reference model commands um (t) are those commands that are the solution to some
unknown set of linear ordinary differential equations. This includes steps, ramps, polynomials, exponentials, and sinusoids. Define the ideal trajectories (xap )∗ (t), such that, if the augmented plant could reach
and move along them, its output would perfectly track the output of the reference model. That is, the
ideal trajectories are targets that the augmented plant tries to reach or at least be close to, in order to have
bounded tracking errors.
Mathematically,
(yap )∗ = Cap (xap )∗ = Cm xm = ym
(12)
The ideal trajectories are defined as
(xap )∗ (t) = X11 xm (t) + X12 um (t)
(13)
and the ideal control is defined as
u∗p (t) = K̃x xm (t) + K̃u um (t)
(14)
Substituting (xap )∗ (t) from (13) into (12) gives a condition for existence of the ideal target trajectories.
Cap X11 xm (t) +Cap X12 um (t) = Cap X11 xm +Cap X12Cv vm
= Cm xm
(15)
(16)
or



 Cap X11
= Cm
(17)


 Cap X12Cv = 0
Solutions for X11 and X12 in (17) exist, in general, because this system consists of more variables than
equations. This implies the existence of some bounded trajectories in the x ap space that the plant needs to
attain for perfect tracking.
6
IV
Adaptive Control
A Algorithm
The adaptive control law is given by
u p (t) = Kx xm (t) + Ku um (t) + Ke [ym (t) − yap (t)]
(18)
where Kx , Ku , and Ke are adaptive gains.
The adaptive gains are concatenated into matrix K(t) defined as
K(t) = [Ke (t), Kx (t), Ku (t)]
(19)
The concatenated gain K(t) is defined as the sum of a proportional gain K p (t) and an integral gain KI (t),
each of which is adapted as follows:
K(t) = K p (t) + KI (t)
(20)
K p (t) = eay (t)rT (t)T¯
(21)
K̇I (t) = eay (t)rT (t)T − σKI
(22)
T
(t), uTm (t)]
rT (t) = [(eay (t))T , xm
(23)
eay (t) = ym (t) − yap (t)
(24)
where T , T¯are time invariant weighting matrices.
The σ term in Eq. (22) was originally proposed by Ioannou and Kokotovic [8] and it is used to avoid
divergence of the integral gains in the presence of disturbances.
7
Define ex = (xap )∗ − xap . The error equation is derived in appendix A and is given by
ėx = Ac ex − Bap αi [K − K̃i ]r − F(t)
(25)
where
Ac = Aap − Bap αi K̃eiCap
(26)
K̃i = [K̃ei , K̃xi , K̃ui ]
(27)
F(t) = (Aap X11 − X11 Am + Bap αi K̃xi )xm +
(Aap X12Cv − X11 BmCv + Bap αi K̃uiCv − X12Cv Av )vm + D
V
(28)
Stability Analysis
In this section, we present two theorems describing the stability of the adaptive controller. Theorem 1
shows that positive definite quadratic Lyapunov functions exist such that their derivatives are negative definite whenever ex or KI is sufficiently large. Theorem 2 describes a sufficient condition for the boundedness
of the Lyapunov functions at the failure instants.
Theorem 1 If
1. the augmented plant Pia is ASPR , i.e., there exist positive definite matrices Pi , Qi , and a gain matrix
K̃ei such that:
Pi (Aap − Bap αi K̃eiCap ) + (Aap − Bap αi K̃eiCap )T Pi = −Qi < 0
(29)
(Cap )T = Pi Bap αi
(30)
8
2. dim[yap (t)] ≤ dim[u p (t)] and T and T¯positive definite symmetric
3. the disturbances are bounded
then a positive definite quadratic Lyapunov function Vi (ex , KI ) can be selected such that its derivative V̇i is
negative definite whenever ex or KI is sufficiently large for t ∈ (Ti , Ti+1 ), i = 0, 1, 2, . . . , q0 .
Proof: see appendix B
Remark 2 We present an overview of the proof of Theorem 1. Choose the positive definite quadratic
Lyapunov function as
Vi (ex , KI ) = eTx Pi ex + tr[(KI − K̃i )T −1 (KI − K̃i )T ]
(31)
then, the derivative of (31) along the trajectories of Eq. (25) is
a
a T a T
T
V̇i = −eTx Qi ex − 2(eay )T eay (eay )T T¯
e ey − 2(ey ) ey [xm T¯
x xm + um T¯
u um ]
−2σtr[(KI − K̃i )T −1 (KI − K̃i )T ]
−2σtr[(KI − K̃i )T −1 K̃iT ] − 2eTx Pi F(t)
(32)
Hence, we have
V̇i < −β1 kex k2 − β2 keay k4 − β3 keay k2 (β4 kxm k2 + β5 kum k2 ) − β6 kKI − K̃i k2 + β7 kKI − K̃i k + β8 kex k
(33)
where β1 , β2 , β3 , β4 , β5 , β6 , β7 and β8 are positive constants.
We conclude that whenever kex k or kKI − K̃i k increase beyond some bound, then the negative quadratic
and quartic terms in (33) will become dominant, and thus V̇i (ex , KI ) becomes negative. The quadratic form
of Vi (ex , KI ) then guarantees that ex and KI are bounded.
9
Theorem 2 Let Vi (t) ≡ Vi (ex (t), KI (t)), i = 1, 2, . . . , q0 . The Lyapunov functions at the failure instants
given by Vi (Ti ), i = 1, 2, . . . , q0 are bounded.
Proof: see appendix C
VI
Feedforward Design
We propose a method which uses the MATLAB R LMI toolbox [9] and the MATLAB R Optimization
toolbox [10] to design a robust feedforward compensator. Each augmented LTI plant P ia , i = 0, 1, . . . , q0
must be ASPR using the same feedforward. However, we do not know in advance which failures will
occur.
Suppose there are j0 possible failure configurations P j , j = 1, . . . , j0 . Each possible failure configuration P j corresponds to one surface failure, or two simultaneous surface failures, or three simultaneous
surface failures, and so on. The P j ’s are parameterized by a range of possible control effectiveness loss.
Thus, the P j ’s are different from the Pi ’s in (1) because the Pi ’s uniquely define a failure whereas the P j ’s
define a possible parameterized failure configuration. Also, let (A p , B0p ,C p ) correspond to the unfailed
plant.
Consider an affine parameter dependent representation of P j given by



 ẋ p = A p x p + B pj (σ)u p
P j (σ) :


 y p = Cpx p
10
(34)
k
j
where B p (σ) = B0p − ∑k=1
σk Bk , j = 1, . . . , j0 and where k j is the number of surfaces of P j (σ) which
j
j
j
j
j
simultaneously fail. Here Bk describes the failure of one surface and σk describes the corresponding range
of control effectiveness loss.
The following two lemmas due to Barkana [6] show that the feedforward compensator for a strictly proper
plant has a simple form.
Lemma 1 Let G(s) be any m × m strictly proper transfer matrix of arbitrary McMillan degree. G(s) is
not necessarily stable or minimum phase. Let K f be a nonsingular constant output feedback matrix such
that the closed loop transfer matrix
Gcl (s) = [I + G(s)K f ]−1 G(s)
(35)
is asymptotically stable. Then, the augmented open loop transfer matrix
Ga (s) = G(s) + K −1
f
(36)
is ASPR.
Lemma 2 Let G(s) be any m × m strictly proper transfer matrix of arbitrary McMillan degree. G(s) is
not necessarily stable or minimum phase. Let H(s) = K f (1 + τs) be a stabilizing closed loop compensator
of G(s). Then, the augmented plant
Ga (s) = G(s) + H −1 (s) = G(s) +
K −1
f
(37)
1 + τs
is ASPR.
11
We want to minimize the norm of K −1
f so that
yap (t) = y p (t) + h p (t) ≈ y p (t)
(38)
while ensuring that K f robustly stabilizes every P j (σ), j = 1, . . . , j0
Let P̃ j (σ) be P j (σ) with feedback K f , j = 1, . . . , j0
Mathematically
min k K −1 kF s.t P̃ j (σ) is robustly stable; j = 1, . . . , j0
f
Kf
(39)
We ensure that the P̃ j (σ)’s are robustly stable by using program “pdlstab” from the MATLAB R LMI
toolbox [9] and program “fmincon” from the MATLAB R Optimization toolbox [10]. At each iteration
of the optimization, we call “pdlstab” and require that the quantity t min be negative. Here tmin is computed
by “pdlstab” where tmin < 0 ensures the existence of a parameter dependent Lyapunov function for P̃ j (σ).
With tmin < 0, every LTI plant which belongs to P̃ j (σ), j = 1, . . . , j0 is stable. We remark that on each
interval t ∈ (Ti , Ti+1 ) we have an LTI plant Pi , i = 1, . . . , q0 . Let P̃i be the LTI plant Pi with feedback K.
Then, every LTI plant P̃i is one of the P̃ j (σ) for some values of j and σ. Thus, every LTI plant P̃i is stable.
Then, it follows that every augmented plant Pia , i = 1, . . . , q0 is ASPR.
A flowchart of the design process is shown in Figure 1. First, “fmincon” and “pdlstab” are used to
find the maximum failure such that K f is a robust stabilizing gain that minimizes k K −1
f k. However, it
is possible that K f may be robust stabilizing for a larger maximum failure. This is because the function
to be minimized and the constraints are quite complicated. Also, there is no guarantee of a global rather
than a local minimum. Therefore, the gain K f from the optimization is tested for larger maximum failures
by using “pdlstab” without optimization. This is done repeatedly until the maximum failure is found for
12
which “pdlstab” can ensure robust stability when using the gain K f . Then, the feedforward gain is K −1
f .
Remark 3 The system in (34) with feedback H(s) = K f (1 + τs) is given by
j
ẋ p = Acl (σ)x p
(40)
where
j
Acl (σ) = A p − B pj (σ)[I + τK f C p B pj (σ)]−1 K f C p (I + τA p )
(41)
j
which we observe is not affine in B p (σ) as required by program “pdlstab”. However, the closed loop
system matrix with constant feedback K f is given by
j
Acl (σ) = A p − B pj (σ)K f C p
(42)
j
which is affine in B p (σ). This is the reason that we use “pdlstab” to compute a stabilizing constant
feedback K f instead of the dynamic feedback K f (1 + τs).
Remark 4 The implementation of a constant feedforward K −1
f yields an algebraic loop which causes a
problem in the computer simulation. Therefore, we first compute a constant gain K f which stabilizes
j
Acl (σ) in (42) for all j and σ by using “pdlstab” and “fmincon”. Then, we implement the feedforward
H −1 (s) = K −1
f /(1 + τs) where τ is sufficiently small. We show in Theorem 3 that such a τ exists so that
j
H(s) = K f (1 + τs) stabilizes Acl (σ) in (41) for all j and σ. Therefore, H −1 (s) = K −1
f /(1 + τs) satisfies
lemma 2.
j
Theorem 3 Suppose Acl (σ) in (42) is stable with a non-defective modal matrix for all j and σ. Then,
j
there exists τ sufficiently small such that Acl (σ) in (41) is stable for all j and σ.
Proof: see appendix D
13
VII
Example
A Aircraft and Reference Model
Consider the linearized lateral dynamics of the F/A-18A aircraft. The rigid body states are lateral velocity
(v), yaw rate (r), roll rate (p), and bank angle (φ). The control surface deflections are asymmetric trailing
edge flaps (δte ), ailerons (δa ), and rudder (δr ). The measurements are sideslip angle (β), washed out yaw
rate (rwo ), and roll rate(p). The unfailed aircraft is described by the triple (A p , B0p ,C p ) where the matrices
A p , B0p , and C p are shown in appendix E. The block diagram of the adaptive control system is shown in
Figure 2.
Let T j , j = 0, 1, . . . , m0 , be the instants at which a bounded input disturbance in the form of a lateral gust
vg (t) occurs, where vg (t) is defined as shown in MIL-F-8785-C [11]




t − Tj < 0
 0



π(t−T j )
vm
vg (t) =
1
−
cos
0 ≤ t − Tj ≤ T
2
T






 vm
t − Tj > T
(43)
The lateral gust magnitude is given by vm . The gust length T is chosen to be the inverse of the natural frequency of the closed loop complex eigenvalue pair of the unfailed aircraft. Here the dutch roll eigenvalues
are λdr = −2 ± j2 so that T = 1/ωn = 0.3536 sec.
The state equations for the lateral dynamics are given by [12]
v̇ = Yv v − rU0 + gφ +Yδa δa +Yδr δr −Yv vg
ṙ =
(44)
Nv0 v + Nr0 r + N p0 p + Nδ0 a δa + Nδ0 r δr − Nv0 vg −
ṗ = Lv0 v + Lr0 r + L0p p + Lδ0 a δa + Lδ0 r δr − Lv0 vg −
14
Nr0g
U0
Lr0 g
U0
v̇g
v̇g
(45)
(46)
where U0 is the trim velocity. Nr0g and Lr0 g are given by [12]
Nr0g
Lr0 g
Nr
2 −1
Ixz
= Nr 1 −
Ix Iz
2 −1
Ixz
Ixz
=
Nr 1 −
Ix
Ix Iz
Ixz
= Nr0 − Lr0
Iz
(47)
(48)
(49)
The inertias Ix , Iz , and Ixz were not available for the F/A-18 aircraft. Therefore, the values for the F16 aircraft [13] are used which are given by Ix = 9496, Iz = 63100, and Ixz = 982. The trim velocity
is 646.9 f t/sec. Using these values, the gust derivative coefficients in (47)-(48) are Nr0g = −0.2578 and
Lr0 g = −0.0266.
The open loop aircraft has a lightly damped dutch roll mode with damping 0.13, a roll subsidence mode at
-2.76, and a spiral mode at -0.0002. An output feedback gain matrix Keig is designed using eigenstructure
assignment for the unfailed aircraft by assigning the dutch roll mode to have a damping of 0.707 and a
natural frequency of 2.83 rad/sec. The roll subsidence mode is assigned to -4. The desired eigenstructure
and the feedback gain matrix Keig are shown in Table 1. The closed loop spiral mode is unstable at 0.009.
However, this instability is within the limits of MIL-F-8785C [11] specifications which specifies that the
spiral minimum time to double amplitude is 12 seconds. Here the default value for TOL is used when
computing the achievable eigenvectors using MATLAB R function pinv. Only rudder and aileron are used
because these two control surfaces are sufficient to achieve desired handling qualities for the unfailed aircraft. However, the adaptive controller will use all three control surfaces in order to accommodate failures.
The closed loop time responses of the reference model to a one degree initial sideslip (not shown) exhibit
both desirable damping and sideslip to bank decoupling.
15
The unfailed aircraft with feedback Keig is not ASPR because there are two transmission zeros at z1 =
0.2028 × 10−13 ; and z2 = −0.0002 × 10−13 one of which is non-minimum phase. The feedforward which
satisfies the ASPR condition will be designed for the unfailed aircraft with feedback Keig . This closed loop
unfailed aircraft is described by the triple (A p − B0p KeigC p , B0p ,C p ). The reference model is chosen to be
the same triple so that Am = A p − B0p KeigC p , Bm = B0p , and Cm = C p . Thus, when there are no failures the
reference model is the aircraft with eigenstructure assignment feedback Keig . Next, it is shown how the
reference model input um is computed.
um = −Keig (ym − yc )





= −Keig 



β
rwo
p − pc
(50)







(51)





K
(1,
1)
K
(1,
2)
K
(1,
3)

 eig
eig
eig



= −

Keig (2, 1) Keig (2, 2) Keig (2, 3) 



β
rwo
p − pc








 Keig (1, 3) 
 pc
= −Keig ym + 


Keig (2, 3)
(52)
(53)
The reference model in Figure 2 is implemented using (53) where p c is the pilot roll rate command.
B Feedforward Computation
A robust feedforward is designed so that the aircraft with feedback Keig and feedforward K −1
f is ASPR for
all possible failures. The affine parameter dependent representation of the aircraft with feedback gain Keig
16
j
j
is described by the triple (A p − B p (σ)KeigC p , B p (σ),C p ). It can be shown that the closed loop system for
j
j
(A p − B p (σ)KeigC p , B p (σ),C p ) with feedback gain K f is given by
j
ẋ p = Acl (σ)x p
(54)
where
j
Acl (σ) = A p − B pj (σ)[Keig + K f ]C p
(55)
The feedback gain K f is computed so that (54) is stable for all possible failures.
The optimization in (39) is performed by using program “fmincon” from the MATLAB R Optimization Toolbox [10] and program “pdlstab” from the MATLAB R LMI toolbox [9]. The Frobenius norm
of K −1
f is minimized by using “fmincon” as shown in (39). The possible failures are chosen to be the
loss of control effectiveness of any one, two, or three surfaces. However, it is not known which control
surfaces will fail nor the amount of control effectiveness loss. Experience indicates that the optimization
should be initialized with a nonsingular stabilizing gain. The initial value for K f , denoted by Kinitial , is
computed by using eigenstructure assignment for (A p − B0p KeigC p , B0p ,C p ). The initial gain Kinitial uses
all three control surfaces because the adaptive controller will use all three control surfaces. The desired
eigenvalues and eigenvectors are the same that were used to compute Keig . An attempt was made to obtain
Kinitial using eigenstructure assignment, but the unassigned eigenvalues were unstable. The (5 × 3) control
distribution matrix B0p has singular values 35.3966, 18.6783, and 0.2101. The small singular value, which
corresponds to a direction of weak control, causes some of the feedback gains to be on the order of 10 4 .
The physical explanation is that even with three control surfaces we can still only generate yawing moment
and rolling moment. To alleviate this design problem, the pseudo-control strategy of Sobel and Lallman
17
[14] is used. The columns of the new control distribution matrix B̃0p are chosen to be the two singular
vectors corresponding to the two larger singular values and eigenstructure assignment is used to compute
a (2 × 3) feedback gain matrix. Then, this (2 × 3) feedback gain is mapped back to the original matrix
B0p to obtain the (3 × 3) gain Kinitial . The designer must be careful when computing the achievable dutch
roll eigenvectors because the pseudo-inverse of an ill-conditioned matrix is required. This ill-conditioned
matrix represents the basis vectors for the achievable dutch roll eigenvector subspace modified by deleting
the rows corresponding to unspecified entries in the desired dutch roll eigenvectors. Its pseudoinverse is
used to calculate the least squares projection of the desired dutch roll eigenvectors onto the achievable
dutch roll eigenvector subspace [15]. The ill-conditioning is caused by the deletion of the rows corresponding to unspecified entries in the desired dutch roll eigenvectors. Prior to the row deletion the four
singular values were all unity indicating that the basis was orthonormal. The ill-conditioned matrix has
singular values given by 1.0000, 1.0000, 0.9998, and 0.0028. The achievable eigenvectors were computed
using MATLAB R function “pinv” with TOL=0.01. This has the effect of discarding the singular value at
0.0028 in the pseudo-inverse computation. Press et. al. [16] state that it is very often better to obtain the
least squares solution by using a singular value decomposition and zeroing the small singular values. If
the singular value at 0.0028 is not discarded, then the initial gain matrix has a norm on the order of 10 −10 .
This would not be a useful initial gain for an optimization whose objective is to minimize the norm of the
inverse gain. The initial gain Kinitial is shown in Table 2.
A feedforward is designed which allows up to 50% loss of control effectiveness in any one, two, or
three control surfaces. The optimization is over K f as shown in (39). The optimization converges with
kK −1
f kF = 0.34415. The optimization is restarted with the final gain as the new initial gain. Then, the
18
optimization exceeds the maximum allowable function evaluations with kK −1
f kF = 0.0099724. The optimization is again restarted with this last gain as the new initial gain. The optimization converges with
the optimal robust stabilizing gain Koptimal shown in Table 2. The Frobenius norm has been reduced from
−1
−1
k Kinitial
kF = 1.2 × 105 to k Koptimal
kF = 0.000629 and robust stability is ensured because tmin is negative.
Next, “pdlstab” is called using the optimal gain Koptimal with increasing values of loss of control effectiveness. Here there is no optimization. It was found that “pdlstab” could ensure robust stability for single,
double, or triple failures of up to 88% loss of control effectiveness using the optimal gain Koptimal . Hence,
−1
Theorem 1 ensures that the aircraft with feedback Keig and feedforward Koptimal
is ASPR for all single,
double, and triple control surface failures of up to 88% loss of control effectiveness. However, dynamics
are added so that the plant is augmented with a dynamic feedforward compensator H −1 (s) =
−1
Koptimal
1+τs
where
τ = 10−16 . This eliminates the algebraic loop associated with the use of a constant feedforward compensator. The maximum failure is chosen to be 88% because ”pdlstab” could not ensure robust stability for
a maximum failure of 89%. Finally, it is noted that the example used version 2.2 of the Optimization
toolbox and version 1.0.8 of the LMI Toolbox on a SUN Ultra 10 workstation. Solutions which use earlier
versions of the toolboxes or which use 32 bit computing may produce different answers for Koptimal . This
just indicates that roundoff error determines to which local minimum the optimization converges. It is not
important whether a global minimum is obtained because only a robust stabilizing gain whose inverse has
a small norm is needed.
C Simulation Results
The pilot command pc is chosen to be a roll rate step of 10 deg/s. The initial conditions on both the
aircraft and reference model are zero. A lateral gust with magnitude of 15 f t/sec is applied at T1 = 2 sec
19
for t ∈ [2, 15]. A control surface failure occurs at t = 6 sec. Therefore, the control surface failure occurs
during a gust condition.
A non-adaptive simulation of the aircraft is performed with the fixed eigenstructure assignment feedback gain Keig for a single 88% failure. This corresponds to the block diagram in Figure 2 with Kx =
0, Ke = 0, Ku = I, and H −1 (s) omitted. The aircraft and reference model responses (not shown) exhibit
poor model following in sideslip angle, yaw rate, and roll rate. The roll rate and yaw rate exhibit model
following errors of 400% and 300%, respectively at 40 seconds when an aileron failure occurs at 6 seconds. The sideslip angle is negative for a positive roll rate which is an undesirable response. The aircraft
should always sideslip into the turn.
Next, simulations are performed of the adaptive system of Figure 2. The adaptive gains are initialized
to be Kx = 0, Ke = 0, and Ku = I. This choice of initial gains causes the reference model and the unfailed
aircraft to be identical with identical inputs. Thus, the time responses of the reference model and the
unfailed aircraft will be identical. The weighting matrices T and T¯are chosen to be T = T¯= 1000I and
σ = 0.01. The aircraft responses (not shown) exhibit perfect tracking in sideslip angle, yaw rate, and roll
rate.
Simulations are performed for an 88% simultaneous failure of any two control surfaces. The sideslip
angle, yaw rate, and roll rate for the non-adaptive and adaptive controllers are shown in Figure 3. The
unfailed aircraft responses (solid line) and reference model responses (solid line) are identical. The nonadaptive controller exhibits an aircraft roll rate that is approximately 20% of the reference model roll rate
at 40 seconds when a simultaneous aileron and rudder failure (dotted line) occurs. The aircraft roll rate is
approximately 25% of the reference model roll rate at 40 seconds when a simultaneous aileron and trailing
edge flap failure (dashed line) occurs. Furthermore, the sideslip is negative for positive roll rate which is
20
an undesirable response. In both of these failure cases, the reduced roll rate causes the yaw rate to be
reduced by approximately 67%. Finally, the aircraft tracking errors in roll rate and yaw rate are small
for a simultaneous rudder and trailing edge flap failure (dashdot line). However, the tracking errors are
increasing with time. Furthermore, the aircraft sideslip angle is twice that of the reference model at 40
seconds. This indicates degraded performance because the sideslip angle should be small for a coordinated
turn. Observe from Figure 3 that the adaptive controller achieves perfect tracking in sideslip angle, yaw
rate, and roll rate.
Simulations are performed for an 88% simultaneous failure of all three control surfaces. The time
responses using the non-adaptive controller (not shown) are similar to the simultaneous aileron and rudder
failure given by the dotted line in Figure 3. The time responses using the adaptive controller (not shown)
achieve perfect tracking in sideslip angle, yaw rate, and roll rate.
VIII
Conclusions
A model reference adaptive controller for multi-input multi-output systems has been extended to include
loss of control effectiveness failures. It has been proven that all signals in the adaptive system are bounded
during the occurrence of a bounded disturbance. A state space approach was introduced for computing the
feedforward compensator that is required by the stability result. The adaptive controller was applied to the
linearized lateral dynamics of the F/A-18 aircraft when one, two or three control surfaces fail with an 88%
loss of control effectiveness during a lateral gust condition. Simulations show that the adaptive controller
exhibits significantly improved model following as compared to a fixed gain eigenstructure assignment
controller.
21
Acknowledgment
The authors thank Dr. I. Barkana of Kulicke & Soffa Industries and Dr. J.E. Piou of MIT Lincoln Laboratory for their contributions to theorems 2 and 3, respectively.
Appendix A: Error Equation
Define ex = (xap )∗ − xap . Then,
ėx = (ẋap )∗ − ẋap
(A.1)
Add and substract Aap (xap )∗
ėx = (ẋap )∗ − ẋap + Aap (xap )∗ − Aap (xap )∗
(A.2)
ėx = X11 (ẋm ) + X12 (u̇m ) − Aap xap − Bap αi u p − D + Aap (xap )∗ − Aap (xap )∗
(A.3)
ėx = X11 Am xm + X11 Bm um + X12Cv Av vm − Aap xap − Bap αi u p − D + Aap (xap )∗ − Aap (xap )∗
(A.4)
Recall u p = Kr, K = KI + K p and obtain
ėx = Aap ex − Bap αi Kr − Aap (xap )∗ + X11 Am xm + X11 Bm um + X12Cv Av vm − D
(A.5)
ėx = Aap ex − Bap αi Kr − Aap (X11 xm + X12 um ) + X11 Am xm + X11 Bm um + X12Cv Av vm − D
(A.6)
ėx = Aap ex − Bap αi Kr − (Aap X11 − X11 Am )xm − (Aap X12 − X11 Bm )um + X12Cv Av vm − D
(A.7)
Add and substract Bap αi K̃i r
ėx = Aap ex − Bap αi Kr + Bap αi K̃i r − Aap (X11 − X11 Am )xm − (Aap X12 − X11 Bm )um + X12Cv Av vm −
Bap αi K̃i r − D
(A.8)
22
Use K̃i from Eq. (27) to obtain
ėx = Aap ex − Bap αi [K − K̃i ]r − (Aap X11 − X11 Am + Bap αi K̃xi )xm − (Aap X12 − X11 Bm + Bap αi K̃ui )um +
X12Cv Av vm − Bap αi K̃ei eay − D
(A.9)
Use um = Cv vm and eay = ym − yap = (yap )∗ − yap = Cap (xap )∗ −Cap xap = Cap ex .
ėx = Aap ex − Bap αi [K − K̃i ]r − (Aap X11 − X11 Am + Bap αi K̃xi )xm − (Aap X12 − X11 Bm + Bap αi K̃ui )Cv vm +
X12Cv Av vm − Bap αi K̃eiCap ex − D
(A.10)
ėx = (Aap − Bap αi K̃eiCap )ex − Bap αi [K − K̃i ]r − (Aap X11 − X11 Am + Bap αi K̃xi )xm −
(Aap X12Cv − X11 BmCv + Bap αi K̃uiCv − X12Cv Av )vm − D
(A.11)
Finally
ėx = Ac ex − Bap αi [K − K̃i ]r − F(t)
(A.12)
where
Ac = Aap − Bap αi K̃eiCap
(A.13)
F(t) = (Aap X11 − X11 Am + Bap αi K̃xi )xm +
(Aap X12Cv − X11 BmCv + Bap αi K̃uiCv − X12Cv Av )vm + D
(A.14)
Appendix B: Proof of Theorem 1
Consider the Lyapunov function:
Vi (ex , KI ) = eTx Pi ex + tr[(KI − K̃i )T −1 (KI − K̃i )T ]
(B.1)
where Pi is an n × n postive definite symmetric matrix, K̃i is an unspecified matrix, T −1 is a positive
definite matrix, and, “tr” denotes the trace. Note that Vi is positive definite in the state variables of the
23
adaptive system, ex (t) and KI (t).
The time derivative of Vi is
V̇i = ėTx Pi ex + eTx Pi ėx + tr[K̇I T −1 (KI − K̃i )T + (KI − K̃i )T −1 K̇IT ]
(B.2)
V̇i = ėTx Pi ex + eTx Pi ėx + 2tr[(KI − K̃i )T −1 K̇IT ]
(B.3)
Use Eqs. (22) and (25) to obtain
V̇i = [Ac ex − Bap αi [K − K̃i ]r − F(t)]T Pi ex + eTx Pi [Ac ex − Bap αi [K − K̃i ]r − F(t)] +
2tr[(KI − K̃i )T −1 Tr(eay )T − σ(KI − K̃)T −1 KIT ]
(B.4)
V̇i = eTx ATc Pi ex + eTx Pi Ac ex − [Bap αi [K − K̃i ]r]T Pi ex −
eTx Pi Bap αi [K − K̃i ]r − F T (t)Pi ex − eTx Pi F(t) +
2(eay )T (KI − K̃i )r − 2σtr[(KI − K̃i )T −1 KIT ]
(B.5)
Use KIT = KIT + K̃iT − K̃iT , eay = Cap ex , and combine terms to obtain
V̇i = eTx (ATc Pi + Pi Ac )ex − 2eTx Pi Bap αi [K − K̃i ]r −
2eTx Pi F(t) + 2eTx (Cap )T KI r − 2eTx (Cap )T K̃i r −
2σtr[(KI − K̃i )T −1 (KI − K̃i )T ] − 2σtr[(KI − K̃i )T −1 K̃iT ]
(B.6)
V̇i = eTx (ATc Pi + Pi Ac )ex − 2eTx Pi Bap αi (KI + K p )r +
2eTx Pi Bap αi K̃i r − 2eTx Pi F(t) + 2eTx (Cap )T KI r −
2eTx (Cap )T K̃i r − 2σtr[(KI − K̃i )T −1 K̃iT ] − 2σtr[(KI − K̃i )T −1 (KI − K̃i )T ]
(B.7)
V̇i = eTx (ATc Pi + Pi Ac )ex − 2eTx Pi Bap αi KI r −
2eTx Pi Bap αi K p r + 2eTx Pi Bap αi K̃i r − 2eTx Pi F(t) + 2eTx (Cap )T KI r − 2eTx (Cap )T K̃i r −
2σtr[(KI − K̃i )T −1 (KI − K̃i )T ] − 2σtr[(KI − K̃i )T −1 K̃iT ]
24
(B.8)
Use Eq. (30) to obtain
V̇i = eTx (ATc Pi + Pi Ac )ex − 2eTx (Cap )T K p r − 2σtr[(KI − K̃i )T −1 (KI − K̃i )T ] −
2σtr[(KI − K̃i )T −1 K̃iT ] − 2eTx Pi F(t)
(B.9)
Use K p = eay (t)rT T¯and eay = Cap ex to obtain
V̇i = eTx (ATc Pi + Pi Ac )ex − 2(eay )T eay rT T¯r − 2σtr[(KI − K̃i )T −1 (KI − K̃i )T ]
−2σtr[(KI − K̃i )T −1 K̃iT ] − 2eTx Pi F(t)
(B.10)
T x + uT T¯u to obtain
a
Finally, use Eq. (29) and r T T¯r = (eay )T T¯
x m
e ey + xm T¯
m u m
a
a T a T
T
V̇i = −eTx Qi ex − 2(eay )T eay (eay )T T¯
e ey − 2(ey ) ey [xm T¯
x xm + um T¯
u um ]
−2σtr[(KI − K̃i )T −1 (KI − K̃i )T ] − 2σtr[(KI − K̃i )T −1 K̃iT ] − 2eTx Pi F(t)
(B.11)
Hence, we have
V̇i < −β1 kex k2 − β2 keay k4 − β3 keay k2 (β4 kxm k2 + β5 kum k2 ) − β6 kKI − K̃i k2 + β7 kKI − K̃i k + β8 kex k
(B.12)
where β1 , β2 , β3 , β4 , β5 , β6 , β7 and β8 are positive constants.
We conclude that whenever kex k or kKI − K̃i k increase beyond some bound, then the negative quadratic
and quartic terms in (B.12) will become dominant, and thus V̇i (ex , KI ) becomes negative. The quadratic
form of Vi (ex , KI ) then guarantees that ex and KI are bounded.
Appendix C: Proof of Theorem 2
The plant before failure is given by
ẋap = Aap xap + Bap u p + D(t)
(C.1)
After the finite change in Bap at t = T1 the plant becomes
ẋap = Aap xap + Bap α1 u p + D(t)
(C.2)
25
So at t = T1 , α1 is bounded; hence, ẋap bounded. Then, it follows that xap (T1− ) = xap (T1+ ). Since, ex (T1 ) =
(xap )∗ (T1 )−xap (T1 ) it follows that ex (T1− ) = ex (T1+ ). And the adaptive gains KI =
Rt
0 vr
T T dt,
K p = vrT T¯are
bounded at t = T1 . Thus V (T1 ) = eTx P1 ex + tr[(KI − K̃1 )T −1 (KI − K̃1 )T ] is bounded. The above argument
is valid for any finite number of failures.
Appendix D: Proof of Theorem 3
The system in (34) with feedback H(s) = K f (1 + τs) is given by
j
ẋ = Acl (σ)x
(D.1)
where
j
Acl (σ) = A p − B pj (σ)[I + τK f C p B pj (σ)]−1 K f C p (I + τA p )
(D.2)
A necessary and sufficient condition for the existence of the inverse in (D.2) is that the eigenvalues λ i
satisfy
1 + τλi {K f C p B pj (σ)} 6= 0
(D.3)
Use the matrix inversion lemma to obtain
j
Acl (σ) = A p − B pj (σ)[I − τK f C p (I + τB pj (σ)K f C p )−1 B pj (σ)]K f C p (I + τA p )
(D.4)
Rearrange (D.4) to obtain
Acl (σ) = (A p − B pj (σ)K f C p ) + [τB pj (σ)K f C p (I + τB pj (σ)K f C p )−1 B pj (σ)K f C p (I + τA p ) − B pj (σ)K f C p τA p(]D.5)
j
j
where K f is computed so that (A p − B p (σ)K f C p ) is stable for all j and σ.
Use the result from Sobel et. al. [17] that the system in (D.1) is stable if
α > K2 (M)β
(D.6)
26
where
α = −max Re[λi (A p − B pj (σ)K f C p )]
(D.7)
j
K2 (M) is the condition number of the modal matrix of (A p − B p (σ)K f C p ).
and
k τB pj (σ)K f C p (I + τB pj (σ)K f C p )−1 B pj (σ)K f C p (I + τA p ) − B pj (σ)K f C p τA p k≤ β
(D.8)
It is clear that β can be made small by choosing τ sufficiently small.
Let αmin be the minimum of α over j, σ; and let K2max be the maximum of K2 (M) over j, σ.
Then, we require
β < αmin /K2max
(D.9)
j
Hence, there exists τ sufficiently small such that (D.9) is satisfied which implies that A cl (σ) in (D.4) is a
stability matrix for all j, σ.
Appendix E: Aircraft State Space Model

Ap
0
 −0.245 −646.9 0.0285 32.189


 0.00849 −0.2460 0.112
0
0



= 
0.73
−2.83
0
0
 −0.0256



0
0
1
0
0



0
0.5
0
0
−0.5
27

















B0p
Cp

0
−2.915 34.909 





 −0.835 −0.896 −3.26 







= 
13.06
13.14
4.4








0
0
0






0
0
0


 1/650 0 0 0 0 




= 
1 0 0 −1 

 0




0
0 1 0 0
References
[1] Boskovic, J.D. and Mehra, R.K., ”An Adaptive Scheme for Compensation of Loss of Effectiveness of
Flight Control Effectors”, 40th IEEE Conference on Decision and Control, Orlando, Florida, December 2001, 2448-2453.
[2] Bodson, M. and Groszkiewicz, ”Multivariable Adaptive Algorithms for Reconfigurable Flight Control”, IEEE Tran. on Control Systems Technology, Vol. 5., No. 2, 217-229, March 1997.
[3] Tao, G., Chen S., Tang, X., and Joshi, S.M., ”Adaptive Control of Systems with Actuator Failures”,
Springer-Verlag London, 2004.
[4] Morse, W.D., and Ossman, K.A., ”Model following reconfigurable flight control system for the
AFTI/F-16”, Journal of Guidance ,Control, and Dynamics, pp. 969-976, 1990.
[5] Sobel, K., Kaufman, H., and Mabius, L., ”Adaptive Control for a Class of MIMO Systems”, IEEE
Transactions on Aerospace, Vol. 18, 1982, 576-590.
28
[6] Barkana, I., “Parallel Feedforward and Simplified Adaptive Control” International Journal of Adaptive Control and Signal Processing, Vol. 1, No. 2, pp. 95-102, 1987.
[7] Kaufman, H., Barkana, I., and Sobel, K.M., “Direct Adaptive Control Algorithms Theory and Applications”, Second Edition, Springer-Verlag New York, Inc., 1998.
[8] Ioannou, P., Kokotovic, P., “Adaptive Systems with Reduced Models”, Springer-Verlag, 1983.
[9] Pascal, G., Arkadi, N., Alan, L., Mahmoud, C., “LMI Control Toolbox For Use with Matlab” The
Mathworks, 1995.
[10] Branch, M.A., and Grace, A., “MATLAB R Optimization Toolbox User’s Guide”,Version 1.5,, The
Mathworks, 1996.
[11] “Military Specifications-Flying Qualities of Piloted Airplanes”, MIL-F-8785C, ASD/ENESS,
Wright-Patterson AFB, Ohio.
[12] McRuer, D., Ashkenas, I., Graham, D., “Robust Adaptive Control”, Princeton University Press,
1973, pp. 541-544.
[13] Brian, L.S., Frank, L.L, “Aircraft Control and Simulation” John Wiley & Sons, Inc. , pp. 584, 1992.
[14] Sobel, K.M., and Lallman, J.F., “Eigenstructure Assignment for the Control of Highly Augmented
Aircraft” J. of Guidance ,Control, and Dynamics, Vol. 12, No. 3, May-June 1989.
[15] Andry, A.N., Jr., Shapiro, E.Y., and Chung, J.C., ”Eigenstructure Assignment for Linear Systems”,
IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-19, No. 5, pp.711-729, September
1983.
29
[16] Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., ”Numerical Recipes: The Art of
Numerical Computing”, Cambridge University Press, 1986.
[17] Sobel, K.M., Banda, S.S., and Yeh, H.-H., 1989, Robust control for linear systems with structured
state space uncertainty, International Journal of Control, 50, 1991-2004.
30
Table 1: Desired Eigenstructure and Feedback Gain Matrix Keig
Dutch Roll
Roll subsidence
λdr = −2 ± 2 j
λroll = −4
1 + jx
0
x + j1
Keig
β
rwo
p
v
-1.8768
0.4583
0.1111
δac
0
r
1.7234
-1.2
-0.0656
δrc
0 + j0
1
p
0 + j0
x
φ
x + jx
0
x5
Table 2: Initial and Optimal Gain Matrices
Kinitial
−1
kKoptimal
k
Koptimal
β
rwo
p
β
rwo
p
1.6456
-0.6527
-0.0067
2242.3
-407.34
882.47
δte
1.7811
-0.6997
-0.0072
601.74
-2184.6
8356.2
δa
-0.9257
0.2858
0.003
49.048
-1792.4
-3891.6
δr
31
6.3×10−4
Use eigenstructure assignment
to compute initial gain Kinitial
Choose maximum failure to be any
single, double, or triple failure up to
maximum failure percent
Call function fmincon to minimize kK −1
f k
subject to tmin < 0
Reduce maximum
failure percent
Call function
pdlstab to
compute tmin
No Converge?
Yes
Call function pdlstab with increasing
values of maximum failure percent
tmin < 0?
Yes
Increase maximum
failure percent by 1%
No
Maximum allowable failure for K f
is current value minus 1%
Feedforward gain is K −1
f
Figure 1: Flowchart for design of feedforward gain using MATLAB LMI and Optimization toolboxes
32
u p (t)
- +i
-
6
- H −1 (s)
h p (t)
- Aircraft
y p (t)
yap (t)
Keig
pc


 Keig (1, 3) 




Keig (2, 3)
Reference Model
- +i
- Unfailed Aircraft Model
6
ym (t)- -?
i
eay (t)
Keig
+i
Kx
Ke
?
Ku
i
- +?
?
6
Figure 2: Block Diagram of Adaptive Control System
33
xm (t)
Non−Adaptive Controller
Adaptive Controller
Sideslip Angle
0.6
0.4
0.2
0
Yaw Rate
−0.2
0
10
20
30
0.4
0.2
0
−0.2
40
8
8
6
6
Yaw Rate
Sideslip Angle
0.6
4
2
0
0
10
20
30
4
2
0
20
30
40
0
10
20
30
40
0
10
20
Time(sec)
30
40
2
6
Roll Rate
Roll Rate
6
10
4
0
40
0
0
10
20
Time(sec)
30
4
2
0
40
Figure 3: Sideslip angle, yaw rate, and roll rate using the non-adaptive and adaptive controllers for reference model (solid), aircraft with 88% aileron and trailing edge flap failures (dashed), aircraft with 88%
rudder and trailing edge flap failures (dashdot),and aircraft with 88% aileron and rudder failures (dotted).
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