American Institute of Aeronautics and Astronautics
1
Control allocation with actuator dynamics for aircraft flight
controls
Hammad Ahmad1, Trevor M Young2 Department of Mechanical and Aeronautical Engineering University of Limerick, Ireland
Daniel Toal 3, and Edin Omerdic 4 Department of Electronics and Computer Engineering University of Limerick, Ireland
This paper addresses the control allocation to several aircraft flight controls to produce
required body axis angular accelerations. Control law is designed to produce the virtual
control effort signals, which are then distributed by solving a sequential least squares
problem using active set method to the flight control surfaces to generate this effort. Two
cases are described: in the first case the control law and allocation for the healthy aircraft is
implemented, and in the second case, jamming of one control surface is introduced at time
zero. In this case, it was shown how the controller and allocation compensate for this failure
without changing the control law. To implement this system it was assumed that there is a
good fault identification system onboard. Normally aircraft are over-actuated and in the
case of a control failure this over actuation is more pronounced due to coupling of aircraft
dynamics. Instead of using one-to-one mapping between control allocator and control
surfaces, actuator dynamics was included in the system. The discrepancy in the optimal
signal from control allocation due to this additional dynamics was compensated using the
scheme mentioned in this paper. Each gain corresponding to the actuator is tuned using
genetic algorithms (GA). The controller and allocation design are implemented on a
nonlinear B747 model with actuator dynamics.
Nomenclature
aorδ = right outboard aileron (deg)
airδ = right inboard aileron (deg)
aolδ = left outboard aileron (deg)
ailδ = left inboard aileron (deg)
eorδ = right outboard elevator (deg)
eirδ = right inboard elevator (deg)
eolδ = left outboard elevator (deg)
eilδ = left inboard elevator (deg)
ihδ = stabilizer (deg)
urδ = upper rudder (deg)
drδ = down rudder (deg)
p = roll rate about body x-axis (rad/s)
q = pitch rate about body y-axis (rad/s)
r = yaw rate about body z-axis (rad/s)
TV = true airspeed (m/s)
1 PhD student, M&AE Dept., University of Limerick / email: [email protected], AIAA student member. 2 Senior lecturer, M&AE Dept., University of Limerick and AIAA Senior member. 3 Senior lecturer, ECE Dept., University of Limerick. 4 Post doctoral fellow, ECE Dept., University of Limerick.
7th AIAA Aviation Technology, Integration and Operations Conference (ATIO)<BR> 2nd Centre of E18 - 20 September 2007, Belfast, Northern Ireland
American Institute of Aeronautics and Astronautics
2
α = angle of attack (rad)
β = side slip angle (rad)
φ = roll angle (rad)
θ = pitch angle (rad)
ψ = yaw angle (rad)
v = virtual control effort (rad/s2)
I. Introduction
odern jet aircraft have many “actuators” required for flight path control (e.g. two or more engines, elevators, rudders, flaps). In essence aircraft are "over-actuated" as they possess control redundancy and the pilot
commanded flight vector can be realized with more than one (often many) different combinations of settings of the actuators. With advanced control schemes this redundancy in the control of actuators can be taken advantage of to enhance aircraft safety in the event of an aircraft malfunction or damage. The research objective is to utilize the multiple redundancies in the control systems in the event of a system failure or other aircraft malfunction to control the aircraft by automatically switching control laws and control allocation techniques. This technique, which is based on online optimization, has recently been explored for use in military air vehicles; however, little research has been undertaken for civil aircraft applications.
The idea of control allocation can be given by a simple example. The lateral and directional dynamics are coupled in aircraft. On certain aircraft (e.g. B747), there are two rudders (i.e. upper and lower) for directional control redundancy. In theory, it is possible to use this redundancy to control the aircraft following a certain type of failure. In the event of a failure affecting lateral control (e.g. aileron jam) it is theoretically possible to still roll the aircraft by moving the two rudders in opposite directions, without yawing of aircraft (i.e. moving the aircraft left or right). In this work optimization is done in two phases in first phase optimal and feasible set of points (i.e. control surfaces positions) is found and in the second phase minimum deflection point is calculated among the points in feasible set from phase 1, which corresponds to a minimum drag point.
Gao and Antsaklis1 have proposed the approach of control law reconfiguration using the Pseudo Inverse Method (PIM), which was successfully accepted in flight simulation by Caglayan et al.
2. The main idea is to modify the feedback gain so that the reconfigured system approximates the nominal system in some sense. They proposed the modified PIM with respect to stability constraint but this method loses optimal sense in dealing with the general multivariable systems. This limitation was overcome using the robust control mixer module method by Yang and Blanke3. The mentioned approaches1,2,3, which are related to control mixer method, deals with configuring the flight control law. In control allocation (CA), Härkegård 4, separated the control design into the following two steps:
Design a control law specifying what total control effort is to be produced (net torque, force, etc.). Design a control allocator that maps the total demand onto individual actuator settings (commanded
aerosurfaces deflections, thrust, forces, etc.). Geometric constrained control allocation was proposed under the assumption that the actuators are linear in their
effect throughout their ranges of motion and independent from one another in their effects5. This was extended to a three moment problem by Durham6. Linear and quadratic programming approaches for control allocation are given by Dale7. Regarding the evaluation of optimization methods for control allocation, Bodson8 discusses a variety of issues that affect the implementation of various algorithms in flight control systems. Comparison between robust servomechanism and CA was done by Burken et al.9 and showed CA working with fault detection system. The concept of static CA was modified to dynamic CA by Härkegård10. Karen and Krishnakumar11 proposed control reallocation strategies with daisy chain CA, optimal CA using linear programming and table look up with blending. Comparison of optimal control versus CA was shown in Härkegård12. Doman et al.
13,14 have worked on dynamic control allocation with non-negligible actuator dynamics. Historically control allocation has been performed by assuming that a linear relationship exists between the control induced moments and the control effectors displacements. However a non-linear relationship leads to non-linear CA, as proposed by Doman et al.15,16.
II. Modelling of B747 100/200
A dynamic rigid body model of the Boeing 747 is considered in this paper. The Simulink model for this aircraft is FTLAB74717,18. In the flight control system of FTLAB747, there is no actuator redundancy utilization and similar control surfaces are considered to be the same (e.g. the four elevators are considered to be one surface). In the design
M
American Institute of Aeronautics and Astronautics
3
of control allocation scheme all actuators redundancies should be exploited. The model was thus modified to replicate the aircraft control redundancies. These actuators are shown in Fig. 1. The sign convention used for control
surface position is “leading edge down” is treated as positive. Input to the aerodynamic model is aerou , which is a
control vector of eleven control signals. Input to the propulsion system is propu .
The aircraft model is trimmed at straight and level flight at a flight condition of 241 m/s true airspeed and
7000 m height, with the flight path angle γ set to zero. The trimmed flight control (radians) and thrust control
American Institute of Aeronautics and Astronautics
4
where nx ℜ∈ is the system state vector, mu ℜ∈ is the control input vector to the system, and py ℜ∈ is the
output vector of the system to be controlled. The state vector is
TTVrqpx ][ ψθφβα=
III. Control allocation
Control allocation is useful for the control of over-actuated systems, and deals with distributing the total control demand among the individual actuators. Using control allocation, the actuator selection task is separated from the regulation task in the control design. To introduce the ideas behind control allocation, consider the following system:
21= uux +
where
x is a scalar state variable, and 1u and 2u are control inputs.
x can be thought of as the velocity of a unit mass object affected by a net force 21= uuv + produced by two
actuators. Assume that to accelerate the object, the net force 1=v is to be produced. There are several ways to
achieve this. One way can be to utilize only the first actuator and select 1=1u and 0=2u or to gang the actuators
and use 0.5== 21 uu . It is even possible to select 12=1 −u and 11=2u , although this might not be very practical.
Which combination to pick is essentially the problem of control allocation. (Today, control allocation is an active research topic in aerospace and marine vessel control.)
The nominal control allocation layout for a healthy aircraft is shown in Fig. 2. Here v is the virtual control
signal to the control allocation part. The control law, which is designed separately from the allocation scheme, is designed using a linear quadratic regulator (LQR) design.
A. Control law design
This control law is based on robust servomechanism design, which is generalization of proportional-plus-integral (PI) design. A PI controller is designed to stabilize the aircraft (stabilization)9. This law is also treated as baseline control law, and in the event of failure the redundant degrees of freedom are utilized to cancel the effect of the jammed surface using control allocation.
The control law for the linear model given in Eq. (1) is designed with virtual control signals of angular
acceleration in roll, pitch and yaw. The input matrix uB is factored into
BBB viru =
where the mkBrank u ≤=)( , and uB is mn × , virB is ,kn × and B is mk × .
The system in Eq. (1) is now given by
vBAxx vir+= (2)
Buv = (3)
Figure 2: Nominal control allocation design
f0 D Control
Allocation Control law x
y
Actuator dynamics Actuator
Constraints
f0 D x
y Ref nonlinear Aircraft model
v u
American Institute of Aeronautics and Astronautics
5
Cxy = (4)
Note that ∆ is removed for simplicity of notation. The controller dynamics are set to be
)(= yrBxAx cccc −+ (5)
where pcx ℜ∈ is the controller states, pp
cA ×ℜ∈ and ppcB ×ℜ∈ .
Consider the open loop system including the plant Eqs. (2-4) and Eq. (5) with 0=r
vDB
B
x
x
ACB
A
x
x
gg B
c
vir
c
A
ccc
−+
−=
0 (6)
with Trqpv ][= and [ ]Ty ψθφ= . The controllability of the augmented system Eq. (6) is checked by
( ) lCrank =0
where
][ 120 g
lggggg BABABABC−= , and pnl += .
The augmented system Eq. (7) is controllable. Hence there exist control laws
cc xkkxv += (7)
such that the closed loop system is stable. The control law can be conveniently found by applying the LQR approach to Eq. (7). In this special case r is a
constant command, therefore ][0= 33×cA and 33= ×IBc , according to their definitions. From controller dynamics
given by Eq. (5), it can be seen that edtdtyrxc ∫∫ − =)(= . Thus control law Eq. (7) is simply a PI control law of
multi input and multi output (MIMO) system. The structural design limitation in terms of load factor for the B747 is considerably smaller than a highly
maneuverable fighter aircraft. So it is better to control the position rather than the rotation rates (i.e. in roll, pitch, and yaw). In this way the aircraft will easily remain inside the design limits.
The control allocation is now designed by solving a sequential least squares (SLS) problem using active set
method to distribute the virtual control effort Trqpv ][= among the control surfaces optimally and feasibly.
B. Active set method for SLS problem
A sequential treatment of Least Squares problems may be preferable for several reasons: It divides a large computing burden into smaller parts to reduce the requirements on both processing
capability and storage, and It is the key to real-time applications.
Active set methods are used in many of today's commercial solvers for constrained quadratic programming, and can be shown to find the optimal solution in a finite number of iterations. In this method the inequality constraints are either disregarded or treated as equality constraints. The algorithm comprises two phases solving sequential least square problem19. The schematic of active set method for control allocation is given in Fig. 3.
American Institute of Aeronautics and Astronautics
6
C. Phase 1
a. At the start 2
= minmax0
uuu
+ and ][=W , otherwise it is the working set of active equality
constraints from previous sampling.
b. Here the post failure dynamics is used in allocation problem, solve 2l - optimal control allocation
problem
2
)(argmin= vuBWu rrvir
ru
−Ω (8)
subject to
vuB rr = (9)
Start of algorithmAssign initial feasible point
Select the inequality constraints satisfying that
point and put them in working set as equalityconstraint
Solve the problem for feasible step p
Calculate Lagrangemultipliers
Solution found.Assigning solution to
aircraft control
vector
Remove mostnegative muliplierconstraint from
working set.(Moving away from
most negative willdecrease objective
function)
True
Take a step indirection of p
If there are blocking
constraint in previousstep.Update the
working set with thatconstraint
False
Setting aircraft control
surfaces
p=0?
multipliersare
positive?
False
True
Controller output
Total demand fromcontroller in terms ofvirtual control input v
Control allocation
Figure 3: Control allocation schematic using active set method
American Institute of Aeronautics and Astronautics
7
maxmin uuu r ≤≤ for all Ir ∈ (10)
where
ru are the remaining control surfaces, rB is the control effectiveness matrix resulting by removing the
column corresponding to jammed surface from B . I is the set of inequality constraints. The problem in Eqs. (8) to (10) can be written as:
maxmin
2min
uuu
bAu
r
r
ru
≤≤
− (11)
where
vWbBWA virrvir ==
2~
)~(min bpuA ir
p
−+ (12)
0=~pBr (13)
where p~ is the optimal perturbation such that moving along p~ from iru , i
rruB does not change because
vuBpuB irr
irr ==)~( +
The iterative algorithm is given in Appendix.
c. If vuBr =Ω , move to phase 2 else stop with Ωuu = .
D. Phase 2
a. Let initial Ωuu =0 and W is the working set from phase 1
b. Solve
maxmin
=
)(argmin=
uuu
vuB
uuWu
r
rr
dru
ru
r
≤≤
−
using the algorithm in Appendix.
The weighting matrices uW and virW are assumed to be non singular. ,uW being non-singular ensures that the
posed optimization problems have a unique optimal solution. In phase 2 of SLS the desired control surface inputs
du are treated as zero to achieve minimum drag performance criteria (which was set for this work).
Control allocation is under the assumption that there is a good fault identification system available. Thrust vectoring (which may be available on certain military aircraft) is not used in this work. The control allocation for the failure case is given in Fig. 4.
American Institute of Aeronautics and Astronautics
8
E. Retrim and stability
Before proceeding with the control allocation for damage adaptation, it is important to determine whether the aircraft can still be retrimmed with a particular aerosurface jammed at a given position. This approach is taken from Ref.9. Rewrite the post failure aircraft model as
δδbuBAxx rr ++= (14)
where δ is the jammed surface position (as before), rB is the post failure B matrix, ru is the remaining
control surfaces, and δb is the control effectiveness vector corresponding to the jammed surface.
Let dy represent the three body angular (roll, yaw, and pitch) rates of the vehicle in body frame. Suppose
that xCy dd = , then
δδbCuBCAxCy drrdd ++= (15)
A necessary condition for retrimming the vehicle with the jammed surface is that the right hand side of the
preceding equation can still be made to vanish at 0=x with ru in its allowable range. For the range of jammed
position of aerosurface δ for which retrimming is possible, the following linear programming (LP) problem is
solved:
δδ
δδ
axu
or
u
r
r
m,
min,
(16)
subject to
0=δδbCuBC drrd + (17)
maxrminmaxrrminr uuu δδδ ≤≤≤≤ , (18)
The solution of the LP problem, expressed by Eqs. (16) to (18), gives the minimum (most negative) or maximum
jammed incremental position of δ that can be balanced at the trim condition by the remaining aerosurfaces ru
within the saturation limits. The resulting range serves as a reasonable estimation within which the reconfigurable
f0 D
Controlallocation
Control lawx
y
Actuator
dynamicsActuator
constraints
f0 D
x
yRef
Fault detection,isolation
and identification
Modified controleffectiveness
matrix
Post failure
dynamics
Figure 4: Control allocation assuming an effective, onboard fault diagnostic system
American Institute of Aeronautics and Astronautics
9
control can still possibly stabilize the system. In the present study, which concerned an inboard aileron jamming at full range (-20 to 20 degrees), it is required that the system can still be retrimmable and stabilizable.
Closed loop stability is assured by constructing a control law, in terms of the virtual control signal, which stabilizes the system (sufficient condition). The control allocator merely distributes the total control demand among the available effectors and (in principle) does not affect the closed loop behavior. However, if the control demand from the control law cannot be fulfilled, closed loop stability cannot be assured but the system does not necessarily become unstable. In this case control demand is fulfilled by the control laws. And the nominal control law does not push the vehicle too hard for performance. Hence, the control laws were designed in terms of virtual control signals.
IV. Simulation results
A. Control design of healthy aircraft
Control law and control allocation are implemented on a high fidelity FTLAB747 nonlinear simulation model of the B747-200. All the simulation results presented in this section are from nonlinear simulations with actuator dynamics. The control law and CA for the healthy aircraft worked well for the required trajectory tracking as shown in Fig. 5. The good thing about this design is that the control law and allocation were designed separately from each other (as mentioned earlier) so that the control allocation is merely the distribution of the control effort among the actuators. However when one of the actuators is jammed there is discrepancy in tracking and disturbance rejection (shown in Fig. 6).
B. Control allocation with jamming of control surface
Damaged is introduced by jamming the left inboard aileron at 20 degrees downward at time zero, this acts as a flap because of the reduced angle of attack. In high speed flight the outboard ailerons are neutral18 (due to the inherent torsional elasticity of the wings). This limitation is neglected under the assumption that the rigid body dynamics is considered. The inboard right aileron will move to maintain symmetry, but the rest of the rolling moment compensation comes from the outboard ailerons and some additional roll is from the rudders (Fig. 7), because of the strong coupling of lateral and directional dynamics. As longitudinal dynamics is only weakly coupled to lateral/directional dynamics, the required pitch attitude is achieved by symmetrical deflection of the elevators (Fig. 8). As can be seen in Fig. 6a a doublet roll maneuver for the damaged aircraft is implemented, the
jammed ailδ tries to resist the roll in the direction of jam. This induces a high load factor (but the load factor is below
the specification laid down by FAR 25.337, which is in the range of 2.5–3). In the direction opposite to jamming the damaged aileron is advantageous of achieving the desired roll angle, as seen in the figure.
0 10 20 30 40 50 60 70 80 90 100-20
0
20
φ (
deg)
0 10 20 30 40 50 60 70 80 90 100-10
0
10
20
θ (
deg
)
0 10 20 30 40 50 60 70 80 90 100-2
0
2
ψ (
deg)
time (sec)
Desired output
Actual output
Figure 5: Control law and CA for healthy
aircraft. Control law used is a baseline
control law for the aircraft
American Institute of Aeronautics and Astronautics
10
V. Adding actuator dynamics
When designing control allocation typically the actuator dynamics are ignored because the bandwidth of the actuators is larger than the frequencies of the rigid body modes of the aircraft. Therefore, the actuator dynamics is ignored and there is only one-to-one mapping between allocator and control surfaces. If there is a case in which actuator frequencies are comparable with the bandwidth of the rigid body modes then the actuator dynamics cannot be neglected. In this case the output of control allocator, poscmd, should match the output of the actuator dynamics, posactual, as shown in Fig. 9. In reality the optimum output of the CA is attenuated due to the presence of non-negligible actuator dynamics. The loss of the information from CA output signal is compensated by the scheme shown in Fig. 9. In the second order dynamics of the actuator the rate could be estimated using a Kalman filter. The vector of gains as shown in the figure is tuned offline using genetic algorithms (GA).
0 10 20 30 40 50 60 70 80 90 100-20
0
20φ
(deg)
0 10 20 30 40 50 60 70 80 90 100-10
0
10
20
θ (
deg)
0 10 20 30 40 50 60 70 80 90 100-10
0
10
ψ (
deg)
time (sec)
Desired output
Actual output
0 50 100-40
-20
0
20
δair (
deg)
time (sec)
0 50 10019
19.5
20
20.5
21
δail (
deg)
jam
med
time (sec)
0 50 100-20
-10
0
10
20
δaor (
deg)
time (sec)
0 50 100-20
-10
0
10
20
δaol (
deg)
time (sec)
(a) (b)
Figure 6: a) Control law is same as the healthy aircraft. CA is to distribute the virtual control among
remaining control surfaces in the damaged aircraft. b) Ailerons position to control roll rate with the inboard
left aileron jammed at 20 degrees
0 50 100-20
-10
0
10
20
δeir (
deg)
time (sec)
0 50 100-20
-10
0
10
20
δeir (
deg)
time (sec)
0 50 100-20
-10
0
10
20
δeir (
deg)
time (sec)
0 50 100-20
-10
0
10
20
δeir (
deg)
time (sec)
Figure 8: Symmetrical elevator movement
to control pitch rate
0 10 20 30 40 50 60 70 80 90 100-2
0
2
δih
(deg)
time (sec)
0 10 20 30 40 50 60 70 80 90 100-20
0
20
δur (
deg)
time (sec)
0 10 20 30 40 50 60 70 80 90 100-20
0
20
δdr (
deg)
time (sec)
Figure 7: Stabilizer to control pitch rate
and redundant rudders to control yaw rate
American Institute of Aeronautics and Astronautics
11
A. Genetic algorithms
Genetic algorithms (GA) have the following features: A GA operates with a population of possible solutions (individuals) instead of a single individual. Thus
the search is carried out in a parallel form. GAs are able to find optimal or suboptimal solutions in complex and large spaces. Moreover, GAs are
applicable to nonlinear optimization problems with constraints that can be defined in discrete or continuous search spaces.
GAs examine many possible solutions at the same time, so there is higher probability that the search converges to an optimal solution.
B. Algorithms
The methodology of GA is as follows: Choose initial population Repeat
Evaluate the individual fitnesses of a certain proportion of the population
Select pairs of best-ranking individuals to reproduce
Apply crossover operator Apply mutation operator
Until terminating condition
In this case the stopping criterion is maximum
number of generations. The generation is defined
transition from a population generationpopulation to
1+generationpopulation as shown in Fig. 10.
C. Simulation results
During simulation, a mixture of actuator dynamics was used. In case of redundant control surfaces diagonal gain matrices were tuned by GA. The control surfaces were approximated by following transfer functions as shown in Table 1.
Control
allocation
Actuator
dynamics
f0 D
Vector of
gains
Compensator
poscmd
posactual
rates
Figure 9: Scheme for compensation due to loss of information by adding the actuator dynamics.
generationPopulation
1generationPopulation +
Figure 10: Representation of executed operation during
a generation.
American Institute of Aeronautics and Astronautics
12
The virtual control signal, v consists of chirps of
amplitude 0.1, 0.15, 0.1 (rad/s2) in roll, pitch and yaw angular accelerations respectively. The frequencies of chirps ranged from 0.1–0.7 Hz in 20 seconds. The flow chart for GA routine is shown in Fig. 11. In the processing of GA routine exception handling was done to avoid breaking GA optimization process. For example if there is an individual (i.e. gains in diagonal matrix) in population that gives division by zero that would eventually break the simulation. This is dealt with in an exception handling block, which will give a penalty to that individual without breaking the simulation. In the next generation that individual would not likely to be selected.
Simulations are done with and without compensation, as shown in Figs. 12 and 13. As can be seen clearly from the results with no compensation there is serious attenuation and mismatch, but as soon as the
compensation is turned on ‘ Buv = ’ is achieved because
there was sufficient control authority existed. Deviations in case of no compensation case means
that the desired control surface positions coming out of control allocator are different from actual position of control surfaces. This interaction between control allocator and actuator dynamics results in serious consequence if the bandwidths of actuators are not high or, in other words, actuators are slow.
Table 1: Aerosurfaces actuator dynamics
Control
surfaces
No. of surfaces Transfer
functions
Elevators 4
6128.0
6128.0
+s
Ailerons 4
497
492 ++ ss
Stabilizer 1
0087.0
0087.0
+s
Rudders 2
497
492 ++ ss
Main GA
function
Calculation of diagonal
gains matrices
Simulating model in
Simulink
Cost
calculation
Penalty
Yes Exception handling
If
Exception generated
No
Figure 11: Flow chart of GA routine.
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
roll
accele
ration
(ra
d/s
ec
2)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
pitch a
ccele
ration
(ra
d/s
ec
2)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
yaw
accele
ration
(ra
d/s
ec
2)
time (sec)
Desired acceleration v
Simulated B*u
Figure 13: Desired acceleration ( v ) and
actual acceleration ( Bu ) in rad/s2 when
compensation is on
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
roll
accele
ration
(ra
d/s
ec
2)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
pitch a
ccele
ration
(ra
d/s
ec
2)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
yaw
accele
ration
(ra
d/s
ec
2)
time (sec)
Desired acceleration v
Simulated B*u
Figure 12: Desired acceleration ( v ) and
actual acceleration ( Bu ) in rad/s2 when
compensation is off
American Institute of Aeronautics and Astronautics
13
VI. Conclusions
This research work mainly focused on flight safety. It can be seen from the results that the control allocation has found the optimal and feasible solution (flight control settings) by solving a SLS problem using the active set method. The control law and the associated control allocation have performed satisfactorily in the case of a failure. The interaction between control allocator and actuator is compensated successfully by tuning gains using GA. The benefit of using this method is that in the case of first order time lag dynamics of the actuators there is no need to know the dynamics of the actuators themselves. But in the case of a second order system, the rates are required in compensation hence for the estimation of these rates the Kalman filter could be used (which required the model of the actuators). GA are used offline and the chirps are used as the excitation signal in the simulation. However, a band limited pseudo random binary signal (PRBS) for this type of identification process could be used as an excitation signal rather than chirps.
Appendix
Iterative algorithm for active set method for solving sequential least square problem20.
[ ]
for end
else
if
else
Set
else
STOP
if
if
for
;W W
W tosconstraint
blocking theof one addingby obtain W
sconstraint blocking are there
;p~αuu
compute
0)p~(
) Wfrom j constraint the(Dropping ;;~uu
;min arg j
;~uusolution with
W i allfor 0λ
Eq.(10).in
sconstraint
active with the and Eq.(9) with associated is .Winsconstraintactive iC
λ
µCBb)(AuA
fromλmultiplierLagrangecompute
0p
Wi wherep~findto(12,13)solve
.0,1,2,3...k
k1k
1k
1k
k
ik
k
r
1k
r
k
k
i
k1
k
r
1k
r
Wj
1k
rr
ki
k0
T
0
T
r
T
i
k
i
k
k
i
k
=
+=
≠
=+=
=
+==
Ι∩∈≥
=−
=
Ι∩∈
=
+
+
+
+
++
Ι∩∈
+
α
λ
λµ
kk
k
i
j
k
i
k
r
WWp
pu
s
Acknowledgments
The authors would like to thank Dr. James F Whidborne (Senior Lecturer, Dynamics, Simulation and Control Group, Cranfield University) for providing a valuable insight into the area of control allocation and flight control reconfiguration.
References 1Gao, Z., and Antsaklis, P. J., "On the stability of the pseudo-inverse method for reconfigurable control systems, "IEEE
Proceedings of the National Aerospace and Electronics Conference, Vol. 1, 1989, pp. 333.
American Institute of Aeronautics and Astronautics
14
2Caglayan, A. K., Allen, S. M and Wehmuller, K., "Evaluation of a second generation reconfiguration strategy for aircraft flight control systems subjected to actuator failure/surface damage." IEEE National Aerospace and Electronics Conference -
NAECON. 1988, pp. 520-529. 3Yang, Z. and Blanke, M., "Robust control mixer module method for control reconfiguration," American Control Conference,
2000, pp. 3407-3411. 4Härkegård, O., "Backstepping and control allocation with application to flight control,” Ph.D. Dissertation, Linköping
Studies in Science and Technology, Vol. 820, 2003. 5Durham, W. C., "Constrained control allocation," Journal of Guidance, Control, and Dynamics, Vol. 16, 1993, pp. 717-725. 6 Durham, W. C., "Constrained control allocation: three-moment problem," Journal of Guidance, Control, and Dynamics,
Vol. 17, 1994, pp. 330-336. 7Enns, D., "Control allocation approaches," AIAA Guidance, Navigation, and Control Conference and Exhibit, Boston, MA,
1998, pp. 4109. 8Bodson, M., "Evaluation of optimization methods for control allocation,” Journal of Guidance, Control, and Dynamics, Vol.
25, 2002 pp. 70311. 9Burken, J. J., Lu, P., Wu, Z. and Bahm, C., "Two reconfigurable flight-control design methods: Robust servomechanism and
control allocation," Journal of Guidance, Control, and Dynamics, vol. 24, 2001, pp. 482-493. 10Härkegård, O., "Dynamic control allocation using constrained quadratic programming," Journal of Guidance, Control, and
Dynamics, Vol. 27, 2004, pp. 1028-1034. 11 Gundy-Burlet, K., Krishnakumar, K., Limes, G., Bryant, D., "Control reallocation strategies for damage adaptation in
transport class aircraft, "AIAA, 2003, pp. 5642. 12Härkegård, O. and Glad, S.T., "Resolving actuator redundancy - Optimal control vs. control allocation," Automatica, 2005,
Vol. 41, pp. 137-144. 13Luo, Y., Serrani, A., Yurkovich, S., Doman, D.B., and Oppenheimer, M., W., "Dynamic control allocation with asymptotic
tracking of time varying control input commands, "Proceedings of the American Control Conference, 2005. 14Luo, Y., Serrani, A., Yurkovich, S., Doman, D.B., and Oppenheimer, M., W., "Model predictive dynamic control allocation
with actuator dynamics, "Proceedings of the American Control Conference, Vol. 2, 2004, pp. 1695. 15Poonamallee, V., L., Yurkovich S., Serrani, A., Doman, D.B., and Oppenheimer, M., W., "A nonlinear programming
approach for control allocation, “Proceedings of the American Control Conference, Vol. 2, 2004, pp. 1689. 16Bolender, M., A. and Doman, D.B., "Non-linear control allocation using piecewise linear functions: A linear programming
approach, "Collection of Technical Papers - AIAA Guidance, Navigation, and Control Conference, Vol. 2, 2004, pp. 1394. 17Marcos,A. and Estban,G. J. B., "A B747-100/200 aircraft fault tolerant and fault diagnostic benchmark, "Technical Report,
Aerospace Engineering and Mechanics Dept., University of Minnesota, 2003. [18]Hanke, C. "The simulation of large transport aircraft, "Technical Report, Vol. 1, June 2003. [19]Härkegård, O., "Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation,
"41st IEEE Conference on Decision and Control, 2002. 20Nocedal J. and Wright, S. J., Numerical Optimization, 2nd ed., Springer-Verlag, New York, 2006, pp. 472.
