Dynamic control allocation using constrained
quadratic programming
Ola H¨arkeg˚ard
Control & Communication
Department of Electrical Engineering
Link¨opings universitet, SE-581 83 Link¨oping, Sweden
WWW: http://www.control.isy.liu.se
E-mail: ola@isy.liu.se
16th February 2004
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LINKÖPING
Report no.: LiTH-ISY-R-2594
Submitted to AIAA Guidance, Navigation, and Control
Conference, 2002
Technical reports from the Control & Communication group in Link¨oping are
available at http://www.control.isy.liu.se/publications.
Abstract
Control allocation deals with the problem of distributing a given con-
trol demand among an available set of actuators. Most existing methods
are static in the sense that the resulting control distribution depends only
on the current control demand. In this paper we propose a method for
dynamic control allocation, in which the resulting control distribution also
depends on the distribution in the previous sampling instant. The method
extends the traditional generalized inverse method by also penalizing the
individual actuator rates. Its main feature is that it allows for different
control distributions during the transient phase of a maneuver and during
trimmed flight. The control allocation problem is posed as a constrained
quadratic programming problem which provides automatic redistribution
of the control effort when one actuator saturates in position or in rate.
When no saturations occur, the resulting control distribution coincides
with the control demand fed through a linear filter which can be assigned
different frequency characteristics for different actuators.
Keywords: control allocation, aircraft control, constrained control
DYNAMIC CONTROL ALLOCATION USING CONSTRAINED
QUADRATIC PROGRAMMING
Ola H¨arkeg˚ard, member AIAA
Div. of Automatic Control, Link¨opings universitet, Sweden
Control allocation deals with the problem of distributing a given control de-
mand among an available set of actuators. Most existing methods are static in
the sense that the resulting control distribution depends only on the current con-
trol demand. In this paper we propose a method for dynamic control allocation,
in which the resulting control distribution also depends on the distribution in
the previous sampling instant. The method extends the traditional generalized
inverse method by also penalizing the individual actuator rates. Its main feature
is that it allows for different control distributions during the transient phase of
a maneuver and during trimmed flight. The control allocation problem is posed
as a constrained quadratic programming problem which provides automatic re-
distribution of the control effort when one actuator saturates in position or in
rate. When no saturations occur, the resulting control distribution coincides with
the control demand fed through a linear filter which can be assigned different
frequency characteristics for different actuators.
1 Introduction
In recent years, nonlinear flight control design
methods, like dynamic inversion
1
and backstep-
ping,
2
have gained increased attention. These
methods result in control laws specifying the mo-
ments, or angular accelerations, to be produced in
pitch, roll, and yaw, rather than which particu-
lar control surface deflections to produce. How to
transform these “virtual” control commands into
physical control commands is known as the control
allocation problem.
With a redundant actuator suite there are sev-
eral combinations of actuator positions which all
give (almost) the same overall system behavior. A
common approach is to pick the combination that
minimizes, e.g., control deflections, drag, wing
load, or radar signature.
3–7
In this paper we
will use the redundancy to let different actuators
produce control effort in different parts of the fre-
quency spectrum. We will refer to this as dynamic
control allocation.
Such frequency division may be desirable for at
least two reasons. First, an actuator can be as-
signed a frequency range according to its intended
operational use, e.g., for high, low, or midrange
frequencies. Second, the high frequency control
distribution, affecting the initial aircraft response
to a pilot command, can be tuned without affect-
ing the steady state control distribution, which
may be designed to minimize, e.g., drag.
The remainder of this paper is organized as fol-
lows. In Section 2 the aircraft and actuator models
to be used are introduced and motivated. Sec-
tion 3 discusses the differences between static and
dynamic allocation. The proposed control allo-
cation method is presented in Section 4 and its
properties, for the case when no actuator satura-
tions occur, are analyzed in Section 5. A design
example can be found in Section 6, and Section 7
contains some concluding remarks.
2 Aircraft Model
Let the aircraft dynamics be given by
˙x = f(x, δ)
˙
δ = g(δ, u)
where x = aircraft state vector, δ = actuator posi-
tions, and u = actuator inputs. To incorporate the
actuator position and rate constraints we impose
that
δ
min
≤ δ ≤ δ
max
(1a)
˙
δ
≤ δ
rate
(1b)
where δ
min
and δ
max
are the lower and upper po-
sition constraints, and δ
rate
specifies the maximal
individual actuator rates.
Even in the case when f and g are linear, it is
nontrivial to design a control law which gives the
desired closed loop dynamics while assuring that
the actuator constraints are met. A common ap-
proach is to split the design task into two subtasks.
To do this, we first use the fact that typically,
control surface deflections primarily produce aero-
dynamic moment, M (x, δ). Second, the actuator
1
American Institute of Aeronautics and Astronautics
dynamics are often very fast compared to the re-
maining aircraft dynamics, and can therefore be
neglected. This gives us
δ ≈ u
˙x ≈ f
M
(x, M(x, u))
The control design can now be performed in two
steps as follows. First, design a control law
M(x, u)=k(r, x)(2)
where r = pilot command, that yields some de-
sired closed loop dynamics,
˙x = f
M
(x, k(r, x))
Second, determine u, constrained by (1) (with δ =
u), that satisfies (2).
The latter step is the control allocation step.
Since modern aircraft use digital flight control sys-
tems we rewrite (1b) in discrete time as
˙u
≈
u(t) − u(t − T )
T
≤ δ
rate
to get the overall position constraints at time t,
u
(t) ≤ u(t) ≤ u(t)(3)
where
u
(t)=max{δ
min
,u(t − T ) − δ
rate
T }
u(t)=min{δ
max
,u(t − T )+δ
rate
T }
and T is the sampling time. To simplify the search
for a feasible solution we will assume the aerody-
namic moment to be affine in the controls. This
gives us
M(x, u)=B(x)u + c(x)=k(r, x)(4)
or, equivalently,
Bu(t)=v(t)(5)
where
v(t)=k(r, x) − c(x)(6)
is the virtual control input. Now, to perform on-
line control allocation we need to find a feasible
solution u(t) at each sampling instant, satisfying
(3) and (5). Figure 1 shows the structure of the
resulting closed loop system.
r
Feedback
law
Control
allocation
Aircraft
x
u
v
Fig. 1 Overview of the modular controller structure.
3 Static vs Dynamic Control Allocation
Several control allocation methods, like direct
control allocation,
8
daisy chaining,
9
redistributed
pseudoinverse,
3
and methods based on constrained
quadratic
10, 11
or linear
7
programming, have been
proposed in the literature, see ref. 12 for a survey.
A common denominator for all these methods is
that they are static in the sense that the physi-
cal control commands computed at time t,only
depend on the virtual control commands at that
time, i.e.,
u(t)=f
v(t)
Using a static mapping, no frequency division
can be made between the actuators. To obtain
a frequency division, and let different actuators
produce control effort in different parts of the fre-
quency spectrum, we need to use a dynamic map-
ping of the form
u(t)=f
v(t),u(t − T ),v(t − T ),
u(t − 2T ),v(t − 2T ),...
With a dynamic mapping, the high frequency con-
trol distribution, affecting the initial aircraft re-
sponse to a pilot command, and the low frequency
control distribution, determining the distribution
at steady state, need not be the same. Using static
control allocation, on the other hand, a trade-off
has to be made between good initial behavior and,
e.g., low drag at trimmed flight.
In the following sections we will develop a strat-
egy for performing dynamic control allocation us-
ing constrained quadratic programming. When no
actuators saturate the relationship between u and
v will be given by a first order linear filter of the
form
u(t)=Fu(t − T )+Gv(t)
Previous efforts in this direction include ref. 13,
where the required pitching moment is distributed
to the tailerons through a low-pass filter and
to the canard wings through a high-pass filter.
This is motivated by the desire to get a fast ini-
tial response, produced by the canards, while the
tailerons are known to produce more pitching mo-
ment, and are therefore used to generate the re-
quired moment at steady state.
2
American Institute of Aeronautics and Astronautics
The difference between our approach and ref. 13
is twofold.
• To handle constraints on actuator positions
and rates, we will perform the control alloca-
tion within a constrained quadratic program-
ming framework. This ensures that (5) is
satisfied whenever possible, since the control
effort is redistributed when one actuator sat-
urates.
• In a complex situation, where the number of
moment generators is large, it is not an easy
task to explicitly design the frequency distri-
bution among the actuators while ensuring
that (5) is satisfied. We propose the use of
weighting matrices to affect the distribution
of control effort, in size as well as in frequency,
among the actuators.
4 Dynamic Control Allocation Using QP
The control allocation algorithm that we pro-
pose can be formulated as a linearly constrained
quadratic programming problem:
min
u(t)
W
1
(u(t) − u
s
(t))
2
2
+
W
2
(u(t) − u(t − T ))
2
2
(7a)
Bu(t)=v(t)(7b)
u
(t) ≤ u(t) ≤ u(t)(7c)
Equation (7a) is the cost function to be mini-
mized under the linear constraints (7b) and (7c).
Equation (7b) specifies which virtual control, v,
to produce. We will assume B to be an n × m
matrix (n<m) with rank n,wheren is the num-
ber of virtual controls (typically n =3)andm is
the number of physical controls available. Equa-
tion (7c) represents the feasible actuator positions
at time t, regarding both the overall position con-
straints and the rate constraints as in (3).
Let us now focus on the cost function in (7a).
·
2
denotes the Euclidean 2-norm, i.e.,
x
2
=
√
x
T
x where x is a column vector. u
s
(t)isthe
desired stationary distribution of control effort
among the actuators and determines the actuator
positions at trimmed flight. We will discuss the
choice of u
s
in Section 5.3. W
1
and W
2
are weight-
ing matrices whose (i, i)-entries specify whether it
is important for the i:th actuator, u
i
, to quickly
reach its desired stationary value, or to change its
position as little as possible. With this interpreta-
tion, a natural choice is to use diagonal weighting
matrices but in the analysis to follow we will allow
arbitrary matrices with the following restriction.
Assumption 1 Assume that the weighting matri-
ces W
1
and W
2
are symmetric and such that
W =
q
W
2
1
+ W
2
2
is nonsingular.
This assumption certifies that there is a unique
optimal solution to the control allocation problem
(7).
The difference between our approach and previ-
ous efforts based on quadratic programming is the
second term in the cost function (7a), which pe-
nalizes the actuator rates. The two terms in the
cost function can be merged into one term (see
Lemma 2) without affecting the solution. Thus,
any QP solver suitable for real-time implementa-
tion
3, 10, 11, 14
can be used to find the solution.
How do the design variables, u
s
, W
1
,andW
2
,
affect the solution, u(t)?
5 The Nonsaturated Case
To answer this question, let us investigate the
case where the optimal solution to (7a)-(7b) is
feasible with respect to (7c). Then the actuator
constraints can be disregarded and (7) reduces to
min
u(t)
W
1
(u(t) − u
s
(t))
2
2
+
W
2
(u(t) − u(t − T ))
2
2
(8a)
Bu(t)=v(t)(8b)
5.1 Explicit solution
Let us begin by stating the closed form solution
to (8).
Proposition 1 Let Assumption 1 hold. Then the
control allocation problem (8) has the solution
u(t)=Eu
s
(t)+Fu(t − T )+Gv(t)(9)
where
E =(I − GB)W
−2
W
2
1
F =(I −GB)W
−2
W
2
2
G = W
−1
(BW
−1
)
†
The
†
symbol denotes the pseudoinverse operator
defined as
A
†
= A
T
(AA
T
)
−1
for an n × m matrix A with rank n ≤ m.
The proposition shows that the optimal solution
to the control allocation problem (8) is given by
the first order linear filter (9). The properties of
3
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