Vol.:(0123456789)
1 3
CEAS Aeronautical Journal (2022) 13:33–43
https://doi.org/10.1007/s13272-021-00549-z
ORIGINAL PAPER
Applying Eigenstructure Assignment toInner‑Loop Flight Control
Laws foraMultibody Aircraft
AlexanderKöthe1· RobertLuckner1
Received: 25 March 2020 / Revised: 23 June 2021 / Accepted: 24 June 2021 / Published online: 21 December 2021
© The Author(s) 2021
Abstract
Unmanned aircraft used as high-altitude platform system has been studied in research and industry as alternative technologies
to satellites. Regarding actual operation and flight performance of such systems, multibody aircraft seems to be a promis-
ing aircraft configuration. In terms of flight dynamics, this aircraft strongly differs from classical rigid-body and flexible
aircraft, because a strong interference between flight mechanic and formation modes occurs. For unmanned operation in the
stratosphere, flight control laws are required. While control theory generally provides a number of approaches, the specific
flight physics characteristics can be only partially considered. This paper addresses a flight control law approach based on a
physically exact target model of the multibody aircraft dynamics rather than conventionally considering the system dynamics
only. In the target model, hypothetical spring and damping elements at the joints are included into the equations of motion
to transfer the configuration of a highly flexible multibody aircraft into one similar to a classical rigid-body aircraft. The
differences between both types of aircraft are reflected in the eigenvalues and eigenvectors. Using the eigenstructure assign-
ment, the desired damping and stiffness are established by the inner-loop flight control law. In contrast to other methods,
this procedure allows a straightforward control law design for a multibody aircraft based on a physical reference model.
Keywords Flight control theory· Control allocation· Multibody aircraft· Highly flexible aircraft structures
1 Introduction
Aircraft operating as so-called High-Altitude Platform Sys-
tems (HAPS) have been considered as a complementary
technology to satellites for several years. These aircraft can
be used for similar communication and monitoring tasks
while operating at a fraction of the cost. Such concepts have
been successfully tested. Those include the AeroVironment
Helios [13] and, in particular, the Airbus Zephyr, with a
proven endurance of nearly 624 hours (26 days) [1]. All
these HAPS aircraft have a single high-aspect-ratio wing
using lightweight construction. In gusty atmosphere, this
results in high bending moments and high structural loads,
which can lead to overloads. Aircraft accidents, for example
observed with Google’s Solara 50 [6] or Facebook’s Aquila
[5], give proof of that fact. Especially in the troposphere,
where the active weather takes place, gust loads occur,
which can lead to the destruction of the structure. Besides
the general challenges regarding long-endurance opera-
tion, specific flight performance characteristics are impor-
tant for HAPS, among which payload capacity is the most
prominent. The Airbus Zephyr, for example, provides only a
very small payload. Thus, it does not fully comply with the
expectations towards future HAPS, a phenomenon typically
observed with single-wing configurations.
To overcome the shortcomings of aircraft with wings of
extreme high aspect ratio and flexibility, so-called multi-
body aircraft are considered to be an alternative. The con-
cept assumes multiple aircraft with wings of low aspect
ratio connected to each other at their wingtips to increase
the wingspan. The idea dates back to the German engineer
Dr. Vogt [12]. In the United States, shortly after the end of
World War II, he experimented with the coupling of manned
aircraft. This resulted in a high-aspect-ratio wing for the
overall aircraft formation. The range of the formation could
be increased correspondingly. The engineer Geoffrey S.
Sommer took up Vogt’s idea and patented an aircraft con-
figuration consisting of several unmanned aerial vehicles
* Alexander Köthe
alexander.koe[email protected]
1 Flight Mechanics, Flight Control andAeroelasticity,
Technische Universität Berlin, Marchstraße 12 - F5,
10587Berlin, Germany
34 A.Köthe, R.Luckner
1 3
coupled at their wingtips [16]. A flight mechanical analysis
(static and dynamic) and the design of flight control laws is
missing in Sommer’s patent.
In the internal TU Berlin project AlphaLink, the flight
mechanic design, the flight dynamic modelling and the
flight control laws for a multibody aircraft configuration
were established. The fundamental differences between the
multibody aircraft and a conventional aircraft that can be
considered as rigid or flexible are the following:
1. A high-aspect-ratio wing is achieved through wing tip
coupling of several individual aircraft with mechanical
joints leading to an aircraft structure with multiple, dis-
tributed flight controls along the wingspan.
2. The number of the degrees of freedom is finite (depend-
ing on the joint configuration and the number of coupled
aircraft), as each individual aircraft is assumed to be
rigid.
3. The coupling equations between the aircraft are non-
linear, but can be expressed mathematically exact.
4. The formation modes that occur due to the mechanical
wing tip connection do not have any mechanical stiffness
or damping and, hence, their eigenvalue and eigenvector
characteristics depend only on the aerodynamics.
These special characteristics have to be considered in the
flight control law design. The multibody aircraft is an over-
actuated multiple input multiple output system. Control
theory provides a number of design methods in the time
and frequency domain including linear quadratic regulation,
optimization or loop shaping. The challenge for all those
methods is the right definition of the design goals and the
control law structure. In classical flight control, the design
goals as well as the flight control law structure shall be
derived from the flight physics. This design philosophy is
also desired for the multibody aircraft. This article makes a
contribution to an inner-loop control law based on a physi-
cally correct target model that modifies the flight dynamics
of the unconventional multibody aircraft to become similar
to the one of a rigid-body aircraft. It is based on former
research carried out in a PhD project on flight mechanics and
flight control of multi-body aircraft [9]. For this purpose,
hypothetical spring element and damping elements at the
joints are introduced in the equation of motion. This converts
the very flexible aircraft configuration into a nearly rigid-
body aircraft, when a high stiffness is used. The eigenvec-
tors of the theoretical rigid-body aircraft are determined and
later on used in an eigenstructure assignment to calculate the
inner-loop control law for the very flexible aircraft without
any spring and damping elements. With this method, the
classical, flight mechanical rigid-body modes and the for-
mation modes are well separated from each other. Hence,
the outer loop has to control the rigid-body motion of the
aircraft only [10].
2 Reference Aircraft Design
For the aircraft design that is used as reference in this paper,
only the most important key facts are mentioned. The main
requirement is the operation of the multibody aircraft as
HAPS. The following design requirements, derived in part
from the U.S. DARPA1 Vulture program [3], are applied:
– Payload capacity shall be 450kg and the required pay-
load power is 5kW.
– The aircraft shall continuously operate for at least one
year in the mission altitude.
– The design operation latitude is specified at
40◦
N/S.
– The single aircraft shall be able to fly to the mission alti-
tude and leave the formation and return to ground on
their own.
– The single aircraft shall be designed as rigid aircraft.
Those requirements are achieved by an aircraft design
with properties that are listed in Table1. For the design, a
planar wing formation is selected, i.e. a configuration where
all individual aircraft have the same pitch angle. Figure1
shows the design of such an aircraft configuration. The
mechanical joint between the single aircraft allows a pitch
and roll degree of freedom and hence it transmits all reaction
forces and the yaw reaction moment. Because of this and
the non-uniform lift distribution, the inner aircraft of the
formation have to partially carry the weight of the outer air-
craft. This causes reaction forces. Considering the free-body
diagram of the single aircraft, those reaction forces are not
equal in magnitude at the left and right wing tip and, hence,
a rolling moment occurs. This moment is amplified by the
non-uniform lift distribution for the single aircraft. The roll
moment balance can be achieved by flap deflection. On one
wing side, the lift is increased while on the other wing side
the lift is decreased. Using this method, the lift distribution
is influenced, and the additional flap deflections lead to drag.
To overcome such difficulties, the center of gravity is shifted
along the wingspan. With this, the lever arms are influenced,
and an equilibrium of moments is established. Because the
battery of the aircraft is nearly the highest single mass com-
ponent of the aircraft shifting it is used to change the lateral
center of gravity position.
1 United States Defense Advanced Research Projects Agency.
35Applying Eigenstructure Assignment toInner-Loop Flight Control Laws foraMultibody Aircraft
1 3
3 Flight Dynamic Model
The flight dynamics of the multibody aircraft that was
designed for operation as a HAPS are now analyzed. The
following assumptions are made:
1. The multibody aircraft consists of multiple single air-
craft, all being individually rigid aircraft. Aeroelasticity
of the single aircraft is not considered due to the struc-
tural design.
2. The aerodynamic forces are modeled using potential
flow theory (vortex lattice method).
3. The engine is ideal. Energy dissipation due to friction is
considered but impacts of propeller rotational speed and
blade pitch angles are not considered.
4. Thrust force acts in x-direction of the body-fixed axes
system without moment about the center of gravity of
the single aircraft.
5. There is no gap between the aircraft. It is assumed that
there is a seal, which prevents flow from the lower to the
upper surface.
6. The joint connections between two single aircraft are
considered to be ideal, i.e. without natural friction,
damping or spring forces.
3.1 Equations ofMotion
The equations of motion are derived using Kane’s method
[8]. They are formulated as:
where
𝐅r
is the vector of the generalized active forces,
𝐅⋆
r
is the vector of the generalized inertial forces in the refer-
ence frame and p is the number of the generalized speeds.
In Eq.1, the denotation “generalized force” includes iner-
tial and active forces as well as inertial and active moments
(translation and rotation) [8]. The generalized inertial force
is determined with
where
N𝐯CG,j
is the velocity of the center of gravity (CG)
of the jth body in the Newtonian frame,
N𝛚B,j
the angular
velocity of the body frame about the Newtonian frame of the
jth body,
ur
are the generalized speeds, and
𝐅k
and
𝐌k
are
force and moment of the jth body decomposed as
(1)
𝐅r
+
𝐅
⋆
r
=0(r=1, …,p)
,
(2)
𝐅
⋆
r=−
l
∑
j=1
N𝐅CG,j
k
𝜕N𝐯CG,j
𝜕ur
−
l
∑
j=1
N𝐌CG,j
k
𝜕N𝛚B,j
𝜕ur
,
Table 1 Selected parameters for the optimized multibody aircraft with planar wing
Span
[m]
210.66
Aspect ratio [1] 55
Total mass
[
kg
]
4509
Total battery mass
[
kg
]
1137
Altitude
[m]
20,000
Airspeed
[
ms
−1]
33.37
Horizontal tail area
[
m2
]
6.05
Vertical tail area
[
m
2]
1.45
Zero drag coefficient [1] 0.008
Available sun energy per day
[
GJ/day
]
11.12
Required sun energy per day
[
GJ/day
]
11.12
Max. engine power
[kW]
11.51
Long. CG position
[m]
− 3.74
Neutral point wing
[m]
− 3.26
Distance wing tail
[m]
11.49
Half span per aircraft
[m]
10.53
AC1 AC2 AC3 AC4 AC5
Angle of attack
[1
◦
]
4.8 4.8 4.8 4.8 4.8
Elevator deflection
[1◦]
− 3.58 − 6.05 − 6.4 − 6.51 − 6.55
Trim engine power
[kW]
5.29 2.04 1.49 1.29 1.22
Lat. CG position
[m]
1.69 2.12 1.68 1.05 0.36
Battery shift
[m]
6.7 8.39 6.65 4.17 1.42
36 A.Köthe, R.Luckner
1 3
with l representing the number of rigid bodies in the system.
The generalized active force is given by
where
𝐅a
and
𝐌a
are the forces and moments acting at the
center of gravity. The coupling within the equations of
motion is carried by motion constrains. Figure2 shows the
free-body diagram for the selected joint configuration that
allows a pitch and roll motion between the aircraft. At the
joints, the nonholonomic motion constraints
are valid with
N𝐯CAB
as body-fixed velocity of the point CAB
(
N𝐯CBA
of point CBA) in the Newtonian reference frame,
𝐞
as unit vector of the Newtonian reference frame and
N𝜔A
of
aircraft A (
N𝜔B
of aircraft B) as angular velocity in the New-
tonian reference frame. For this configuration, the non-linear
differential equations of motions that describe the dynamic
behavior of the multibody aircraft can be expressed by a
first-order non-linear differential equation system that con-
sists of
– 12 non-linear first-order differential equations for the
rigid-body motion for the first aircraft at which the others
shall couple (velocity, position, rotation rates and Euler
angles) and
– 5 non-linear first-order differential equations for each
coupled aircraft (roll and pitch rate as well as Euler
angles).
For the flight dynamic analysis, the navigation differen-
tial equations (position and yaw angle for the rigid-body
(3)
N
𝐅CG
k=m
(
dB𝐯CG
dt +N𝛚B×B𝐯CG
)and
N
𝐌CG
k
=𝐈N𝜔 B+N𝜔B×
(
𝐈N𝜔B
)
,
(4)
𝐅
r=
l
∑
j
=
1
N𝐅CG,j
a
𝜕N𝐯CG,j
𝜕ur
+
l
∑
j
=
1
N𝐌CG,j
a
𝜕N𝛚B,j
𝜕ur
,
(5)
(
N𝐯CAB −N𝐯CBA
)
𝐞x,g =0,
(
N𝐯CAB −N𝐯CBA
)
𝐞y,g =
0,
(
N𝐯CBA −N𝐯CAB
)
𝐞
z,g
=0,
(
N𝛚A−N𝛚B
)
𝐞
z,g
=0,
motion as well as yaw angle for every coupled aircraft) can
be neglected. This reduces the number of differential equa-
tions to eight for the rigid-body motion and to four for every
coupled aircraft. In the case of the reference aircraft (ten
coupled aircraft), 44 first-order differential equations remain.
As external forces, the aerodynamic forces in a body-fixed
reference system
𝐑A,b
, thr ust
𝐓b
in a body-fixed system and
weight in the geodetic reference systems
𝐖n
have to consid-
ered for every aircraft. The active force at the jth aircraft in
the body-fixed reference frame is then determined as
where
𝐓b,n
is the transformation matrix from the Newtonian/
geodetic reference frame (index n) to the body-fixed refer-
ence frame. Applying the introduced assumptions, the aero-
dynamic moment in the body-fixed system
𝐌A,b
is the only
generalized external moment. Hence, the active moment of
the jth aircraft in the body-fixed reference frame is
The aerodynamic forces and moments are calculated using
the vortex lattice method for every aircraft [7]. The intro-
duced formulations are sufficient to describe the flight
dynamic model of the rigid-body aircraft.
As described in the introduction, the later used eigen-
structure assignment requires a flight dynamic model with
hypothetical spring and damping elements at joints. Com-
pared to flexible aircraft, these elements provide the very
flexible multibody aircraft with structural stiffness and
damping. For every joint connecting an aircraft j with an
aircraft
j+1
, the effect of the spring on the roll motion
(bending) is modeled using the moment
Ms,𝛷
and the effect
on the pitch motion (torsion) using the moment
Ms,𝛩
with
(6)
b
𝐅
CG,j
a
=𝐑
A,b,j
+𝐓
b,j
+𝐓
b,n,j
𝐖
n,j ,
(7)
b
𝐌
CG,j
a
=𝐌
A,b,j .
(8)
M
s,𝛷=k𝛷
(
𝛷j−𝛷j+1
)
Ms,𝛩
=k
𝛩(
𝛩
j
−𝛩
j+1),
Fig. 1 Illustration of the reference aircraft configuration
Aircra A
Aircra B
Mz,ACB
Mz,ACA
Fy,ACA
Fy,ACB
Fx,ACA
Fx,ACB
Fz,ACA
Fz,ACB
CAB
CBA
Fig. 2 Reaction forces and moments for a joint with pitch and roll
degree of freedom between two aircraft
37Applying Eigenstructure Assignment toInner-Loop Flight Control Laws foraMultibody Aircraft
1 3
where
k𝛷
and
k𝛩
are spring constants and
𝛩
and
𝛷
are the
pitch and bank angle. The same procedure is carried out
for the damping moments using the roll rate p and the pitch
rate q. The damping moments for the rolling and pitching
moment are
These moments act in the Newtonian reference frame and
are added to the active moments of Eq.7 with
where
𝐓b,n
is the transformation matrix from the Newto-
nian/geodetic reference frame (index n) to the body-fixed
reference frame. Eq.10 represents a general formulation for
the case that the considered aircraft is coupled with other
aircraft on the left and right wing tip. If there is only a one-
sided coupling, only one damping and spring moment for
pitch and roll motion must be considered.
3.2 Non‑linear Simulation Model andLinearization
The non-linear equations of motion, which include the com-
putation of the aerodynamics with the vortex lattice method,
and the kinematic relations must be solved numerically. This
leads to a system
with
𝐱
as state vector,
𝐮
as input vector,
𝐳
as disturbance vec-
tor and
𝐲
as output vector. The non-linear function
𝐟(𝐱,𝐮,𝐳)
represents the dynamic behavior, while the function
𝐠(𝐱,𝐮,𝐳)
maps state, input and disturbance variables to desired out-
puts. All elements are integrated into a Simulink model. The
reference design of ten aircraft has 44 integrators (neglect-
ing position of the formation and yaw angle of the coupled
aircraft). The fifth aircraft is selected as reference aircraft.
Thus, the state vector comprises the following 44 states:
(9)
M
d,𝛷=d𝛷
(
pi−pi+1
)
Md,𝛩
=d
𝛩(
q
i
−q
i+1).
(10)
b
𝐌
CG,j
a=𝐌A,b,j +…
𝐓
b,n,j
⎡
⎢
⎢
⎣
Md,𝛷,j−1+Ms,𝛷,j−1−Md,𝛷,j−Ms,𝛷,j
Md,𝛩,j−1+Ms,𝛩,j−1−Md,𝛩,j−Ms,𝛩,j
0
⎤
⎥
⎥
⎦
(11)
𝐱=𝐟(𝐱,𝐮,𝐳)
𝐲
=
𝐠
(
𝐱,𝐮,𝐳)
(12)
𝐱
=
�
ukf,AC5,vkf,AC5,wkf,AC5,pkf,AC5,qkf,AC5,
…
rkf,AC5,𝛩AC5,𝛷AC5,𝐱AC1,𝐱AC2,𝐱AC3,…
𝐱AC4,𝐱AC6,𝐱AC7,𝐱AC8,𝐱AC9,𝐱AC10�T
with
𝐱ACi=⎡
⎢
⎢
⎢
⎣
qACi
𝛩ACi
pACi
𝛷ACi
⎤
⎥
⎥
⎥
⎦
T
and i=1, …10 .
with
ukf,AC5
,
vkf,AC5
and
wkf,AC5
as flight path velocities in the
body-fixed reference frame, p as roll rate, q as pitch rate, r as
yaw rate,
𝛷
as bank angle and
𝛩
as pitch angle. In case of the
reference aircraft, the classical flight mechanics states shall
be used, where the three generalized speed components are
replaced by the airspeed
𝐕A
, angle of attack
𝛼
and sideslip
angle
𝛽
. The relations are given in [2]. It follows an output
vector with
The elevator deflection
𝜂
, the left
𝜉left
and right
𝜉right
aileron
deflections, the rudder
𝜁
and the thrust F of every aircraft
are used as input variables. In summary, 50 input variables
are available. The wind is considered as disturbance. It is
assumed that a vertical and horizontal wind component can
act at each aircraft. This leads to 20 disturbance variables.
To investigate the dynamic behavior, the non-linear model
is linearized with Matlab using numerical perturbation. The
state-space equation
follows, representing a system of linear first-order differen-
tial equations with
𝐀
as system matrix,
𝐁
as input matrix,
𝐄
as disturbance matrix,
𝐂
as output matrix,
𝐃
as feedforward
matrix and
𝐅
as feedforward disturbance matrix [11].
4 Flight Dynamic Analysis
The differences in the flight dynamics are now investigated
for the uncontrolled (passive) multibody aircraft with (i)
joints that have hypothetical pitch and roll spring elements
and (ii) joints with free motion in the final configuration.
For the first case, the relation between the spring constant
k𝛩
and the spring constant
k𝛷
is used with
which is similar to the relation between shear modulus G and
Young’s modulus E for isotropic materials [14]. To establish
a rigid-body aircraft configuration, a very high roll stiffness
of
k𝛷
=
200 GNm rad−1
is used.
(13)
𝐲
=
�
qAC5,𝛼AC5,VAC5,𝛩AC5,rAC5,𝛽AC5,pAC5,
…
𝛷AC5,𝐲AC1,𝐲AC2,𝐲AC3,𝐲AC4,…
𝐲AC6,𝐲AC7,𝐲AC8,𝐲AC9,𝐲AC10�T
with
𝐲ACi=⎡
⎢
⎢
⎢
⎣
qACi
𝛩ACi
pACi
𝛷ACi
⎤
⎥
⎥
⎥
⎦
T
and i=1, …10 .
(14)
𝐱
(
t
)=
𝐀𝐱
(
t
)+
𝐁𝐮
(
t
)+
𝐄𝐳
(
t
)
,
𝐲(t)=𝐂𝐱(t)+𝐃𝐮
(t)+𝐅𝐳
(t)
(15)
k
𝛩=
1
2.6
k𝛷
,
38 A.Köthe, R.Luckner
1 3
4.1 Artificial Model forMultibody Aircraft Dynamic
The resulting eigenvalues of the linearized state-space
system with high stiffness are shown in Fig.3. The low-
frequency eigenvalues belong to the rigid-body flight
dynamics and the other ones to the formation modes. The
mode identification is carried out with eigenvectors. In
the case of the rigid-body modes, the additional coupling
degrees of freedom (pitch angle, bank angle, pitch rate
and roll rate) have the same phase and magnitude like the
reference aircraft. The high-frequency formation shows
different phase angles and magnitudes in the entries of
the eigenvector. The resulting form of the multibody air-
craft formation corresponds to a flexible aircraft structure.
Fig.4 shows the mode shapes of the formation modes with
the lowest frequency. Due to the large frequency difference
between rigid-body modes and formation modes, a clear
separation is possible.
4.2 Multibody Aircraft Dynamic (without spring
elements)
Figure5 shows the eigenvalues of a linearized system with
no spring (second case) and, in addition, the identified rigid-
body modes of the reference case (
k𝛷
=
200 GNm rad−1
).
The rigid-body modes are identified with the help of the
eigenvectors. The pitch mode, phugoid and spiral eigenval-
ues can be detected, while an identification of the roll mode
and the Dutch roll is not unequivocal possible. Eigenvalues
of formation modes and rigid-body modes are close together.
In contrast to the reference case, the system with no spring
elements has eight complex conjugate eigenvalues (four
modes) on the right-hand side. The interference between
rigid-body modes and formation modes also becomes clear
in simulation studies. Using the same inputs that lead to
a roll maneuver, the bank angle response is illustrated in
Fig.6a for a formation with multibody dynamics with hypo-
thetical springs (target model) and in Fig.6b without spring.
While in the case with spring elements all bank angles have
the same magnitude, differences occur in the case without
spring and a roll maneuver seems to be impossible.
5 Inner‑Loop Flight Control Law Design
The flight dynamic investigation showed that the very
flexible multibody aircraft has some unstable poles and
an interference between formation modes and rigid-
body modes occurs. This behavior was not observed in
the case with spring elements and high stiffness. In the
control law design, the mechanical stiffness (as well as
damping) between the aircraft is established by an inner
loop designed with eigenstructure assignment. The desired
eigenvalues result from the target model with hypothetical
spring elements and damper. Before applying the actual
design method, another issue has to be addressed. The
number of available inputs is higher than the number of
outputs. Such a system is called over-actuated [4]. As the
eigenstructure assignment can only deal with the same
number of inputs and entries to be modified in the eigen-
vector, the control allocation method has to be applied.
Fig. 3 Eigenvalues in the com-
plex plane for a roll stiffness of
k𝛷
=200 GNm rad−
1
(Kinds
of motion: RM roll mode, PM
pitch motion, RYM roll-yaw
motion, SP spiral mode, PH
phugoid)
39Applying Eigenstructure Assignment toInner-Loop Flight Control Laws foraMultibody Aircraft
1 3
5.1 Control Allocation
In Sect.3 it was explained that a formation of ten air-
craft with joints that do not transmit rolling and pitch-
ing moments has 24 degrees of freedom (3 translational
degrees of freedom and 21 rotational degrees of freedom).
Every translational degree of freedom is affected by a
force while a rotational movement is caused by a moment.
Thrust as well as aerodynamic surfaces lead to forces and
moments. In total, the multibody aircraft has 50 inputs (cf.
Sect.3.2). That means that there are more inputs available
than required to influence the degrees of freedom. Such
over-actuated systems are handled using control allocation.
The main idea of aircraft control allocation is as follows.
The control design is not carried out by directly using
the aerodynamic surfaces or thrust. Rather, inputs of the
aircraft are expressed (indirectly) by moments and forces
or their equivalent accelerations and rotational accelera-
tions acting on the aircraft. Those inputs are referred to
as virtual inputs
𝐯∈ℝn
. The inputs of the aerodynamic
surfaces or thrust are denoted as
𝐮∈ℝm
with m as num-
ber of real inputs. To establish a relation between the two
types of inputs, a mapping is applied:
𝐁a
transfers the real
inputs to the virtual ones by [4]
The control law design is carried out using the virtual inputs.
In case of the multibody aircraft, the derivatives of the
generalized speeds (acceleration and rotational accelera-
tion) are used as virtual inputs because they are equivalent
to forces and moments. The derivative of the state vector
(cf. Eq.12) contains those 24 derivatives of the general-
ized speeds. Considering Eq.14, the mapping matrix
𝐁a
(16)
𝐯
=
𝐁a𝐮with 𝐁a
∈ℝ
n×m.
(a) (b)
(c) (d)
Fig. 4 Selected eigenvectors of the formation modes for every kind of
motion and a for a roll stiffness of
k𝛷
=200 GNm rad−
1
Fig. 5 Eigenvalues for the joint without spring elements and rigid-
body eigenvalues of the reference case with a spring stiffness of
k𝛷
=200 GNm rad−
1
(a)
(b)
Fig. 6 Non-linear step response for a roll maneuver
40 A.Köthe, R.Luckner
1 3
is part of the matrix
𝐁
. Using this allocation, 24 virtual
inputs are available for 24 degrees of freedom.
The control law design is carried out using the virtual
inputs. By applying the transformation of Eq.16 to the
state-space system of Eq.14 (without disturbances),
follows as new state-space system for the eigenstructure
assignment. After the control law design, the inverse matrix
mapping the virtual inputs to the real inputs has to be cal-
culated by
Control allocation is thus solving Eq.16 for
𝐮
[4]. Because
m>n
, the inverse of
𝐁a
does not exist and hence the solu-
tion of
𝐏
is not trivial. There are two types of methods to
solve the problem: on-line and off-line solutions. So-called
on-line solutions are calculating the allocation from the vir-
tual inputs to the real control inputs in real time, while off-
line solutions are pre-computed. For the multibody aircraft,
the off-line solution is used and described within Sect.5.4.
The block diagram for the control law is shown in Fig.7.
5.2 Eigenstructure Assignment fortheInner‑Loop
Control Law
The application of the eigenstructure assignment requires
controllability [11]. This condition is fulfilled for the sys-
tem. The desired eigenvalues are taken from the model
with a roll stiffness of
k𝛷=200 GNm rad−1
. To establish
damping, a value of
12.8 10
6kg m
2
s
is used for both the pitch
d𝛩
and roll damping
d𝛷
coefficient. This leads to nearly
rigid-body aircraft with well separated and well damped
formation modes. The eigenvalues still have to be modi-
fied. Due to the high stiffness, the resulting frequencies of
the formation modes are very high. It is known from classi-
cal flight control theory that a high desired frequency leads
to high gains caused by high aerodynamic surface deflec-
tions or thrust [2]. Hence, the frequencies of the formation
modes have to be reduced. Since every eigenvalue belongs
to a certain eigenvector, the frequency reduction cannot
be conducted in an arbitrary way. Therefore, the real and
imaginary parts of the eigenvalues are changed using the
same relation. The flight mechanics modes (eigenvalues
and eigenvectors) should be maintained with the excep-
tion of the unstable spiral mode. This eigenvalue is shifted
to
𝜆SP =−0.01
. Hence, the roll mode with
𝜆RM =−1.57
is the eigenvalue with the highest magnitude. The for-
mation mode with the lowest frequency should have an
undamped frequency (magnitude of the eigenvalue) that
is five times higher than the roll mode. This leads to a
(17)
𝐱
=𝐀𝐱+
𝐁𝐯
𝐲=𝐂𝐱+
𝐃𝐯
(18)
𝐮=𝐏𝐯.
desired frequency for the first formation mode (FOM) with
𝜔
0,1st FOM =8
rad
s
. The differences in the natural frequencies
of the formation modes are selected using
𝛥𝜔
0,FOM =1
rad
s
.
Thus, the highest natural frequency of the last formation
mode is
𝜔
0,16th FOM =24
rad
s
. Using this selection, a sepa-
ration of formation modes and flight mechanics modes is
achieved and the desired eigenvectors and eigenvalues for
the eigenstructure assignment are selected.
The origin of the eigenstructure assignment is the
eigenvalue equation
with
𝐀
as dynamic matrix of the state-space system,
𝜆i
as
eigenvalue and
𝐗i
as corresponding eigenvector. According
to Fig.7, the control law for the state feedback is defined
with
Linking Eqs.19, 20 and the state-space differential equation
of Eq.17 leads to
The product of controller and eigenvector is substituted by
and inserted into Eq.21. After a rearrangement, the result is
transformed into matrix notation with
The eigenvalue
𝐗i
has 44 elements, but only 24 control
inputs are available. This is not an issue, as no input can
directly influence the entries of the Euler angles in the eigen-
vector. Hence, a reduced eigenvector
𝐗
is used that contains
only the pitch and roll rates of every aircraft (20 elements)
as well as the yaw rate, airspeed, angle of attack and side-
slip angle of the formation. In sum, the reduced eigenvec-
tor has 24 elements, which is equivalent to the number of
(19)
(
𝐀−𝜆
i
𝐈
)
𝐗
i
=
𝟎
(20)
𝐯=−𝐊𝐱.
(21)
(
𝐀−𝜆
i
𝐈
)
𝐗
i
=
𝐁𝐊𝐗
i.
(22)
𝐫i=𝐊𝐗
i
(23)
[
𝐀−𝜆i𝐈
𝐁
][
𝐗i
−𝐫
i]
=𝟎
.
Mul-Body
Aircra
Dynamics
x = Ax + Bu
Feedback
Controller
K
Control
Allocaon
Matrix
P
x
vu
Fig. 7 Control law structure for the design of the inner loops with vir-
tual inputs
41Applying Eigenstructure Assignment toInner-Loop Flight Control Laws foraMultibody Aircraft
1 3
virtual inputs. A mapping matrix
𝐌
is used to express the
full eigenvector as reduced eigenvector by
This relation is inserted into Eq.23, which yields
Eq.25 is now solved for
𝐛i
with
This procedure can be applied to every eigenvalue and the
corresponding eigenvalue. Every solution
𝐛i
comprises the
vectors
𝐗i
and
𝐫i
. The use of Eq.22 and all
n=44
solutions
for
𝐛i
leads to
that is used to compute the gains of the control law with
5.3 Design Results
After selecting the desired eigenvalues and eigenvectors,
the methodology of eigenstructure assignment is applied to
the linearized state-space system of the multibody aircraft.
Eigenvalues of the modified flight dynamics are shown in
Fig.8. The eigenvalues meet the desired values and the sys-
tem is nominally stable and the flight mechanics modes and
the formation modes are separated from each other.
5.4 Solving theControl Allocation Problem
So far, the feedback controller was designed using virtual
inputs. These virtual inputs have to be transferred to the real
inputs using Eq.18. A frequently used solution is the Moore-
Pensorse pseudo-inverse [4]. This pseudo-inverse reduces
the 2-norm of the control vector
‖𝐮‖2
. It arises from
as a solution of the control allocation with
𝐏
as pseudo-
inverse [4]. Instead of minimizing
𝐮T𝐮
, a weighting matrix
can be used to take different control efforts into account. The
use of a diagonal matrix with
(24)
𝐗=𝐌𝐗 with 𝐌∈ℝ24×44 .
(25)
[
𝐀−𝜆i𝐈
𝐁
𝐌𝟎
][
𝐗i
−𝐫i
]
���
𝐛i
=
[
𝟎
𝐗i
].
(26)
𝐛
i=
[
𝐀−𝜆i𝐈
𝐁
𝐌𝟎
]
−
1[
𝟎
𝐗
i].
(27)
[
𝐫1𝐫2…𝐫n
]
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
𝐑
=𝐊
[
𝐗1𝐗2…𝐗n
]
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
𝐗
(28)
𝐊=𝐑𝐗
−1.
(29)
𝐮
=𝐁T
a
[
𝐁a𝐁T
a
]−1
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
𝐏
𝐯
reduces the
𝐮T𝐖T𝐮𝐖
. For the aerodynamic surfaces, a
maximum value of
30◦
and for the engine power of
11.51 kW
is used. The solution of the control allocation problem is
now given with
Based on the introduced virtual inputs, the subsequent outer
loops shall command a pitch and roll rate derivative for the
complete formation to influence the rigid-body dynamics.
This can be established using a generalized pitch rate deriva-
tive
qgen, input
for all pitch rate derivatives in the virtual con-
trol input
𝐯
. This is expressed by
with
qkf, ACi, input
as virtual pitch rate derivative input in the
control allocation matrix and
qkf, ACi, input, IL
as pitch rate
derivative of the inner loops. The same approach is used
for the bank angle control law. Now, a generalized roll rate
derivative
pgen, input
is used as a common input for all virtual
inputs of the roll rate derivatives with
The control allocation problem is solved accordingly
and the combination of control law
𝐊
and control alloca-
tion matrix
𝐏
is tested in non-linear simulations. Figure9
shows a non-linear pitch angle response for a step input
in the generalized pitch rate derivative of
0.1
◦
s−2
. Fig-
ure10 shows the bank angle response for a step input in
the generalized roll rate derivative of
0.1
◦
s−2
. In contrast
to the open-loop results for the multibody aircraft dynam-
ics without spring elements (cf. Fig.6), the pitch and bank
angles for all aircraft are equal. This shows that the inner
loop successfully separates the formation modes from the
rigid body modes. The missing stiffness at the joints is
mimicked by the control law.
6 Conclusion andOutlook
This paper shows an approach for the inner-loop control laws
of a multibody aircraft based on an artificial but physically
exact target model. A multibody aircraft is a very flexible
aircraft configuration that strongly differs from conventional
aircraft. The flight dynamics show a strong interference
(30)
𝐖
=diag
(
1
|umax|)
with 𝐖∈ℝn×
n
(31)
𝐮
=𝐖−1𝐁T
a
[
𝐁a𝐖−1𝐁T
a
]−1
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
𝐏
𝐯
.
(32)
q
kf, ACi, input
=q
gen, input
−…
q
kf, ACi, input, IL
∀i∈[1, 10]
.
(33)
p
kf, ACi, input
=p
gen, input
−…
p
kf, ACi, input, IL
∀i∈[1, 10]
.
42 A.Köthe, R.Luckner
1 3
between classical, flight dynamic rigid-body modes and
formation modes that are caused by the joint connection
between the aircraft. To separate the modes, a suitable con-
trol method is required. This opens the question regarding
the right position of the desired eigenvalues and eigenvec-
tors. Using hypothetical spring and damping elements at the
joint transforms the highly flexible aircraft into a system
similar to a rigid-body aircraft. The resulting eigenvector
and scaled eigenvalues stem from an artificial, but physically
correctly motivated model and are successfully applied to
the design of the inner loops using eigenstructure assign-
ment. A clear definition of the design goals becomes possi-
ble, providing an advantage in comparison to other methods
like the loop shaping or the linear quadratic regulator.
In further investigations, the proposed method can be
used for aircraft with aeroelastic deformations. Using modal
transformation, a physically exact target model with high
stiffness can be used to define the design goals for inner
loops. So far, the non-linear effects of the plant were not
considered. Using an eigenstructure assignment with uncer-
tainties could increase the robustness of the inner loops.
Funding Open Access funding enabled and organized by Projekt
DEAL.
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included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
(a)
(b)
Fig. 8 Eigenvalues of the multibody aircraft’s flight dynamics after
applying eigenstructure assignment
Fig. 9 Non-linear response of all pitch angles for a step input in the
generalized pitch rate derivative of
0.1
◦
s
−
2
Fig. 10 Non-linear response of all bank angles for a step input in the
generalized roll rate derivative of
0.1
◦
s
−
2
43Applying Eigenstructure Assignment toInner-Loop Flight Control Laws foraMultibody Aircraft
1 3
the article's Creative Commons licence and your intended use is not
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