AIAA Atmospheric Flight Mechanics Conference and Exhibit AIAA 2008-6701

18 - 21 August 2008, Honolulu, Hawaii

MODELLING, SIMULATION AND VALIDATION OF DIFFERENT CONTROL STRATEGIES

FOR A LAUNCH VEHICLE WITH A NON NEGLIGIBLE ROLL RATE
G. Baldesi1,2,3, S. Monaco1, C. Berard2, P. D. Resta3
1
Dip. di Informatica e Sistemistica, Univ. “La Sapienza”, Rome Italy
2
ISAE-ONERA, System Control and Flight Dynamics, Toulouse, France
3
European Space Agency

This paper is mainly focused on a comparison between different autopilots for launch
vehicles having a non negligible roll rate.
After a brief introduction on the important of mathematical model’ s selection, two
main design models for a launch vehicles have been introduced. Each of them will be
used by a dedicated control strategies. The first one is based on the nonlinear feedback
linearization control and the second one on the robust eigenstructure assignment
using multi-model approach. These techniques were selected since they allow to enable
coupling effects due to the non negligible vehicle roll rate and also the nonlinear
characteristics of the systems (variable mass, … ).
The performances of each different autopilot will be evaluated using a complex flight
dynamic simulator, which was built at ESA in the frame of support activities for
VEGA launch vehicle. This simulator is based on the multi-body dynamic simulation
software DCAP (Dynamic and Control Analysis Package) and allows full three
dimensional, trajectory simulation with the launch vehicle structural characteristics
modelled with time varying properties during the flight evolution.

1 – Nomenclature
Symbol Unit Definition
LV - Launch Vehicle
LILA3D - LInear LAUncher 3D simulator
NOLILA3D - NOLInear LAUncher 3D simulator
SB - Body reference frame
SI - Inertial reference frame
0 - Generic steady-state value
[ψ, θ, φ] [rad] “Yaw-Pitch-Roll” angles
BI
T - transformation matrix from SI to SB
ω
[rad/s] body frame angular velocity in SB
(P, Q, R)
(p, q, r) [rad/s] body frame angular velocity “disturbance” terms in SB
(U, V, W) [m/s] mass element velocity in SB
(u, v, w) [m/s] mass element velocity “disturbance” terms in SB
I
(Ixx, Iyy, Izz, [kg m2] inertia dyadic
Ixy, Iyz, Ixz)
Mi [kg] mass of the mass element i
φi (x) [m] normalized mode shape of the ith mode

ξ i (t ) [-] generalized coordinate due to elasticity for the ith mode on zB-axis

κ i (t ) [-] generalized coordinate due to elasticity for the ith mode on yB-axis

δ [rad] nozzle total deflection around yB-axis

ε [rad] nozzle total deflection around zB-axis
ζi [-] relative damping coefficient of the ith elastic mode
ωi [rad/s] natural frequency of the ith elastic mode
NE - degrees of freedom required to satisfactorily model the structural dynamics

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 g [m/s2] gravity acceleration
ηCG [m] LV centre of gravity location on the LV symmetric axis
Ts [N] nonswivelled thrust
Tc [N] swivelled thrust
lC [m] distance between pivot point and the LV centre of mass
α [rad] angle of attack
β [rad] side-slip angle
Ρ [kg/m3] atmospheric density
2
SR [m ] reference surface used for the aerodynamic coefficients
lF [m] distance between the centre of aerodynamic pressure and the LV centre of mass

2 – Introduction
As a standard practice, the control design for a launch vehicle is based on decoupled dynamic models. Several Thrust
Vector Control (TVC) law designs, which assume that an action on the pitch plane produces no effect on the yaw plane,
can be easily found in the literature [Ref. 1]. Coupling effects [Ref. 2] in the dynamics can be due to the external forces,
but also to the state variables for the velocity linear components and the angular rate components. As Roux and
Cruciani [Ref. 3] have presented, there can be cases where a non negligible roll rate can be foreseen. Therefore in order
to keep valid the assumption of two uncoupled axes, a dedicated roll control system is added to the initial control design
to reduce the roll rate. This additional subsystem has an obvious important effect on the mass performance of the whole
launch vehicle.
After a brief introduction on the important of mathematical model’ s selection, two main design models for a launch
vehicles have been introduced in the section 3. Moreover, a detailed launcher flight simulator, which is current used at
ESA in the frame of support activities for VEGA launch vehicle, is briefly presented. In the section 4, two different
control strategies are presented. The first one is based on the nonlinear feedback linearization control and the second
one on the robust eigenstructure assignment using multi-model approach. These techniques were selected since they
allow to enable coupling effects due to the non negligible vehicle roll rate and also the nonlinear characteristics of the
systems (variable mass, … ). In the section 5, the different performances achieved by the two control strategies are
reported in different off-nominal conditions. In order to exploit the results of this study, the main advantages and
drawbacks of the two “no classic” control techniques - namely robust modal control (RMC) and nonlinear feedback
linearization control (NFLC) - are hereby summarized in section 6.

3 – Launch Vehicle Modelling

As it was already highlighted in the previous paragraph, the aerospace applications are often investigated by means of
mathematical models, particularly during a design phase. Therefore there is the need to represent the real system with
different dedicated models/simulators depending on the particular analysis, which is required. With this purpose, two
different categories can be identified:

™ Models used in the design phase

In design phase, it is not important to model the whole system in details. Several assumptions are introduced at
this stage, in order to have a model as simple as possible, which is able to reproduce the main dynamic
behaviour of the real system as already presented in the Ref. 5. The most difficult task, which any designer
have to face, is to identify which system behaviour should be taken into account and which can be neglected.
In fact, based on the particular analysis or operation condition, different hypotheses might be introduced during
the formulation of the mathematic models. Therefore the design should identify which specific mathematic
model should be used for his purposes. Once this model is selected, all the secondary terms, which have been
neglected, have to be investigated and evaluated by dedicated analysis. Moreover these effects will be taken
into account during the validation phase, where a more complex simulator is used.

™ Simulator used in the validation phase

Once a control law has been designed based on a simplified model, it should be verified on the real system.
Generally, it is very expensive, and therefore impossible, to perform the validation loop on the real hardware.
Therefore a new more complex model, simulator, where all the single subsystem is reproduced as close as
 possible to the real one, has to be built. In other words, the main goal is to reduce to zero all the simplifications
introduced in the design models in order to estimate the real system performance. A simulator, therefore, is
quite demanding in terms of computer power, required to model the “real” launch vehicle behaviour.

This study is mainly focused on VEGA launch vehicle, which is currently being developed within a European Program
organised under the aegis of ESA. The launcher’s prime contractor role has been assigned to ELV S.p.A. a joint
company of Avio and the Italian Space Agency (ASI). The VEGA launch vehicle is basically a four-stage vehicle.
Three solid propellant stages and a bi-propellant (UDMH/NTO) upper fourth stage, the Attitude and Vernier Upper
Module (AVUM). The total launch vehicle length is 30 meters for an external diameter of 3 meters, and a lift-off mass
of 135 tons. VEGA is tailored to carry small scientific spacecraft and other lighter-weight payloads, targeted on a
payload lift capability of 1500kg at 700km polar orbit.

Figure 3- 1 Expanded view of VEGA. Launch Vehicle

3.1 Models used in the design phase

A “design” model has to be able to reproduce only the main characteristics of the systems for the particular flight
condition, which is investigated. Any minor dynamics, which is neglected at this stage, will be taken into account in the
launcher simulator during the validation phase. In general, different mathematical models are built, since it is not
always straightforward to evaluate the influence of each single dynamic effect on the whole system behaviour a priori.
In this work, different linear 3D dynamic models (LILA3D) and nonlinear one (NOLILA3D) were built from the
equations of motion derived on the Ref. 5, in order to have a wide overview of the possible design models for a launch
vehicle.

3.1.1 Non-Linear Simulator (NOLILA3D)

When important manoeuvres have to be modelled, it is important to use a more generic simulator, which allows to
represent the complex dynamics of a body in the space not only in the neighbourhood of a trim flight condition. In this
case, important manoeuvres can be required. Therefore the non-linear behaviour of both the launcher dynamics and the
aerodynamics forces are playing a significant role and they can not be neglected anymore.

0 Vrel
0
α
XB YB
XB
- dFA(η)
α dFAero(η) dη
dη
Vrel
c lC
c
Ts
ε θ0 mg
φ ' (η )
mg φ (ηT ) η
Tc
ηT
Tc σ
ZB η

Figure 3- 2 Sketch of an “elastic” launch vehicle in Pitch Plane Figure 3- 3 Sketch of a “rigid” launch vehicle in Yaw
Plane
Equations of translation:
 
m(U + QW − RV ) = −mg
XI
+ (TC + TS ) + ...
2 2 2
 X I + YI + Z I

 1  L
∂C A (η ) 
... − ρV rel S R C A0 + ∫ α ' dη 
2
 2 ∂α
  0 
 
m(V + RU − PW ) = TC ε + (TC + TS )∑ κ i (t ) ⋅ φ ' i (η T ) + ...
NE

 i
 Eq. 3-1
1
L
∂C N (η ) YI
 ... − ρV rel S R ∫
2
β (η )dη − mg
 2 0
∂α 2 2
X I + YI + Z I
2

 NE
m(W + PV − QU ) = −TC δ + (TC + TS )∑ ξ i (t ) ⋅ φ ' i (η T ) + ...
 i
 1
L
∂C N (η ) ZI
 ... − ρV rel S R ∫
2
α (η )dη − mg
 2 ∂α 2 2
X I + YI + Z I
2
 0

Equations of rotation:

  (
 I xx P + I zz − I yy )QR = M x
  NE
 NE
 I yy Q + (I xx − I zz )PR = l C  − TC δ + (TC + TS )∑ ξ i (t ) ⋅ φ ' i (η T )  − (TC + TS )∑ ξ i (t ) ⋅ φ i (η T ) +...
  
 i  i

 1
L
∂C N (η )
 ... + ρV rel S R ∫
2
(η CG − η )α (η )dη Eq. 3-2
 2 0
∂ α
  NE
 NE
 I zz R + (I yy − I xx )PQ = −l C  TC ε + (TC + TS )∑ κ i (t ) ⋅ φ ' i (η T )  − (TC + TS )∑ κ i (t ) ⋅ φ i (η T ) + ...
  
 i  i

 1
L
∂C N (η )
 ... − ρV rel S R ∫
2
(η CG − η )β (η )dη
 2 0
∂α
where the angle of attack (α) and the side-slip angle (β) can be expressed as follows neglecting any aeroelastic
terms:
w + wwind η CG − η
α (η ) = − q
Uo Uo
Eq. 3-3
v + v wind η CG − η
β (η ) = + r
Uo Uo

Trajectory equations:
X I  U 
d   IB 
YI = T V  Eq. 3-4
dt  
 Z I  W 

Euler rates of the frame


ϕ = p + sin ϕ tan ϑ ⋅ q + cos ϕ tan ϑ ⋅ r

ϑ = cos ϕ ⋅ q − sin ϕ ⋅ r Eq. 3-5
 sin ϕ cos ϕ
ψ = ⋅q + ⋅r
 cos ϑ cos ϑ

In addition, the following NE elastic equations should be considered as well, where the aeroelastic
contributions and roll rate effect have been neglected:
  
(κ )∫ m( x)φ
NE

i + 2ς i ω i κ i + ω i2 κ i i
2
( x)dx = − TC ∆ε + (TC + TS )∑ κ i (t ) ⋅ φ ' i (η T ) φ i (η T )
L Eq. 3-6
 i 
for i =1,.., NE

 
(ξ + 2ς ω ξ )∫ m( x)φ
NE

i i i i + ω i2 ξ i i
2
( x)dx =  TC ∆δ − (TC + TS )∑ ξ i (t ) ⋅ φ ' i (η T ) φ i (η T )
L Eq. 3-7
 i 
for i =1,.., NE

The main differences in the dynamic response between a rigid versus a flexible NONLILA3D model are shown in the
following figure.

-3 Nozzle deflection (δ) [rad]

x 10
3

-1

-2

NOLILA3D flex
-3 NOLILA3D rigid

-4
0 2 4 6 8 10 12 14 16 18
Time [s]

Figure 3- 4 NOLILA3D results comparison between rigid vs flexible model

3.1.2 LILA3D (LTI Model)

A launch vehicle dynamic model is generally based on the short-period equations of motion for the atmospheric flight.
Indeed, in this particular flight phase, the main constraint for a launcher is to minimize the angle of attack, which
generates a lift force acting on the lateral direction of vehicle. Therefore no important manoeuvres are commanded and
the launcher dynamics can be described using linear time invariant (LTI) models.
When the launch vehicle has a not negligible roll rate, an action on the pitch plane produces an effect also in the yaw
plane. Although these equations of motion are still linear, there is a coupling effect between the two axes.

The translation equations are:

 NE

 m 
v = m ( − rU 0 + P 0 w ) + T C ∆ ε + (T C + T )
S ∑ κ i (t ) ⋅ φ ' i (η T ) + ...
 i

 1
L
∂C N (η )
... − ρV rel S R ∫ ∆β (η )dη + mg cos ϑ 0 ⋅ ∆ϑ
2

 2 0
∂α
 NE
Eq. 3-8
mw = m(− P v + U w ) − T ∆δ + (T + T ) ξ (t ) ⋅ φ ' (η ) + ...
S ∑ i
 0 0 C C
i
i T


 1
L
∂C N (η )
ρ R∫ ∆α (η )dη − mg sin ϑ 0 ⋅ ∆ϑ
2
 ... − V S
∂α
rel
 2 0
The rotation equations are:
  NE

 I yy q = −(I xx − I zz )P0r + lC  − TC ∆δ + (TC + TS )∑ ξi (t ) ⋅ φ 'i (ηT )  + ...
  i 
 NE
1
L
∂ C (η )
 ... − (TC + TS )∑ ξi (t ) ⋅ φi (ηT ) + ρVrel S R ∫ N (ηCG − η )∆α (η )dη
2

 i 2 0
∂α
 Eq. 3-9
 I r = −(I − I )P q − l  T ∆ε + (T + T ) E κ (t ) ⋅ φ ' (η )  + ...
N

 zz yy xx 0 C C C S ∑ i i T 

  i 
 NE
1
L
∂C (η )
... − (TC + TS )∑ κ i (t ) ⋅ φi (ηT ) − ρVrel S R ∫ N (ηCG − η )∆β (η )dη
2

 i 2 0
∂α
where the angle of attack (∆α) and the side-slip angle (∆β) can be expressed as follows:
w + wwind ηCG − η
∆α (η ) = − q
Uo Uo
Eq. 3-10
v + vwind ηCG − η
∆β (η ) = + r
Uo Uo
and the relative cinematic equations for the vehicle attitude:
∆ϑ = q

 1 Eq. 3-11
∆ψ = cosϑ ⋅ r
 0

The longitudinal axis dynamics is not taken into account since we are mainly focusing on the short-period dynamics. In
addition, only the propulsion force is taken into account in the structural dynamics as it is shown in the following
structural dynamic equations, one for the yaw plane and the other for pitch plane:
T φ (η ) (T + TS )φ i (η T ) N '
κi + 2ζ i ω i κ i + ω i2 κ i = − C i T ∆ε R − C
M Gi M Gi
∑j =1
φ j (η T )ξ j
Eq. 3-12
for i =1,.., 3
T φ (η ) (T + TS )φ i (η T ) N '
ξi + 2ζ i ω i ξ + ω i2ξ i = C i T ∆σ R − C
M Gi M Gi
∑j =1
φ j (η T )ξ j
Eq. 3- 13
for i =1,.., 3
The actuation dynamics is modelled as the sum of a rigid motion, which is modelled as 2nd order system:
ωa 2
∆δ TR ( s ) = ∆δ TC Eq. 3-14
s 2 + 2ζω a s + ω a
2

The commanded nozzle deflections (∆δC, ∆εC) and the wind velocity (vwind, wwind) represent the system inputs:
u = [ ∆δ C wwind ]
T
∆ε C vwind Eq. 3-15
The states, which are more in order to simulate the additional dynamics, are the lateral launcher velocities (w, v), yaw
and pitch rate (q, r), the yaw and pitch angle (∆ψ, ∆θ), the nozzle deflection (∆δR, ∆εR), the realized rigid and elastic
nozzle rate ( ∆δTR , δTE ) and the elastic modal coordinates and their velocity.
T
x =  v r ∆ψ w q ∆ θ ∆δ R ∆ δR ∆ε R ∆εR κ1 ... κ3 ξ1 ... ξ3  Eq. 3-16

In order to investigate the effect of the roll rate on the system dynamics, different models were generated for the
different roll rate (0, 0.1, 0.5, 1, 10, 20, 50) deg/s, and the different poles are reported in the Figure 3- 6. Although the
real part of the low frequency poles doesn’t vary, the main effect is on the imaginary one. In other words, for a higher
roll rate, the rigid dynamics is characterized by a pair of poles with a higher imaginary part. Based on the LILA3Ds
equations of motion, the main influence of the launcher roll rate is on the rigid vehicle dynamics, as shown in Figure 3-
6. Neither the actuation dynamics or the elastic modes, except for high roll rate as mentioned in [Ref. 6], are influenced
by the launcher spinning velocity.
 System Poles
100

80

60

40

Imag. Part [rad/s]

20

-20

-40

-60

-80

-100
-60 -50 -40 -30 -20 -10 0 10
Real Part [rad/s]

Figure 3- 5 System open-loop poles vs roll rate (P0) Figure 3- 6 LILA3D flexible poles for different roll rate
values

3.2 Simulators used in the validation phase

During the validation phase, a quite detailed launcher simulator is needed, since each subsystem should be modelled as
close as possible to his real behaviour. The main goal is to reduce to zero all the simplifications, which were introduced
during the design phase.
In order to study the performance of generic controlled dynamic systems, it is essential to have a dedicated tool, which
allows the user to model, in a short time, the complex behaviour of the dynamic systems and their interactions with the
control. In fact, some systems require a model with more than one body in order to take into account relative dynamics
due to different motions of system component such as fluids in the tank or engine movement. This task is pretty
complex and requires to dedicate quite some time to derive the equations of motion and to validate the dynamic
behaviour of the system. To fulfil this task different approaches, which are mainly based on Euler-Newton or Lagrange
dynamics formulations, are well described in literature. Hereby, a new generation of launcher flight simulator, which is
based on multibody software, is presented.

3.2.1 Multibody software

A lot of research has gone into the development and the improvement of multibody software, with the aim of reducing
the time of modelling a system and the computation time required to run an analysis [Ref. 7]. Multibody software
involves the derivation of the equations of motion for multibody systems, which are systems characterized by several
bodies connected by hinges that allow relative motion across them. Robots, launchers and spacecrafts including
articulated appendages such as solar arrays are typical examples for such systems [Ref. 8]. In particular, for a flight
simulator the task is quite cumbersome since it requires to model the environment of the planet and the complex
guidance, navigation and control system as well [Ref. 9].
Early approaches of the dynamics formulation for multibody systems lead to the equations of motion, for open-loop tree
topologies, of the form:

M q = F Eq. 3-17

where, n is the number of degree of freedom, M is an (n x n) mass matrix, q = [q1 q2 … qn]T is an (n x 1) column matrix
representing the generalized coordinates and F is the column matrix containing the contributions from centrifugal,
Coriolis and external forces.
For a numerical simulation of such a system, the mass matrix must be inverted. Since the inversion of an (n x n) matrix
involves operations proportional to the cubic power of n, this is called an Order(n3) approach. As the number of degrees
of freedom increases, this matrix inversion for every integration step, becomes computationally expensive. Thus,
researchers have sought methodologies to circumvent the mass matrix inversion and to improve computational
efficiency. The research into improvements in formulations that increase computational speed resulted in - what are
today called - Order(n) algorithms. The reason for this nomenclature is that the computational burden in these schemes
increases only linearly with n. More details have already been presented in [Ref. 10].
3.2.2 VEGA-DCAP-sim modelling
In the Structures section (TEC-MCS) at Estec it was decided to use DCAP to develop a launcher flight simulator, which
can allow to model different configuration of launch vehicle with minimal workload taking into account the control
interactions [Ref. 11]. In fact, DCAP provides the user with an outstanding capability to model, simulate and analyze a
complex multi-body system made up of coupled rigid and flexible structures with time-varying mass characteristics
including the control systems. The software interfaces directly with several other software such as Nastran,
Simulink/Matlab and CATIA.
The VEGA-DCAP-sim, already presented in [Ref. 12], is briefly described in order to demonstrate the simulators
capabilities and emphasizes the available options.

3.2.2.1 Dynamic model

As shown in the topology sketch in Figure 3- 7, the VEGA problem consists of two bodies, launcher and nozzle, which
are connected by a hinge with two degrees of freedom. Those correspond to the nozzle deflection angles that allow the
nozzle to rotate with respect to the whole launch vehicle in order to deflect the thrust. On top of that, springs were
introduced to characterize the stiffness and damping of the flexible joint (hinge 2). An additional hinge, which has six
degrees of freedom, is needed to locate the whole system in the inertia frame.
The nozzle, which is modelled as a rigid body, corresponds to the first body and the remaining launch vehicle, flexible
body, is the second one. In order to take into account the variation of mass properties and, therefore, the structural
properties (such as stiffness, damping, modes shape) during the flight, different finite element models are automatically
interpolated with respect to the time.
Two electro-mechanic actuators are used to point the nozzle in the desired orientation (see Figure 3- 8). and all sensor
needed for the analysis (such as accelerometers, gyros) are directly defined in DCAP.

Launcher
Hinge 2
2 dof Body 2
Nozzle
Body 1

Hinge 1
6 dof

Inertial frame

Figure 3- 7 VEGA-DCAP-sim topology for VEGA problem Figure 3- 8 VEGA nozzle and pistons model

3.2.2.2 Environment model

The environmental disturbances consist mainly of aerodynamic loads, gravity gradient, magnetic field interaction and
solar pressure. These produce forces and moments, which need to be accounted for in the generalized forces and
moments on the right hand side of the equations of motion. DCAP already includes all of these types of disturbances,
which the user can easily apply and tune for his other particular application. It is quite important to underline that
DCAP takes into account whether or not the analysis involves flexible bodies applying the loads in an appropriate
manner.
Since this work is mainly focused on the atmospheric flight of a launch vehicle, DCAP magnetic field and solar
pressure model are not presented. More information can be found in the DCAP manual.
The aerodynamic force computation requires the atmospheric density and sound speed with respect to the altitude in
order to compute the Mach number of the launcher during flight. These are physical inputs derived from the standard
launch site atmospheric data. In addition there are some atmospheric disturbances such as wind and gust, which can be
applied. The wind (see Figure 3- 9) represents an additional velocity of the air, with respect to the assumption of air-
fixed with the Earth, generally for the whole atmospheric flight. On the other hand, the gust (see Figure 3- 10)
represents a locally strong, abrupt rush of wind, acting at a particular altitude. In view of the fact that the unknown
disturbances can act in different directions, DCAP allows the user to choose where the particular wind profile should be
applied.
 Gust
H
velocity

Wind velocity time

Figure 3- 9 Typical wind profiles Figure 3- 10 Typical gust profile

3.2.2.3 Control model

The control system represents a crucial component of a launcher, since it always requires autonomous
operation during flight. Indeed, by means of the control system the vehicle can be steered along the desired
flight path and it can autonomously counteract the disturbances of an internally unstable system.
The preliminary Guidance, Navigation and Control part is modelled as shown in Figure 3- 11. Since a
completed description has been presented in detail [Ref. 10], the attention is focused on the modelling only in
the aerodynamic disturbances, which has been updated, and on the analysis in some different DCAP
capability.

Figure 3- 11 Launcher GNC description

Existing multi-body softwares, chosen for describing the intricate dynamic part, have some troubles to model
the control system. Some of them allow the user to code it in some common computer language such as
Fortran or C. Especially complex algorithms can require quite some time to program. With the aim of
reducing this phase, DCAP has the capability to avoid this step. Based on the fact that nowadays a significant
number of control designers use Simulink/Matlab environment, it was decided to import the DCAP dynamics
into Simulink environment as a block. To this end, it is possible to automatically create a dedicated Simulink
S-function, which describes both the dynamics and the DCAP environment model. This s-function, as any
other Simulink block, can be linked directly to the control model (see Figure 3- 12). It makes the modelling
of the control systems much easier and efficient.
 Figure 3- 12 VEGA-DCAP-sim Simulink model

3.2.2.4 Conclusions
As quick recap the DCAP and VEGA-DCAP-sim combines all of the following:
• flexible structural characteristics directly importing Nastran Finite Element Models
• environment and disturbances for flexible bodies flexible or rigid bodies both with time varying mass and
inertia
• complex nonlinear dynamics in Simulink format
• multiple bodies to model separation and nozzles
• MonteCarlo capabilities
• Internal loads computation
Among the different dynamics formulation a comparison between the Order(n) and Lagrange approach was presented.
The Order(n) is able to perform the same simulations with the same results in less than 7% of the total computational
time with respect to the Lagrange one, at least for these examples.

4 – Control Techniques
Once the two main design models (LILA3D and NOLILA3D), dedicated control techniques are investigated in this
section. The first one is based on the nonlinear feedback linearization control and the second one on the robust
eigenstructure assignment using multi-model approach. These techniques were selected since they allow to enable
coupling effects due to the non negligible vehicle roll rate and also the nonlinear characteristics of the systems (variable
mass, … ).

4.1 Nonlinear feedback linearization control

The equations of motion for a launch vehicle can be nonlinear depending on the particular manoeuvres/conditions. With
this regard, the problem can be solved using the nonlinear feedback linearization control technique, which has been
widely studied and applied within the past 25 to 30 years. Theoretical advances essentially completed the background
for ensuring the feedback control laws that make prescribed outputs independent of important classes of inputs; namely,
disturbances and decoupled control effectors. These two vital aspects of control theory – noninteracting control laws
and the transformation of nonlinear systems into equivalent linear systems – are embodied in what is often called DI.
Falb and Wolovich [Ref. 13] considered noninteractions as a facet of linear systems theory. Isidori and his colleagues
[Ref. 14] contributed significantly to DI theory by using mathematical notions from differential geometry.
4.1.1 Method overview
Feedback linearization theory, more discussion can be found in [Ref. 15], is summarized in this section. Consider an n-
dimensional multivariable nonlinear system with m inputs ui and m outputs yj, and is affine in the input u:
x = f ( x) + g ( x)u
Eq. 4-1
y = h( x )
where x ∈ X ⊂ R , u ∈ U ⊂ R , and y ∈ Y ⊂ R .
n m m

The functions f(x) = [f1(x), …, fn(x)]’ and h(x) = [h1(x), …, hm(x)]’ are assumed to be continuously differentiable on X
and the functions g(x) = [g1(x), …, gm(x)]’ ∈ Rn x m are continuous function of x. A multivariable nonlinear system of the
form of Eq. 4-1 has a vector relative degree {r1,…,rm} at a point x0 if
(i) L g Lkf hi ( x ) = 0
j
Eq. 4-2
for all 1 ≤ j ≤ m, for k < ri -1, for all 1 ≤ i ≤ m, and for all x in a neighbourhood of x0.
 Lg1 Lρf1 −1h1 ( x) " Lgm Lρf1 −1h1 ( x) 
 
(ii) A( x)=  # % #  Eq. 4-3
 Lg1 Lρf m −1hm ( x) " Lgm Lρf m −1hm ( x) 
 
is non-singular at x = x0.
The system has a vector of relative degree [ρ1, …, ρm]’ and the total relative degree ρ = ρ1 + … + ρm. Then there exists a
state feedback control law defined as
u = ϕ ( x ) + ϑ ( x )ν Eq. 4-4
which results in a closed-loop linear input-output behaviour between the new input υ and the output y. The vector φ(x)
and matrix θ(x) are defined as:
ϕ ( x) = − A −1 ( x )l ( x ) Eq. 4-5
ϑ ( x ) = A −1 ( x ) Eq. 4-6
where
 Lρf1 h1 ( x) 
 
l ( x) =  #  Eq. 4-7
 Lρf m hm ( x) 
 
ρ ρ −1
and where L f j h j (x) and Lgij h j (x) are the Lie derivatives of the scalar functions hj(x) with respect to the vectors f(x)
and gi(x), with j,i =1 to m. If the matrix A(x) is nonsingular, the control law is well the control law is well defined and a
coordinate transformation,
ξ 
Φ (x ) =   Eq. 4-8
η 
define as a local diffeomorphism based on the calculation of the relative degree of each output, yields the closed-loop
system.
ξ = Aξ + Bν Eq. 4-9
y = Cξ Eq. 4-10
η = z (ξ ,η ,ν ) Eq. 4-11
with the state matrices A, B, and C in Brunovsky block canonical form and the new input vector υ. The new state
vectors ξ and η are of dimension ρ and n-ρ, respectively. The vector z contains the nonlinear internal dynamic, which
only appears when the total relative degree ρ is smaller than the original state dimension n (partly feed-back linearized
system). The zero dynamics, defined as the internal dynamics when the input is chosen such that the output is and
remains zero [z(0,η,υy=0)], should be stable to ensure close-loop stability.

4.1.2 Launch Vehicle application

As already mentioned, the nonlinear control techniques allow the designer to use directly a nonlinear model for the
launch vehicle without introducing any limitation such as linearization. Therefore to investigate the nonlinear control
capabilities, it is possible to use directly the 3D nonlinear mathematical model (NOLILA3D sim).

4.1.2.1 Rigid Launch Vehicle model

During the atmospheric flight phase, the attitude control function ensures launcher attitude stability and control as well
as load and trajectory drift limitation. Therefore, our attention is focused on the rotational motion of a rigid body, which
were presented for NOLILA3Dsim:
 
 I xx P + (I zz − I yy )QR = M x

  1
 I yy Q + (I xx − I zz )PR = −TC l C δ + ρVrel S R C Nα l F α
2
Eq. 4-12
 2
 
 I zz R + (I yy − I xx )PQ = −TC l C ε − 2 ρVrel S R C Nα l F β
1 2

and on the Euler rates of the frame:


ϕ = p + sin ϕ tan ϑ ⋅ q + cos ϕ tan ϑ ⋅ r

ϑ = cos ϕ ⋅ q − sin ϕ ⋅ r Eq. 4-13
 sin ϕ cos ϕ
ψ = ⋅q + ⋅r
 cos ϑ cos ϑ
Although the aerodynamic loads play an important effect on the launch vehicle dynamics (mainly responsible for the
vehicle unstable behaviour), it is also important to mention that the angle of attack is kept very close to zero. As a first
approximation, the aerodynamic moments can be neglected and considered as external disturbance. Therefore the
launch vehicle model, which is going to be used for the control law design, can be expressed following as:
− (I zz − I yy ) I xx QR 
 
 − ( I xx − I zz ) I yy PR 
− (I − I ) I PQ 
 yy xx zz

f ( x) =  p + sin ϕ tan ϑ ⋅ q + cos ϕ tan ϑ ⋅ r  Eq. 4-14
 
cos ϕ ⋅ q − sin ϕ ⋅ r 
 sin ϕ cos ϕ 
 ⋅q + ⋅r 
 cos ϑ cos ϑ 
 −TC lC I xx 0 0 
 0 − T l I 0 
 C C yy 
 0 0 −TC lC I zz 
g ( x)=   Eq. 4-15
 0 0 0 
 0 0 0 
 
 0 0 0 
where the x = [P, Q, R, φ, θ, ψ] is state and u =[γ, δ, ε] is the input of the system. Assuming that the current attitude of
the vehicle can be measured by the inertial measurement unit, the outputs are:
ϕ 
h(x)= ϑ  Eq. 4-16
ψ 
NOTA: The fictitious input M x = TC lC I xx γ was introduced in order to have the same number of inputs and
outputs. This actuator will not be active during the simulations and, therefore, the roll angle will no be controlled.

4.1.2.2 Relative degree computation

Applying the relative degree definition:
Lg h1 ( x) = [ 0 0 0 1 0 0] g ( x) = 0 Eq. 4-17
Lg h2 ( x ) = [ 0 0 0 0 1 0] g ( x) = 0 Eq. 4-18
Lg h3 ( x ) = [ 0 0 0 0 0 1] g ( x ) = 0 Eq. 4-19
L f h1 ( x ) = [ 0 0 0 1 0 0] f ( x) = p + sin ϕ tan ϑ ⋅ q + cos ϕ tan ϑ ⋅ r Eq. 4-20
L f h2 ( x) = [ 0 0 0 0 1 0] f ( x) = cos ϕ ⋅ q − sin ϕ ⋅ r Eq. 4-21
sin ϕ cos ϕ Eq. 4-22
L f h3 ( x) = [ 0 0 0 0 0 1] f ( x) = ⋅q + ⋅r
cos ϑ cos ϑ
 Tc lc  Eq. 4-23
− I 
 xx 
 0 
 cos ϕ ⋅ q − sin ϕ ⋅ r   Tl
Lg 2 L f h1 ( x) = 1 sin ϕ tan ϑ cos ϕ tan ϑ cos ϕ tan ϑ ⋅ q − sin ϕ tan ϑ ⋅ r 0  0  = − c c
( cos ϑ )
2
I xx
   0 
 
 0 
 0 
 Tclc Tl Eq. 4-24
Lg2 L f h1 ( x) = − sin ϕ tan ϑ ; Lg3 L f h1 ( x ) = − cos ϕ tan ϑ c c ;
I yy I zz
 Tc lc  Eq. 4-25
− I 
 xx 
 0 
 
Lg1 L f h2 ( x) = [ 0 cos ϕ − sin ϕ − sin ϕ ⋅ q + cos ϕ ⋅ r 0 0]  0  = 0
 0 
 
 0 
 0 
Tc lc Tc lc Eq. 4-26
Lg2 L f h2 ( x) = − cos ϕ ; Lg 3 L f h2 ( x) = sin ϕ
I yy I zz
 Tc lc  Eq. 4-27
− I 
 xx 
 0 

Lg1 L f h2 ( x) =  0
sin ϕ cos ϕ cos ϕ ⋅ q − sin ϕ ⋅ r ( sin ϕ ⋅ q + cos ϕ ⋅ r ) sin ϑ 
0  0  = 0

cos ϑ cos ϑ cos ϑ ( cos ϑ )
2
  
0 
 
 0 
 0 
sin ϕ Tclc cos ϕ Tc lc Eq. 4-28
Lg 2 L f h3 ( x ) = − ; Lg 3 L f h3 ( x ) = −
cos ϑ I yy cos ϑ I zz
Using the Eq. 4-3, the A matrix can be derived as :
 
1 I sin ϕ tan ϑ I yy cos ϕ tan ϑ I zz 
 xx 
A( x) = −TC lC  0 cos ϕ I yy − sin ϕ I zz  Eq. 4-29
 
 0  sin ϕ   cos ϕ 
  I yy ⋅   I zz ⋅ 
  cos ϑ   cos ϑ  
It is no singular
lC  1 
det( A) =   Eq. 4-30
I xx I yy I zz  cos ϑ 
but it is not defined when θ = π/2 + k π (k=0,..1).
Therefore, the relative degree is [ρ1, ρ3, ρ3]’= [2, 2, 2]’and the total relative degree ρ = 2 +2+2 = 6. Consequently,
since ρ = n, no zero dynamics is needed and the nonlinear coorditates transformation is hereby presented :
0 0 0 1 0 0
ϕ     p
ϕ  1 sin ϕ tan ϑ cos ϕ tan ϑ 0 0 0   
  0 q
0 0 0 1 0  
ϑ   r 
x = Φ ( x) ⇒   =  0 cos ϕ − sin ϕ 0 0 0   Eq. 4-31
 
ϑ    ϕ 
ψ   0 0 0 0 0 1  
 ϑ
   sin ϕ cos ϕ  
ψ   0 0 0 0  ψ 
 cos ϑ cos ϑ 

The non linear terms:
 cos ϕ ⋅ q − sin ϕ ⋅ r 
1 sin ϕ tan ϑ cos ϕ tan ϑ cos ϕ tan ϑ ⋅ q − sin ϕ tan ϑ ⋅ r 0
( cos ϑ )
2
 
L2f h( x) = 0 cos ϕ − sin ϕ − sin ϕ ⋅ q + cos ϕ ⋅ r 0 0  f ( x) Eq. 4-32
 
 sin ϕ cos ϕ cos ϕ ⋅ q − sin ϕ ⋅ r ( sin ϕ ⋅ q + cos ϕ ⋅ r ) sin ϑ 
0 0
cos ϑ cos ϑ cos ϑ ( cos ϑ )
2
 
Recalling Eq. 4-6, the control output, input of the dynamic system is
  v1 ( x)   Lρf1 h1 ( x)  
  
u = − A −1 ( x)   #  +  #  Eq. 4-33
  v ( x)   Lρm h ( x)  
 m   f m 
Following the tracking problem results, the control input v can be expressed, in the generic formulation, as:
  v1 ( x)  ρ
 #  = y ( ρi ) − i c ξ i − y ( j −1)
  iR ∑
j =1
j −1 j iR ( ) Eq. 4 -34
 vm ( x) 

4.1.2.3 Results
First the control law is tested on a nominal case using the NOLILA3D mathematical models (see Figure 4- 1).
Following the mission requirements, the control parameters (cj) are set in order to have the linearized dynamic
characterized by a frequency of 0.5 Hz and relative damping (ζ = 0.7) for each channel. This frequency value is
constrained by the presence of the elastic properties of the launcher, which will be addressed later.

05-Dec-2007 01:23:41 || D:\PhD\LVsim\Control_Design\DINV || TRAJ: 875.out 05-Dec-2007 01:28:51 || D:\PhD\LVsim\Control_Design\DINV || TRAJ: 875.out
LV ANGULAR Error [deg] LV ANGULAR Error [deg]
0.2 1 Roll
0 Pitch
0.5
Yaw
-0.2
0
-0.4 Roll
-0.6 Pitch -0.5
Yaw
-0.8 -1
10 20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80 90
Time [s]
Time [s]
Angle of Attack [deg]
5 Angle of Attack [deg]
α 5
β
α'
0
0
α
β
-5 α'
0 10 20 30 40 50 60 70 80 90 -5
Time [s] 0 10 20 30 40 50 60 70 80 90
TVC deflection [deg] Time [s]
TVC deflection [deg]
5 δ on z-axis
ε on y-axis 5
δ on z-axis
ε on y-axis
0
0

-5
-5
0 10 20 30 40 50 60 70 80 90
Time [s] 0 10 20 30 40 50 60 70 80 90
Time [s]

Figure 4- 1 Nonlinear feedback linearization results on a rigid Figure 4- 2 Nonlinear feedback linearization results on a
LILA3D rigid LILA3D with an external roll torque disturbance

Furthermore, in order to investigate the decoupling performance, the same controller was tested for a worst case
condition, when an external roll torque is acting on the vehicle symmetry axis. (see Figure 4- 2).

4.1.2.4 Control Improvement

First of all, it is important to underline that the control design was performed without taking into account the elastic
properties of the vehicle. As already discussed in the previous chapters, this effect should be carefully analyzed. In this
study, already good performances are obtained only by taking into account an additional filter as shown Figure 4- 3.
This filter has two functions: the first one is to cancel the elastic dynamic responses, measured by the sensors, and the
second one is to roll off in order to reduce the influence of measurement noise.
Bode Diagram
0

-10
Magnitude (dB)

-20

-30

-40
0

-45
Phase (deg)

-90

-135
-1 0 1 2 3
10 10 10 10 10
Frequency (rad/sec)

Figure 4- 3 Filter added to compensate the launch vehicle flexibility

First of all, it is important to underline that the control design was performed without taking into account the elastic
properties of the vehicle. As already discussed in the previous chapters, this effect should be carefully analyzed. In this
study, already good performances are obtained only by taking into account an additional filter as shown Figure 4- 3.
This filter has two functions: the first one is to cancel the elastic dynamic responses, measured by the sensors, and the
second one is to roll off in order to reduce the influence of measurement noise.
As it is shown in Figure 4- 1 and Figure 4- 2, the angle of attack has a no zero angle for the whole trajectory. This result
was expected since no information was included in the model, which was used in the control design, about the
aerodynamics. In order to reduce the angle of attack, an external loop was added to the nonlinear feedback linearization
control. This additional feedback is assumed proportional to the lateral drift velocity along the transversal axes.

4.1.2.5 Control validation

Once the control law and filters have been designed and tested on a mathematical model (NOLILA3D), it is important
to validate them on the high-fidelity simulator VEGA-DCAP-sim (see Figure 4- 4.).
.
02-Dec-2007 23:52:50 || DCAP model: IT3F_ELV_875 & CTRL LAW: DINV_ch6_run0 || TRAJ: 875.out
LV ANGULAR Error [deg] LV TRAJECTORY Error [m]
100
Yaw Xinertial
0.2 Pitch Yinertial
0
Roll
Zinertial
0
-100

-0.2
-200

-0.4
-300
0 20 40 60 80 100 0 20 40 60 80 100
Time [s] Time [s]
5
Angle of Attack [deg] x 10 Dynamic Pressure x Angle of Attack [Pa*deg]
10 3

5
2

1
-5

-10 0
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Time [s] Time [s]
TVC deflection [deg] TVC Consumption (integral TVC deflection) [deg*s]
10 0.2
Yaw Yaw
Pitch Pitch
5 0.15

0 0.1

-5 0.05

-10 0
0 20 40 60 80 100 0 20 40 60 80 100
Time [s] Time [s]

Figure 4- 4 Nonlinear feedback linearization results for the nominal condition

4.2 Modal Robust Control

It is often believed that pole placement cannot deal with robustness. The Robust Pole Placement method proposed by de
Larminat [Ref. 16] is a first step to show the adequacy between robustness and pole assignment.
An iterative modal multi-model pole assignment technique is proposed in [Ref. 17] and [Ref. 18], which is based on the
first order perturbation. As for all other methods based on first order approximation, efficiency is subject to a good
initialization.

4.2.1 Multi-model modal control

Multi-model eigenstructure assignment [Ref. 19] is done by simultaneously assigning triplets Ti for several models
which reduces to solve a set of equality constraints of type ([Ref. 20]). The choice of the models to treat with and the
triplets to assign is determined by an analysis of the stability and/or performance robustness
 Procedure: (Mu- µ )-iteration

Step A.1 Elaborate a first initial design on a nominal model. All kinds of synthesis methods can be applied at
this step ( H ∞ control, LQG optimal control, mu -synthesis, etc...). In the case of initial non-modal
approaches, look for eigenstructure assignment having the same characteristics as the initial controller.
Step B.1 Proceed to a multi-model analysis of the pole map and/or time-responses and/or real µ -analysis like
proposed in [Ref. 21]. If the initial design is satisfactory for all models or all values of uncertainties,
then stop. Otherwise identify the worst-case model, determine its critical triplet Ti and continue with
Step B.2
Step B.2 Improve the behaviour of the worst-case model by replacing the triplet Ti by Ti ∗ respecting the
specifications while preserving the properties of all models treated before. Return to Step B.1.
Remark: See [Ref. 22] and for some general rules on multi-model eigenstructure assignment, for example to
avoid incompatible assignments, we should treat models as ‘far’ as possible from each other in the
considered parameter space and/or relax some constraints on models treated before.

Figure 4-5 (Mu-µ)-iteration

4.2.2 Multi-model modal self-scheduling

Gain-scheduling approach is very attractive to design global control law covering wide operative range. The relevant
literature can be divided into two main parts: papers about gain-scheduled design for a wide variety of specific
applications, and formal reviews such as [Ref. 23] that provides useful overview of the different existing methods. But
the used theoretical results and design procedures, which are not explicitly mentioned in the surveys, proved to be not
only very efficient but also easily reusable. The traditional scheduling methods used in industry consist of designing a
set of controllers associated to the system operating points and of interpolating a posteriori those feedbacks, see for
examples [Ref. 24] and [Ref. 25]. The proposed technique does not belong to this category but is based on a priori
interpolation. In a nutshell, the scheduling parameters are directly taken into account during the synthesis of a unique
controller. This choice can be guided by physical constraints or previous experiments.
 ∆'

u y

M(s)
w z

∆
Figure 4-6 “M-∆ form” for µ-analysis of system controlled by a self-scheduled

Let us for example take a scheduling w.r.t measurable parameter ∆’ = δ and an interpolation formula

K s (δ ) = K 0 + δ K1 + δ 2 K 2 Eq. 4-35
The determination of such a self-scheduled controller is equivalent to the synthesis of a multi-model modal controller

K =  K 0 K1 K2  Eq. 4-36

with respect to the augmented system

  C   D 
    
 A, B,  δ C  ,  δ D   Eq. 4-37
  δ 2C   δ 2 D  
   

As it can be seen, the problem boils down to increasing the number of outputs of the original system ( A, B, C , D )
from p to 3 p . The augmentation of the output number also offers additional degrees of freedom necessary to achieve
desired robustness properties.

4.2.3 Launch Vehicle applications

As already presented in [Ref. 26], the Robust Modal Control technique, which is applied to a rigid launcher
mathematical model, is briefly recalled. Moreover, it was adjusted by means of “ad-hoc” filter is designed in order to
guarantee satisfactory performance also in presence of the elastic structural dynamics of the vehicle.

4.2.3.1 Coupling effect

For systems with important coupled dynamics, it is important that the control law is able also to assure a good
decoupling between the controlled axes. More in detail, the decoupling objective is formulated as follows: a command
on θ should not affect ψ and reciprocally. Indeed, when a roll rate (p) is not zero, θ and ψ are naturally coupled. More in
detail, if the “classic” control law is used for LILA3D (see par. 3.1.2), the results satisfied the specification only when
the roll rate has very low values (Figure 4- 7).
 K eig 01 x 4 
K=
K eig 
Eq. 4-38
 01 x 4
 Classic Eigenvalues Approach : LILA3D model at T = 55s with Po = 0°/s Classic Eigenvalues Approach : LILA3D model at T = 55s with Po = 20°/s
YAW angle [deg] PITCH angle [deg] YAW angle [deg] PITCH angle [deg]

1.2 1.2 1.2 1.2

1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0

-0.2 -0.2 -0.2 -0.2

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s] Time [s] Time [s]

YAW angle [deg] PITCH angle [deg] YAW angle [deg] PITCH angle [deg]

1.2 1.2 1.2 1.2

1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0

-0.2 -0.2 -0.2 -0.2

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s] Time [s] Time [s]

Figure 4- 7 Classic Eigenvalue approach : LILA3D model Figure 4- 8 Classic Eigenvalue approach: LILA3D model at
with Po = 0°/s with Po = 20°/s

4.2.3.2 Control design

The controlled outputs are the pitch angle θ and the yaw angle ψ. In addition, to the standard tracking specifications, the
controller will have to limit the coupling effects. At last, the controller must be robust to any roll rate variations between
0°/s and 30°/s and be effective all along the takeoff phase. The available measurements are v, r, ψ, w, q, θ. We add two
integrals on ψ and θ in order to ensure zero steady-state errors on these variables and perturbation robustness.
The two inputs ∆σ and ∆ε will make the axes decoupling possible. So there are 16 gains that must be computed in order
to satisfy the requirements. The gains will be denoted as follows:
 K v→∆σ K r →∆σ Kψ →∆σ K ψ →∆σ K w→∆σ K q →∆σ K θ →∆σ K θ →∆σ 
K = ∫ ∫  Eq. 4-39
 K v→∆ε K r →∆ε Kψ →∆ε K ∫ψ →∆ε K w→∆ε K q →∆ε K θ →∆ε K ∫ θ →∆ε 
 

ƒ Step 0: Preliminary study on the decoupling effects

The first encountered difficulty is the choice of the closed-loop eigenvalues and eigenvectors. Indeed, the
choice of eigenvalues and eigenvectors is not as straightforward as the coupling depends on the roll rate (P0).
A trial and error approach did not lead to the selection of adequate eigenvalues and eigenvectors. In order to
solve the issue, a LQR design is then considered with
Q = diag ( 0 0.1 0.1 1 0 0.1 0.1 1)
Eq. 4-40
R = diag ( 0.001 0.001)
where ∫θ and ∫ψ are the more weighted states. The complex pair of eigenvalues λ1,2 , which is located nearby a
multivariable zero, seems to be critical in the decoupling effect.
LQR Approach : LILA3D model at T = 55s with Po = 20°/s
YAW angle [deg] PITCH angle [deg]

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 9 LQR Control : LILA3D model at T = 55 s with P0 = 20°/s

ƒ Step 1: Initial static controller
An initial static controller is then designed on the LILA3D model T = 55 s with P0 = 20 °/s. For decoupling
reasons, λ1,2 , λ3,4, λ5 are affected to θ and λ6,7, λ8 are affected to ψ. Figure 4- 10 shows that the requirements are
satisfied (settling time, overshoot and decoupling).
Eig Struct Approach : LILA3D model at T = 55s with Po = 20°/s
YAW angle [deg] PITCH angle [deg]

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4
Time [s] Time [s]

Figure 4- 10 Eigenstructure approach : LILA3D model at T = 55s with P0 = 20°/s

using the initial static controller
Nevertheless, by testing this controller for other roll rate values, oscillations tend to appear for
P0 = 30 °/s and decoupling is not so good. The idea is to introduce P0 as a scheduling parameter to improve the
performances.

ƒ Step 2: Self scheduling controller wrt P0

The roll rate P0 is normalized as |P0| = -15+15 δP0. In order to improve the performances following the
different p values, the controller is scheduled as:
K = K 0 + K1δP0 Eq. 4-41
The self-scheduled controller is computed by assigning the eigenvalues of the augmented system:
 C  D 
 A∆ , B∆ ,  ∆ ,  ∆   Eq. 4-42
 
 δP0 C ∆  δP0 D∆  
YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

0 0.2

-0.2 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

1 0.8

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0

0 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 11 LILA3D models from T = 5 s to T = 55 s any roll rates in [-30°/s, 30°/s]

using scheduled controller wrt P0
 Using these new degrees of freedom, the two former eigenstructures (LILA3D model at T = 55s with
P0 = 0 °/s and P0 = 20 °/s) are placed on the augmented system.
A quick study of the time responses reveals that this scheduled controller performs very well for all the
LILA3D models from T = 5 s to T = 55 s any roll rates in [0°/s, 30°/s] Figure 4- 11. Nevertheless the
behaviour tends to be quite oscillatory and not satisfactory for the LILA3D models from T = 65 s to
T = 95 s with rates above 10°/s Figure 4- 12.
YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

0 0.2

-0.2 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

1 0.8

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0

0 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 12 LILA3D models from T = 65 ss to T = 95 s with P0 = 20°/s

using scheduled controller wrt P0

ƒ Step 3: Self scheduling controller wrt P0 and T

Depending on the time, the performances of the former scheduled controller are not acceptable. The idea is
now to add the time as a supplementary scheduling parameter. The new scheduled controller has the form:
K = K 0 + K1δP0 + K 2δt Eq. 4-43

The total time is normalized as T = 55+40 δt and δt ∈ [0, 1] (for LILA3D model till T = 55 s, δt = 0 and for
LILA3D model at T = 95 s, δt = 1). The self-scheduled controller is obtained by assigning the eigenvalues of
the augmented system:
 C ∆   D∆  
    
 A∆ , B∆ , δP0 C ∆ , δP0 D∆   Eq. 4-44
 δtC ∆  δtD∆  

YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

0 0.2

-0.2 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

1 0.8

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0

0 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 13 LILA3D models from T = 5 s to T = 95 s with all P0 [-30°/s, 30°/s]

using the “final” scheduled controller
 The former models are still considered with the same assignments. The LILA3D model at T = 85 s with P0 =
20 °/s is added. Again λ1,2 , λ3,4, λ5 are affected to θ and λ6,7, λ8 are affected to ψ. Figure 8 shows that time-
responses are now quite satisfactory for all models and roll rates. Nevertheless, there are still light oscillations,
which become unacceptable for P0 = ± 50 °/s (Figure 4- 14), and the decoupling could still be improved. One
way to improve this behaviour could be to treat also model 1 at T = 5 s during synthesis. This would mean to
add an additional term K3 to the controller of Eq. 4-43 with an additional parameter δ3 (for example δt2) in
order to create the necessary degrees of freedom for the control design. Here, a simpler way was chosen.
YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2
0.8
1
0.6
0.8
0.4
0.6
0.2
0.4
0

-0.2 0.2

-0.4 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

0.8
1
0.6
0.8
0.4
0.6
0.2
0.4
0
0.2 -0.2

0 -0.4
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 14 LILA3D models from T = 5 s to T = 95 s with all P0 [-50°/s, 50°/s]

using the “final” scheduled controller

ƒ Step 4: “Improved” self scheduling controller wrt P0 and T

By speeding up the fast poles of LILA3D models from T = 55 s to T = 95 s (instead of
λ5,8 = −6, we take λ5,8 = −20 on LILA3D models at T = 55 s and λ5,8 = −30 on LILA3D models
at T = 95 s), one can obtain much better time-responses for P0 ∈ [-30°/s, 30°/s] (Figure 4- 15) and
P0 ∈ [-50°/s, 50°/s] (Figure 4-16). But an overshoot can be observed on θ time-responses of LILA3D models at
T = 5 s and at T = 15 s. As these models were not considered in the eigenstructure assignment, they are not
entirely satisfactory but still remain acceptable. Nevertheless, gains are higher compared to the former
scheduled controller.
YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

0 0.2

-0.2 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

1 0.8

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0

0 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 15 LILA3D models from T = 5 s to T = 95 s with all P0 [-30°/s, 30°/s]

using the “improved” scheduled controller
 YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

0 0.2

-0.2 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

1 0.8

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0

0 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4-16 LILA3D models from T = 5 s to T = 95 s with all P0 [-50°/s, 50°/s]

using the “improved” scheduled controller

4.2.3.3 Extension for a flexible launch vehicle

Now it is important to test the control, which is designed on a rigid model, on the LILA3D flexible models. The
“improved” controller, which is characterized by faster poles, significantly perturbs the structural dynamics of the
vehicle. Therefore it was decided to use the “final” scheduled control, for which a filter (see Figure 4- 17) was easier to
design in order to cope with the elastic properties of the launcher.

Bode Diagram
0

-5
Magnitude (dB)

-10

-15

-20

-25

-30
0
Phase (deg)

-45

-90
1 2 3 4
10 10 10 10
Frequency (rad/sec)

Figure 4- 17 Filter designed for the “final” scheduled controller

Now the LILA3D flexible models are stable for different roll rates and for different flight instants, while a still good
decoupling performance are kept (see Figure 4- 18). Certainly, the vehicle elasticity has degraded the overall system
performances.
 YAW angle [deg] PITCH angle [deg]
1.2 1.4

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

0 0.2

-0.2 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

YAW angle [deg] PITCH angle [deg]

1.4 1.2

1.2 1

1 0.8

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0

0 -0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [s] Time [s]

Figure 4- 18 LILA3D flexible models from T = 5 s to T = 95 s with all P0 [0°/s, 30°/s]

using the “final” scheduled controller

4.2.3.4 Control validation

Once the control law and filters have been designed and tested on different mathematical models, it is important to
validate them on the high-fidelity simulator VEGA-DCAP-sim.
09-Dec-2007 09:43:44 || DCAP model: IT3F_ELV_875 & CTRL LAW: EIG_IFAC2008_run0 || TRAJ: 875.out
LV ANGULAR Error [deg] LV TRAJECTORY Error [m]
100
0.4 Yaw Xinertial
Pitch Yinertial
0.2 0
Roll
Zinertial
0 -100

-0.2 -200

-0.4
-300
0 20 40 60 80 100 0 20 40 60 80 100
Time [s] Time [s]
5
Angle of Attack [deg] x 10 Dynamic Pressure x Angle of Attack [Pa*deg]
10 3

5
2
0

1
-5

-10 0
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Time [s] Time [s]
TVC deflection [deg] -4 2
x 10TVC Consumption (integral TVC deflection) [deg*s ]
10 8
Yaw Yaw
Pitch Pitch
5 6

0 4

-5 2

-10 0
0 20 40 60 80 100 0 20 40 60 80 100
Time [s] Time [s]

Figure 4- 19 RMC results for the nominal condition

5 – Control Techniques Comparison
Once each single control law technique has been presented and implemented on a dedicated mathematical model, it is
also important to validate them using a high-fidelity simulator. Therefore, all the control laws have been coded in the
VEGA-DCAP-sim and they have been tested in different flight conditions. It is important to identify the most
significant parameters, which can summarize the whole launch vehicle dynamics, but also which allow comparing
different control techniques. For these reasons, it is compulsory to introduce a definition of merit figures, which are
called performances.

5.1 Selection of the “key” parameters

In order to properly investigate the effectiveness of a generic control system, it is necessary to monitor the commands,
which are requested by the control law, and the objectives, which represent the controlled dynamic variable of the
system.

Reference s Objectives

Control System
Commands

Measurements

Figure 5-1 Typical feedback control representation

For the specific application of a launch vehicle, it is straightforward to identify the command variables. The only
actuation system, which is considered in this study, is the thrust vector control by means of nozzle deflection in the
pitch and yaw plane. Two interesting parameters are the request on the maximum nozzle deflections (TVC max defl)
and the consumption needed for nozzle activity (TVC cons). For what concerns the system objectives, or controlled
variables, there are different significant variables depending on the task that the system should achieve. In our specific
case, in the atmospheric flight, the launch vehicle should mainly follow the desired attitude and should reduce as much
as possible the loads, which are generated by the aerodynamic pressure. Therefore, the system objective variables which
are considered, are the maximum angular nozzle deflection (Max Ang Err), which is required along the whole
simulation, the maximum angular error (Ang Err at Tf) and trajectory error at the end of the flight (Traj Err at Tf) and
the maximum aerodynamic loads, which are expressed by the dynamic pressure times the angle of attack (Max Q* α).
Even if these values well summarize the global system dynamics, it has been decided to reduce the two main synthetic
indexes, which give a general overview of the system in order to facilitate the control strategies comparison. The idea is
to merge these control command data into a “Consumption” index and system objectives into an “Inaccuracy” index.
Obviously, a high Consumption value corresponds to a control technique, which requires important actuation activity.
On the other side, a low Inaccuracy value is obtained for a control law, which achieved global excellent system
objectives. These two main indexes are obtained based on the following formula:
ai
K= ∑w a
i =1, n
i Eq. 4- 5-1
i0
Where
K index value
ai ith parameter value
ai0 ith parameter reference value (or maximum allowable value from the requirements)
wi weighting factor

Although this approach proposes a standard procedure to identify these indexes, different subjective values can be used
for the weighting factor. In this study, it has assumed that the nozzle deflection consumption is more important than the
maximum nozzle deflection (0.7 and 0.3) for computing the Consumption index and both angular (maximum and at
Tfinal) and trajectory error (at Tfinal) more than the aerodynamic load (0.3, 0.3, 0.3 and 0.1) for the Inaccuracy index.

5.2 “Preliminary” Robustness Analysis

In order to provide a preliminary assessment on how robust each control technique is with respect to model
uncertainties, few simulations were run varying different critical parameters. Based on the nominal results (Figure 4- 19
and Figure 4- 4), the most important launch vehicle parameters and external disturbances were included following a
“step-by-step” approach. As reported in Table 5-1, 7 different degraded scenarios were identified in addition to the
nominal case (Run 0). Assuming the simplest approach (min-max), the worst variation of the launch vehicle dynamic
parameter is assumed for each simulation run based on physical considerations. For instance, the Run 1 assumes that no
external disturbance is present and all the system parameters are characterized by their nominal value except the
aerodynamic coefficients, which were assumed to be 20 % higher. It supposes an increase of the aerodynamic
coefficients, since they represent the main cause of the instability for a conventional launcher in the atmospheric flight.
For the Run 2 and Run 3, it is obvious that the worst conditions due to the elastic properties are generated from lower
elastic frequencies and for lower damping values. In Run 4, the external disturbances have been considered by adding a
specific wind profile in the flight direction along the whole trajectory and a local gust disturbance around the maximum
dynamic pressure in a perpendicular flight direction. An external torque, which might be caused by the solid propellant
motors, is added in the Run 5. This scenario is quite central since it is characterized by a not negligible launch vehicle
roll rate. In these conditions, the control law has to cope with a coupled pitch-yaw dynamics system. Both
misalignments and offsets of the thrust force are taken into account in Run 6. Finally the Run 7 simulates one of the
most critical scenario, since it assumes not only a roll torque, which produces a not negligible roll rate, but also a thrust
vector offset/misalignment including wind and gust disturbances.

Table 5-1 Different scenarios for the Robustness Analysis

Elastic properties External disturbance
ID Aero. Coeff. Thrust Offset
ω ζ wind/gust roll
0 - - - - - -
1 + 20 % - - - - -
2 - - 15 % - - - -
3 - - - 20 % - - -
4 - - - X - -
5 - - - - X -
6 - - - - - X
7 - - - X X X

In the Figure 5-2, Inaccuracy versus Consumption map is reported for the two presented control techniques: modal
robust control and nonlinear feedback linearization control for the different scenarios (see Table 5-1). It is interesting to
point out that the Run 4 and Run 7, which considered wind and gust external disturbances, represented the worst cases
for both controllers. Moreover, the nonlinear feedback linearization is able to achieve better results in terms of
Inaccuracy but it requires more actuation energy. Furthermore, a quite similar response is obtained for the Run 1, which
is characterized by an increase of the aerodynamic coefficients.
The last comment concerns the robust modal control results for the Run 2, which simulates the launch vehicle having an
elastic frequencies -15 % with respect to the nominal value. This result shows that, even if the control strategy allows to
guarantee a good accuracy level, the consumption is much higher than the other cases and is quite close to Run 4. This
deficiency should be solved when the elastic vehicle properties are taken into account during the design phase.

Inaccuracy vs Consumption
RMC - Run 0
1.40 RMC - Run 1

RMC - Run 2
1.20
RMC - Run 3

RMC - Run 4
1.00
RMC - Run 5

RMC - Run 6
0.80
Inaccuracy

RMC - Run 7

NFLC - Run 0
0.60
NFLC - Run 1

0.40 NFLC - Run 2

NFLC - Run 3

0.20 NFLC - Run 4

NFLC - Run 5
0.00 NFLC - Run 6
0.09 0.59 1.09 1.59 2.09 2.59
NFLC - Run 7
Consum ption

Figure 5-2 Inaccuracy versus Consumption map for RMC & NFLC for different runs
6 – Conclusion
In order to exploit the results of this study, the main advantages and drawbacks of the two “no classic” control
techniques - namely robust modal control (RMC) and nonlinear feedback linearization control (NFLC) - are hereby
summarized. Based on these preliminary results, some considerations can be drawn, in particular, with respect to the
launch vehicle typical criticalities. A clear summary table is hereafter reported:
Table 6-1 RMC vs NFLS: advantage and drawbacks for launch vehicle application

LV
RMC NFLC
Criticalities
Flexibility ↑ ↔
Time Varying ↔ ↑
Performance ↔ ↑
Decoupling ↑ ↑
Robustness ↑ ↔
Analysis Tool ↑ ↓
Missionazation ↔ ↑

where the following convention is assumed:

¾ The arrow-pointing upwards (↑) represents a positive effect of the advance control techniques wrt the
old one
¾ The arrow-pointing downwards (↓) represents a potential problem caused by the advance control
techniques wrt the old one
¾ The arrow-from left to right (↔) represents that no significant improvement are introduced by the
advance control techniques wrt the old one

™ Robust Modal Control (RMC)

¾ Advantages:
This technique has shown good results in handling launch vehicle with non negligible roll rate. In fact, using
the eigenstructures assignment it is possible to assure a good decoupling performance. Although it was not
directly addressed in this study, elastic characteristics can be taken into account in the design phase.
Furthermore, the technique was developed in order to robustify the standard eigenstructures assignments. In
fact, this technique is based on real µ-analysis and modal Multi-model design is proposed.
¾ Drawbacks:
Despite the traditional a posteriori scheduling methods, the proposed technique is based on a priori
interpolation. In fact, this technique permits to proceed to multi-model synthesis of complex systems and to
validate the closed loop behaviours on a continuum of models, it has no theoretical guarantee of convergence.
The overall performances are good but worse than the ones obtained by the nonlinear feedback linearization.
Moreover, the control law, which was decided for this study, is specific for a particular trajectory. In fact, the
different mathematical models are derived from a selection of equilibrium points along a specific trajectory.
For a different payload, which changes the vehicle mass properties, and for different flight path and high roll
rate (higher than 30 deg/s), a new control has to be redesigned.. To prove it, an additional run was used to
evaluate the effect of an “amplified” external roll torque on the worst case condition (Run 7). As shown in
Figure 8 10, the RMC control is not able any more to assure good performances, since the roll rate of launch
vehicle is not anymore within the design range [-30 deg/s, 30 deg/s] . As shown in Figure 6-1 RMC plot results –
Run ID 7 with an “amplified” external roll torque, this is not the case for the nonlinear feedbabk linearization
control.
 31-Mar-2008 10:04:18 || DCAP model: IT3F_ELV_875 & CTRL LAW: DINV_ch6_run8 || TRAJ: 875.out
LV ANGULAR Error [deg] LV TRAJECTORY Error [m]
100
Yaw Xinertial
0.2 Pitch Y inertial
0
Roll
Zinertial
0
-100

-0.2
-200

-0.4
-300
0 20 40 60 80 100 0 20 40 60 80 100
Time [s] Time [s]
Angle of Attack [deg] 5 Dynamic Pressure x Angle of Attack [Pa*deg]
x 10
10 3

5
2

0
1
-5

-10 0
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Time [s] Time [s]
TVC deflection [deg] -3 2
x 10TVC Consumption (integral TVC deflection) [deg*s ]
10 2
Yaw Yaw
Pitch Pitch
5 1.5

0 1

-5 0.5

-10 0
0 20 40 60 80 100 0 20 40 60 80 100
Time [s] Time [s]

Figure 6-1 RMC plot results – Run ID 7 with an “amplified” external roll torque

™ Nonlinear Feedback Linearization (NFLC)

¾ Advantages:
The application of nonlinear feedback linearizzation techniques in flight control has been popular in recent
years. This design technique uses the information about the nonlinear dynamics of the launch vehicle aircraft in
the process of designing the control law. The resulting nonlinear controller is valid for the whole of the flight
envelope without having the need to apply gain scheduling techniques between for example various linear
controllers. Other attractive features of this design technique are as follows:
ƒ the decoupling of the chosen variables to be controlled after the inversion is applied, which allows the
decoupling of the longitudinal from the lateral dynamics;
ƒ the independent assignment of closed-loop dynamics on each output channel
ƒ the simplicity in designing the controller and the simple structure of the controller, which is based on
state feedback and allows the designer to have an insight on how the controller behaves.
¾ Drawbacks:
The main disadvantage of nonlinear feedback linearizzation is that it does not provide a priori any robustness
guarantees for the closed-loop system, thus making the flight clearance procedure a crucial step in the
development of the control law. The clearance procedure demonstrates to the aviation authorities that a control
law will function in a satisfactory way for a given flight envelope and in the presence of failure conditions and
uncertainty. Major sources of uncertainty in modeling rigid aircraft dynamics are caused by:
ƒ variations in parameters such as the mass and the moments of inertia resulting from different store
configurations
ƒ uncertainty in the aerodynamic data resulting from a lack of knowledge regarding the effects of the
nonlinear unsteady aerodynamics.
Most analysis techniques for nonlinear feedback linearizzation flight control laws are based on linear methods.
The methodology is to linearize the nonlinear closed-loop system at various points in the flight envelope and
apply linear methods to investigate the robustness of the control law.

Each control technique, which is described in a dedicated chapter, is implemented in a “high-fidelity” simulator
(VEGA-DCAP-sim). A policy, which should be used to compare different control law performances, was presented as
well as important indexes (Inaccuracy and Consumption). Moreover different scenarios were simulated and the results,
reported and commented. Finally the two advance control techniques, robust modal control and nonlinear feedback
linearization control, have been reviewed in order to highlight their advantages and drawbacks for the specific VEGA-
like launch vehicle control application.
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