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Optimal design of a hybrid control system for the mitigation of the wind-induced
torsional response of tall buildings
Ilaria Venanzi, Filippo Ubertini, Annibale Luigi Materazzi
Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti 93, 06125 Perugia, Italy
email: [email protected], [email protected] , [email protected]
ABSTRACT: In this paper a general procedure is developed for the optimization of a hybrid control system for tall buildings
subjected to wind-induced vibrations. The control system is conceived to mitigate the flexural and torsional response and satisfy
serviceability limits. The hybrid control system is made of one active tuned mass damper with two longitudinal and one
torsional degrees of freedom. The passive device, that is, the active tuned mass damper without active control actuators, is
preliminarily designed to minimize the response of the flexural and torsional modes. The introduction of the control actuators
improves the performances of the device especially for the torsional response. The feedback information necessary to compute
the control forces is provided by a limited number of accelerometers arranged over the building’s height. The Kalman observer
provides the estimate of the states of the system known the measured accelerations and the covariances of the wind forces. The
reduction of the top corner’s accelerations and the control forces are chosen as competing targets of the multi-objective
optimization problem. The design variables are the accelerometers’ positions over the building height and the weighting
matrices of the LQR performance index. To reduce the computational effort, subsequent optimization sub-procedures are
employed that take advantage of the genetic algorithm to find the solution of the nonlinear, constrained optimization problems.
As an illustrative example, a control system is optimized for the response mitigation of a square tall building 180 m high. Wind
forces are obtained by wind tunnel tests on a rigid model of the building. Results highlight that the procedure is effective in
finding the optimal balance between the need of energy saving and the structural response reduction.
KEY WORDS: Hybrid control system; Optimization; Tall buildings; Wind-excited vibrations.
1
INTRODUCTION
Many recent studies demonstrated that the wind-induced
rotational motions in high-rise buildings can be relevant in
some cases [1]. In particular, the increased accelerations near
the building perimeter may cause inhabitants discomfort also
compounded by the increased awareness of the motion for the
occupants, provided by the visual cue of the rotating horizon
[2]. Moreover, the current tendency towards the increase of
the buildings’ height and towards the realization of structures
with irregular shapes makes the serviceability conditions a
crucial problem for the design [3].
In order to satisfy serviceability limits, i.e. to reduce
structural accelerations, many high-rise building are equipped
with passive, semi-active or hybrid control systems [4-6]. The
most common ones are passive devices, such as tuned mass
dampers (TMDs) [7], that do not require power supply.
However, these systems are well-known to be subjected to
frequency mistuning which might strongly weaken their
effectiveness in practical applications. Multiple tuned mass
dampers can partially circumvent this drawback [8], but their
application in tall buildings is not free from a degree of
complexity. A quite promising approach, able to effectively
control both flexural and torsional motions in tall buildings, is
to use hybrid control systems [9], such as active tuned mass
dampers (ATMDs), which share the advantages of the active
control, needing a lower actuation power with respect to the
purely active systems, with the capability of working as
passive systems when power supply is missing.
Although it is true that semi-active and hybrid control [10]
allow, in principle, to solve most of the drawbacks that have
limited the applications of active systems until today, some
aspects still need to be investigated. On this respect, one main
concern in the field is the overall reliability of the control
systems which must properly operate even in the case of
extreme events. In this viewpoint, the role of physical limits,
such as actuators saturations and limited stroke extensions of
inertial actuators, cannot be neglected, as they were usually in
the past. Two are the most common approaches in the
literature to deal with these issues: (i) to account for physical
limitations directly in the control algorithms; (ii) to design the
control systems in some evolved way considering the physical
limits as design constraints. The former approach, followed
for instance in [11-15] using the theory of nonlinear control, is
probably more suited for earthquake engineering applications
where the input severity is hardly predictable and there is the
need of conceiving a system which is required to be effective
also when physical limitations are reached in order to prevent
the structure from attaining an ultimate limit state. The latter
approach [16], on the contrary, is probably preferable when
the control system is designed to prevent serviceability issues,
which is the case, for instance, of wind-excited vibration
mitigation in tall structures. In this case, the optimal design of
the control system plays a crucial role, while the control
algorithm can be chosen with some freedom among the classic
ones of linear control theory [17].
In this paper, a methodology for the optimal design of a
hybrid control system using a three degrees of freedom
(DOFs) ATMD for the flexural/torsional response mitigation
of tall buildings is presented. The reduction of the top corner’s
accelerations and the control forces are the competing targets
of the multi-objective optimization problem. In a general
framework of incomplete number of measurements [18], the
design variables are the accelerometers’ positions over the
building height. Also the LQR performance index is
optimized to balance the need of energy saving and structural
response reduction. To reduce the computational effort,
subsequent optimization sub-procedures are employed that
take advantage of the genetic algorithm to find the solution of
the nonlinear, constrained, multi-objective optimization
problem.
2
THE HYBRID CONTROL SYSTEM
The hybrid control system considered, without loss of
generality, in this paper, is made of an ATMD with 3 DOFs
(Figures 1,2). The device is composed by a rigid body with
translational and rotational mass, placed on the top floor of a
tall building. The structure is schematized as a simplified
dynamic system having 3 DOFs for each floor. The total
number of DOFs, including those of the ATMD, is 3n+3
where n is the storeys number. The control forces produced by
the four actuators in the x and y directions, named as
u1x , u1y , u2 x , u2 y , result in two translational forces Fx, Fy and a
torsional moment Mt applied at the elastic center, G, of the
ATMD which coincide with the elastic center of the structural
storey.
Figure 2. Schematic representation of the ATMD location at
the top floor of the tall building.
The control forces are:
u1x  Fx 2  M t e

u2 x  Fx 2  M t e
u  u  F 2
2y
y
 1y
(1)
where e is the distance between each control device acting
along the x direction and the elastic center G of both the
structural storey and the ATMD.
Under the hypothesis of neglecting aeroelastic effects, the
classic equations of motion of the structure-ATMD system are
written in second-order form as:
M s q  Cs q  Ks q  f  B0u
(2)
where q is the vector of generalized displacements of the
structural-TMD system, having dimension 3n+3, M s , Cs and
K s are the mass, damping and stiffness matrices, respectively,
f is the vector of wind loads, u is the vector of control
forces which, in the present case, is defined as:
u   Fx
M t  T
Fy
(3)
B0 is a convenient collocation matrix and a dot denotes time
derivative.
The state space formulation of the equation of motion of the
actively controlled system is stated from Equation (2) as
follows:
z  Az  Bu  Hf
(4)
where z  q q is the state vector, A is the system matrix, B
T
Figure 1. Schematic representation of the ATMD’ location
over the tall building’s height.
and H are the location matrices for the vectors u and f ,
respectively.
Owing to the common availability of accelerometers as
monitoring sensors, tracking of the state by means of a state
observer using only acceleration measurements is here
considered. In particular, a limited number of storeys are
instrumented with 3 sensors per floor in order to measure the
alongwind, acrosswind and torsional accelerations. The
output, y , thus results in a linear combination of generalized
nodal accelerations, as:
y  Ca q
(5)
where Ca is a convenient matrix that selects the monitored
DOFs.
After well-known computations, Equation (5) can be
rewritten in terms of state vector and control forces as:
y  Cz  Du  Hf 
(6)
where:
C  Ca  M s 1 K s
M s 1Cs 
D  Ca M s 1 B0
(7)
and v is the vector of measurement noise.
The linear optimal control algorithm is used for the problem
at hand. The linear quadratic performance index can be
written as:
J    zT Qz  uT Ru 
t
0
histories and from the real-time measurement of the mean
wind velocity and direction at the reference height.
In this way, the wind process is correctly reproduced in the
numerical calculations, which allows an indirect check of the
effectiveness of the Kalman observer for tracking the state in
wind excited vibrations of tall buildings.
3
OPTIMAL DESIGN OF THE CONTROL SYSTEM
3.1
Statement of the problem
In a general framework of incomplete number of
measurements and limited power supply, the optimization
problem is aimed at designing the hybrid control system
which gives the best trade off between structural response
reduction and energy saving, with due account given to the
physical limitations of the actuators.
In order to minimize the computational effort, this goal is
achieved through subsequent optimization steps (Figure 3):
1) Optimal TMD tuning;
2) Optimal calibration of the LQR performance index;
3) Optimal choice of the sensors’ location.
(8)
where Q and R are the weighting matrices of the state vector
and the control forces vector respectively. By application of
the classic LQR algorithm the optimal gain matrix K, which
allows minimizing the performance index J in Equation (8), is
computed and the feedback is calculated as u   Kz .
To provide an estimate, zˆ , of the state from the incomplete
measurement set, a classic Kalman filter is used. Accordingly,
the equation of the state observer reads as:
zˆ  Azˆ  BKzˆ  L  y  Czˆ  DKzˆ 
(9)
where L is the optimal Kalman gain matrix. In Equation (9),
calculation of the feedback using the state estimate should be
noticed. For fast time integration, Equations (4), (6) and (9)
are readily converted in compact form using the augmented
state  z
zˆ  as:
T
z  A
 
 zˆ   LC
 BK
  z   Hf 

A  BK  LC   zˆ   0 
(10)
The computation of the Kalman filter gain matrix L ,
requires the hypothesis that both measurement and process
vectors are realization of white Gaussian stochastic processes.
Here, the measurement noise, v , is assumed to satisfy such
hypothesis and its covariance matrix Rn  E vvt  is directly
assigned. The wind process, on the contrary, is non-Gaussian
and non-white but, for the purpose of computing matrix L in
Equation (9), the hypothesis of a white Gaussian wind process
is retained. Given the practical difficulty of directly measuring
the cross-correlation of the wind forces f, due to the high
number of anemometers required, the covariance matrix
Qn  E  f f t  may be obtained from simultaneous wind
tunnel measurements of the pressure coefficients time
Figure 3. Outline of the optimization procedure.
3.2
Optimal TMD tuning
The parameters of the passive TMD are adjusted to
minimize the response of the controlled modes. In particular,
according to the solution proposed by Warburton [19] for
i
random excitation, the optimal tuning ratio,  opt
, of the i-th
mode is:
i
 opt

i
TMD

Si
1  i / 2 
(11)
1  i
and the corresponding damping ratio is:
i
 opt

i 1  3i / 4 
1  i 1  i / 2 
(12)
i
where TMD
is the i-th TMD’s angular frequency, si is the i-
i
th angular frequency of the structure, i  mTMD
M Si is the
i
mass ratio, mTMD
is the i-th generalized mass of the TMD and
M Si is the participating mass of the selected mode. In order to
control the flexural mode, the stiffness kTMD and the damping
1
1
and  opt
for a fixed
cTMD of the TMD are computed from  opt
value of the mass ratio 1 .
To control the structural rotations, the TMD must also be
tuned to the torsional mode with an iterative procedure as
follows:
- a trial value of the rotational mass ratio 3j is fixed;
-
3
the tuning ratio  opt
, j is computed according to
-
Warburton (11);
3
the angular frequency TMD
, j and the mass moment
j
of inertia JTMD
 kTMD
 
2
3
TMD
are computed;
 i  1, 2
j  x, y 
(15)
where ui j are the control forces, umax is the upper bound of
the control forces, qTMD, ij are the strokes of the TMD in the
directions x and y, qTMD,max is the upper bound of the TMD
strokes. The upper bounds umax and qTMD ,max depend on the
technical characteristics of the selected control devices. To
keep into account the constraints, a penalty function P  q, f 
is added to the objective function when the constraints are
violated so that the solution may be discarded.
-
the mass ratio 
-
the procedure is repeated with the new value 3j 1
3.4
until 3j 1  3j  tol , with tol assigned tolerance;
The third phase of the optimization procedure leads to the
optimization of the sensors locations along the height of the
building.
In this optimization sub-problem the multi-objective
function, f 2  p  , to be minimized is:
-
j 1
3
 JTMD M
3
S
is computed;
3
the damping ratio  opt
is computed from Equation
(12) in correspondence of the final value 3j 1 .
3.3

ui j  umax


qTMD ,i j  qTMD ,max
Optimal choice of the sensors locations
Optimal calibration of the LQR performance index
The optimal calibration of the weight matrices R and Q
applied to the state vector and the control forces in the LQR
performance index, Equation (3), is achieved through an
optimization procedure. In this optimization sub-problem the
multi-objective function, f1  k  , to be minimized is:
  q 
  u 
f2  p  
  q 
i  x, y
i
observed
  q 
i  x, y
i
 P  q, u 
(16)
active
where the design variable p is a (m x 1) location vector with m
number of instrumented floors,  i  x , y   qi  observed is the sum
 P  q, u  (13)
of the standard deviations of the top corner accelerations of
the actively controlled system instrumented with a limited
number of sensors,  i  x , y   qi  active is the sum of the
where C1, C2 are the weighting coefficient of the objective
function,  i  x , y   qi  active is the sum of the standard
standard deviations of the top corner accelerations of the ideal
actively controlled system.
The location vector p contains numbers from 1 to n, with n
equal to the total number of storeys, to identify which are the
instrumented floors:
f1  k   C1 
i
i  x, y
  q 
i  x, y
i
active
passive
 C2 
i  x, y
j 1,2
ij
  e 
i  x, y
ij
deviations of the top corner accelerations of the actively
controlled system,  i  x , y   qi  passive is the sum of the
standard deviations of the top corner accelerations of the
passively controlled system,  i  x , y   uij  is the sum of the
j 1,2
standard deviations of the control forces,

i  x, y
  eij  is the
sum of the standard deviations of the elastic restoring forces
of the TMD and P  q, f  is the penalty function.
The matrix Q is assumed to be equal to the identity matrix
while the matrix R is the product between the identity matrix
and a coefficient k that is the design variable of this subproblem. In order to reduce the computational effort, a lower
and an upper bounds are set to the design variable:
kmin  k  kmax
(14)
where the bounds kmin and kmax are assigned on the basis of a
preliminary sensitivity analysis.
The non-linear constraints to the sub-problem are the
following:
1  pi  n
i  1,..., m
(17)
Three accelerometers are arranged over each instrumented
floor to measure accelerations along the 3 DOFs.
During the optimization process, the trial components of the
vector p which are real numbers are rounded to the closer
integer.
The non-linear constraints to the sub-problem are
expressed by Equation (15).
3.5
Optimization algorithm
The problem at hand is a nonlinear constrained optimization
problem for which the objective function cannot be written as
an explicit function of the design variables. For this reason, it
is not possible to analytically compute the first and second
derivatives of the function to be minimized and gradient based
methods cannot be used.
The proposed procedure is based on the use of the genetic
algorithm that does not require any information about the
gradient of the objective function but is based on random
generation of trial solutions [20, 21].
In particular, the genetic algorithm is an optimization
method that is based on natural selection, the process that
drives biological evolution. It repeatedly modifies a
population of individual solutions selecting at each step
individuals at random from the current population to be
parents and using them to produce the children for the next
generation. Over successive generations, the population
evolves toward an optimal solution using techniques inspired
by natural evolution such as selection, crossover and
mutation.
4
4.1
NUMERICAL EXAMPLE
Description of the structure
The proposed optimization procedure was applied to a square
tall building 180 m high with an aspect ratio of 6.
The structure is made of steel with central cores and
systems of bracings in both the principal directions. Floors are
reinforced concrete slabs capable of warranting a rigid inplane behavior. The structure was modeled as a simplified
dynamic system having 3 DOFs for each floor obtained by
static condensation from a finite element model of the
structure (Figure 4).
profiles and turbulence intensity corresponding to suburban
terrain conditions.
To make the pressure time histories measured in the wind
tunnel representative of the real phenomenon, it was chosen to
respect the similitude criterion on the reduced frequency, as it
is usually done for tall buildings.
4.3
Results
In the present section the following cases will be analyzed:
uncontrolled case, passively controlled case, ideally actively
controlled case and control-with observer case. In the ideal
active case, the entire state is assumed to be known, while in
the control-with observer case it is estimated by means of the
state observer presented in Section 2.
The first step of the optimization process was the optimal
tuning of the passive control system to the first flexural and
torsional modes as explained in Section 3.2. The optimal
values of the mass, tuning and damping ratio are reported in
table 1.
Table 1. TMD optimal parameters.
Mode
No.
1 (Flexural x)
2 (Flexural y)
3 (Torsional)
Mass ratio

Tuning ratio
 opt
Damping
 opt
0.020
0.019
0.001
0.985
0.986
0.999
0.070
0.069
0.866
The modal characteristics of the uncontrolled structure and
the passively controlled structure are summarized in Table 2.
Table 2. Modal characteristics of the uncontrolled structure
and the structure with TMD.
Mode
1
2
3
Figure 4. Structural model of the analyzed system.
4.2
Wind load modeling
The forcing functions representing the wind load were
obtained from synchronous pressure measurements carried out
in the wind tunnel. Experimental tests were carried out in the
boundary-layer wind tunnel operated by CRIACIV (Interuniversity Research Center on Buildings Aerodynamic and
Wind Engineering) in Prato, Italy. The rigid 1/500 scale
model of the building was instrumented with 120 pressure
taps, 30 for each side. Tests were carried out with wind speed
Frequency (Hz)
uncontrolled
0.208
0.215
0.287
Frequency (Hz)
with TMD
0.192
0.195
0.222
The second step of the optimization procedure was the
calibration of the LQR weighting coefficients, that is, matrices
Q and R in Equation (8). In this case, the choice of the control
actuators and the maximum extensions of the strokes of the
ATMD are the main control constraints to be satisfied. Indeed,
in principle, there would be essentially no upper limit to the
control effectiveness of the system if these constraints were
not accounted for. Nevertheless, the obvious counterparts of a
large control effectiveness are large strokes extensions and
large control forces. Hence, the aforementioned physical
limitations dictate, in practice, the maximum achievable
control effectiveness. The control constraints considered in
this work are summarized in Table 3.
Table 3. Constraints of the hybrid control system.
Maximum control forces
[kN]
Maximum strokes
[m]
u1x , u2 x
u1 y , u2 y
qxATMD
q yATMD
800
800
3.0
3.0
It was decided to consider unit weights for all generalized
structural displacements and their time derivatives. Moreover,
a preliminary analysis showed the convenience of applying
small penalties also to the displacements of the ATMD, in
order to satisfy the stroke limitation.
The weighting coefficients of the objective function C1 and
C2 expressed by Equation (13) are assumed equal to 1 and 0.2
respectively, in order to privilege the response reduction with
respect to the control force limitation.
Then, by applying the optimization procedure described in
Section 3.3, the optimal value of the coefficient k appearing in
Equation (14) was obtained. The results are summarized in
Table 4.
J
  qcon     quncon 
  quncon 
where   qcon  is the standard deviation of the top corner
acceleration obtained with the controlled system and
  quncon  is the standard deviation of the top corner
acceleration obtained with the uncontrolled system. The
performance indexes are reported in Table 5 with reference to
the top corner’s accelerations in x and y directions and to:
- the passively controlled system;
- the ideal actively controlled system;
- the actively controlled system with state observer.
Table 4. Results of the calibration of the LQR coefficients.
qxATMD
q yATMD
qrATMD
k
0.6
0.7
0.6
10-12
1
qx [m/s2]
0.5
0
:
-0.5
-1
200
300
t [s]
400
500
600
1
0.5
qy [m/s2]
Acceleration x
(%)
- 26.5
- 38.3
- 37.7
Passive
Active ideal
Active with observer
In Figure 5 are shown the time histories of the x and y
components of the acceleration at the corner of the top floor
for the cases of uncontrolled structure, passively controlled
structure and ideal actively controlled structure.
100
Table 5. Performance indexes J .
Control
penalty
State penalties
0
(18)
0
Acceleration y
(%)
- 28.9
- 48.4
- 47.9
It can be noticed that the passive solution proves to be quite
effective in reducing the structural response. However, as it is
well-known, such a result is not robust in the sense that the
almost unavoidable mistuning of the device would strongly
impair its control effectiveness. Hence, the need for an
upgrading towards the hybrid approach clearly points out. The
slightly improved performance of the hybrid system with
respect to the passive one should therefore be interpreted in
this viewpoint. It should be also observed that the technical
difficulties in making the TMD devices effective both in
translation and in torsion may lead to the design of a control
system in which the torsional response is mitigated only by
the active forces.
A minor remark concerns the effectiveness of the active
system using acceleration information from an incomplete
measurement set. Indeed, it was observed that, already with a
relatively small number of sensors, the effectiveness of the
control system was essentially similar to the ideal case. On
this respect, Figure 6 shows a comparison between the ideal
active response and the one with state observer considering an
instrumented storey every four storeys.
:
-1
0
100
200
300
t [s]
400
500
600
Figure 5. Accelerations in the x and y directions at the corner
of the top floor: uncontrolled (gray), passively controlled
(black), actively controlled (red).
In order to quantitatively represent the effectiveness of the
control system in reducing the structural accelerations, a
performance index J was defined as follows:
qy [m/s2]
-0.5
0.5
0
-0.5
0
100
200
300
t [s]
400
500
600
Figure 6. Comparison between ideal active response (red) and
response with state observer (black).
It should be noticed that the overall optimization of the
control parameters pursued in this example depends upon the
severity of the external input. In other words, different weight
coefficients would be obtained under different levels of
external excitation. This aspect could be solved by means of a
gain scheduling approach, as proposed, for instance, in [14]
for earthquake engineering applications: a similar approach
goes beyond the purposes of the present investigation but shall
represent its natural future development.
Table 6. Optimal position of the accelerometers along the
building’s height.
Sensor no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
ATMD x stroke [m]
3
2
1
0
-1
-2
-3
0
100
200
300
t [s]
400
500
600
ATMD y stroke [m]
3
2
1
0
-1
-2
-3
0
100
200
300
t [s]
400
500
600
100
200
300
t [s]
400
500
600
Actuator x force [kN]
300
200
100
0
-100
-200
-300
0
Actuator y force [kN]
200
100
0
-100
-200
-300
0
100
200
300
t [s]
400
500
600
Figure 7. ATMD strokes along the x and y directions (top)
and corresponding actuators forces (bottom).
Heigth (m)
6
18
21
42
51
72
75
87
108
111
120
135
144
147
153
156
165
168
171
180
183
In Figure 7 are shown the strokes of the ATMD along the
principal x and y directions and the corresponding forces of
one actuator. It can be noted that the limits summarized in
Table 3 are never exceeded although stroke limitation
appeared to be a rather more severe constraint than actuator
force saturation.
The final step of the proposed design methodology is to
optimize the position of the monitoring sensors by applying
the procedure presented in Section 3.4. This analysis, in the
present case, led to the results presented in Table 6. It should
be noticed that, in the optimal solution, sensors are placed
almost uniformly along the height to represent the motion of
the first three modal shapes which are, for the symmetric
building chosen as case study, nearly linear along the height.
The genetic algorithm showed to be effective and fast in
providing the optimal set of design variables and the solution
proved to be quite stable with respect to the initial guess
population.
5
300
Floor no.
2
6
7
14
17
24
25
29
36
37
40
45
48
49
51
52
55
56
57
60
61
CONCLUSIONS
A general procedure for the optimal design of a hybrid
control system for the mitigation of the wind-induced
flexural/torsional response of tall buildings has been presented
in this paper.
The considered control actuator is represented by an ATMD
with 2 longitudinal and 1 torsional DOFs.
The chosen control algorithm is the classic LQR
complemented with a Kalman observer for state tracking
using only acceleration measurements.
The proposed formulation allows to optimize the weight
parameters of the controller as well as the position of the
monitoring sensors, under appropriate constraints represented
by the physical limitations of the control actuators. Indeed, the
choice of handling these limitations offline, i.e. as “a priori”
optimization constraints, was found to be quite appropriate for
applications where the structure must be preserved from
serviceability issues.
The proposed methodology has been also applied to a
numerical example in order to show its effectiveness.
The following are the main observations of this work:
- The adopted 3 DOFs passive TMD, if properly
tuned, is seen able to strongly reduce the wind
excited structural response;
- The improvement of control effectiveness achieved
by means of the hybrid approach, for realistic values
of the control constraints, with respect to the optimal
passive case is significant although not extraordinary.
However, this improvement should be regarded as
substantial in the sense that the hybrid control
solution is more robust and does not suffer from
mistuning issues;
- In the optimal design procedure, stroke limitation
appeared to be a rather more severe constraint than
actuator force saturation;
- Although the wind forces acting on the structure are
neither Gaussian nor white (in the present case they
were extracted from wind tunnel tests), the classic
Kalman state observer is seen to be quite effective
for the chosen application;
- The proposed procedure is seen to be a quite
effective tool for the optimal design of the control
system. Particularly, its ability and in yielding the
optimal positions of the sensors is noteworthy;
- As the main future developments of this work,
different levels of input severity
shall be
incorporated in the formulation using a gain
scheduling approach. Also, the robustness against
random variations of structural parameters shall be
investigated.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the collaboration of
Davide Pauselli in performing the numerical analysis
presented in this work.
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