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Aerospace 2024, 11, 493. https://doi.org/10.3390/aerospace11060493 www.mdpi.com/journal/aerospace
Article
A Smart Wing Model: From Design to Testing in a Wind
Tunnel with a Turbulence Generator
Ioan Ursu 1, George Tecuceanu 1,†, Daniela Enciu 1,*, Adrian Toader 1, Ilinca Nastase 2, Minodor Arghir 1
and Manuela Calcea 1
1 National Institute for Aerospace Research “Elie Carafoli”–INCAS Bucharest, 061126 Bucharest, Romania;
ursu.ioan@incas.ro (I.U.)
2 CAMBI Research Center, Building Services Department, Technical University of Civil Engineering
Bucharest, 66 Avenue Pache Protopopescu, 021414 Bucharest, Romania
* Correspondence: enciu.daniela@incas.ro
† Eng. George Tecuceanu has passed away.
Abstract: The paper concerns the technology of the design, realization, and testing of a flexible smart
wing in a wind tunnel equipped with a turbulence generator. The system of smart wing, described
in detail, consists mainly of: a physical model of the wing with an aileron; an electric servomotor of
broadband with a connecting rod-crank mechanism for converting the rectilinear motion of the
servoactuator into the aileron deflection; two transducers: an encoder for measuring the deflection
of the control aileron and an accelerometer mounted on the wing to measure its bending and
torsional vibrations; a procedure for determining the mathematical model of the wing by
experimental identification; a turbulence generator in the wind tunnel; implemented
and
LQG algorithms for active control of vibrations. The aenuation experimentally obtained for the
aeroelastic vibrations of the wing, but also for those accentuated by the turbulence, reaches values
of up to 50%.
Keywords: climatic changes; smart wing; electric broadband servomotor; turbulence generator;
experimental identification; wind tunnel tests; active vibration control;
robust control; LQG
control
1. Introduction
1.1. Directions for Aircraft Wing Vibrations Aenuation and Flight Envelope Expansion
The tendency of modern aviation is to create lighter and more flexible aircraft to
accomplish the new regulations regarding green technology
(hps://www.tandemaerodays19-20.eu/ accessed on 18 June). These specific requirements
come at some cost: the stability of the flexible structures, as in the case of an airplane wing,
is strongly influenced by becoming more vulnerable to the action of forces. As a response
to these potentially destabilizing issues, solutions to mitigate the vibrations induced by
turbulence and to expand the flight envelope across the limit of fluer occurrence, as an
indirect consequence, are in the aention of the engineering community.
A main debating point present on the agenda of major aeronautical forums, see the
aforementioned Tandem AERO days 19–20 Congress, is represented by the more and
more accentuated climatic changes that put pressure in the aerospace industry [1]. Rising
temperatures, especially in the tropopause area where jet streams meet, the destruction of
the ozone layer, and the odd changes in weather paerns affect aircraft infrastructure and
performance, leading to flight cancellations, delayed or rerouted flights, increased fuel
consumption, and high costs dedicated to aircraft maintenance. For example, hundreds
of millions of dollars per year were spent only in the USA due to this issue [2–4]. The
Citation: Ursu, I.; Tecuceanu, G.;
Enciu, D.; Toader, A.; Nastase, I.;
Arghir, M.; Calcea, M. A Smart Wing
Model: From Design to Testing in a
Wind Tunnel with a Turbulence
Generator. Aerospace 2024, 11, 493.
hps://doi.org/10.3390/
aerospace11060493
Academic Editor: Rosario Pecora
Received: 20 May 2024
Revised: 14 June 2024
Accepted: 18 June 2024
Published: 19 June 2024
Copyright: © 2024 by the authors.
Submied for possible open access
publication under the terms and
conditions of the Creative Commons
Aribution (CC BY) license
(hps://creativecommons.org/license
s/by/4.0/).
Aerospace 2024, 11, 493 2 of 26
worrying effect that accompanies climate change is the increased occurrence of areas with
atmospheric turbulence, especially of Clear-Air Turbulence (CAT) [5,6]. This is the most
dangerous type of turbulence, both for passengers and cabin crew, as well as for the
aircraft, because it is found in areas without clouds, areas that do not announce such an
atmospheric phenomenon, which is an element of surprise for pilots. They are generally
present at very high altitudes in the tropopause region, where jet streams are encountered
and most civil aircraft fly [2]. Accidents, not necessarily catastrophic but causing severe
panic and even injuries to passengers and cabin crew, caused by the unexpected entry of
the plane into CAT, are more and more common in civil aviation. In fact, no less than 91
CAT events were recorded between 01 April 2018 and 25 June 2021 [7].
Recent efforts are focused on the discovery of atmospheric paerns through which
CAT areas can be identified. Unfortunately, at present, this problem is still unresolved, as
CAT cannot be identified by using LIDAR or other known methods [8,9]. However, there
are some promising results after a long research for over 50 years (see [10]). Note that in
the literature there are some references regarding actions to prevent the injuries of
passengers and cabin crew and less related to the technical part of counteracting the effects
induced by the strong vibrations produced by the turbulence; see also [11, 12].
Another significant benefit related to vibration aenuation is that the presence of an
active control system allows the calculation of an increased flight envelope right from the
design phase, which means an increased aircraft performance, whether it is commercial
or military.
A phenomenon that can produce strong vibrations with everything that comes from
here, even catastrophes, is the fluer, usually that of control surfaces. It consists of self-
sustaining, unstable oscillations whose amplitude grows strongly in a short time,
accumulating energy in the structure. Beyond a threshold, the vibrations can no longer be
dampened, the amplitude increases rapidly, and fluid energy is accumulated in the
structure, leading to extremely violent reactions such as the breaking down of some
aircraft parts or severe injuries to the pilot. The unpredictable, nonlinear, and chaotic
character of the fluer makes the study of this phenomenon a complex and difficult
process to approach. This unstable vibration has been carefully studied since the origins
of aviation, but the causes are neither fully identified nor fully controlled; see [13]. For the
history of the fluer phenomenon, see [14, 15].
The authors would like to mention, based on their own experience in the field, the
risk of triggering the fluer if the impedance (also called dynamic stiffness) of the
hydraulic servomechanism in the flight control chain has negative damping [16–19]. On
24 November 1977, an IAR 93 Eagle aircraft crashed as a result of the left elevator breaking
off due to tail fluer (see [20]). The negative evaluation of the aeroservoelastic
compatibility of the hydraulic servomechanism BU-51MS retrieved from aircraft MIG 21
out of use had already been expressed in an internal report [21], but it was ignored by the
decision makers, given the lack of specific requirements about impedance in the Aviation
Publication (AvP) 970 dated 1959. (As an irony of fate, the express requirement on the
impedance function of the servomechanisms from the primary flight controls was
introduced a few years later in the updated version of that Regulation). The consequence
of the catastrophe was to replace the initial hydraulic servomechanisms of the flight
controls, which were improperly designed from an aeroservoelastic point of view, with
hydraulic servomechanisms manufactured in collaboration with Dowty Group [22].
It should be added that the weakness of the BU-51MS servomechanism configuration
was not repeated in the case of the Romanian SMHR servomechanism that equips the
ailerons of the IAR 99 Hawk aircraft, according to [23]. The lesson of IAR 93 was well
learned.
There is a huge bibliography related to the active control of the fluer and vibrations
of the aircraft in general, the influence of servoactuators, the gust loads and gust load
alleviation, the active control techniques, the programs and tests in wind tunnels etc., [24–
27]. A comprehensive presentation of the state-of-the art and technology maturation needs
Aerospace 2024, 11, 493 3 of 26
in the field of aircraft active fluer suppression is given in detail in the paper [28]. It should
be noted that just a few of the 703 references concern vibration aenuation. Instead, most
of the 126 references in [29] concern active vibration suppression of plate-like structures,
representative of aerospace structures, with piezoelectric actuators.
Another phenomenon that can cause catastrophes is the phenomenon of pilot-
induced-oscillations (PIO) in the closed loop with the aircraft controls. PIO is
homologated as factual if there is at least one measurable state of the aircraft that is 180°
out of phase with at least one measurable pilot control input [30].
This paper presents the results of a complex project that was aimed at achieving a
demonstrator test in the Subsonic INCAS wind tunnel (WT) [31,32] for active vibration
control. The controlled system, tested in WT in a turbulent environment by a turbulence
generator (TG), is an elastic smart wing with aileron (Figure 1). A smart system
incorporates functions of sensing, actuation, and control in order to describe and analyze
a situation and make decisions based on the available data in a predictive or adaptive
manner [33]. A fine distinction must be made between a smart system and an artificial
intelligence-based system, or simply, an intelligent system, which must contain one or
more of the following techniques: artificial neural networks, fuzzy logic, and genetic
algorithms. For applications in aviation systems, see [34,35].
The control exercised by means of the aileron is executed with a broadband electric
servo actuator. The approach is quite different from those existing in the literature. Thus,
paper [24] is the one that introduces a series of control-active applications in which a
piezoelectric V-stack actuator is used. Active control, thought of as positive position
feedback, is applied to a rigid wing supported on springs, simulating the pitch and plunge
degrees of freedom of the wing system. In the paper [36], the control law is given by the
same positive position feedback, a totally atypical approach compared to the mainstream
in control that has become prevalent since the mid-80s and that is based on the already
classical package MATLAB (Version 10.4 ) (R2018a)Control Toolbox of the LQG robust
control generations. Until, in last few years, the scene has begun to be dominated by
artificial intelligence techniques. The approach for increasing the fluer speed presented
in [37] is similar to that in [36] by using the same type of actuator and positive position
feedback and obtaining a 20% increase in the fluer speed.
There is a long dispute in the field of active control between the supporters of
piezoelectric actuators and the supporters of electric actuators. This is also the case with
aircraft flight controls, with fans of hydraulic servomechanisms and those who vote for
green aviation and all-electric aircraft, with electromechanical servomechanisms.
In order to be able to evaluate the arguments a bit, let us say that, on the one hand,
we have the actual benefits of using piezoelectric actuators for aeroelastic vibration
aenuation: increased control bandwidth, mechanical simplicity, lack of control lag, and
the nonintrusive nature of flat piezoelectric actuators. However, their major drawback is
the additional heavy, bulky, and expensive hardware required to power piezoelectrics,
that is, amplifiers of the order of hundreds of volts. On the other hand, the usage of
electrical actuators, such as DC motors, will bring together multiple benefits, such as their
lightweight, the fact that they can be easily integrated into a system, and the necessity of
less power than any smart materials, such as piezoelectric actuators.
In the present paper, it is shown how a competitive bandwidth with that of a
piezoactuator is obtained by adding an internal feedback loop PD (Proportional-
Derivative) to a coil linear actuator.
Aerospace 2024, 11, 493 4 of 26
Figure 1. Up: left: elastic smart wing with aileron and encapsulated electric actuator and
accelerometer; middle: turbulence generator (TG) mounted in the wind tunnel and right: fluer test
in WT; down: layout and TG dimensions; right: hybrid model for the study of turbulent flow: Stress-
Blended Eddy Simulations [38].
1.2. Motivation of the Research
There were several reasons for writing this paper, as follows:
Airlines frequently report turbulence during flights, with or without injuries to
passengers, cabin crew, or damage to aircraft.
Costs of maintenance, fuel, infrastructure, cancellation, and rerouting of the flight
increase as turbulence often occurs.
The complex problem of turbulence remains an unresolved issue and still brings
challenges in the field of scientific research [39].
Lack of method for identifying CAT leads to the development of ways to combat the
effects induced by turbulence by controlling and reducing vibrations.
The desire of the aeronautical companies is to make long-distance flights in the
shortest possible time, which can be achieved by increasing the flight envelope
inclusively through vibration control.
In general, the design of aviation wind tunnels was based on the outdated idea that
aircraft usually fly at altitudes of kilometers where the degree of turbulence is very
low, which is true only if the phenomenon of CAT is ignored.
Also, the paper is based on results obtained over time in projects and comes as a
development of those results [40–42].
The goal assumed in the project was to apply simple and efficient solutions, both in
terms of hardware and software.
1.3. Contributions
Elaboration of a complex procedure for active vibration control of an elastic model of
the wing with aileron in the presence of turbulence generated in the wind tunnel,
based on a methodology of simple experimental identification of the open loop
system in the frequency domain
Designing an elastic physical wing model displaying a given set of basic natural
(modal) frequencies
Designing an electric servo actuator consisting of a moving coil linear actuator and a
crank-type mechanism
Aerospace 2024, 11, 493 5 of 26
Developing an algorithm for tuning the PD internal feedback loop of a servo actuator
to increase the bandwidth
Designing a passive turbulence generator in the wind tunnel, with the important
property that the achieved degree of turbulence does not depend on the value of the
air speed V upstream of the generator
Developing a procedure for system mathematical model identification
Reaching a vibration reduction of about 18 dB on the basic 5 Hz modal frequency for
both control laws LQG and a competitive performance with other achievements
described in the literature of the field.
The paper is organized as follows: Section 2 describes in detail the hardware
components of the smart wing system: the physical model of the wing with aileron, the
synthesis of the electric servo actuator with increased bandwidth, the transducer system,
the synthesis of the turbulence generator, and the LQG and active vibration control laws.
Section 3 presents a software-type component of the smart wing in subsonic WT, namely
the procedure for determining the mathematical model of the wing by experimental
identification. Section 4 reviews the active control laws LQG and static weights. Section 5
details the results of the active control tests. Section 6 ends the work with some concluding
remarks.
2. Smart Wing System with Active Control
The smart wing system described in this paper consists of (1) a physical model of a
wing with aileron, (2) an electric broadband servoactuator with a connecting rod-crank
mechanism (CR-CM) for converting the rectilinear motion of the servoactuator into
aileron deflection, (3) two transducers: an encoder for measuring the deflection angle of
the aileron, and a wing-mounted accelerometer so as to capture the bending and torsional
vibrations, (4) a Turbulence Generator (TG) in the wind tunnel (WT) required to test the
proposed procedure, Figure 1, right, (5) a procedure for determining the mathematical
model of the wing by experimental identification, and (6) LQG and
with static
weights active control laws of vibrations.
In the following, an essential description of the elements composing the smart wing
system is given. Sections 3 and 4 are reserved for elements (5) and (6).
(1) The wing physical model has a special design in view of obtaining a realistic
elastic wing that, in principle can reproduce a set of vibration modes of an aircraft wing
in the preliminary, conception phase. This goal can be achieved relatively simply through
a trial-and-error procedure, including ANSYS analysis applied to the geometry of the wing
model. This approach differs radically from those widespread in the literature, where the
model is represented by a rigid wing supported on two springs to simulate the first two
vibration modes of bending and torsion [43]. Thereby, the wing is composed of a longeron,
Figure 2, covered by an aerodynamic layer (profile NACA 0012). The longeron is a 1 mm-
thick rectangular tube of aluminium (1200 × 120 × 25 mm3 ), provided with notches to
control its stiffness. At one end of the wing there is the aileron, and at the other end there
is a flange whose role is to fix the wing in the WT.
The elements defining the aerodynamic surface were accomplished from woodchips
or resin ROHACELL 71S. One design requirement was to ensure the occurrence of fluer
for the uncontrolled system in INCAS subsonic WT [32]. The tests performed in WT
established that the fluer speed was 41 m/s and the fluer frequency was 5.8 Hz [41].
Thus, the fluer speed of 41 m/s and the fluer frequency of 5.8 Hz were experimentally
measured, as the effect can be seen in Figure 1 (see also work [41]). These values were
neither “designed” nor intended as realistic values for any aircraft. The purpose of the
fluer tests was to develop an active vibration control methodology. Therefore, firstly, we
had to prove that fluer was actually a risk in our field of vibration damping tests, and
secondly, we had to prove that we could, under an actively controlled regime, rise above
this speed with the tests. In another context of interpretation, it is well known that fluer
Aerospace 2024, 11, 493 6 of 26
is an aeroelastic phenomenon, and its appearance is proof that the wing is elastic, therefore
realistic. Fluer cannot, in principle, appear on a rigid wing, as many vibration-damped
models in the literature take into account. The tests reported in the paper show that
vibration damping is convincingly illustrated, and these vibrations are simultaneously
aeroelastic and turbulent.
The modal frequencies of the wing model, measured in the laboratory, were 5.1 Hz
(bending), 17.4 Hz (torsion), 23.14 Hz, 41.85 Hz, and 49.41 Hz. In a previous stage, instead
of the electric actuator, a piezoelectric actuator was used for tests. With some minimal
details different from the current smart wing, the results from Table 1 were obtained.
Table 1. Natural frequencies determined by CATIA and measured, respectively, see Figure 2.
Mode
Number
Longeron Frequency [Hz]
CATIA Model
Wing Frequency [Hz]
CATIA Model
Wing Frequency [Hz]
Experimental Tests
1 13.59 6.23 5.93
2 43.42 10.21 11.70
3 44.12 20.83 22.73
4 59.14 26.32 -
5 78.91 29.83 -
Figure 2. Up: CATIA modal analysis; in the middle: laboratory modal tests (left: wing longeron;
middle: instrumentation setup for modal tests; right: wing); down; spectral analysis of wing.
Aerospace 2024, 11, 493 7 of 26
(2) The electric broadband servoactuator consists of (a) a moving coil linear actuator
NCC05-18-060-2PBS connected with (b) an atypical rod-crank mechanism converting the
linear motion of the actuator into an incomplete rotational movement of the aileron hinge
of about 20 degrees in both directions (in short, a connecting rod-crank mechanism, CR-
CM, Figure 3), and (c) an internal PD (Proportional-Derivative) feedback loop (Figure 4)
to transform the actuator into an angular displacement tracking actuator.
The connecting rod-crank mechanism. For proper operation, the CR-CM meets
several essential conditions: symmetry of movement and efforts in both directions of
movement; minimal clearances to avoid discontinuity of movement when changing
direction; minimal wear during operation. The disadvantages of a classic CR-CM are
eliminated by an atypical constructive solution, as follows: With reference to Figure 3, to
reduce the overall size of the assembly and maintain the linearity of the movement, firstly,
the connecting rod 1 (Figure 3) does not have a plane-parallel movement as in classic cases,
but only a translational back-and-forth motion. Secondly, given the fact that, through
mounting, the connecting rod 1 is stiffened with the actuator CR-CM at the opposite end
of the actuator 4, it is replaced by a slider that gives the crank 2 a rotational movement in
both directions. Pair 1–2 supports high contact pressures, being built on radial bearing
rollers; therefore, if a cambered aerofoil is used instead of the NACA0012, the load
asymmetry on the aileron will not pose problems to the CR-CM and the smart wing
system.
Figure 3. The connecting rod-crank mechanism (CR-CM). Up: details with the actuator installed;
middle left: actuator-CR-CM assembly: rod 1; crank 2; actuator shaft 3; actuator 4; middle right:
connecting rod assembly with crank rollers 1 (translational slider); contact point “K” between the
driving roller and the driven rollers; down left: the mechanism of transforming linear motion into
rotation; down right actuator assembly (CATIA).
Aerospace 2024, 11, 493 8 of 26
Figure 4. Block diagram of the controlled smart wing; c
the servo actuator input signal generated
by the implemented control;
the deflection of the wing aileron;
C
implemented Control,
PD Proportional-Derivative,CS Controlled System. ( )
u
H s
incorporates LCAM and CR-CM;
see Figures 3.
Fourthly, the crank 2, integral with the flange shaft 5, in turn transmits the movement
to the wing, without the existence of other joints and supports other than those strictly
necessary for the operation of the assembly.
Fifthly, for the slider, in order to avoid backlash and ensure its reliability and
sustainability, a constructive solution was adopted in which the parts in contact on the
connecting rod and crank elements are bearing rollers (Figure 3, middle and Figure 3)
down, of current execution, specially intended for intensively demanded rotating joints
and which ensure minimal functional clearances.
In order to avoid clearances and ensure the reliability and sustainability of the crank
guideway, the solution from Figure 3 down left was adopted. The contact parts on the CR-
CM elements are bearing rollers, of current manufacturing, specially designed for the
highly demanded rotating joints and which ensure minimum functional clearances. In the
adopted solution, the bearing rollers are used for linear motion. Two identical rollers
were used, pressed into the base material of the crank (drive part). By the correct execution
of the holes in this part and the observance of the distance “D” between the axles, the
desired distance between the rollers is ensured. This distance must coincide with the
diameter of the roller “d” (Figure 3, middle right), provided to be pressed in the
connecting rod element, the driving element. The pair of rollers is chosen from the series
production of the bearing manufacturers. The axles of these two linear bearings allow
relative movement; the contact point “K” between the driving roller and the driven rollers,
not being fixed, is able to move along the rollers 1 (Figure 3 middle right). Thus, the
oscillation radius of the crank “R” is modified (Figure 3 down), which is necessary for the
operation of the mechanism.
The internal Proportional-Derivative (PD) feedback loop. A simplified scheme for
calculating CR-CM dynamics is shown in Figure 5. Note that point C slides in the OD
direction and rotates around point O in the plane
X,Y .
By A
F and R
F were noted
the active force, called also electromagnetic force, and the resistance force, respectively.
The two forces act on the aileron by means of a lever OC of variable length. As seen in
Figure 5, the perpendicular distance from the line of action of the force to the pivot is b,
0.02615m,
b as the mechanism was designed. The active force
A
F
is developed by
the linear actuator and is expressed as
A M
F K i
(1)
where
M
K
is the constant force of the actuator,
27.8 N A,
M
K and i represents the
intensity of the electric current in the actuator coil.
a
i K u
(2)
with u
the electrical voltage applied to the voltage-current amplifier (VCA) considered
here as a control variable and
1
a
K
is the amplification factor established in the design.
Aerospace 2024, 11, 493 9 of 26
VCA is included in the block diagram of the controlled system, CS, Figure 4. It follows
from (1) and (2) that
A a M
F K K u
.
(3)
R
F
is the resistance force consisting of inertia force and viscous friction force. This
force is associated with the rectilinear motion of a moving coil linear actuator and is
expressed as
1
R
F mx f x
(4)
where
m
is the mass of the movable actuator assembly,
0.12 kg
m and
1
f
is a
conventional viscous friction coefficient. The resistance moment associated with the
rotational movement of the aileron around the hinge shaft,
R
M
, is given by
2R
M I f k
(5)
with
I
the moment of inertia of the aileron,
2
f
the viscous friction coefficient (of
mechanical and aerodynamic nature), and
k
the modulus of elasticity of the elastic
aerodynamic force. The equation of the dynamic equilibrium of the moments with respect
to the hinge axle is
0
A R R
F b F b M
(6)
or, taking into account (3)–(5)
1 2
0.
a M
b K K u mx f x I f k
(7)
Consider the kinematic equation in the usual hypothesis of small angles
,
.x b
(8)
Therefore,
,
x b x b
(9)
and substituting in (7), it results in
2 2
2 1
.
a M
I b m k f b f bK K u
(10)
By defining the state variables 1
:
x
, 2
:
x
and noting
2 2
2 1
: , : ,F f b f I I b m
(11)
system (10) is rewrien in the form of a system of first-order differential equation
11
2
2
0 1 0
.
a M
xx
u
k F bK K
x
xI I I
(12)
From (12), the open loop transfer function from
u
to 1
x
is obtained (Figure 4)
2 2
( ) : .
a M
u
bK K K
H s
Is Fs k Is Fs k
(13)
A suitable internal PD loop is required to increase the servoactuator bandwidth
(Figure 4).
The servoactuator has the following closed loop transfer function from
:
c
u
to
2 2 2
/ /
( ) : .
/ / 2
P P
u
D P n n
K K I K K I
H s
s F K K s I k K K I s s
(14)
Aerospace 2024, 11, 493 10 of 26
The determination of the parameters
( , )
P D
K K
follows an analytical-experimental
identification. In the first step, the initial estimated values
1 1 1
, ,k I F
are considered. The
choice of poles
2
1,2 0 0 1s i
takes into account the achievement of a
bandwidth of at least 30 Hz. This is equivalent to imposing values
2 30 rad/s
n
and
1 2,
which leads to the identification of intermediate values for
,1 ,1
( , ).
P D
K K For a
beer understanding, if
0
k
and
1 2,
the aenuation in the closed loop (14), at
the frequency
,
n
is exactly 3 dB, which corresponds to the definition of the
bandwidth. In a subsequent step, the control law of the internal loop of the servoactuator
is implemented with these intermediate values
,1 ,1
( , )
P D
K K , in order to control the wing
system in WT. The system is excited by a chirp signal
c
t
, i.e., a sinusoidal signal with
a constant amplitude (corresponding to an expected angular aileron displacement of 2
degrees) and a frequency that is linearly variable in time in the range of interest of the
frequencies
0.1 60 Hz Hz
. The deflection angle of the aileron
t
provided by an
encoder is recorded. Then, the experimental transfer function
;
i
c
H
exp will be a
result of the Fast Fourier Transform (FFT) of the two experimental time signals
t
and
c
t
. The next step is to get, based on MATLAB System Identification Toolbox functions,
a convenient approximation of this response
;
i
c
H
exp by a rational transfer function
(i.e., a ratio of two polynomials in the complex variable
i
s
)
;
i
cy
H
idt , in other
words, to obtain
; ;exp
i i
c c
y y
H H
idt . Then, the relation (14) is reformulated in this
way.
,1 0
,idt
2 2
,1 ,1 1 0
/
( ) : ( )
/ /
c
P
u
D P
K K I b
H s H s
s F K K s I k K K I s a s a
(15
)
and, as a result, three algebraic equations are obtained, with the solutions
2 2 2
, , .k I F
These values are substituted in the equations
2
2 2 2 2 2
/ 2 ; / .
D n P n
F K K I k K K I
(16)
from which are deduced the values of the second iteration
,2 ,2
( , )
P D
K K . Starting with
these values, the parameters
( , )
P D
K K
will be experimentally tuned through a “trial and
error’’ procedure. The following iterations can be continued by repeating the previous
steps. After 3 iterations, the desired values of the PD controller gains are obtained:
12.786,
P
K
0.069.
D
K The diagram of the tuning algorithm of the PD controller of the
servoactuator is given in Figure 6.
Figure 5. Sketch for calculating the transfer matrix of CR-CM.
Aerospace 2024, 11, 493 11 of 26
The successful synthesis of a high-performance servoactuator for vibration
aenuation, whose signal bandwidth is over 30 Hz, is aested by Figure 7 and by the
vibration aenuation results described below in Section 5. In fact, the frequency
characteristic recorded online in WT denotes a 3 dB bandwidth of about 36.6 Hz.
Therefore, an indirect but essential condition, required by Shannon’s sampling theorem,
is met: the bandwidth is at least twice as large as the second modal frequency of the wing,
identified experimentally. This is because only the first two modal frequencies, about 5
Hz (bending) and 17 Hz (torsion), are subject to active control, as shown in the following
sections of the paper.
Figure 6. Algorithm for tuning the PD controller.
Figure 7. Aileron response in the frequency domain (V = 25 m/s); bandwidth of about 36.6 Hz.
(3) The two used transducers are a Winkel MOT 13 encoder (Megatron Elektronik
GmbH & Co. KG, München, Germany) and a capacitive accelerometer 4394-S (Brüel &
Kjær, DK-2830 Virum, Denmark).
(4) Turbulence generator (TG). The aviation WTs are specially built so as to ensure
the lowest possible degree of turbulence, which is relevant for aerospace tests. However,
for certain aerospace applications, or even for some civil engineering applications, certain
degrees of turbulence intensity are required. It was established that the optimal solution
in terms of cost-quality for increasing the degree of turbulence in WT is to introduce a
passive grid upstream of the experiment chamber with a square mesh (M) with
dimensions of 0.15 m × 0.15 m and a distance between neighboring meshes of 0.05 m. A
degree of turbulence (the ratio of the standard deviation of fluctuating wind velocity to
the mean wind speed) of 8% obtained with TG designed and manufactured by INCAS,
Aerospace 2024, 11, 493 12 of 26
located 30 cm from the physical wing model, represents a high turbulence case, causing
abrupt variations in altitude and aitude for the airplane in that field and panic and
injuries for passengers [44].
Using relations (38) from [45], the degree of turbulence I is calculated, which is, as
can be seen, independent of the velocity from infinity upstream U0, thus:
222
1 2 2
0 0
1.25 1.25
1.25
3
1 2
3 21 3.5 3 20 3.5
1.27 1.27
1 31 1 .
3 70 3.5
1.27
uuu
IU U
X X
M M
X
M
u
(17)
Taking into account the fact that the grid and the WT in [45] are not identical to those
from INCAS, a correction factor of 1/4.63 was introduced to obtain the same degree of
turbulence at the distances X = 3 m and M = 0.15 m.
Therefore, for the INCAS grid, the relationship is recommended.
'
0.625
0
1 31 1 1 .
3 70 4.63
3.5
1.27
u
IUX
M
(18)
With this relationship, the following results are obtained: see Table 2.
Table 2. The evolution of the turbulence intensity depending on the distance to the grid.
X (m) 1 1.5 2 2.5 3 3.5 4 4.5 5
I 0.0338 0.019 0.0142 0.0116 0.01 0.0089 0.008 0.0073 0.0068
The evolution of the turbulence intensity depending on the distance to the grid, as
well as the graph in Figure 8, is described in percentages.
Figure 8. Left and middle: the evolution of the degree of turbulence mediated in the transversal
planes, along the flow; right: the turbulence intensity curve as a function of the distance to the grid;
the grid is positioned at the coordinate 0x.
3. Tests on the Smart Wing in Subsonic WT for Mathematical Model Identification
In the case of a classical control law synthesis, two problems must be solved: that of
the mathematical modeling and that of compatibility between the mathematical
methodology used for control law synthesis and the obtained mathematical model [46,47].
The conventional approach follows an analytical path, e.g., [48], or a numerical one by
using the finite element method (FEM) [49]. As will be seen below, herein will be avoided
the determination of a structural mathematical model, respectively, of the matrices M, C,
K (mass, damping and stiffness matrices) by an online (in process) identification,
Aerospace 2024, 11, 493 13 of 26
considered much safer than any analytical or FEM approach, even more in this particular
case of the wing having such an atypical longeron.
The identification procedure takes place in the mounting set-up shown in Figure 1
right, at various air speeds, and is sequentially carried out as follows:
(a) A chirp signal
c
t
(Figure 4) is applied to the actuator; the signal has constant
amplitude (corresponding to an expected angular aileron displacement, for example,
2 degrees, 4 degrees, etc.) and a linearly variable frequency in time in the band [0 Hz;
60 Hz], which sufficiently covers the interest field of the first two modal frequencies
of the wing.
(b) The signal
y t
(Figure 4), corresponding to wing displacement in the normal
direction on the wing, and provided by the accelerometer, is recorded by integrating
the acceleration twice; the accelerometer is mounted on the wing (Figure 2) so as to
react simultaneously to the bending and torsional movements corresponding to the
first two modes of vibration.
(c) The experimental frequency response, defined by the
c
yδ , ,
arctan Im i / Re i
c
j y j
H H
exp exp − phase-
,i
c
y j
H
exp − aenuation-
frequency characteristics, and frequency characteristics,
1,..., ,j M
i 1,
associated with the transfer function
;exp
i
c
y
H
is estimated; the laer is obtained
by comparing (dividing) the Fast Fourier Transform (FFT) of the two experimental
time signals
y t
and
c
t
; therefore
;exp
i
c
y
H
consists of a sequence of
complex numbers, of length M, indexed with values of the circular frequencies
j
(d) A convenient approximation of this response by rational transfer functions is sought
(i.e., a ratio of two polynomials in the complex variable
i
s
),
; ;exp
i i
c c
y y
H H
idt ; for this purpose, functions from the MATLAB System
Identification Toolbox are available. For the air speed in WT of 25 m/s, the
experimental
;exp
i
c
y
H
and identified
;idt
i
c
y
H
transfer functions are
represented in the graphs in Figure 9.
;idt
i
cy
H
is obtained with an accuracy of
estimation of 81.58% (see Figure 10) and is given analytically below as a rational
expression of two polynomials, with two zeros and two pairs of complex-conjugated
poles ( :
c
u
, see Figure 4). The estimation accuracy of over 80% of the transfer
function
2 4 6
4 3 4 2 5 7
611.7 4.986 10 2.533 10
( )
94 1.327 10 1.286 10 1.232 10
s s
yu
H s
s s s s
(19)
It proved to be excellent during the online vibration dampening process in the wind
tunnel.
Figure 9. Experimental and identified transfer function
,u y
see Figure 4.
Aerospace 2024, 11, 493 14 of 26
Figure 10. The accuracy of transfer function estimation
; ;exp
i i
c c
y y
H H
idt .
(e) The identification of the frequency domain is followed by converting the transfer
function (19) into the state space system
2 2
, , ,0
A B C
; for this purpose, the MATLAB
function tf2ss is used.
2 2
, 0 .x x u y x u A B C
(20)
Since the interest is the active control of vibration modes, the initial matrices A
and
2
B
are subjected to an intermediary state transformation in order to be brought to a
„modal form” to highlight the first modal (angular) frequencies
i
and dampings
,
i
1,2i of each mode of A
22
1 1 1
2
2 2 2
T
2
0 0 1 0
2.29
0 0 0 1
2.29 ,0 2 0
301.18
912.86 0 0 2
0 0 1 0 1
0 0 0 1 1
, .
1024.93 0 2.92 0 0
0 11975.1 0 91.08 0
B A
A C
(21)
The two modal frequencies are 5.1052 Hz (bending mode) and 17.4168 Hz (torsion
mode), and the damping factors are 0.0455 and 0.4162, respectively; these values derive
from poles ‒1.459 ± 32.044i, ‒45.542 ± 99.506i of
( )
yu
H s
. The synthesis of the control laws
starts from with triplets
2 2
, ,
A B C
corresponding to WT air speeds 115 m/s,V
2
20 m/s,
V 3
25 m/s,
V 3
30 m/s,
Vand so on.
4. Brief Presentation of Active Control Laws
Two control laws, LQG (Linear Quadratic Gaussian) and
will be implemented
with the purpose of having reciprocal comparison for vibration aenuation performance.
4.1. Standard LQG Control Synthesis
The stochastic framework of the LQG problem requires the formal presence of a
white Gaussian noise component on the right sides of Equation (20). This is because the
stochastic content of the turbulence phenomenon is already well known [50]. Therefore,
the following system is used as a basis for the control law synthesis of a LQG standard
optimal problem (which has its origin in the works of R. E. Kalman from 1960 [51,52])
1 2 1 2
( ) ( ) ( ) ( ), ( ) ( ), ( ) ( ) ( ).t t w t u t z t t y t t t x Ax B B C x C x
(22)
Aerospace 2024, 11, 493 15 of 26
A Gaussian component
1
w t
B representing the disturbance introduced as
aerodynamic turbulence is considered and
1
B
is entered as equal with
2
B
.These
equations herein characterize a SISO (Single-Input-Single Output) system
w t z t
.
( )tx
is the state vector,
( )z t
is the quality output,
( )y t
is the measured output, and
( )u t
is the control input.
( )w t
and
( )t
are white Gaussian noises on state and measured
output, respectively. The role of control LQG is to diminish the influence of the unknown
input disturbance
w t
on the quality output
z t
. The state vector is given by the
displacements and velocities of the two modes highlighted in (21).
T
1 2 1 2
( ) ( , , , )
t x x x xx
(23)
(by
T
( )
is noted the transposed matrix of
( )
).
The statement of LQG control synthesis: find the control law
( )u t
that stabilizes the
system (18) and minimizes the cost function.
T
0
T T
1 1
0
0
lim ( ) 0
0( )
lim ( ) ( ) , : .
0( )
( )
( ) ( )
TJ
LQG TJ
T
J
TJ
Q
J u t dt
R
t
t u t dt Q
Ru t
z t
z t u t
Qx
x Q C C
E
E
(24)
E is the expectation operator [53].
J
Q
and
J
R
are weights on the cost function,
which is thought to be a trade-off between the quality output
( )z t
and control
( )u t
achieved by manipulating these weights. For example, the decrease of the weight
J
R
has
as a consequence the increase of the control variable
( )u t
and, implicitly, the decrease of
the quality output variable
( )z t
, which represents, after all, the essential objective of
active control, that of diminishing as much as possible the influence of the disturbance
w
in the system. Unfortunately,
'su
growth is physically limited by saturation, and the
laer, if it appears, is accompanied by a dangerous behavior of the entire system, called
windup [54]. The solution to the LQG problem is given by a controller and a state-
estimator, e.g., a Kalman filter providing an estimate of the internal state of the system
from measurements. The state estimator is described by
2 2
ˆˆ
ˆ
( ) ( ) ( ) ( ) ( )
f
x t t u t y t t
Ax B K C x
(25)
and the controller is given by
ˆ
( ) ( )
R
u t t
K x
(26)
The two gains
f
K
and
R
K
are obtained by first solving the decoupled algebraic
Riccati equations.
T 1 T T T 1 T
2 2 2 2 1 1
0, 0
J w
R Q Q
A P PA PB B P Q AS SA SC C S B B (27)
1 T
2
,
R
K R B P
T 1
2
.
f
Q
K SC (28)
The noise matrices
w
Q
and
Q
are described by the relation
0
( )
( ) ( ) ( )
0
( )
w
Q
w t w t t t
Q
t
E (29)
which characterizes the independence of the two noises
w t
and
.t
( )
t
is the
Dirac distribution. Substituting the controller (26) in the first equations (22) and in (25),
we obtain the closed-loop system.
Aerospace 2024, 11, 493 16 of 26
1 2
2 2 2
ˆ
( ) ( ) ( ) ( )
ˆ
ˆ
( ) ( ) ( ) ( ).
R
f f R f
t t w t t
x t t t t
x Ax B B K x
K C x K A B K K C x
(30)
The LQG optimal compensator, with input
y t
and output
,u t
is
2 2
ˆ ˆ
ˆ
ˆ ˆ
ˆ( ) ( ) ( ); ( )
ˆˆ
ˆ
; ; .
cp cp cp
cp R f cp f cp R
x t t y t u t t
A x B C x
A A B K K C B K C K
(31)
Essentially, the solution to the LQG problem is ensured by the conditions of
stabilitystabilizability of the pair
2
,
A B
and detectability of the pair
2
,
A C
[53]. The
fulfillment of these conditions is verified in Section 5, before proceeding to the synthesis
of the compensator (31).
It is worth noting that almost all reference books and papers give an endless variation
of “technical conditions” for solving the LQG problem, considered equivalent to the
problem (see, for example, [55], Chapter 14, where three more conditions appear that “can
be relaxed”), so that the proverb “many men, many minds” is once again true.
4.2.
. Synthesis, Static Weights
A simplified version of output feedback
control concerns the system [55].
1 2 1 12 2 21
, ,
t t t u t t t u t y t t t
x Ax B w B z C x D C x D w
, .x t x t y t u t x t
A B C
(32)
The top dynamic system (the first three equations) is the plant (the controlled system)
,s
G and the boom one is the compensator
,s
K in standard notations,
s
is the
Laplace variable. Equations in the time domain (32) are seen as realizations of transfer
matrices
s
G and
.s
K Herein
4
x
is the state vector,
3
z
represents the
quality output, which includes the control variable
,u
T
1 1
,
x x uz
T
2
,w w
are the exogenous disturbances on state
x
and on the measured output
,y
respectively,
1
y
is the measured output.
There are thousands of papers, workshops, sessions at conferences, MATLAB
toolboxes, and numerous books since the theoretical foundations of the
topic were
laid by G. Zames [56,57]. We try to present, in what is essential, a formulation and a
solution to the suboptimal problem
as approached in this paper.
Obviously, the compensator must have the sine qua non property that it internally
stabilizes the system. The ensemble
,G K
(we give up the writing of the variable) is
called internally stable if the origin
, : ,
x x
0 0
is asymptotically stable; in other
words, the closed loop spectrum of the system
,x x
, defined by the eigenvalues of the
closed loop matrix
e
A
of the extended system.
T
T T
( ) ( )+ ( )
( ) ( )+ ( )
,
e e e e
e e e
e
t t t
t t t
t t t
x A x B w
z C x D w
x x x
(33)
must be located in the left open half-plane. There is also the matrix
1
11 12 22 21
:
zw
T G G K I G K G
(the calculation of this matrix involves the partitioning
of the matrix
s
G into four blocks; the notation was simplified by giving up the writing
of the Laplace variable
s
) which characterizes the stability and magnitude of the transfer
input
w
output
.z
This matrix has the set of poles included in the set of eigenvalues
Aerospace 2024, 11, 493 17 of 26
of the matrix
e
A
. Therefore, internal stability guarantees input-output stability, with the
reciprocal not being true without some additional assumptions. These are given next:
(i) The pairs
1
( , )A B
, 1
( , , )C A
are stabilizable, respectively, detectable.
(ii) The pairs
2
( , ),A B
2
( , ,)C A
are stabilizable, respectively, detectable.
(iii)
1
T T
12 1 12 21
21
,
B
O
D C D O I D
D
I
.
Remark 1 [55]. Assumption (ii) is necessary and sufficient for a controlled system
G
to be
internally stabilizable, but is not needed to prove the equivalence of internal and input-output
stabilities. This equivalence does not occur if the controlled system
G
or compensator
K
is not
stabilizable and detectable.
Internal stability is a concept stronger than input-output stability (more precisely,
bounded input-bounded output stability). Indeed, some internal modes of system
response may not be seen in the input-output transfer function. In the case of the linear
feedback system, internal stability is ensured if the characteristic closed-loop polynomial
is stable and if any pol-zero simplifications appearing in the KG loop take place only in
the left open half-plane.
Proposition 1 (Corollary 16.3, [55]). Suppose that assumptions (i) and (iii) hold. Then a
controller
K
is an internally stabilizing compensator if and only if the system is input-output
stable (in other words, if
zw
T
has all the poles in the left open half-plane.
Remark 2. We have preferred here to refer to the assumptions in Proposition 1, much simpler to
verify than those in Lemma 16.1 [55], which provide the same necessary and sufficient condition
given in Proposition 1.
Proposition 2 (Lemma 16.1, [55]). Suppose that the realizations for
G
and
K
are both
stabilizable and detectable. Then the feedback connection
zw
T
of the realizations for
G
and
K
is
(a) detectable if
2
1 12
A I B
C D
has full column rank for all
Re 0;
(b) stabilizable if
1
2 21
A I B
C D
has full row rank, for all
Re 0.
Moreover, if (a) and (b) hold, then
K
is an
internally stabilizing compensator if and only if the system is input-output stable.
Optimal
control synthesis: find all compensators
K
that internally stabilizes the
system (32) and such that infinite norm
( )s
zw
T of the transfer function
( )s
zw
T is
minimized.
Suboptimal
control synthesis: given
0,
find all compensators
K
that
stabilize internally the system, if there are any, such that
( ) : sup ( ) .
s j
zw zw
T T
The first problem is extremely difficult and has no unique solution [55]. We now
present the solution of the suboptimal problem, using as a reference Theorem 16.4 [55].
The problem of suboptimal synthesis will be approached for System (32) (first three
equations), for which assumptions (i)–(iii) are true.
Proposition 3. If assumptions (i)–(iii) are true and the realization of
s
K is stabilizable and
detectable, then it is true the equivalence of internal and input-output stabilities.
Aerospace 2024, 11, 493 18 of 26
Proof. According to Proposition 2, it is enough to show that the matrices
2
1 12
A I B
C D
,
1
2 21
A I B
C D
are full column rank, and full row rank, respectively, for all
Re 0.
We
assume the opposite for the first matrix. Then there is a
C
with
Re 0
and
T
T
0
u
x such that
2
1 12
A I B
C D
u
x
= 0, or, otherwise wrien, 2
,
u
Ax B x
1 12
0.
u
C x D It immediately follows that
T
12 1 12
0
D u
C x D , hence, with assumption
(iii) it results that
0
u
; 2 1 12
, 0
u u
Ax B x C x D is rewrien
1
0,
A I x
C
therefore, according to assumption (i),
0,
x which together with
0
u
involves
T
0
u
x, which is absurd. Similarly, it is shown that
1
2 21
A I B
C D
is full row rank,
for all
Re 0.
□
The demonstration also shows that in assumption (iii) values other than the identity
matrix
I
do not affect the validity of Proposition 1.
Proposition 4 (Theorem 16.4 [55]). There exists a compensator
K
that internally stabilizes the
system (32) and so that ( )s
zw
T if and only if the spectral radius
2
,
X Y where
,X Y
are the semi-positively defined solutions (if any) of the Riccati equations
T T 2 T T
2 2 1 1 1 1
T T 2 T T
2 2 1 1 1 1
0
0.
A X XA X B B B B X C C
AY YA Y C C C C Y B B
(34)
When these conditions hold, one such compensator is
1
2 T T 2 T
1 1 2 2 2 2
1
2 T T
2 2
,
: ,
: , :
cp cp cp
cp
cp cp
t x t y t u t x t
x A B C
A A B B X B B X I YX YC C
B I YX YC C B X
(35)
with
,X Y
solutions of algebraic Riccati Equations (34).
The Equation (35) or the System (32) characterizes a MIMO (Multi-Input-Multi-
Output) system
w z
.
Remark 3. It is risky to determine a priori which is the best solution to a given problem. Often,
the best solution does not actually exist, as shown in [18,46,47]. As an example, we refer to a
multitude of control laws applied to solve an active control problem. It turned out that all methods
of control synthesis, LQG,
LQG/LTR,
preview control, receding horizon,
,
robust
,
sliding mode, backstepping, neural control, fuzzy logical control, neuro-fuzzy control etc., were
competitive in giving substantially similar results.
A justification for the optimal
control resides in the min-max nature of the problem, with the
argument that minimizing the “peak” of the transfer
w z
necessarily renders the magnitude of
zw
T
small at all frequencies. Otherwise stated, minimizing the
-norm of a transfer function
is equivalent to minimizing the energy in the output signal due to the inputs with the worst possible
frequency distribution. This improvement of the “worst-case scenario” has a direct correspondent
in the active vibration control problem and seems particularly aractive for light structures with
embedded piezoelectric actuators.
Aerospace 2024, 11, 493 19 of 26
Remark 4. In the problem
,
unlike the LQG problem, the variables
w
and
z
are vectors by
concatenation:
w
concatenates the disturbances on input and output, and
z
concatenates an
actual output of quality and control variable
.u
5. Results of Active Control Tests
The experimental setup for smart wing active control in WT is shown in Figure 11. The
user interface runs on the PC-type computer (1), from which the start and stop of the
experiment are commanded and the signals from the sensors are monitored. The control
algorithms PD, LQG, and
are implemented in the same LabVIEW project, which is
compiled and downloaded on a real time computing system type PXI-1082 (2). The PXI
system is equipped with a PXI-6225 data acquisition board (3) with analog and digital
input channels. A voltage source (5) supplies the driver (6) of the linear motor (7), which
drives the wing aileron (10), with the mechanical system CR-CM, which converts the
linear motion into rotational motion. A capacitive accelerometer (8) measures the
acceleration of the wing. A signal conditioner/load amplifier (9) takes the signal from the
accelerometer and converts it into a voltage signal proportional to the displacement of the
wing. To measure the deflection angle of the aileron, a Winkel MOT 13 encoder (7)
connected to the connector board (4) of the acquisition board (3) is used, which feeds the
angular transducer (7).
The mathematical model of the LQG control law described in Section 4 must be
completed with the numerical values of the key parameters, which are the weights in the
Riccati equations. The numerical simulations outlined values for these weights, which
were thus established online:
0.000001,
Q
5,
w
Q
1,
J
Q
0.1.
J
R The influence
matrix
1
B
of the perturbation is chosen
1 2
B B
. The quality output selection matrix is
1
1 0 0 0 ,
C as active vibration control focuses on bending modal displacement
1 1
: ,x q
and the matrix
2
1 1 0 0
C indicates that the accelerometer collects
information about both modal displacements
1 2
, .x x
Figure 11. Equipment connection in the experimental setup: (1) computer; (2) PXI-1082; (3) PXI-6225;
(4) connector SC-68; (5) source TR9158; (6) amplifier LCAM 5/15 H2W; (7) linear actuator H2W
Technology NCC05-18-060-2PBS with Winkel MOT 13 encoder; (8) accelerometer 4394-S capacitive;
(9) load amplifier TYPE 2635; (10) wing model.
Unlike the case of the LQG mathematical model, the
mathematical model (32)
will be structured by introducing static weights, as illustrated in the block diagram in
Figure 12 (see also [58]), with the benefit of manipulating vibration aenuation
performance and increasing the robustness of the system [59]. The quality output
T
1 2
x x u
z is replaced by the weighted quality output
1 2
T
T
1 2 3 1 2
: ;
z z u
e e e W z W z W u
e the disturbance
T
w
w is replaced by
the weighted (by software) disturbance
T
T:x
w W w W
w
. The equations of
the controlled system in (32) are rewrien as follows:
Aerospace 2024, 11, 493 20 of 26
1
2
1 2 1 12 2 21
1
2
1 21
3
4
1 12 2
, ,
0
0
0, 0
0
0
0
0 0 0 0
0 0 0 , 0 , : 1 1 0 0
0 0 0 0
x
z
z
u
t t t u t t t u t y t t t
b
W
bW
W
b
b
W
W
W
x Ax B w B e C x D C x D w
B D
C D C
(36)
where
, 1,...,4
i
b i are the entries in the matrix
2
B
. Again, the numerical simulations
outlined values for these weights, which were then established online:
1 2
0.5; 0
z z
W W and 0.3
u
W are weights on the quality output (first mode
displacement, second mode displacement, and control), 3
x
W and 0.01W are
weights on the state perturbation and on the output perturbation, respectively. These
weights characterize the compensator called „strong” below. With
0.01
W and
1.2
u
W we have a so called “standard” compensator. Decreasing further
u
W
to
0.1
u
W , we have the compensator called “superstrong” with accentuated vibration
aenuation properties, as a result of increased control. The names “standard”, “strong”,
and “superstrong” are given in correlation with the maneuvers performed on the weights
but are perfectly proven, as will be seen, by the experimental results.
Figure 12. Block diagram of the augmented system, static weights. CS–controlled system.
Before proceeding to the synthesis of the compensator
for the system governed
by the matrix quartet (21), the fulfilment of the conditions from the Proposition 1 and the
Proposition 2 is verified making replacements 1 1,B B
1 1,
C C
12 12 ,D D
21 21.
D D
As shown in Remark 2, in condition (iii) in the first equality instead of I we
obtain 2
u
W and in the second 4x
b W W.
As “smart”, a wing system can only be a feedback system in which information
about the output behavior
y t is taken with the sampling frequency 1,
s
where
is the sample time, i.e., a scalar interval between samples; in the project, was taken
at 0.001 s. The retrieved information on
y t
is processed, generating the control
variable
u t
, in accordance with the assumed control law,
or LQG. Note that the
structure of the control law
u t
has been synthesized off line, in continuous time, and
that it must be implemented in discrete time at times n. Therefore, the displacement
information
y n
is received from the accelerometer (Figure 11), it is calculated based on
this information, the control,
,u n
which will be applied at times
1
n
, remains
constant throughout the interval
1 , 2n n
. This is the so-called zero-order hold
Aerospace 2024, 11, 493 21 of 26
discretization. We add that discretization is also intuitively justified by the low value of
.
To exemplify discretization in the case of the law (31), the calculations start with the
general solution of the second equation in (32)
0
0
ˆ ˆ
0
.
ˆ
ˆ
ˆ( ) ( )
cp cp
t
t t t s
cp
t
x t e t e y s ds
A A
x B (37)
Substituting
0
, 1
t n t n
and taking into account the choice of zero-order
hold
,for , 1y t y n t n n
yields
1
ˆˆ1
.
ˆ
ˆ
ˆ1 ( )
cp cp
nn s
cp
n
x n e n e dsy n
A A
x B
(38)
Changing in integral the Laplace variable s by
1 ,S n s
discrete LQG
compensator equation results
ˆˆ
, ,
0
ˆˆ
1
, , 4
ˆ
ˆ ˆ
ˆˆ
ˆ1 ( ) ( )
ˆ ˆ ˆ
ˆˆˆ
: , :
, 1 1
cp cp
cp cp
cp cp D cp D
cp D cp D cp cp cp
x n e n e dSy n n y n
e e I u n n
A A
A A
x B A x B
A B A B C x
(39)
We take as a starting point the matrix quartet (21). All the data is now gathered to
numerically write the discrete LQR compensator matrices.
,
,
0.0488 0.0586 0.00003 0.00004
0.0484 0.0581 0.00003 0.00004
ˆ
1.6021 1.5811 0.0085 0.0011
4.0657 4.1820 0.0023 0.0063
5.8799
4.8346
ˆ, 1.8145 0.00 0.0725
158.6613
408.0732
cp D
cp D cp
A
B C
0.00 .
(40)
Proceeding similarly for the discrete
compensator, the matrices are obtained.
0.7212 0.0233 0.0056 0.00001
0.3040 0.9850 0.0079 0.0009
,30.2510 0.8736 0.2353 0.0004
78.8723 14.3941 2.0692 0.9059
0.0004
0.0002 ; 74616.25 0.00 1792.40 0
, ,
0.0163
0.0554
cp D
cp D cp D
A
B C
.00 .
(41)
Relevant results for the performance of vibration active control are summarized by
the graphs in Figures 13–15. In Figure 13, the simulation of mathematical models with the
two laws, LQG and
,
aests, at V = 25 m/s, a reduction of about 18 dB at the basic
modal frequency of 5 Hz for both laws compared to the uncontrolled vibration regime
(UC). In literature, for an experiment carried out also in a wind tunnel, a reduction of
about 6 dB of the modal frequency 7 Hz is shown in [60,61]. The aenuation of
y
displacement in WT is illustrated in the time domain, in pure air, Figure 14, top, and in
turbulent regime, Figure 14, middle. By superposing the graphs, one can see the efficiency
of the control. Figure 14, down, highlights the efficiency of the electric broadband servo
actuator: the movement of the aileron
follows the control
u
very well for the
experiment in WT. Figure 15 shows the net faster extinction of the transient vibration at
rest, at the suspension of the excitation, when the active control is present.
Aerospace 2024, 11, 493 22 of 26
Next, the AC (Active Control) versus UC (UnControlled) results are presented
quantitatively in Tables 3–5. The vibration aenuation coefficients are calculated with the
relation
std std
std
j i
ij
i
AC UC
CUC
(42)
in which the standard deviation (in mm) is noted with std. For a motion signal recorded
by the accelerometer as in Figure 14, with random evolution, the quantification given by
the relation (42) is used naturally.
Table 3. Pure air, V = 25 m/s.
Control Law # UC AC Aenuation
normal
(−37.04%)
1 0.096 0.052 −45.83%
2 0.085 0.061 −28.24%
LQG
(−41.49%)
1 0.061 0.045 −26.23%
2 0.085 0.044 −48.24%
3 0.080 0.040 −50.00%
Table 4. Turbulence, V = 25 m/s.
Control Law # UC AC Aenuation
strong
(−24.61%)
1 0.642 0.537 −16.36%
2 0.746 0.524 −29.76%
3 0.722 0.522 −27.70%
super-strong
(−36.04%)
1 0.815 0.447 −45.15%
2 0.749 0.498 −33.51%
3 0.662 0.467 −29.46%
LQG
(−26.75%)
1 0.784 0.558 −28.83%
2 0.635 0.529 −16.69%
3 0.726 0.447 −34.71%
Table 5. Aenuation, comparison, pure air.
Control law V = 25 m/s V = 33 m/s
robust −37.04% −30.16%
Figure 13. Aenuation of about 18 dB at 5 Hz,
, left and LQG, right, V = 25 m/s.
Aerospace 2024, 11, 493 23 of 26
Figure 14. Recording of 2 superimposed regimes (for visual comparison) of vibration at V = 25 m/s;
left: y displacement, without control versus with active control (AC), LQG, pure air; middle:
similar, with turbulence,
“superstrong”; right: turbulent mode,
“superstrong”
:
c
u
and aileron displacement .
Tables 3 and 4 show a comparison of vibration aenuation at 25 m/s from a double
perspective: pure air versus turbulence and
versus LQG regimes. It was specified in
the synthesis that the introduced static weights, according to the block diagram in Figure
12, allow a choice regarding the relaxation or intensification of the control variable, with
consequences for the relaxation or intensification of the vibration aenuation. In this
context, the regimes marked “strong” and “superstrong” were also launched in the
process. Table 5 shows a comparison regarding the vibration aenuation in pure air at two
air velocities by applying a
control law. The percentage aenuations in Tables 3–5
can be interpreted as aenuation values in dB. For example, an aenuation of −45.83%
would mean a vibration aenuation of about 33 dB.
Figure 15. Left: comparative graphs of some overloaded vibration regimes, UC and AC, followed
by damping after extinguishing the excitation; right: zoom on the chart above.
6. Concluding Remarks
There are several particularities of this paper that deserve to be emphasized. An
elastic wing model has been designed and used, whose resonant frequencies can in
principle be achieved in a predetermined sequence, starting from the geometry of the
longeron. The result of active vibration control proves to be more relevant than in the case
of a rigid wing supported on two springs that simulates only two resonant frequencies.
Although the mathematical model of the wing system included only two modes, no
harmful, spillover type high-frequency oscillating phenomena were observed in the WT,
neither in the absence nor in the presence of air turbulence. Spillover is an instability of a
closed-loop system caused by the observation or excitation of unmodeled dynamics by
sensors or actuators; see [62]. A second aspect is to highlight the effect of turbulence
generated in the WT. The level of vibration in the absence of GT is relatively low, as shown
Aerospace 2024, 11, 493 24 of 26
by the comparison of Tables 3 and 4. In the presence of disturbances, the level of
(controlled) oscillations increases by an order of magnitude.
It is known that applying active control to the wing reduces the load on the wing and,
implicitly, reduces the bending moments of the wing. Consequently, the designer can
extend the wing span or reduce the structural weight of the wing. Increasing the span for
a given wing area improves the aerodynamic efficiency of the wing; that is, it increases
the lift-to-drag ratio. In addition, more importantly, the active control of vibrations
prevents the appearance of fluer, eventually leading to the expansion of the flight
envelope.
The use of weights, even only static ones, in the
synthesis can ensure the
robustness of the system and gives the possibility to manipulate these weights in order to
obtain a more severe reduction of the vibrations, as shown in Tables 4 and 5. In fact, the
problem of static weights in the
model can be treated in terms of robustness, with
the variables newly introduced on the diagram in Figure 11 being called fictitious inputs
and outputs [18,63–66].
Author Contributions: Conceptualization, I.U. and D.E.; methodology, I.U.; software, G.T. and A.T.;
validation, I.U. and G.T.; formal analysis, I.N.; investigation, M.A.; writing—original draft
preparation, I.U.; writing—review and editing, D.E.; visualization, M.C.; project administration,
I.U., A.T., I.N., and D.E. All authors have read and agreed to the published version of the
manuscript.
Funding: This work was supported by grants of the Romanian Ministry of Research, Innovation,
and Digitization, CNCS-UEFISCDI, project number PN-III-P2-2.1-PED-2021-2265, project number
PN-III-P2-2.1-PED-2021-4147, and by NUCLEU Programme project codes PN 23-17-02-03, and PN
23-17-07-01, Ctr. 36 N/12.01.2023.
Data Availability Statement: The dataset is available on request from the author.
Acknowledgments: The colleagues from INCAS, Mihai Leonida Niculescu, and Dumitru Pepelea,
and Ionel Popescu are gratefully acknowledged for their support in CFD simulations.
Conflicts of Interest: The authors declare no conflicts of interest.
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