# Flow separation control design with experimental validation

T. Arnoult  Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-
France, UMR 8520 - IEMN - Institut d’Electronique de Microélectronique et de
Nanotechnologie, F-59000 Lille, France.  G. Acher  Univ. Lille, CNRS, ONERA,
Arts et Metiers Institute of Technology, Centrale Lille, UMR 9014, Laboratoire
de Mécanique des fluides de Lille-Kampé de Fériet, F-59000 Lille, France.  V.
Nowinski  Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology,
Centrale Lille, UMR 9014, Laboratoire de Mécanique des fluides de Lille-Kampé
de Fériet, F-59000 Lille, France.  P. Vuillemin  ONERA - The French Aerospace
Lab, F-31055 Toulouse, France.  C. Briat  D-BSSE, ETH Zürich, Switzerland.  P.
Pernod  Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-
France, UMR 8520 - IEMN - Institut d’Electronique de Microélectronique et de
Nanotechnologie, F-59000 Lille, France.  C. Ghouila-Houri  Univ. Lille, CNRS,
Centrale Lille, Univ. Polytechnique Hauts-de-France, UMR 8520 - IEMN -
Institut d’Electronique de Microélectronique et de Nanotechnologie, F-59000
Lille, France.  A. Talbi  Univ. Lille, CNRS, ONERA, Arts et Metiers Institute
of Technology, Centrale Lille, UMR 9014, Laboratoire de Mécanique des fluides
de Lille-Kampé de Fériet, F-59000 Lille, France.  E. Garnier  Univ. Lille,
CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, UMR 8520 - IEMN -
Institut d’Electronique de Microélectronique et de Nanotechnologie, F-59000
Lille, France.  C. Poussot-Vassal  ONERA - The French Aerospace Lab, F-31055
Toulouse, France.

###### Abstract

Flow control aims at modifying a natural flow state to reach an other flow
state considered as advantageous. In this paper, active feedback flow
separation control is investigated with two different closed-loop control
strategies, involving a reference signal tracking architecture. Firstly, a
data-driven control law, leading to a linear (integral) controller is
employed. Secondly, a phenomenological/model-driven approach, leading to a
non-linear positive (integral) control strategy is investigated. While the
former benefits of a tuning simplicity, the latter prevents undesirable
effects and formally guarantees closed-loop stability. Both control approaches
were validated through wind tunnel experiments of flow separation over a
movable NACA 4412 plain flap. These control laws were designed with respect to
hot film measurements, performed over the flap for different deflection
angles. Both control approaches proved efficient in avoiding flow separation.
The main contribution of this work is to provide practitioners simple but yet
efficient ways to design a flow separation controller. In addition, a complete
validation campaign data-set is provided.

##  1 Introduction

###  1.1 Forewords on flow separation objective

Flow separation over an aircraft flap is characterized by a decrease in the
lift coefficient, an increase in the drag coefficient and can occur during the
critical take-off and landing phases. Most aircraft circumvent this issue
using slotted flaps, which however add structural weight and complicate the
maintenance. Therefore, one solution would be to simplify these structures
into plain flaps with integrated flow control devices to avoid flow
separation.

Flow control consists in modifying a flow general behavior with a space
localized perturbation, in order to reach a flow configuration considered as
favorable. In that sense, flow control can help reducing noise radiation,
delaying the laminar-turbulent boundary layer transition or can help avoiding
flow separation [12]. Flow control methods can be categorized as passive or
active methods. Passive methods do not require any external source of energy
to act on the flow. One may mention vortex generators, which have been widely
used to prevent flow separation on aircraft wings. Vortices generated by these
devices help re-energizing the boundary layer and therefore prevent its
separation. However, they act permanently on the flow, even at off-points
design. In that sense, an active flow control method can be employed in a
closed-loop strategy as considered in this study. The actuators command can
therefore be adapted depending on the needs.

As discussed by Pastoor et al. [22], closed-loop flow control strategies have
been applied to different canonical flow configurations such as the flow
around a cylinder, the flow over backward facing steps, the flow over open-
cavities or separated flows over airfoils. These flows are described by
different dynamics and different actuating strategies may be required to
implement their closed-loop control. Considering the control of flow
separation over airfoils, one may aim at modifying the mean flow properties.
Several studies have focused on this case with different control methods and
objectives. One control method consists in turning the actuators when a
threshold value measured by the sensors is exceeded. This triggering method
was used in several studies. For instance, Packard and Bons [21] and Rethmel
et al. [25] studied the flow separation control over wings based on NACA
airfoils shape. In these studies, hot films sensors are placed on the models.
The RMS (Root Mean Square) value of the hot films voltage is used to define
the threshold value, according to which the flow is separated. In a similar
way, Lombardi et al. [19], Tewes et al. [27] and Benard et al. [2] propose a
triggering control method based on pressure measurements. In [19] the
triggering control criterion is based on unsteady pressure measurements
performed on the model. A spectrum analysis of the pressure measurements is
employed to determine the energy at a characteristic frequency indicating the
onset of flow separation. If this energy value is overshot, then the plasma
actuator placed at the wing leading edge is turned on. Regarding the studies
[27] and [2], the triggering control methods are coupled with an hysteresis
effect. Instead of reattaching a separated flow over the considered wings,
both study maintain the flow attached and therefore reduce the energy required
to control the flow. In [27], the threshold value is defined relatively to the
leading edge pressure coefficient value, while in [2] the threshold value is
set with respect to the pressure coefficient RMS computed at the leading edge.
A second control methodology is a model free approach based on gradient
methods such as extremum seeking and slope seeking. For instance, Benard et
al. focus on flow reattachment over NACA 0015 airfoil with a slope seeking
algorithm, which aims at maximising the lift coefficient. In a similar way,
Becker et al. [1] tend to maximize the lift coefficient of a high-lift
configuration composed of a NACA 4412 main airfoil associated with a NACA 4415
flap, firstly with a SISO (Single Input Single Output) extremum seeking
scheme, then with a SISO slope seeking algorithm, finally extended to a MIMO
(Multiple Input Multiple Output) slope seeking approach. This study was
carried out based on the pressure coefficient derived from unsteady pressure
measurements. A third control approach is based on the use of black box models
identifying the system’s input and output transfer. Recently, Sanchez et al.
[26] explored a sliding mode approach and applied it on a numerical example.
King et al. [17] consider in their study the control of flow separation on a
wing of a 1:2 scale model proposed by Airbus and expand their experiments on a
full scale glider. In both cases, they identify a model using PRBS (Pseudo-
Random Binary Signals) to drive the pulsed jets. From the black box model, the
$\mathcal{H}_{\infty}$ synthesis is employed to design a robust controller.
Different type of controllers can be synthesized based on this black box model
approach (see _e.g._ [18]).

###  1.2 Contribution statement

The contribution of the paper is to deploy and evaluate in an experimental
wind tunnel facility, involving a NACA 4412 plain flap airfoil, two active
closed-loop control design strategies to drive the flow separation phenomena.
The first one is a (model-free) _linear data-driven_ approach, while the
second one is a _positive nonlinear phenomenological/model-driven_ strategy.
The data-driven rationale is extensively detailed in [16], while the positive
strategy is discussed in [4], initially considered for the control of
biological systems in [5]. Both control methods are based on the use of on/off
solenoid valves as actuators and on hot film sensors. Both control structure
involve the same reference signal to track, which value is also discussed.
Each strategy is validated in a wind tunnel facility (see Figure 1) and leads
to a lift increase, and cancelled / reduced flow separation.

Figure 1: Wind tunnel facility view (Onera, Lille, France). The commanded
horizontal wing is in between the two vertical structures. The flow is
longitudinally travelling from the back of the photo.

As a glimpse of this paper result, Figure 2 illustrates the lift performances
with and without flow separation control. In addition, Figure 3 illustrates
the closed-loop frequency response (obtained with a frequency sweeping
reference signal) for varying flap angles, with respect to the reference
objective transfer. The rest of this note details the process for reaching
such performances and derives a generic but yet simple approach to design and
validate two flow separation feedback control laws. We believe that the
proposed rationales are sufficiently simple to be applied on a variety of
similar setups.

Figure 2: Evolution of the lift coefficient ${C_{L}}$ without (blue curve) and
with (red curve) control against the flap deflection angle ${\delta}$ for
${U_{\infty}=\ 34.5}$ m/s. Figure 3: Frequency-domain responses of the
controlled system for ${U_{\infty}=\ 34.5}$ m/s. Coloured solid lines
(response for different flap deflection angle ${\delta}$) and reference
(dashed black).

###  1.3 Notations and paper organisation

After recalling the flow separation problem in section 1, section 2 presents
the considered experimental setup. Both linear data-driven and nonlinear
positive model-driven flow control designs are recalled and validated in
section 3. Conclusions and outlooks are gathered in section 4.

By $\mathbb{R}$, $\mathbb{Z}$ and $\mathbb{N}$ we indicate the sets of real,
integer and natural (positive integer) numbers, respectively. The LTI
dynamical system $\mathbf{K}$ pencil is denoted $\Lambda_{\mathbf{K}}$. We
denote with $h\in\mathbb{R}_{+}$, the sampling-time, with $I$ the identity
matrix and $\imath$ ($\sqrt{\imath}=-1$) the complex variable. The time
average of a quantity is denoted $\langle\rangle$. Regarding the model,
$c_{\text{flap}}$ and $c_{\text{tot}}$ respectively stand for the flap chord
length and the model total length. The flap deflection angle is denoted
$\delta$. The freestream velocity is noted $U_{\infty}$ and the Reynolds
number based on the model total length is computed as
$Re=U_{\infty}c_{\text{tot}}/\nu$, with $\nu$ the air kinematic viscosity.
Considering the actuators, $f$, $f^{+}$ and $\alpha$ refer to the actuation
frequency, its reduced form and to the duty cycle. The momentum coefficient
$C_{\mu}$ is derived from $q_{\text{jet}}$, $U_{\text{jet}}$, $\rho_{\infty}$
and $A_{\text{ref}}$ respectively the actuators mass flow rate, actuators
outlet velocity, freestream density and reference area chosen as the flap
area. Concerning the sensors setup, the pressure coefficient $C_{p}$
calculation is based on the static pressure $p$ and the freestream pressure
$p_{\infty}$. The flap lower and upper surfaces pressure coefficients are
denoted $C_{p_{\text{lower}}}$ and $C_{p_{\text{upper}}}$. These coefficients
help computing the lift coefficient $C_{L}$ as detailed in the following. In
some of the following figures considering the controlled case, zones without
and with actuation are distinguished. They respectively denote cases in which
the control is applied but valves are not actuated or actuated with a duty
cycle fixed by the controller.

##  2 Experimental control setup description

###  2.1 Setup overview and wing surface properties

Flow control experiments were carried out in the L1 wind tunnel facility at
ONERA, Lille. It is characterized by a test section diameter of 2.40 m. The
model is composed of a 867 mm long flat plate, stabilizing the boundary layer,
followed by a plain flap of chord length $c_{\text{flap}}=220$ mm, yielding a
model total length $c_{\text{tot}}=1087$ mm. The plain flap design is based on
a NACA 4412 airfoil. As depicted in Figure 1, two lateral panels separated by
800 mm are placed inside the wind tunnel test section, next to the model
borders to avoid side effects and ensure that the flow developing over the
model is bi-dimensional. The flow separation control over this model has
already been studied by Chabert et al [7, 8], who implemented a slope seeking
control algorithm. The boundary layer developing over the model is turbulent.
Its transition is triggered by a carborundum line placed at the flat plate
leading edge. Velocity measurements upstream the model are performed with
Pitot tubes. During the experiments, $U_{\infty}$ is fixed to 34.5 m/s,
yielding $Re=2.39\times 10^{6}$. The turbulence level inside the wind tunnel
is about $1.3\ \%$. The model is placed inside the test section with a
$0^{\circ}$ angle of attack. The motorized flap can be deflected downward at
an angle $\delta$ varying between $2^{\circ}$ and $37^{\circ}$. Note that the
flap is not an actuator used in the control loop but a way to modify the
configuration accounting for uncertainties and allowing exhibiting the
separation phenomena (Figure 2).

###  2.2 Actuators and sensors description

####  2.2.1 Actuation system

The actuation setup is composed of 7 slots spanning along the flap leading
edge at a location of $x/c_{\text{flap}}=0.08$. Separated by 7 mm from each
other, the actuators slots are 90 mm long and 0.25 mm thick each. They cover
80% of the flap span and are supplied by on/off Festo MHE2 fast response
solenoid valves, fed with pressurized air up to 7 bar. The slot outlet
velocity is inclined by $30^{\circ}$ relatively to the flap local tangent. The
valves mass flow rate and the actuation frequency are fixed respectively to
$21$ g/s and to $f=100$ Hz. The reduced actuation frequency referred as
$f^{+}$ is therefore:

$f^{+}=\dfrac{f\ c_{\text{flap}}}{U_{\infty}}\cong 0.64$ (1)

The added momentum to the flow can be characterized by $C_{\mu}$ defined as:

$C_{\mu}=\dfrac{q_{\text{jet}}U_{\text{jet}}}{\dfrac{1}{2}\rho_{\infty}U_{\infty}^{2}A_{\text{ref}}},$
(2)

Considering the conditions detailed above, the constant blowing momentum
coefficient has a value of 1.6%.

As detailed in [6], for a fixed mass flow rate supplied to the actuators, the
momentum coefficient can be extended to pulsed blowing according to the
following equation:

${\langle C_{\mu}\rangle=\dfrac{1}{\alpha}\dfrac{\rho_{\text{jet}}\langle
U_{\text{jet}}^{2}\rangle
A_{\text{jet}}}{\dfrac{1}{2}\rho_{\infty}U_{\infty}^{2}A_{\text{ref}}}},$ (3)

As the valves considered here are driven with square signals with a duty cycle
$\alpha$, the momentum coefficient can be simplified as:

${\langle
C_{\mu}\rangle=\dfrac{1}{\alpha}\dfrac{q_{jet}U_{jet}}{\dfrac{1}{2}\rho_{\infty}U_{\infty}^{2}A_{ref}}=\dfrac{1}{\alpha}C_{\mu}}.$
(4)

The $\alpha$ variable then denotes the control signal used in the loop
control. During the flow control experiments $\langle C_{\mu}\rangle$ used
reached a maximum value of 4.6%.

####  2.2.2 Sensing system

Regarding the sensors setup, 51 pressure taps are dispatched on both the flat
plate and flap upper and lower surfaces. From these pressure measurements both
the pressure coefficient at each tap location and the global lift coefficient
can be computed. The pressure coefficient is defined according to the
following equation:

${C_{p}=\dfrac{p-p_{\infty}}{\dfrac{1}{2}\rho_{\infty}U_{\infty}^{2}}},$ (5)

The lift coefficient is derived from the pressure coefficient computations
according to the following formula:

${C_{L}=\int_{0}^{1}(C_{p_{\text{lower}}}-C_{p_{\text{upper}}})\,{d}\dfrac{x}{c_{\text{tot}}}},$
(6)

Both quantities $C_{p}$ and $C_{L}$ are used to assess the control effects,
comparing cases of uncontrolled and controlled flows.

####  2.2.3 Experimental strategy

To monitor the flow separation over the flap and implement the control part,
eight Senflex® hot films are placed along the flap chord-wise direction.
Connected to two Dantec® Streamlines units, hot films signals are recorded at
a sampling frequency of $1.25$ kHz over 3 minutes for each measurement points.
The control tracking value is defined with respect to the fifth hot film
measurements, located at the dimensionless abscissa $x/c_{\text{flap}}=0.511$,
taking the coordinate origin at the flap leading edge. Control scripts,
written in LabVIEW Real-Time 2011 via a PXIe-8102 controller, are embedded in
a PXI chassis. Both the actuators and sensors setups are sketched in Figure 4.

Figure 4: Scheme of the model placed in the wind tunnel with the actuators
command and hot films positions.

###  2.3 Performance characterisation toward specifications

First measurements focused on the unforced flow characterization. The
freestream velocity was fixed to $U_{\infty}=34.5$ m/s and the flap was
deflected from $2^{\circ}$ down to $37^{\circ}$. The evolution of the lift
coefficient is presented in Figure 5 and similar results to the one presented
in [8] are obtained.

Figure 5: Evolution of the unforced flow lift coefficient $C_{L}$ against the
flap deflection angle $\delta$ ($U_{\infty}=34.5$ m/s).

Following Figure 5 and as described by Hoerner [15], four zones can be
distinguished. The first zone (I), for $\delta$ between $2^{\circ}$ and
$12^{\circ}$, describes a linear evolution of $C_{L}$ against $\delta$. The
second zone (I)-(II) corresponds to a slower increase in the lift coefficient
and spreads between $12^{\circ}$ and $20^{\circ}$, indicating the development
of the flow separation over the flap. The zone denoted (II) corresponds to a
plateau of $C_{L}$ due to the recirculation bubble entirely developed over the
flap. Finally, the zone (III) denotes a zone of non-linear increase in the
lift coefficient. The non-linear evolution of $C_{L}$ in (III) would be better
observed with higher deflection angles, as pointed out by Hoerner [15].

The development of the recirculation area over the flap can also be observed
in the pressure coefficient. Figure 6 highlights the evolution of $C_{p}$
against the dimensionless abscissa $x/c_{\text{flap}}$ for a deflection angle
of $18^{\circ}$. The pressure gradient between 0.19 and 0.46 is followed by a
plateau of $C_{p}$ between 0.46 and 0.65, indicating the flow separation. The
longer the plateau is, the longer the flow recirculation area is. Therefore,
as the flap deflection angle is increased, the $C_{p}$ plateau spreads over a
larger area.

Figure 6: Evolution of the unforced flow pressure coefficient $C_{p}$ against
the dimensionless abscissa $x/c_{\text{flap}}$. ($\delta=18^{\circ}$ and
$U_{\infty}=34.5$ m/s). The two curves represent the profile upper and lower
coefficients.

Signals of the 8 hot film sensors were also recorded during these tests. For
each deflection angle, hot films time series have been averaged and normalized
according to $U^{\ast}=(\langle U\rangle-U_{min})/(U_{max}-U_{min})$, where
$U^{\ast}$ is the dimensionless hot film voltage, $\langle U\rangle$ the hot
film mean voltage, $U_{min}$ and $U_{max}$ respectively are the hot film
minimum and maximum voltages. Figure 7 depicts the evolution of $U^{\ast}$ for
the fifth hot film on the flap, which is located at $x/c_{\text{flap}}=0.511$.
As the flap angle is increased the hot film normalized voltage decreases. This
trend indicates a decreasing wall shear stress, which reaches a local minimum
for $\delta=20^{\circ}$. The first minimum at this flap angle, points out the
flow separation at this location. When the flap is further deflected, the hot
film voltage increases as the flow recirculation bubble intensifies and
reaches a local maximum. The voltage reaches a global minimum for an angle of
$\delta=~{}32^{\circ}$. This second minimum may be due to the apparition of a
second recirculation zone occurring for high deflection angles, as observed in
[6].

Figure 7: Evolution of the normalized voltage of the fifth hot film against
the deflection angle $\delta$ ($U_{\infty}=34.5$ m/s).

###  2.4 Reference signal and control architecture

Following these observations, a reference value for the fifth hot film sensor
is defined, such that the flow separation over the flap is avoided. The
normalized objective value $U_{\text{obj}}^{\ast}$ is fixed to $0.3903$. In
Figure 8, this reference value is superimposed with the evolution of the fifth
hot film normalized voltage. The controller aim is therefore to maintain the
hot film normalized voltage to the reference value. Therefore, two different
zones are defined in this chart. One for deflection angles
$\delta<13.8^{\circ}$ and the second one for $\delta>13.8^{\circ}$. The first
one corresponds to deflection angles for which actuators do not add momentum
to the flow, as the hot film voltage is above the reference value. The second
one defines angles $\delta$ for which the Festo valves are cyclically actuated
and aim at maintaining the hot film voltage to the reference value. Based on
this reference value, from now on denoted $\mathbf{r}$, ability of both
control strategies (either linear data-driven or positive model-driven) to
maintain this value is investigated.

Figure 8: Evolution of the normalized voltage of the fifth hot film (blue
curve) and the reference value (red curve) against the deflection angle
$\delta$ ($U_{\infty}=34.5$ m/s). The black curve separates areas without and
with actuation.

Based on the above considerations, a _reference signal tracking feedback
control architecture_ can be set up. With reference to Figure 9, one aims at
designing a $h$-sampled control law aiming at ensuring that the output signal
$\mathbf{y}(t_{k})$, measured by the fifth hot film, tracks
$\mathbf{r}(t_{k})$, the reference level previously defined. The controller
provides a sampled-time continuous order $\mathbf{u}(t_{k})$, transformed in
an on-off one $\overline{\mathbf{u}}(t_{k/N})$, leading to the controlled duty
cycle $\alpha$ applied by the PFA (Pulsed Fluidic Actuator), sampled $N$ times
faster (see also [23] for details on the PFA). The control problem boils down
to a _reference tracking one_.

Controller$h$PFA$h/N$Measurement$h$${\mathbf{u}(t_{k})}$$\overline{\mathbf{y}}(t_{k/N})$$\mathbf{r}(t_{k})$$\mathbf{e}(t_{k})$$\overline{\mathbf{u}}(t_{k/N})$$\mathbf{y}(t_{k})$

Figure 9: Overview of the considered closed-loop architecture. The controller,
sampled at frequency $h$, feeds a series of PFA acting along the wing span.
The system is illustrated by the setup photo, and the measurement is achieved
by the hot films located along the wing flap. The orange block is the overall
system.

######  Remark 1 (Sensor location)

In the considered experimental case, the fifth sensor is selected. However,
other locations or even multiple sensors may be considered. The impact of this
placement/selection may be considered in future works. The choice for this
sensor was dictated by the following considerations: first, the quality of the
signal was good, second, it was located far enough to actually see the
separation phenomenon.

######  Remark 2 (Control architecture extensions)

Similarly, in the considered experimental context, a single input, single
output controller is sought. The rest of the section sticks to this
configuration. However, extensions to multi-input and single-output are
possible. Extensions to multiple actuators are also possible but would lead to
considerably more complex analysis.

##  3 Flow separation control tuning

Considering the control architecture in Figure 9, this section details the
design and tuning of the controller. Two different strategies are implemented
and evaluated. First, the linear Loewner Data-Driven Control (L-DDC) [16]
(section 3.1) and second, a phenomenological-driven nonlinear positive control
[4] (section 3.2). Both configurations performances are commented and
illustrated in section 3.3.

###  3.1 L-DDC design

####  3.1.1 Idea and principle

The L-DDC belongs to the so-called data-driven reference model
approaches111DDC methods have a long history dating to the proportional,
integral, derivative (PID) tuning method by Ziegler-Nichols in early 40’s or
the self tuning regulator by Åström in the 90’s (see _e.g._ [29] for more
details and references). This field remains still very active (see _e.g._
[11]). The L-DDC procedure boils down to two steps: first deriving the _ideal
controller_ denoted $\mathbf{K}^{\star}$, and second, the _controller
identification_ via interpolation in the Loewner framework [20, 13].

$\mathbf{K}(z)$$\mathbf{H}(z)$$\mathbf{M}(z)$${\mathbf{u}}$${\mathbf{r}}$$\mathbf{e}$${\mathbf{y}}$$\boldsymbol{\varepsilon}$

Figure 10: Data-driven control problem formulation. $z$ denotes the complex
variable either in the continuous or sampled-time.

We recall the main steps in the SISO case and with the considered reference
tracking architecture. Following Figure 10, the objective is to find an LTI
controller with transfer function
$\mathbf{K}:\mathbb{C}\backslash\Lambda_{\mathbf{K}}\rightarrow\mathbb{C}$
that minimizes the transfer difference between $\mathbf{r}$ and
$\boldsymbol{\varepsilon}$, _i.e._ between the resulting closed-loop and a
user-defined reference model
$\mathbf{M}:\mathbb{C}\backslash\boldsymbol{\Lambda}_{\mathbf{M}}\rightarrow\mathbb{C}$.
This is made possible through the definition of the ideal controller
$\mathbf{K}^{\star}$, being the LTI controller that would have given the
desired reference model behaviour if inserted in the closed-loop. The latter
is defined as
$\mathbf{K}^{\star}=\mathbf{H}^{-1}\mathbf{M}(I-\mathbf{M})^{-1}$, where
$\mathbf{H}:\mathbb{C}\backslash\boldsymbol{\Lambda}_{\mathbf{H}}\rightarrow\mathbb{C}$
is the model of the system to control. In the data-driven case, when
$\mathbf{H}(z)$ is not explicitly known but may be evaluated at some frozen
values $z_{k}\in\mathbb{C}$, this definition may be recast as a set of
$k=1,\dots,N$ equations:

$\mathbf{K}^{\star}(z_{k})=\boldsymbol{\Phi}_{k}^{-1}\mathbf{M}(z_{k})(I-\mathbf{M}(z_{k}))^{-1},$
(7)

where $\boldsymbol{\Phi}_{k}=\mathbf{H}(z_{k})\in\mathbb{C}$ is the evaluation
of the unknown model at $z_{k}$. In an experimental context, one usually
considers sampling $\mathbf{H}$ at $z_{k}=\imath\omega_{k}$
($\omega_{k}\in\mathbb{R}_{+}$). In this case, $\boldsymbol{\Phi}_{k}$ is the
frequency response of the open-loop system at $\omega_{k}$. Then, the couple

$\\{z_{k},\mathbf{K}^{\star}(z_{k})\\}_{k=1}^{N},$ (8)

is referred to as the _raw data_ for our controller design. Finding a
controller $\mathbf{K}$ that _fits_ (8) can be considered to be an
identification/interpolation problem which may be solved by many approaches.
The Loewner framework [20] allows constructing both a function $\mathbf{K}$
with minimal McMillan degree and realization order $n\leq N$, satisfying
conditions (7) or an approximation of it with a realization of order $r<n$.

######  Remark 3 (Advantages of the L-DDC)

L-DDC is a combination of determining the ideal controller from frequency-
domain data via a reference model and the use of the Loewner framework to
construct a reduced order controller. Such an interpolatory-based data-driven
control design solves problems faced by practitioners: (i) the controller
design is directly obtained using open-loop raw data (8) collected on the
experimental setup, (ii) without any optimization process, only linear algebra
manipulation, (iii) and without any prior controller structure or order
specification (these latter may be automated by a rank revealing
factorization). This approach has proven to be effective for digital control
[28] and on experimental application [23]. [13, sec. 4] provides practical
details and didactic examples including infinite dimensional systems.

######  Remark 4 (Limitations of the L-DDC)

As a linear controller design, it embeds regular linear limitations. One
important has been highlighted during the experiments. It concerns the fact
that it does not handle actuator limitations, which is in this case works
on/off only and are only able to blow air. In presence of an integral action,
this may result in stability issues. This point is discussed in sections 3.2
and 3.3.

####  3.1.2 Application

The frequency-domain response describing the separated flow dynamics over the
flap is first needed. The operating point for this step is a deflection angle
$\delta=24^{\circ}$ and $U_{\infty}=34.5$ m/s, operating point for which the
flow is indeed separated at the considered measurement position. The actuators
command signal $\mathbf{u}(t_{k})$ consists in a logarithmic frequency sweep
applied to the duty cycle with frequencies ranging from $0.01$ Hz to $10$ Hz
over $180$ s. From this input, the fifth hot film response $\mathbf{y}(t_{k})$
is collected. The discrete frequency-domain transfer data from
$\mathbf{u}(t_{k})$ to $\mathbf{y}(t_{k})$ is obtained and denoted
$\\{\imath\omega_{k},\boldsymbol{\Phi}_{k}\\}_{k=1}^{N}$, where
$\omega_{k}\in\mathbb{R}_{+}$ is the pulsation and
$\boldsymbol{\Phi}_{k}\in\mathbb{C}$ is the SISO transfer response of the
system (orange block in Figure 9) and $N\in\mathbb{N}$ is the length of the
FFT. The data-driven Bode-like diagram is presented in Figure 11. It exhibits
a gain drop around 1 rad/s and a decay in the phase, characteristic of delayed
and fractional systems222Note that from this point, model identification may
be done in order to apply model-driven method. Here we skip this step to
directly go to the control design..

Figure 11: Frequency response gain and phase diagrams of the data
$\boldsymbol{\Phi}_{k}$ collected during the open-loop experiments.

Simultaneously with the previous step, the objective closed-loop transfer
function $\mathbf{M}$ is defined as a first order model
$\mathbf{M}(s)=1/(s/w_{0}+1))$ where $\omega_{0}=2\pi$ rad/s, is the natural
cut-off frequency. We refer to the black dashed lines of Figure 3, given in
the introduction. $\mathbf{M}$ mainly aims at ensuring no steady-sate error
(static gain objective set to one)333Note that the selection of a first order
reference model may be subject to discussions. Here we aim at illustrating the
most simple settings. For more details on the choice of $\mathbf{M}$, reader
is invited to refer to [16].. With reference to equation (7) we are now ready
to compute the ideal controller $\mathbf{K}^{\star}$ as well as its exact
interpolation $\mathbf{K}_{n}$, where $n=128$ is automatically selected by the
rank revealing factorization embedded in the Loewner process, and its
approximations $\mathbf{K}_{r}$ with an order $r=1$. After time-domain
discretisation ($h=1/100$ s), Figure 12 illustrates the controller frequency
response gains. The implemented linear controller is a pure sampled-time
integrator with gain $k=66.19$ (_i.e._ $\mathbf{K}_{r}(z)=66.19/(z-1)$).
Obviously, a proportional integral action model may also be identified with a
better accuracy. Here we stick to the integral action in view of the nonlinear
integral control analysis analysed in the next section.

Figure 12: Bode gain diagrams of the ideal controller data
$\mathbf{K}^{\star}$ (7), its exact interpolated sampled-time controller
$\mathbf{K}_{n}$ and its approximation $\mathbf{K}_{r}$ with an order $r=1$.

Figure 12 well illustrates that the exact interpolation perfectly matches the
data and the approximation with an order $r=1$ preserves the integral term.
Such an observation, coupled with the knowledge of the system input-output
positivity property (input blows air and output measures positive values
only), motivates the use of a more involved control strategy discussed in the
next section.

###  3.2 Nonlinear positive control design

####  3.2.1 Idea and principle

Observing that the considered system is stable (indeed, the configuration is
an amplification one, but without any instability) and input-output positive
(_i.e._ for any nonnegative input $\mathbf{u}$, the output $\mathbf{y}$ is
nonnegative), it seems interesting to exploit this property for control
purposes. Although not recent [10], positive systems have recently attracted a
lot of attention due to their surprising properties; see e.g. [3, 9]. In
particular, the theory of (linear) positive systems is playing an essential
role in the modeling, the analysis and the control of compartmental systems
which include biological, physiological, epidemiological and ecological
systems as special cases [14]. Recently, a novel type of integral controller –
the Antithetic Integral Controller (AIC) – was introduced in the context of
biological control and chemical reaction networks [5]. The rationale for
introducing such a controller was, among others, the derivation of a
controller having a positive system representation that could always return a
nonnegative control input. It was later proved in [4] that this nonlinear
integral controller enjoyed certain interesting properties which are absent
from its linear (_i.e._ non-positive) counterpart. It is also worth mentioning
that other nonlinear positive integral controllers exist [4], but the AIC
exhibits a lot of the desirable behavioral properties of the usual integral
controllers and this is the reason why it is considered here. Indeed, [4, Thm.
3.6] provides a stability proof of the closed-loop interconnection if the
original underlying model is a linear positive one. In fact, those stability
conditions coincide, in the worst-case, with the stability conditions of the
standard integral controller, which indicates that using the AIC is not more
constraining than using a linear integral controller.

######  Remark 5 (Closed-loop stability)

In the L-DDC setting, no stability proof can be guaranteed a-priori. This may
be checked afterward with specific data-driven techniques (see _e.g._ [16,
chap. 7] or [24]). However, one important feature of the positive design by
Briat [4, equation 2.1] is that the closed-loop control of a stable positive
system with an AIC ensures local exponential stability, while respecting input
signal constraints, under very mild conditions.

####  3.2.2 Application

The original AIC is given in [4, equation 2.1] with the following equation
set,

$\begin{array}[]{rcl}\dot{z}_{1}(t)&=&\mathbf{r}(t)-\eta z_{1}(t)z_{2}({t})\\\
\dot{z}_{2}(t)&=&\mathbf{y}({t})-\eta z_{1}(t)z_{2}(t)\\\
\mathbf{u}({t_{k}})&=&kz_{1}(t)\end{array}.$ (9)

The discretized version (using the backward method) takes the form ($h=1/100$
s):

$\begin{array}[]{rcl}z_{1}(t_{k}+h)&=&z_{1}(t_{k})+h\big{(}\mathbf{r}({t_{k}})-\eta
z_{1}({t_{k}})z_{2}({t_{k}})\big{)}\\\
z_{2}(t_{k}+h)&=&z_{2}(t_{k})+h\big{(}\mathbf{y}({t_{k}})-\eta
z_{1}(t_{k})z_{2}(t_{k})\big{)}\\\
\mathbf{u}({t_{k}})&=&kz_{1}(t_{k})\end{array}.$ (10)

Implemented in the real-time environment, user then tunes the values of
$k\in\mathbb{R}$ and $\eta\in\mathbb{R}_{+}$ according to the desired
controller. As (10) aims at reproducing the integral action, in our setting,
gain $k$ has been set equal to the integral term obtained with the L-DDC
approach; _i.e._ $k=66.19$ (section 3.1) and $\eta=300$, used to tend to a
pure integral action. We refer to [4] for further details.

###  3.3 Flow control experimental results

To validate the proposed reference feedback (linear and nonlinear) integral
control, four different type of experiments are carried out. In §3.3.1, the
lift coefficient $C_{L}$ with feedback is computed and compare to the
uncontrolled one (notice that this is the utilate objective of the control).
In §3.3.2, the robustness of the control with the deflection angle is
analysed. In §3.3.3, the frequency response of the controlled flap is computed
and compared to the expected objective. Finally in §3.3.4, some considerations
on the nonlinear positive controller are discussed.

####  3.3.1 Lift coefficient gain evaluation ($C_{L}$)

Both linear and non-linear integral controllers have been applied for the same
flow conditions. As expected for deflection angles below $13.8^{\circ}$,
valves are not opened as the hot film voltage is above the reference value
$\mathbf{r}$. When the hot film voltage tends to stand below the reference
value, valves are opened with a duty cycle determined by the controller (this
is typically the case when flap angle is increased). As an effect, the hot
film voltage is maintained at the reference value thanks to the feedback
control action. As presented in the introduction, Figure 2 highlights the
benefit of control on the lift coefficient $C_{L}$ increase.

For $\delta<13.8^{\circ}$, both uncontrolled and controlled flow present the
same lift coefficient, as in both cases valves are not opened. However, for
$\delta>13.8^{\circ}$, curves for the uncontrolled and controlled cases do not
superimpose anymore. The lift coefficient in the controlled case is higher
than the one of the uncontrolled case. Regarding the controlled case, the
linear evolution of $C_{L}$ is extended up to $\delta=24^{\circ}$. Between
$\delta=24^{\circ}$ and $\delta=26^{\circ}$, $C_{L}$ is reduced drastically.
This phenomenon is due to the apparition of flow separation at the flap
trailing edge. This flow separation does not spread over the entire flap as
the control counters it and hold it at the flap trailing edge444This
observation motivates future investigations involving additional sensors and a
more involved multi-input control.. These results can also be observed in the
analysis of the pressure coefficient evolution in both the uncontrolled and
controlled flow cases. As both the linear controller and the non-linear
positive controller have been applied with the same reference value on the hot
film, both control cases yielded the same results on the lift and pressure
coefficients.

####  3.3.2 Robustness to flap angle deflection ($\delta$)

In order to test the controllers robustness against the deflection angle,
measurements were also performed with varying angles. In a given experiment,
flap deflection angle was varied from $\delta=8^{\circ}$ to
$\delta=18^{\circ}$ and to $\delta=24^{\circ}$. The linear controller shows
its limits due to its linear integral behaviour and the fact that it does not
handle the positiveness of the system. As valves are not opened for
$\delta=8^{\circ}$, the linear controller takes into account the error and
therefore accumulates an integral error. In that sense, when the deflection
angle increases to $\delta=18^{\circ}$ for which valves have to be open, the
controller effect is in the wrong direction. As observed in Figure 13, once
the linear controller has overcome the accumulated error, the controller is
robust to the deflection angle variation from $18^{\circ}$ to $24^{\circ}$.

Figure 13: Evolution of the flap deflection angle against time (left axis) and
evolution of both the $5^{\text{th}}$ hot film voltage and reference value
(right axis) against time for the linear controller.

As such an error could not be tolerated in real application, a way to
circumvent it would be to implement an anti-windup on the linear controller.
This may be done at a price of more involved calculus, usually involving a
model, while here, the pure data-driven setup is employed. However, we proved
in these experiments than an other simple and efficient way to deal with this
integral behaviour is to implement a non-linear positive controller instead.
As described in Figure 14, the delay resulting from the integral error issue
totally vanishes considering this controller. In addition to that, in this
setup, the input output stability is formally guaranteed.

Figure 14: Evolution of the flap deflection angle against time (left axis) and
evolution of both the $5^{\text{th}}$ hot film voltage and reference value
(right axis) against time for the positive controller.

####  3.3.3 Frequency-domain responses

In addition, to compute the frequency response of the closed-loop control and
ensure that the closed-loop performances meet the reference model $\mathbf{M}$
objective, a sinus around the tracking value, with frequency sweep signal is
given as reference. The time-domain responses of the output and control
signals are reported in Figure 15.

Figure 15: Linear controller action for frozen flap deflection angles
$\delta$. Top: reference tracking performances. Bottom: produced control
signals.

The top frame allows computing the frequency response diagram given in the
introduction, illustrating that the performances are very close to the
expected one fixed by $\mathbf{M}$ (see Figure 3). On the bottom frame, the
control signal of the linear controller shows to reach the saturation quite
often, which is an other motivation for the nonlinear positive control.

####  3.3.4 Further remark on the nonlinear positive control

Finally, as the nonlinear positive control law seems to be the best solution,
a time-domain experiment where the reference is fixed but the flap deflection
angle $\delta$ travels from $34$ degrees to $0$ degree with a speed rate of
$0.5$ deg/s is performed (we compare here both strategies). Figure 16 (top)
shows the control signal actuation for both the linear and nonlinear
(positive) controllers. First it illustrates the fact that the positive
controller avoids the saturation while the linear one tends to often reach
then. In the same frame, for the positive control law, we compare the
experimental control signal (solid orange) with the reconstructed theoretical
continuous (9) (dashed yellow) and sampled (10) (dashed violet) when fed by
the experimental data. Both lead to a perfect match which confirms the good
implementation. Then Figure 16 (bottom) illustrates the positive sampled-time
controller internal states, which both remain, as expected, positives.

Figure 16: Experiment where flap angle $\delta$ travels from 34 degrees to 0
degree. Top: control signal of the linear and nonlinear experimental
controllers and theoretical nonlinear continuous and sampled control. Bottom:
sampled-time positive controller internal states

##  4 Conclusions

This paper presents experimental validations of active closed-loop control of
flow separation over a plain flap. We numerically and experimentally
demonstrate that flow separation may be improved by mean of a (SISO) reference
signal tracking feedback control architecture, involving a controller with
integral action. We also proposed two control laws: first _(i)_ a linear one
where the integral gain is computed via a direct data-driven approach, and
second, _(ii)_ a nonlinear positive controller to account for the system
limitations, using the very same gain.

Both strategies enabled to maintain a reference voltage value on the objective
hot film placed at the flap mid-chord. The latter reference, being calculated
based on the flow separation observation. Both controllers efficiency were
assessed through lift coefficient calculations derived from pressure
measurements (Figure 2). Their robustness to the flap deflection angle was
also tested through experiments, during which the flap angle was continuously
varied. Additionally, we demonstrate that the expected theoretical
performances were recovered experimentally. The most significant advantage of
these control techniques lies in the simplicity of their application, which is
of importance for practitioners in view of experimentation and implementation.

Application of such controllers could be extended to other flow control
problems in future works, as well as more detailed validations.

## Acknowledgments

This work was funded by the French National Research Agency (ANR) in the
framework of the ANR ASTRID MATURATION CAMELOTT-MATVAL Project. It is
supported by the regional platform CONTRAERO in the framework of the CPER
ELSAT 2020 Project. The Defence Innovation Agency (DIA) has also financially
sustained this work. The authors also thank RENATECH, the French national
nanofabrication network, and FEDER. It has also been financed by the ONERA
research project FluiDyCon, Fluid Dynamical Control.

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