# Identification of a plasma actuated open-cavity under flow control

Sami Tliba, Luca Greco, Mohamed Yazid Rizi, Luc Pastur, François Lusseyran, Lionel Mathelin

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Sami Tliba, Luca Greco, Mohamed Yazid Rizi, Luc Pastur, François Lusseyran, et al. Identification of a plasma actuated open-cavity under flow control. 2025 Joint IFAC Conference SSSC, TDS, COSY, CentraleSupélec, Jun 2025, Gif-sur-Yvette, France. 10.1016/j.ifacol.2025.09.526. hal-05148991

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# Identification of a plasma actuated open-cavity
under flow control *

S. Tliba * L. Greco * Y. Rizi ** L.R. Pastur **** F. Lusseyran ***
L. Mathelin ***

* Université Paris-Saclay, CNRS, CentraleSupélec, L2S,
F-91190, Gif-sur-Yvette, France

(e-mails: {sami.tliba,luca.greco}@l2s.centralesupelec.fr)

** Valeo Siemens eAutomotive

*** LISN, CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

(e-mails: {mathelin,francois.lusseyran}@lisn.fr)

**** ENSTA-Paris, Institut Polytechnique de Paris, IMSIA, F-91120,
Palaiseau, France (e-mail: luc.pastur@ensta-paris.fr)

**Abstract:** This paper addresses an input-output identification problem for a simulated incompressible open cavity flow. Despite the highly nonlinear and infinite-dimensional nature of the system, we look for a low complexity, linear, finite-dimensional model still capable of capturing the most salient features of the system. The inherent system instability, leading to the formation of its characteristic limit cycle, is a major obstacle in the application of classical identification schemes. To overcome such difficulty, we adopt a closed-loop identification method, which requires a preventive stabilization via an empirically designed controller. Being the controller perfectly known, an indirect approach is pursued. It consists of identifying the closed-loop transfer function and then reconstructing the unstable open-loop model by loop inversion. Extensive experiments led us to opt for a frequency domain identification based on an Output Error model structure. The identified model proves to be consistent with the infinite-dimensional one and, in particular, to reproduce its main instabilities.

**Keywords:** Flow control; closed-loop identification

## 1. INTRODUCTION

Fluid mechanics offers challenging problems for the design of efficient and robust controllers. One of the main difficulties is the very large – in fact, infinite – number of degrees of freedom required for representing the state vector. This is due to the fact that the governing (Navier-Stokes) equations are partial differential equations acting on a spatially-distributed quantity such as the velocity field, which makes it very challenging for designing a controller aimed at working in real-time applications. Another challenging difficulty is the intrinsic nonlinearity of the governing equations. Although in some applications the problem can be linearized about the (unstable) operating point, for instance when the flow is convectively unstable and behaves like a noise amplifier, in most applications, the system instead behaves like a nonlinear oscillator whose dynamics result from the non-linear saturation of the initial instability; see for instance Sipp and Schmid [2013] or Fabbiane et al. [2014] for a detailed discussion of the subject.

In this paper, we consider an incompressible open-cavity flow, which belongs to the second class. Such flows are known to develop self-sustained oscillations, which are the source of a powerful aeroacoustic noise in several applications, for instance in high-speed trains or aircraft during landing and takeoff. The challenge is to design a controller able to suppress the self-sustained oscillations of the shear-layer and stabilize the

steady base flow, by the design of an accurate reduced-order model able to capture the arising dynamics of the cavity flow, while remaining convenient to implement experimentally and achieving real-time performances.

Over the past twenty years, several strategies have been proposed in open-loop and closed-loop plant designs, with actuators that disrupt the flow, either placed at the bottom of the cavity (see Fig. 1 for a sketch of the configuration), as in Yoshida et al. [2005], or at the upstream edge where the flow is the most sensitive to velocity changes, as exemplified in Kegerise et al. [2002]. Proper Orthogonal Decomposition (POD), combined with Galerkin projection, has been the most widely used method to obtain reduced-order models, see for instance Rowley et al. [2004] among others. However, although the resulting POD modes are *optimally controllable*, their observability is not considered in constructing the reduced order model. Yet, the so-called *balanced POD (BPOD) method* is aimed at balancing both the controllability and the observability in the modal decomposition of a system governed by linear equations. Unfortunately, BPOD cannot be applied to experiments since it requires the adjoint state to be determined, which can only be accessed from numerical simulations, as shown in Barbagallo et al. [2011], Sipp and Schmid [2013].

In experimental applications, this problem can be overcome by the introduction of the so-called Eigen-Realization Algorithm (ERA), first introduced by Juang and Pappa [1985]. Recently, Ahuja [2009] has established an equivalence between BPOD and ERA. ERA allows a significant reduction in computation

<footnote>* This work was partially supported by the Labex LaSIPS and the ICODE institute, research project of the Idex Paris-Saclay.</footnote>
cost and memory resources compared to the original system description. However, ERA applies to linear systems and, when applied to unstable systems, generally requires the prior knowledge of a controller which maintains the system in the vicinity of the unstable equilibrium point. Once the system has been stabilized, ERA can be used in a closed-loop configuration to extract the unstable dynamics, as shown by Juang and Phan [1994], Phan et al. [1994]. Recently, this method has been successfully used to identify a model of a flow past a cavity, both in a subsonic regime [Illingworth et al., 2012] and in the incompressible limit [Rizi et al., 2014], after the unstable equilibrium point has been stabilized using an appropriate — but empiric— controller. The resulting optimal controller could efficiently suppress the self-sustained oscillations and stabilize the steady base flow when the loop is closed. A review of control strategies, and associated reduced-order modeling, for the control of fluid flows can be found in Brunton and R. [2015].

Inspired by these results and especially by those in [Hervé et al., 2012] and [Atam et al., 2017], we follow here a similar procedure for the design of a finite-dimensional input-output dynamical model for the open-cavity flow, equipped with a plasma actuator and a pressure sensor. This model is derived from a closed-loop identification of a simulated infinite-dimensional model of the cavity system, stabilized by an empirical controller based on a delayed proportional control law in continuous time. This identification is performed in discrete time through an *Out-put Error* noisy model, obtained from frequency domain data. By using a good discrete-time approximation of the continuous-time delay operator, based on a Thiran Fractional Delay Filter [Thiran, 1971], a linear model of reduced order is obtained to describe the dynamical behaviour of the uncontrolled open-cavity system.

The outline of the paper is as follows. In section 2, the open-cavity system is described. In section 3, we present the identification problem and the features of the empirical controller, designed to stabilize the steady-state base flow around an equilibrium operating point. In section 4, we present the results of this identification approach and then conclude this work with some future developments.

## 2. CONFIGURATION OF THE SYSTEM

### 2.1 Cavity flow

Cavity flows are primarily characterized by their impinging shear-layer, which is known to be unstable against *Kelvin-Helmholtz modes*, as early reviewed in Rockwell and Naudascher [1978]. The vortices generated by this instability are eventually advected downstream of the shear-layer until they reach the cavity trailing edge, causing a pressure perturbation which is instantaneously fed back to the cavity leading edge, in the incompressible regime. The feedback reinforces the shear-layer instability, until a limit cycle is reached in which the shear-layer oscillations are self-sustained and strongly energetic at very well-defined frequencies, see for instance Basley et al. [2011].

The study is carried out with direct numerical simulations of a two-dimensional incompressible flow over a rectangular cavity, governed by the Navier-Stokes equations:

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic description of an open cavity flow.</figcaption></figure>

$$ \left\{ \begin{array}{l} \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\frac{1}{\rho} \nabla p + \frac{f}{\rho} + \mathbf{v} \Delta \mathbf{v}, \\ \nabla \cdot \mathbf{v} = 0, \end{array} \right. \qquad (1) $$

where $\mathbf{v}$ is the discretized velocity field, $p$ the pressure field, and $f$ a body force accounting for the action of the actuator. The kinematic air viscosity is $\nu = 16 \times 10^{-6}$ (m$^2$/s) and density $\rho = 1.23$ (kg m$^{-3}$). The cavity aspect ratio is $L/H = 2$, where $L = 0.1$ (m) and $H$ are the cavity length and depth, respectively. A Cartesian coordinate system $(X, Y)$, for streamwise and crosswise directions, respectively, is set at the top corner of the upstream cavity wall (see Fig. 1).

The numerical domain is meshed with $296 \times 128$ nodes, among which $96 \times 64$ are devoted to the cavity. The mesh is particularly refined close to the walls and at the cavity top in order to improve the spatial resolution of boundary layers and shear layers. The mesh resolution has proven to be dense enough to capture all the phenomena arising on this open cavity flow. Usual no-slip boundary conditions are applied to the walls. The incoming flow velocity $\mathbf{v} = (v_\infty, 0)$ is uniform, $v_\infty = 1.2$ (m/s), and the Reynolds number is $Re = v_\infty L/\nu$ based on the cavity length $L$. The momentum thickness of the laminar boundary layer, upstream of the cavity, is $\theta = 12.1 \times 10^{-4}$ (m). In this configuration, the cavity flow oscillates at a single dominant frequency $f_0 \approx 13.0$ (Hz), and harmonics, corresponding to the Strouhal number $St = f_0 L/v_\infty \approx 1$. All basic features of the cavity flow configuration are summarized in Fig. 1. For more details about the numerical schemes, see Rizi et al. [2014].

### 2.2 Actuator and sensor

A plasma actuator, located at the upstream edge of the cavity, produces the bulk force $f := (f_x, f_y)$ in Eq. (1), where $f_y$ is zero. Such a force induces an acceleration to the fluid particles in the small rectangular area where the plasma actuator lies. Note that while the actuator brings momentum to the flow, it also injects a zero-net-mass flux.

A pressure sensor, located near the downstream edge, provides a local measure of pressure denoted $p_s$. For identification and control purposes, the open cavity flow is regarded as a dynamical system with one scalar control input, denoted $u$ and corresponding to $f_x$, and one scalar output, denoted $y$ and corresponding to $p_s$.

Figure 2 shows a simulated response of the pressure sensor when the system is running. After the transient response, one can observe the establishment of the limit cycle, introduced in Subsection 2.1, that is periodic with frequency $f_0$.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Simulated time response of the measured output y, corresponding to a relative pressure.</figcaption></figure>

### 3. IDENTIFICATION PROBLEM

The uncontrolled cavity-flow system is characterized by the presence of self-sustained oscillations corresponding to a limit cycle in the variable $p_s$. The main control objective is to suppress these oscillations and to stabilize the steady-state base flow, with a simple enough controller to be implemented in real time. It is advisable that the design of such a controller should be based on a simpler model than the one given in (1), which is of infinite dimension and highly nonlinear. To this aim, we propose to build a low complexity, linear, and finite-dimensional input-output model by identifying the behavior of the cavity flow system. The first hurdle we encounter on this path is that well-known identification techniques require the systems to be stable. The cavity, on the contrary, is clearly affected by an instability, which leads to the generation of self-sustained oscillations by saturation phenomena. This calls for a preventive stabilization of the system by means of an empirically designed controller. As a consequence, the identification procedure has to be performed in a closed loop.

#### 3.1 Delayed feedback control

Let us consider the feedback control structure in Fig. 3, where the cavity-flow system is in closed-loop with the empirical controller. Denote by $T_{uy}$ the sought transfer function from $u$ to $y$ in continuous time, by $w$ an exogenous signal positively added to $u$, and by $T_{wy}$ the closed-loop transfer function from $w$ to $y$ in continuous time.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Feedback control of the cavity-flow system.</figcaption></figure>

As mentioned in Rizi et al. [2014], the time-delay proportional feedback controller

$$C(s) = ae^{-\tau s}, \quad (2)$$

where $s$ is the Laplace variable, stabilizes the flow to a desired operating point. It corresponds to the body force

$$f = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} a p_s(t - \tau) \\ 0 \end{bmatrix}, \quad (3)$$

where the proportional gain $a > 0$ determines the magnitude of the applied force, and must be carefully chosen, in order to avoid driving the cavity flow to an undesired flow

regime. An ideal value for the time delay $\tau$ was found to be $\tau \approx 12.9 \times 10^{-3}$ (s) (roughly corresponding to $\frac{1}{6f_0}$), with a proportional gain $a = 10$.

Figure 4 shows a longer excerpt of the same simulation in Fig. 2. It clearly displays the stabilizing action of the feedback control when it is turned on at 20 (s).

<figure><img src="image_5.png" /><figcaption>Fig. 4. Time response of the measured output y.</figcaption></figure>

#### 3.2 Continuous-time simulation and discrete-time identification

The simulated data has been produced in closed-loop by using the feedback controller (2) with the settings of Subsection 3.1. The simulation has been performed in continuous time with the numerical scheme detailed in Rizi et al. [2014] and with a simulation step $\Delta_t$ equal to $10^{-4}$ (s). Apparently, it allows to capture the dynamics of the main phenomena, including the self-sustained oscillations at low frequencies. For the sake of simplicity, the identification procedure will be performed in discrete time. It will be assumed that the effect of sampling is negligible. Conversion from discrete to continuous is made by using a Tustin approximation. The sample time $T_s$ has been chosen small enough to be consistent with the bandwidth of interest, ie around the frequency $f_0$ of the self-sustained oscillations, say [0.01 50] (Hz), so that $T_s = 100\Delta_t = 10^{-2}$ (s). As a consequence, the data has been suitably decimated to match the sample time considered.

In the sequel, we will use the same notation for the sampled signals $u$ and $y$ as for their continuous-time counterparts. We denote by $\tilde{T}_{uy}$ the discrete time transfer function from $u$ to $y$ and by $\tilde{T}_{wy}$ the discrete time transfer function from $w$ to $y$. We also denote by $\hat{C}$ the discrete-time transfer function equivalent to $C$.

#### 3.3 Closed-Loop Identification procedure

The identification of closed-loop systems comes with specific issues and limitations [Ljung, 1999]. They are often related to the correlation between noise sources and control input. There exist two well-known approaches to deal with these specific issues in closed-loop identification, known as the direct and the indirect approach. Both approaches require the introduction of an exogenous input $w$ (see Fig. 3), which has to have a sufficiently rich frequency content to excite the main modes of the cavity system. This means that the magnitude of the power spectrum of $w$ must be high over a wide frequency range. Usually, $w$ is taken as a white noise or a chirp signal that sweeps the considered frequency range. In this work, the Pseudo Random Binary Sequence (PRBS) has proven to be the most effective one. Such a signal has been applied at $t = 35$ (s) during 40 (s), for all the different tests performed.
The direct identification approach consists of identifying $\tilde{T}_{uy}$ by taking into account the input data $u$ and the output data $y$ in the identification algorithm. Note that, in this case, it is the signal $u$ that is required to have good spectral properties. However, even if $w$ is suitably designed, the signal $u$ can lose much of its frequency richness because of the closed-loop connection. This issue can significantly impair the reliability of the identified model. Moreover, in general, the direct approach suffers from the correlation created by the closed-loop structure between the measurement noises and the computed control $u$. As a consequence, this can introduce a bias in the parameters' estimation.

The indirect approach, on the other hand, avoids the aforementioned problems by identifying the transfer $\tilde{T}_{wy}$, which makes it more suitable for realistic problems, as emphasized in Eperlein et al. [2015]. The design of an empirical controller to stabilize such a sensitive system proved to be a nontrivial task, demanding an extensive trial-and-error phase. The ensuing proportional-delayed controller provides a relatively simple structure, but shows poor spectral properties, which make the direct approach not so viable an option. For all these reasons, we adopt the indirect approach.

According to this approach, the desired transfer $\tilde{T}_{uy}$ is then obtained by inverting the closed-loop relation

$$ \tilde{T}_{wy} = \frac{\tilde{T}_{uy}}{1 - \hat{C}\tilde{T}_{uy}} \quad (4) $$

as follow

$$ \tilde{T}_{uy} = \frac{\tilde{T}_{wy}}{1 + \hat{C}\tilde{T}_{wy}} \quad (5) $$

The drawback of this approach is that it requires a very good knowledge of $\hat{C}$, the discrete-time controller equivalent to the empirical controller $C$ in continuous time. Here we find a second hurdle: the controller is infinite dimensional, and we seek a finite-dimensional model for $\tilde{T}_{uy}$ of reduced order. A solution is to approximate, in discrete time, the infinite-dimensional delay operator with a finite-dimensional transfer function similar to the Padé approximation, and able to cope with a fractional delay of the step time. The Thiran Fractional Delay Filter (TFDF) [Thiran, 1971] is known to be a good answer to this issue. Denote by $z$ the variable of the $z$-transform with sample time $T_s$ related to the Laplace variable $s$ by the relation $z = e^{T_s s}$. The $z$ transfer function of the TFDF has the form

$$ H(z) = \frac{\alpha_N z^N + \alpha_{N-1} z^{N-1} + \dots + \alpha_1 z + 1}{z^N + \alpha_1 z^{N-1} + \dots + \alpha_{N-1} z + \alpha_N}, $$

where the coefficient $\alpha_1, \dots, \alpha_N$ are given by

$$ \alpha_j = (-1)^j \binom{N}{j} \prod_{l=0}^{N} \frac{D-N+l}{D-N+j+l} \quad \forall j = 1, 2, \dots, N $$

with $D = \tau/T_s$ and $N = \text{ceil}(D)$ (round towards plus infinity). Here, we have $D \simeq 1.29$ and $N = 2$. Thus, we adopt a 2nd-order TFDF to approximate the controller $C$, leading to the discrete-time controller

$$ \hat{C}(z) = a \frac{\alpha_2 z^2 + \alpha_1 z + 1}{z^2 + \alpha_1 z + \alpha_N}, $$

where $\alpha_1 = -\frac{2\varphi}{\varphi+3} \simeq 0.62$, $\alpha_2 = \frac{\varphi(\varphi+1)}{(\varphi+3)(\varphi+4)} \simeq -27.33 \times 10^{-3}$ and $\varphi = D-N = -0.71$.

### 3.4 Frequency domain approach with OE noisy model

We have to select an appropriate model structure for the main system dynamics within a bandwidth of interest. Several noisy model structures are available, such as, for instance, ARX, ARMAX, Output Error (OE), or Box & Jenkins [Ljung, 1999]. It is worth noting that, even if simulated data are not corrupted by noise, a noise model is chosen to capture all the dynamics outside the bandwidth of interest, given a desired order for the model. In our case, the best results were obtained with an OE model applied to frequency domain data. An OE model in discrete time has the following form for the closed-loop identification procedure:

$$ y(k) = \frac{B(q^{-1})}{A(q^{-1})} w(k-d) + e(k) \quad (6) $$

where $k$ is the discrete time variable, $d$ an unknown delay to identify, $e$ is a stochastic disturbance of the system (noise), $q^{-1}$ is the unit delay operator, $B$ and $A$ are the polynomials to identify that have the following form:

$$ \begin{aligned} B(q^{-1}) &= b_1 q^{-1} + \dots + b_{n_B} q^{-n_B}, \\ A(q^{-1}) &= 1 + a_1 q^{-1} + \dots + a_{n_A} q^{-n_A}. \end{aligned} $$

The degrees $n_A$ and $n_B$ should be sufficiently large to capture all the dynamics in the bandwidth of interest and small enough to lead toward a low complexity identified model. The discrete time transfer function $\tilde{T}_{wy}$ can easily be derived from these polynomials by applying a $z$-transform to relation (6). Then, the sought discrete time transfer function $\tilde{T}_{uy}$ of the uncontrolled cavity data is obtained using the relation (5).

<figure><img src="image_6.png" /><figcaption>Fig. 5. Frequency domain data: spectral estimation of the discrete-time closed-loop transfer function $\tilde{T}_{wy}$.</figcaption></figure>

The frequency domain data are derived from time domain data by computing the Power Spectrum Density (PSD) of the input/output signals, using Welch's algorithm, with suitable settings. Our time data contains $n = 5500$ input-output pairs of samples. These data are divided into sections equal in length to floor($n/8$) = 687 (round toward minus infinity). Each section is windowed with a Kaiser Window set to give acceptable sidelobe attenuations. The sections are overlapped using 98% of the length of a section, say 673 pairs of samples. Finally, the periodogram is computed with zero-padding since the number of points to compute the PSD is chosen equal to 8192. Figure 5 shows the resulting frequency domain data of the open-cavity
system in closed-loop. A good confidence can be granted to the frequency data in the range [8 22] (Hz) which encompasses the frequency $f_0$ of the self-sustained oscillations. On the other hand, outside the frequency range, it is difficult to determine if the peaks correspond to some relevant dynamics or not.

The identification criterion is based on the minimization of the prediction error variance with a constraint of stability. This constraint is necessary since the collected data comes from the stabilized cavity system with the delayed proportional controller. The System Identification Toolbox of MATLAB [Ljung, 2014] has been used since it implements several efficient algorithms of identification, for any structure and with several tunable algorithms of optimization.

## 4. RESULTS

The purpose of this work was to obtain a linear model of order as low as possible satisfying the following unavoidable constraints:

* the identified closed-loop transfer function $\tilde{T}_{wy}$ should be stable;

* the corresponding open-loop transfer function $\tilde{T}_{uy}$, derived from (5), should contain unstable modes, to be consistent with the limit cycles observed in Fig. 4;

* this transfer function, once in closed-loop with the initial empirical controller in (2), should give a stable system;

* finally, the frequency response of the identified model should match with the data, in gain as well as in phase, within the frequency range of interest.

Taking all of this into account, the identification procedure has led to a transfer function $\tilde{T}_{wy}$ of order $n_A = 13$, with a direct feedthrough term and no delay $d$, which means that $n_B = n_A = 13$ and $d = 0$. The Bode plot of $\tilde{T}_{wy}$ is compared with the corresponding estimated one in Fig. 6.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Comparison of the Bode diagrams of the transfer function $\tilde{T}_{wy}$ estimated from data (blue) and the identified one (dashed red).</figcaption></figure>

The transfer function $\tilde{T}_{uy}$ derived from $\tilde{T}_{wy}$ is of order 16 after reduction by truncation and matching DC gain. The reduction concerned one discrete-time pole, stable and fast, real and negative, which has no physical equivalent in continuous time and

thus, no physical meaning. It is worth noting that the identified transfer function $T_{\tilde{u}y}$ in [Rizi et al., 2014] was of order 23 (after model reduction). In contrast, the model proposed in this work is of order 16. Moreover, in [Rizi et al., 2014], the authors have said that the feedback with the delayed proportional controller did not stabilize one of the open-loop unstable poles of their identified model. This is not the case in our work since we have enforced an identification of a stable transfer function $\tilde{T}_{wy}$.

The comparison of the Bode plots for both computed $\tilde{T}_{uy}$, with a focus on the frequency range of interest, is shown in Fig. 7. This figure shows that the self-sustained oscillations of frequency $f_0$ seem to be captured by an unstable open-loop mode at roughly 12.6 (Hz) and another one at roughly 17.6 (Hz) with a higher peak of resonance.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Comparison of the Bode diagrams of the transfer function $\tilde{T}_{uy}$ estimated from data (blue) and the identified one (red).</figcaption></figure>

**Model validation:** the obtained open-loop model $T_{\tilde{u}y}$ has been validated in closed-loop with the controller $\hat{C}$ by simulating the time response of the closed-loop and comparing with the output data. More precisely, we have applied the input data $w$ to this closed-loop and we have computed the relative error between the response $y$ and the output of the identified model in closed-loop with the controller $\hat{C}$. We have observed a maximum relative error of 5%, as shown in Fig. 8. This error is computed by the relation $\frac{y-y_m}{\|y\|_\infty}$, where $y_m$ is the time response of the identified model to input $u$. Other test signals $w$, such as chirp signals or other PRBS signals, have been used to check the validity of the identified model presented in this paper. This model appeared to be efficient with maximum relative errors less than 7%. The figures of such tests have not been presented due to a lack of space.

## 5. CONCLUSION

In this paper, we have addressed the identification problem of a simulated incompressible open-cavity flow, equipped with a plasma actuator and a pressure sensor. The purpose of this work was to obtain a linear model of low complexity that could capture the most salient features of a system that is originally highly nonlinear and of infinite dimension. Such a model could be used to design low complexity controllers that can be implemented in real-time in an experiment. We have developed a closed-loop identification method, in discrete-time, based on
<figure><img src="image_9.png" /><figcaption>Fig. 8. Relative error between the closed-loop response of the identified model and the output data, to the same input *u*. The error is displayed from the instant $t = 35$ (s) when the PRBS signal *w* has been applied.</figcaption></figure>

the identification of an Output Error noisy model with frequency domain data. This method could be considered thanks to a feedback control based on a delayed proportional controller, which was empirically set up. A discrete-time approximation of this controller, using a 2^nd^-order Thiran approximation of the delay, permits the reconstruction of the unstable open-loop model by loop inversion. The model we have identified fixes the issue with the closed-loop unstable pole that affected the one proposed in Rizi et al. [2014]. Moreover, it is of a lower order. Applying this method to real experimental data would be very challenging, and this can be considered as a perspective for this work.

## ACKNOWLEDGEMENTS

The authors would like to thank the Labex LaSIPS of Université Paris-Saclay for supporting this work.

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