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# Reactive control of $2^{nd}$ Mack mode in a supersonic boundary layer with
freestream velocity/density variations

Pierre Nibourel1<EMAIL_ADDRESS>Colin Leclercq1  Fabrice Demourant2
Eric Garnier1  Denis Sipp1  1ONERA-DAAA, Université Paris-Saclay, 8 Rue des
Vertugadins, 92190 Meudon, France 2ONERA-DTIS, 2 Avenue Edouard Belin, 31000
Toulouse, France

###### Abstract

We consider closed-loop control of a two-dimensional supersonic boundary layer
at $M=4.5$ that aims at reducing the linear growth of second Mack mode
instabilities. These instabilities are first characterized with local spatial
and global resolvent analyses, which allow to refine the control strategy and
to select appropriate actuators and sensors. After linear input-output reduced
order models have been identified, multi-criteria structured mixed
$H_{2}$/$H_{\infty}$ synthesis allows to fix beforehand the controller
structure and to minimize appropriate norms of various transfer functions: the
$H_{2}$ norm to guarantee performance (reduction of perturbation amplification
in nominal condition) and the $H_{\infty}$ norm to maintain performance
robustness (with respect to sensor noise) and stability robustness (with
respect to uncertain free-stream velocity/density variations). Both
feedforward and feedback setups, i.e. with estimation sensor placed
respectively upstream/downstream of the actuator, allow to maintain the local
perturbation energy below a given threshold over a significant distance
downstream of the actuator, even in the case of noisy estimation sensors or
free-stream density variations. However, the feedforward setup becomes
completely ineffective when convective time-delays are altered by free-stream
velocity variations of $\pm 5\%$, which highlights the strong relevance of the
feedback setup for performance robustness in convectively unstable flows.

###### keywords:

##  1 Introduction

Transition to turbulence in a boundary layer results in increased wall
friction, penalizing aircraft drag. At high speeds, the generated heat is
significant and becomes a major concern for the design of
supersonic/hypersonic vehicles (Juliano & Borg, 2015). The transition to
turbulence in boundary layers is initiated by amplification of external
disturbances of various kinds (roughness, sound waves, freestream turbulence,
etc.) and several paths to transition are possible depending on the nature and
intensity of incoming disturbances (Morkovin, 1969). With low levels of
disturbances, their growth is described by linear stability theory. The
stability of a supersonic boundary layer has been widely studied in the
literature (Mack, 1984; Malik, 1989; Ma & Zhong, 2003; Bugeat et al., 2019,
and many others). For sufficiently high Mach numbers, this configuration is
characterised by the presence of two distinct inviscid instability mechanisms:
a generalised inflection point for the first Mack mode (Mack, 1984) and a
region where the streamwise base-flow velocity relative to the disturbance
phase velocity is supersonic for the second Mack mode implying that acoustic
noise is trapped in this region (Mack, 1984; Fedorov, 2011). A classical
approach for relating instability to transition is precisely based on this
linear framework and is called the $N$-factor method (Smith & Gamberoni,
1956), wherein transition is assumed to occur when a perturbation has been
amplified by a factor of $e^{N}$, which defines an energy threshold depending
on the disturbance environment.

Numerous studies addressed the problem of transition delay in the supersonic
boundary layer flow using active control: Gaponov & Smorodsky (2016) injected
heavy gas through porous wall to reduce surface friction and heat transfer,
Sharma et al. (2019) resorted to the generation of streaks to counter
transient instabilities, Yao & Hussain (2019) investigated the impact of
spanwise wall oscillation on the drag of a supersonic turbulent boundary layer
and Jahanbakhshi & Zaki (2021) took advantage of the sensitivity of the Mack
modes to temperature to delay transition to turbulence. However, all the
aforementioned studies employed predetermined active strategies which do not
exploit any real-time measurement and may therefore be less cost effective and
robust to changes in operating conditions than a reactive control strategy
(Gad-el-Hak, 2000). To the best of our knowledge, reactive control of
convective instabilities in the supersonic boundary layer flow has not yet
been considered.

Contrary to oscillator flows (Barbagallo et al., 2009; Schmid & Sipp, 2016)
which are, by definition, linearly globally unstable (Huerre & Monkewitz,
1990) and have intrinsic dynamics, noise-amplifier flows like the supersonic
boundary layer are extremely sensitive to external disturbances, which are
amplified downstream as they are convected by the flow (hence the name
convective instabilities). In this context, the purpose of reactive control is
to cancel out noise-induced perturbations (Bagheri et al., 2009; Barbagallo et
al., 2012) by producing destructive interference with an actuator. This task
is difficult for mainly two reasons: a) the detection of the time-delay
associated with the convection of perturbations which may trigger out of phase
action with respect to the incoming perturbations, b) the wide spatially
evolving range of amplified frequencies along the plate, from higher
frequencies upstream to lower ones downstream.

###  1.1 Historical dominance of feedforward/LQG for the control of noise-
amplifier flows

Controller synthesis is only feasible for models of small dimensions, of the
order of $10^{2}$ degrees of freedom at most, because of the computational
cost and storage requirements of currently available tools (Utku & Garba,
1989). Therefore, most fluidic control problems require the identification of
reduced order models (ROMs), using for instance the eigensystem realization
algorithm (ERA) on impulse response data. This popular tool, introduced by
Juang & Pappa (1985), has already been used in many control studies for noise-
amplifier flows (Belson et al., 2013; Dadfar et al., 2013; Sasaki et al.,
2020, and many others). Once ROMs are obtained, the control law is built with
classical tools of control theory which are mathematically well-established in
a linear framework and thus perfectly suited for controlling the linear growth
of small perturbations.

In noise amplifier flows, there is no synchronization of the dynamics at a
global scale, perturbations from an actuator $u$ are rapidly damped in the
upstream direction, hence the control setup changes fundamentally depending on
the position of the estimation sensor $y$ relative to $u$. When $y$ is placed
upstream, actuator-induced perturbations are not observable and the
configuration is termed ”feedforward” (Bagheri et al., 2009; Semeraro et al.,
2011; Hervé et al., 2012; Juillet et al., 2013; Morra et al., 2020). On the
other hand, when $y$ is placed downstream, the sensor measures the
superposition of noise-induced and actuator-induced perturbations, hence the
term ”feedback” (Barbagallo et al., 2012; Belson et al., 2013; Semeraro et
al., 2013b; Vemuri et al., 2018; Tol et al., 2019). In this case though, there
may be a significant time-delay before the effect of actuation may be seen by
the sensor, because perturbations are convected at a finite rate by the
underlying base flow: the farther downstream $y$ is, the longer the delay.

The literature on noise-amplifier control is dominated by the linear-
quadratic-Gaussian (LQG) synthesis (Semeraro et al., 2011; Barbagallo et al.,
2012; Juillet et al., 2013; Sasaki et al., 2018a; Tol et al., 2019, and many
others), a synthesis method dating back to the 1960s (Kalman, 1964). Despite
being theoretically optimal with respect to a performance criterion, this
method comes with no guarantees on stability margins (Doyle, 1978). In other
words, tiny errors in the model may end up in an unstable feedback loop when
$y$ is placed downstream of $u$ (feedback setup), which represents a major
drawback for practical applications. Using the loop-transfer-recovery (LTR)
method, it is in some cases possible to overcome this lack of stability
robustness by overwhelming the control signal entering the estimator
(Kwakernaak, 1969; Doyle & Stein, 1981). This procedure has for example been
successfully used by Sipp & Schmid (2016) to improve the stability robustness
of their controller in the case of a flow over an open square cavity
(oscillator flow). The recovery procedure works by inverting the plant
dynamics in order to obtain ultra-fast estimators. This procedure leads to an
unstable closed-loop in the case of systems with time-delays, because they
possess right-half plane zeros which are converted into right-half plane poles
(Zhang & Freudenberg, 1987; Skogestad & Postlethwaite, 2005; Sipp & Schmid,
2016). As a result, this method is not suitable for noise-amplifier flows in
general, and in particular, the supersonic boundary layer flow. Contrary to
the feedback structure, the feedforward design is unconditionally stable and
its implementation via LQG is not a problem. Therefore, feedforward
configurations combined with LQG syntheses dominate the noise-amplifier flow
control literature, particularly in the incompressible boundary layer control
studies (Bagheri et al., 2009; Semeraro et al., 2011, 2013a, 2013b; Dadfar et
al., 2013, 2014; Sasaki et al., 2018a, 2020; Morra et al., 2020; Freire et
al., 2020).

###  1.2 Feedforward ”Achilles heel”: performance robustness

However, the use of a feedforward setup raises the problem of robustness to
performance, which can be defined as the control law’s ability to remain
efficient in terms of perturbation amplitude reduction despite modelling
errors or free-stream condition variations around the reference case. This
problem has been little addressed in the boundary layer control literature,
despite the advent of robust synthesis, introduced by Doyle et al. (1989). So
far, these modern methods have been mainly used in the case of oscillator
flows (Flinois & Morgans, 2016; Leclercq et al., 2019; Shaqarin et al., 2021)
to have some stability guarantees, because using a feedback setup is mandatory
to stabilize a globally unstable flow.

To improve performance robustness compared to a simple fixed-structure LQG
feedforward controller, Erdmann et al. (2011); Fabbiane et al. (2014, 2015)
used an adaptive feedforward method for boundary layer control, based on the
filtered-X least-mean-squares (FXLMS) algorithm, where the controller
structure is adjusted according to the variations of the flow conditions
through real-time measurements. However, this method is not robust to abrupt
changes in inflow conditions because the controller coefficients are adjusted
in a quasi-static fashion. Due to its natural ability to be robust to unknown
disturbances or uncertainties on the model (Skogestad & Postlethwaite, 2005),
feedback design appears to be a promising alternative for performance
robustness on short time scales. Barbagallo et al. (2012) employed a feedback
structure combined with an LQG synthesis to control instabilities over a
backward-facing step and emphasized the importance of placing the estimation
sensor close to the actuator to obtain a reasonable performance. Doing so
increases the controllable bandwidth indeed, as it is limited in feedback
setup by the convection delay of the disturbances from the actuator to the
estimation sensor. However, some of their feedback controllers turned out to
be unstable on the real plant (the full linearized Navier–Stokes equations),
because of the poor stability robustness of LQG to tiny erros in the ROM. Tol
et al. (2019) also obtained some unstable controllers when trying to control
Tollmien–Schlichting (TS) waves in an incompressible two-dimensional boundary
layer using LQG on a feedback setup. Belson et al. (2013) are among the first
to demonstrate the feasibility of a feedback setup with stability and
performance robustness for the same flow, using a simple proportional-integral
(PI) controller that was tuned by hand. However, the simple structure of the
PI controller did not allow to obtain a satisfactory performance for the
chosen actuator/sensor pair, forcing the authors to change it, despite the
good performance obtained with LQG on the ROMs with the same actuator/sensor
pair. A similar approach was used by Vemuri et al. (2018), in order to cancel
out TS waves in an experimental setup. The authors tuned a proportional
controller by hand to optimise the controller gain in closed loop while
ensuring robust stability of their feedback configuration. Such loop-shaping
approaches provide guarantees on stability robustness but are far from optimal
from a performance viewpoint. And perhaps more importantly, they are very
limited in the sense that they cannot be applied to more complex controller
structures in a systematic way.

###  1.3 Designing robust controllers: structured mixed $H_{2}$/$H_{\infty}$
synthesis techniques

In contrast, modern tools for robust multi-criteria synthesis, such as the
structured mixed $H_{2}$/$H_{\infty}$ synthesis (Apkarian et al., 2014), allow
to optimize complex control laws. The structured mixed $H_{2}$/$H_{\infty}$
synthesis is able to treat different kinds of mathematical criteria
simultaneously, contrary to the LQG method which minimizes a single quadratic
criterion based on performance and cost. Furthermore, structured synthesis
(Apkarian & Noll, 2006) has the advantage to limit the controller order and to
impose its structure beforehand (e.g. state-space model of order 10, PID
controller, etc.), unlike methods that solve Riccati equations, such as LQG
(Freire et al., 2020), $H_{\infty}$ (Flinois & Morgans, 2016) or $H_{2}$ (Tol
et al., 2017) optimal controls, which lead to high-order controllers (of the
same order as the plant augmented by weighting functions). These are often too
expensive to use in real-time applications and require reducing the controller
order in a post-processing step. Performing this reduction optimally while
maintaining stability and performance guarantees on the closed-loop remains an
open problem (Chen et al., 1994; Goddard & Glover, 1995). The possibility of
working with both $H_{2}$ (an integrated gain over all frequencies) and
$H_{\infty}$ (the maximum gain over all frequencies) criteria ensures
performance, robustness to stability and robustness to performance (Apkarian
et al., 2010). Indeed, the use of $H_{\infty}$ criteria on some transfer
functions allows to respect stability margins on the feedback design (what was
missing within the LQG synthesis) despite modelling errors and to desensitize
the controller on certain frequency ranges, allowing optimal performance to be
maintained despite the presence of, for example, noise on the estimation
sensor. The use of $H_{2}$ criteria makes it possible to have a performance
objective of disturbance rejection during the synthesis (which was sometimes
lacking in previous feedback studies).

###  1.4 Objective and outline of the paper

In the present paper, we will consider a supersonic boundary layer at $M=4.5$
and focus on two-dimensional, i.e. spanwise invariant, and linear
perturbations. We will not be dealing with oblique modes or finite-amplitude
perturbations, even if they often do play a significant role in transition in
practice. Hence the present work is only a first step in learning how to
design robust control laws for the problem of transition in the supersonic
boundary layer. One key question we wish to address before introducing more
physical complexity is how do the feedforward and feedback setups compare on
this noise-amplifier flow, using modern robust synthesis tools? With the help
of multi-criteria structured $H_{2}$/$H_{\infty}$ controller synthesis, can we
design a feedback setup which outperforms the often-used feedforward/LQG with
regards to performance robustness to realistic changes in operating
conditions, i.e. velocity and density variations?

The paper is organized as follows. Sections §2 and §3 provide a description of
the flow configuration and numerical methods. In §4, local and global linear
stability tools are used to define appropriate closed-loop specifications,
i.e. determining the actuators, sensors and performance criterion to be
optimized. Section §5 is devoted to ROM identification from impulse responses
using the Eigensystem Realization Algorithm (ERA), with special emphasis on
the problem of time-delays in such noise-amplifier flows. Next, we formally
introduce the multi-criteria structured mixed $H_{2}$/$H_{\infty}$ synthesis
and the associated constraint minimization problem we wish to solve. In §6 we
compare the results obtained on and off-design (noisy sensors, density and
velocity variations) for the feedforward and feedback setups. Conclusions are
drawn in §7.

##  2 Flow configuration

A two-dimensional compressible ideal gas flowing over a flat plate is
considered. The flow is governed by the Navier–Stokes equations:
∂ρ∂t+∇⋅(ρu)&=0,
∂ρu ∂t+∇⋅(ρu⊗u)=-∇ p + ∇⋅τ,
∂ρE∂t+∇⋅(ρEu)=∇⋅(- pu+τ⋅u-θ), where $\rho$ is the fluid density, $\mathbf{u}$
the velocity vector, $p$ the static pressure,
$E=\frac{p}{\rho(\gamma-1)}+\frac{\mathbf{u}\cdot\mathbf{u}}{2}$ the total
energy, $\boldsymbol{\tau}$ the viscous stress tensor and
$\boldsymbol{\theta}$ the heat flux vector. The viscous stress tensor and the
heat flux vector are given by:

$\boldsymbol{\tau}=\mu(\boldsymbol{\nabla\otimes\mathbf{u}}+(\boldsymbol{\nabla\otimes\mathbf{u})}^{T}-\frac{2}{3}(\boldsymbol{\nabla}\cdot\boldsymbol{\mathbf{u}})\mathcal{I}),\\\
$ (1) $\boldsymbol{\theta}=-k\boldsymbol{\nabla}T,$ (2)

with $\mathcal{I}$ the identity tensor, $k$ the thermal conductivity and $\mu$
the dynamic viscosity which is deduced from the local temperature $T$ via
Sutherland’s law,

$\mu=\mu_{\mathrm{ref}}\left(\frac{T}{T_{\mathrm{ref}}}\right)^{\frac{3}{2}}\frac{T_{\mathrm{ref}}+S}{T+S}.$
(3)

The parameters of Sutherland’s law are taken as:
$\mu_{\mathrm{ref}}=1.716\times 10^{-5}\ \mathrm{Pa.s}$,
$T_{\mathrm{ref}}=273.15\ \mathrm{K}$ and $S=110.4\ \mathrm{K}$. The gas
considered being air, we have $\gamma=1.4$, $r=287\ \mathrm{J.K^{-1}.kg^{-1}}$
and $Pr=\frac{\mu\gamma r}{k(\gamma-1)}=0.725$. The free-stream flow
conditions are very close to those used experimentally by Kendall (1975) and
in the simulations of Ma & Zhong (2003), i.e.: $T_{\infty}=65.149\
\mathrm{K}$, $U_{\infty}=728.191\ \mathrm{m.s^{-1}}$ and $p_{\infty}=728.312\
\mathrm{Pa}$. Thus, the free-stream Mach number of the simulation is
$M_{\infty}=\frac{U_{\infty}}{\sqrt{\gamma rT{\infty}}}=4.5$.

The computational domain is represented in figure 1a: it consists of a
rectangular domain where the lower boundary is an adiabatic flat plate of
length $L_{x}=2002.1\delta_{0}^{*}$, with $\delta_{0}^{*}=3.2656\times
10^{-4}\ \mathrm{m}$ the compressible displacement thickness at the inlet of
the domain (defined as $\delta_{0}^{*}=\int^{\infty}_{0}(1-\frac{\rho
u}{\rho_{\infty}u_{\infty}})dy$), which results in
$Re_{\delta_{0}^{*}}=\frac{\rho_{\infty}U_{\infty}\delta_{0}^{*}}{\mu_{\infty}}\simeq
2121$.

(a)  Figure 1: a) Diagram of the computational domain. Inputs and outputs of
the control problem are in red and blue, respectively. b) Boundary layer
profile used for the inlet condition.

(b)

A far-field and a supersonic exit conditions are respectively applied at the
top ($y=275\delta_{0}^{*}$) and at the outlet of the computational domain.
Furthermore, a sponge area is used downstream and in the upper part of the
domain to minimize reflections. This sponge area consists of stretching the
mesh in the longitudinal direction for the downstream boundary
($L_{\mathrm{sponge}}=91.9\delta_{0}^{*}$ and 30 cells in the streamwise
direction) and adding a source term in (2) on the last 10 cells to bring the
flow back to its equilibrium point. A supersonic inlet condition is imposed at
the upstream boundary where the complete state is prescribed and matches a
zero-pressure gradient laminar boundary layer profile (see figure 1b) computed
with the ONERA boundary layer code CLICET (see for instance Olazabal-Loume et
al. (2017)). It corresponds to a profile taken at a distance of
$19\delta_{0}^{*}$ from the leading edge. The beginning of the numerical
domain has been chosen to be in a stable area for all frequencies according to
local linear stability theory (see §4.1). The boundary layer thickness
(denoted $\delta$) at the end of the domain of interest leads to
$Re_{\delta}\simeq 35081$. Overall, the useful numerical domain (i.e. not
counting the length of the sponge area) extends from $4\times
10^{4}<Re_{x}=\frac{\rho_{\infty}U_{\infty}x}{\mu_{\infty}}<4.1\times 10^{6}$.

##  3 Base-flow and spatial stability analyses

###  3.1 Base flow and linearized DNS

Direct numerical simulations (DNS) are performed using the finite volume code
elsA (Cambier et al., 2013). An upwind AUSM + up scheme (Liou, 2006)
associated with a third order MUSCL extrapolation method (Leer, 1979) is used
for the spatial discretization of the convective fluxes. The viscous fluxes
are obtained by a second-order centered scheme. The semi-discretized Navier-
Stokes equations then read:

$\frac{\partial\mathbf{q}}{\partial
t}=\mathcal{N}(\mathbf{q})+\mathcal{P}\mathbf{f},$ (4)

where $\mathbf{q}=[\rho,\rho\mathbf{u},\rho E]^{T}$ and
$\mathcal{N}(\mathbf{q})$ is the discretized compressible Navier–Stokes
equations (including the boundary conditions). The momentum forcing
$\mathbf{f}$ may either represent a noise source or the effect of an actuator.
The matrix $\mathcal{P}$ represents the prolongation operator that transforms
the momentum forcing into a full state-vector forcing by adding zero
components. The laminar base-flow $\mathbf{\bar{q}}$, defined as

$\mathcal{N}(\mathbf{\bar{q}})=0,$ (5)

is obtained by time-stepping the unforced unsteady equations (4) with an
implicit time-stepping method based on a local time step, up to convergence of
the residuals. The unsteady simulations for the development of instabilities
are performed with an implicit second-order Gear scheme (Gear, 1971) with 4
sub-iterations and a time-step $dt$ ensuring a CFL number lower than $1.4$ in
the whole domain. For these unsteady simulations, the amplitude of the forcing
$\mathbf{{f}}$ is chosen sufficiently small to ensure that the induced
perturbation $\mathbf{{q}^{\prime}}=\mathbf{{q}}-\mathbf{\bar{q}}$ remains in
the linear regime until the end of the computational domain. The time step and
the number of sub-iterations of the temporal method have been validated by
comparing transfer functions from the linearized DNS and those determined from
the frequency-domain resolvent approach (defined in §3.2).

A resolution of 3200$\times$220 cells for the useful domain was chosen. The
mesh is uniform in the $x$ direction while a geometric law is used in the $y$
direction to resolve strong gradients near the wall. The base-flow and linear
growth rates have been verified against linearized DNS results of Ma & Zhong
(2003), allowing to validate the resolution and the numerical schemes.

###  3.2 Global resolvent analysis

For purposes of controlling instabilities, the choice of the type and position
of the actuator/sensors will play an essential role. This choice is guided by
resolvent analysis, which characterizes the noise-amplifier behaviour from an
input-output viewpoint. The method is briefly detailed in this section.

The purpose of control is to reduce the amplitude of disturbances which
naturally develop in the boundary layer, and thus to maintain the flow as
close as possible to its equilibrium $\mathbf{\bar{q}}$. By injecting the
ansatz $\mathbf{{q}}=\mathbf{\bar{q}}+\mathbf{{q}^{\prime}}$ into (4) and by
considering only small-amplitude forcing $\mathbf{f}$, we obtain after
linearization:

$\frac{\partial\mathbf{q^{\prime}}}{\partial
t}=\mathcal{A}\mathbf{q^{\prime}}+\mathcal{P}\mathbf{f},$ (6)

where $\mathcal{A}$ is the Jacobian matrix defined as
$\mathcal{A}=\frac{d\mathcal{N}}{d\mathbf{q}}\big{|}_{\mathbf{\bar{q}}}$. In
our configuration, all the eigenvalues of $\mathcal{A}$ have a negative real
part and the flow is therefore globally stable. Switching to the frequency
domain, a direct relation between the spatial structure of a harmonic forcing
$\mathbf{f}(x,y,t)=\widetilde{\mathbf{f}}(x,y)e^{i\omega t}$ and its flow
response $\mathbf{q^{\prime}}(x,y,t)=\widetilde{\mathbf{q}}(x,y)e^{i\omega t}$
is established:

$\widetilde{\mathbf{q}}=\mathcal{R}\widetilde{\mathbf{f}},$ (7)

where $\mathcal{R}=(i\omega\mathcal{I}-\mathcal{A})^{-1}\mathcal{P}$ is the
resolvent operator and $\omega=2\pi f\in\mathbb{R}$ is the angular frequency.
For a given frequency and among all the possible forcings, we examine the one
which maximizes the gain
$\widetilde{g}^{2}(\omega)=\underset{\widetilde{\mathbf{f}}\neq
0}{\sup}\frac{||\widetilde{\mathbf{q}}||^{2}_{E}}{||\widetilde{\mathbf{f}}||^{2}_{F}}$
where $||.||^{2}_{E}$ and $||.||^{2}_{F}$ respectively denote the Chu energy
norm and the energy of the momentum forcing (Bugeat et al., 2019). The Chu
energy is defined as

$E_{\mathrm{Chu}}=\frac{1}{2}\int_{\mathcal{V}}(\overbrace{\underbrace{\bar{\rho}(|{u^{\prime}}|^{2}+|{v^{\prime}}|^{2}}_{E_{u^{\prime}}})+\underbrace{r\frac{\bar{T}}{\bar{\rho}}|\rho^{\prime}|^{2}}_{E_{\rho^{\prime}}}+\underbrace{\frac{r}{\gamma-1}\frac{\bar{\rho}}{\bar{T}}|T^{\prime}|^{2}}_{E_{T^{\prime}}}}^{E_{tot}})dV,$
(8)

it contains terms relative to thermodynamic perturbations in addition to the
kinetic one, and is therefore commonly used to study the global behaviour of
compressible flows (Hanifi et al., 1996; Bugeat et al., 2019). For a given
frequency, the fields $\widetilde{\mathbf{f}}$ and $\widetilde{\mathbf{q}}$
corresponding to the optimal gain $\widetilde{g}$ are respectively called
optimal forcing and response modes. Determining the optimal gain amounts to
computing the largest eigenvalue of a positive generalised eigenvalue problem
with the Arnoldi algorithm (ARPACK library, Lehoucq et al., 1998) using a
sparse LU solver (MUMPS library, Amestoy et al., 2001) for linear system
solution. This global analysis tool developed in previous work (Beneddine et
al., 2015) was validated on the supersonic boundary layer results of Bugeat et
al. (2019). In our study, the domains involved in the definition of
$||.||^{2}_{E}$ and $||.||^{2}_{F}$ correspond both to
$x\in[0;1910.2\delta_{0}^{*}]$ and $y\in[0;92\delta_{0}^{*}]$.

###  3.3 Local stability analysis

The primary aim of the local linear stability theory (LLST) for the present
study is to classify the mechanisms involved in our DNS and resolvent analysis
by associating local modal mechanisms from the LLST with those observed in our
purely non-modal DNS and global resolvent study. Indeed, the flow being
globally stable, the growth of disturbances is only due to non-modal
phenomena. These non-modal effects are a consequence of the non-normality of
$\mathcal{A}$ (Schmid, 2007). The non-normal effects can be cast in two
categories for open-flows: the component-type non-normality and the
convective-type non-normality (Sipp et al., 2010). Component-type non-
normality is characterized by a component-wise transfer of energy between the
forcing and response fields like in the Orr or lift-up mechanisms (Bugeat et
al., 2019) (but note the latter is absent here since lift-up is three-
dimensional). Convective-type non-normality is caused by modal amplification
on the local scale and is characterized by a separation of the spatial
supports of the forcing and response fields.

In LLST, we consider perturbations which are evolving very rapidly in the $x$
direction compared to the base flow. At each streamwise position, the base
flow is considered frozen with respect to the perturbations
$\phi^{\prime}=[\rho^{\prime},u^{\prime},v^{\prime},T^{\prime}]$, therefore
the latter can be sought in the form

$\phi^{\prime}=\widetilde{\phi}(y)e^{i(\alpha x-\omega t)},$ (9)

where in general the wavenumber $\alpha$ and the frequency $\omega$ are
complex numbers. Plugging this ansatz in the linearized Navier–Stokes
equations with frozen base flow profile leads to a different dispersion
relation $D(\alpha,\omega;x)=0$ for each value of $x$. In the spatial
stability framework, we consider real angular frequencies $\omega$ and solve
for the complex wavenumber $\alpha=\alpha_{r}+i\alpha_{i}$, where $\alpha_{r}$
is the wavenumber and $-\alpha_{i}$ is the spatial growth rate along $x$. All
perturbations are assumed to vanish at the free-stream boundary $y\to\infty$
while on the flat plate $y=0$, $\tilde{u}=\tilde{v}=0$ and
$\mathrm{d}\tilde{\rho}/\mathrm{d}y=\mathrm{d}\tilde{T}/\mathrm{d}y=0$
(adiabatic plate). Equations are discretized along the wall-normal direction
$y$ using a Chebyshev collocation method. For each values of $x$ and $\omega$,
an eigenvalue problem is solved, using the LAPACK library, in order to
determine the complex eigenvalue $\alpha$ and corresponding eigenvector
$\widetilde{\phi}=[\tilde{\rho},\tilde{u},\tilde{v},\tilde{T}]$. The analysis
is performed using an in-house code fully detailed in Saint-James (2020) and
validated here on the linear local growth rates of the supersonic boundary
layer Ma & Zhong (2003).

##  4 Noise-amplifier behaviour and control setup

###  4.1 Characterisation of instabilities

The local spatial stability diagram of spanwise-invariant perturbations is
displayed in figure 2a, with $F=2\pi f\delta_{0}^{*}/U_{\infty}$ the
dimensionless frequency. It is characterized by two distinct instability
regions (i.e. where the spatial growth rate is positive $-\alpha_{i}>0$): one
for the first Mack mode and one for the second Mack mode. For each mode, the
instability domain (depicted by the red solid line) for a given frequency is
located between branch I (convectively stable/unstable boundary) and branch II
(convectively unstable/stable boundary). Each frequency is therefore amplified
only on a certain portion of the domain: high frequencies are amplified
upstream while low frequencies are found further downstream. Compared to the
first mode, the unstable frequencies of the second mode are higher and are
associated with higher growth rates. Transition to turbulence is often
predicted from LLST using the $N$-factor (Smith & Gamberoni, 1956)

$N(\omega)=\int_{x_{cr}}^{x}-\alpha_{i}(\omega)dx=\ln\left(\frac{|\phi^{\prime}|}{|\phi^{\prime}|_{cr}}\right),$
(10)

with $x_{cr}$ the location of branch I for the considered frequency and
$|\phi^{\prime}|_{cr}$ the amplitude of the mode at this location. The
$N$-factors for different frequencies are represented in figure 2b. Although
the instability range of the first Mack mode is larger, the $N$-factors of the
second mode are greater all along the domain due to their higher growth rates.
Transition is often assumed to occur when the quantity
$\widetilde{N}=\underset{\omega}{\mathrm{max}}\ N(\omega)$ (red solid lines in
figures 2(LABEL:sub@fig:Nfactor,LABEL:sub@fig:Nfactor_controlled)) reaches a
threshold value $N_{t}$ (dashed lines in figures
2(LABEL:sub@fig:Nfactor,LABEL:sub@fig:Nfactor_controlled), arbitrarily placed
for the explanation). This criterion means that the transition process begins
when a perturbation has been amplified by a factor of $e^{N_{t}}$. Thus, in
order to delay transition to turbulence, a control action should transform the
quantity $\widetilde{N}$ obtained without control into the quantity
$\widetilde{N}^{c}$ (blue line in figure 2c) with control, such that
$\widetilde{N}^{c}<N_{t}$ (see figure 2c). The dominant frequency being
different at each streamwise location of the domain, a large frequency
bandwidth needs to be controlled, which complicates the design of the control
law. The $\widetilde{N}^{c}<N_{t}$ criterion could be directly translated into
a $H_{\infty}$ criterion, because this would mean that the maximum
amplification over the entire frequency spectrum must not exceed a threshold
over the entire domain, exactly as in the $N$-factor method. However, this
method may be considered conservative as it is based on the worst
perturbation, which is purely harmonic and therefore not quite realistic
(Mack, 1977). Fedorov & Tumin (2022) recommended instead the use of a
criterion based on both the $N$-factors and the entire frequency spectrum of
the incoming disturbance $|\phi^{\prime}|_{cr}$, which amounts to considering
an $H_{2}$ norm rather than an $H_{\infty}$ norm. We follow this
recommendation and choose a performance objective based on an $H_{2}$ norm.
More precisely, our objective will be to maintain the spatially-integrated
amplification below a given threshold along the plate, and this integrated
amplification will be quantified using an $H_{2}$ norm (see figure 2d).

(a)  (b)

(d)  (c)

Figure 2: a) Stability diagram; red solid lines represent isolines
$\alpha_{i}=0$. b) Calculation of the $N$-factors (black solid lines) for
transition prediction based on LLST: transition occurs when
$\widetilde{N}>N_{t}$ (notional diagram). c) Performance objective for closed-
loop control based on the $N$-factor criterion. d) Modification of the
N-factor criterion using the $H_{2}$ norm, in order to reduce conservatism.
The quantity $F=2\pi f\delta_{0}^{*}/U_{\infty}$ represents the dimensionless
frequency.

The global stability results based on resolvent analysis complement those
previously obtained from LLST. The optimal energy gain $\tilde{g}$ as a
function of the forcing frequency $F$ is represented in figure 3a.

|
---|---

(a)

(b)

(c)

(d)  (e)

(f)  (g)

Figure 3: a) Optimal resolvent gain as a function of the dimensionless
frequency $F$. According to LLST, red and green dashed areas represent the
unstable frequency range of first and second Mack modes, respectively. The
region where both modes are unstable corresponds to an area where the first
mode is unstable over a tiny distance. Real part of the streamwise component
of the optimal forcing (b) and its associated streamwise velocity response (c)
at $F=0.237$. d) Evolution at $F=0.237$ of the forcing density and the
different contributions to the Chu’s energy density normalized by their
maximum values. The position of the branch I and II from LLST are symbolized
by vertical dashed lines. e) Comparison of $-\alpha_{i}$ and
$-\widetilde{\alpha}_{i}$ at $F=0.237$. Profiles of the optimal forcing
components at $x=867.2\delta_{0}^{*}$ (f) and response at
$x=1766.7\delta_{0}^{*}$ (g) at $F=0.237$. The black dashed and dashed-dotted
lines in (b), (c), (f) and (g) represent the generalised inflection point
position and the limit of the region of supersonic instabilities
($\widehat{M}>1$ below this line), respectively.

This curve displays two peaks at $F\approx 0.118$ and $F\approx 0.237$, which
correspond respectively to the first and second Mack modes identified in LLST.
Global resolvent analyses are consistent with those of the local approach,
since the optimal energy gain is closely related to $N$-factors (Sipp et al.,
2010; Beneddine et al., 2015).

For the frequency $F=0.237$ leading to the highest gain, the real parts of the
streamwise optimal forcing and velocity response are shown in figures
3(LABEL:sub@fig:resolvent_Momentumf,LABEL:sub@fig:resolvent_reponseu). The
spatial structure of the forcing is located upstream of the domain while that
of the response is located further downstream. This separation of the spatial
supports, related to the convective-type non-normality of the Jacobian
operator, implies a time-delay between actuation upstream and sensing
downsteam, making the design of a robust control law even more complex. Figure
3d shows that the peak of the forcing density
$d_{E_{f}}(x)=\int_{0}^{y=92\delta_{0}^{*}}|\widetilde{\mathbf{f}}|^{2}dy$
(resp. Chu’s energy density
$d_{E_{Chu}}(x)=\int_{0}^{y=92\delta_{0}^{*}}E_{tot}\,\mathrm{d}y$) is not
very far from the position of branch I (resp. II) from LLST (Sipp et al.,
2010). The energy of the response is dominated at each abscissa by the
thermodynamic quantities $E_{T^{\prime}}$ and $E_{\rho^{\prime}}$, while
quantity $E_{u^{\prime}}$ has a smaller contribution. A comparison between the
spatial amplification rates $-\alpha_{i}$ from LLST (red dashed line) and
$-\widetilde{\alpha}_{i}=\frac{1}{|\widetilde{u}(x,y=1.7\delta_{0}^{*})|}\partial_{x}|\widetilde{u}(x,y=1.7\delta_{0}^{*})|$
from resolvent analysis (black dashed line) is depicted in figure 3e. The
quantity $-\widetilde{\alpha}_{i}$ represents the slope of
$\ln{|\widetilde{u}|}$ (black solid line) and can therefore be compared to a
growth rate. The growth of the resolvent mode within
$x\in[0;1078\delta_{0}^{*}]$ is due to the optimal forcing that is non-zero in
this region (see figure 3d) and that induces the response. The inclined
pattern in the forcing field (see figure 3b) indicates that the response also
takes advantage of the Orr mechanism (Orr, 1907) and more generally of non-
modal local interactions. After this initial growth region induced by the
forcing, both $-\alpha_{i}$ and $-\widetilde{\alpha}_{i}$ exhibit similar
values in the region between $x\in[1200\delta_{0}^{*};1730\delta_{0}^{*}]$,
which indicates that transient growth is then dominated by the convective
instability associated to the second Mack mode.

To maximize the amplification of the second Mack mode, the forcing field (see
figures 3(LABEL:sub@fig:resolvent_Momentumf,LABEL:sub@fig:profile_forcage))
must be localised near the generalised inflection point $y_{g}$ (denoted in
figures
3(LABEL:sub@fig:resolvent_Momentumf,LABEL:sub@fig:resolvent_reponseu,LABEL:sub@fig:profile_forcage,LABEL:sub@fig:profile_reponse)
with a dashed line), defined as
$\left.\partial_{y}\left(\bar{\rho}\partial_{y}\bar{u}\right)\right|_{y_{g}}=0$.
A region of supersonic instabilities (below the dashed-dotted line in figures
3(LABEL:sub@fig:resolvent_Momentumf,LABEL:sub@fig:resolvent_reponseu,LABEL:sub@fig:profile_forcage,LABEL:sub@fig:profile_reponse)
), defined as
$\widehat{M}=\frac{\left|\bar{u}-\frac{\omega}{\widetilde{\alpha}_{r}}\right|}{\sqrt{\gamma
r\bar{T}}}>1$ with $\widetilde{\alpha}_{r}$ the global resolvent streamwise
wavenumber computed as
$\widetilde{\alpha}_{r}=\partial_{x}\arg(\widetilde{u})$ where $\arg$ stands
for the argument of a complex number (see Beneddine et al., 2015), is detected
close to the wall (see figure 3c). This confirms that the optimal response
mode at $F=0.237$ corresponds to a second Mack mode (Mack, 1984). Note that
the critical layer, where $\bar{u}=\frac{\omega}{\widetilde{\alpha}_{r}}$, is
not shown here as it is similar to the generalised inflection point; indeed,
the phase velocity of an inflectional neutral wave in the LLST is equal to the
mean velocity at $y_{g}$ (Mack, 1984).

Finally, we observe in figure 3g that the different components of the second
Mack mode peak at different locations in the wall-normal direction $y$.
Hydrodynamic perturbations (velocity and pressure) peak close to the wall and
seem trapped in the region $\widehat{M}>1$ whereas thermodynamic quantities
(density and temperature) peak near the generalised inflection point. This
observation is in complete agreement with the qualitative results of Bugeat et
al. (2019).

###  4.2 Control setup

External perturbations are modelled using a random time signal $w$ (see figure
1a) that multiplies a time-independent volume force field. In the case of
small amplitude noise considered in this paper, the dynamics is linear and
will take advantage of the various instability mechanisms described in the
previous section. If we consider several performance sensors $z_{i}$ measuring
the flow perturbations along the plate, the transfer functions
$T_{z_{i}w}=z_{i}(s)/w(s)$, with $s\in\mathbb{C}$ the Laplace variable,
provide an accurate prediction of the downstream perturbation level without
control. The reactive control setup is depicted in figure 4. An upstream
actuation $u$ generates small-amplitude perturbations that take again
advantage of the instability mechanisms to grow and eventually cancel the
fluctuations at the downstream measurements $z_{i}$. The phase of the
generated perturbations is therefore important and needs to be tuned with
respect to the incoming perturbations that are governed by $w$. For this, we
introduce an upstream sensor $y$ and design a controller $K$, that actually
corresponds to the transfer function $K=T_{uy}$, and which transforms the
noise measurement $y$ into an actuation signal $u$. It is straightforward to
show that, in the presence of control, the transfer functions from $w$ to
$z_{i}$, denoted with the superscript $c$, become:

$T_{z_{i}w}^{c}=T_{z_{i}w}+T_{z_{i}u}K(1-T_{yu}K)^{-1}T_{yw}.$ (11)

The design of $K$ therefore requires additional transfer functions: $T_{yw}$
characterizes the influence of noise on the upstream measurement $y$,
$T_{z_{i}u}$ characterizes the influence of the actuator on the downstream
performance sensors and, for feedback setups only, $T_{yu}$ characterizes the
influence of the actuator on the upstream sensor. In the following, we will
assume that $w$ is a white-noise input and will seek to reduce the expected
power of the measurements $z_{i}$. This expected power, normalized by the
intensity of the white-noise input, is measured by the $H_{2}$ norm of
$T_{z_{i}w}^{c}$. For any stable SISO transfer function $G$, the $H_{2}$ norm
is defined as

$||G||_{2}=\left(\frac{1}{2\pi}\int_{-\infty}^{+\infty}|G|^{2}d\omega\right)^{1/2}.$
(12)

Figure 4: Block diagram for noise-amplifier flows for feedforward and feedback
configurations in an ideal case (with quantities in blue and black) and in a
realistic setup (with quantities in red and black). The quantities in black
are common to the ideal and realistic cases. The red dotted square therefore
represents the system used with the aim of an experimentally feasible
synthesis. In a feedforward setup, $T_{yu}=\Delta=0$.

However, determining the transfers coming from the noise $w$ is not possible
in realistic cases because the noise environment is unknown (it depends on the
characteristics of the wind tunnel or the free-stream turbulence on
airplanes). An experimentally feasible control design must therefore not be
based on $T_{z_{i}w}$ and $T_{yw}$. Following Hervé et al. (2012), the
solution proposed here is to introduce an artificial transfer function,
$T_{z_{i}\tilde{y}}$, which is intended to predict the downstream measurements
$z_{i}$ from the upstream measurement $y$ in the absence of a control action.
This apparent transfer function ($y$ is not a source) is defined as
$T_{z_{i}\tilde{y}}=T_{z_{i}w}T_{yw}^{-1}$ (observability requires that
$T_{yw}\neq 0$ when $T_{z_{i}w}\neq 0$) (Sasaki et al., 2018a, b). In real
applications, we can identify this transfer function from uncontrolled
$(y,z_{i})$ data. In the following, we will consider $\tilde{y}=T_{yw}w$ as
the new exogeneous input of the system. We are therefore led to the modified
block diagram framed by the red dotted square in figure 4, where in case of
actuation, the upstream measurement reads: $y=\tilde{y}+T_{yu}u$ (+$n$, which
is a measurement noise). In such a case, the controlled transfer function
becomes:

$T_{z_{i}\tilde{y}}^{c}=T_{z_{i}\tilde{y}}+T_{z_{i}u}K(1-T_{yu}K)^{-1}.$ (13)

The ideal and realistic control schemes shown in figure 4 are related through:

$||\ T_{z_{i}w}^{c}\ ||_{2}=||\ |T_{yw}|\ T_{z_{i}\tilde{y}}^{c}\ ||_{2}.$
(14)

The term $|T_{yw}|$ therefore corresponds to a frequency weighting function
and can be replaced by $W_{y}=\sqrt{PSD_{y}(\omega)}$ (where $PSD_{y}$ is the
power spectral density of the estimation sensor $y$ in the absence of a
control action); it represents the fact that the new system input $\tilde{y}$
is no longer a white noise as $w$ but a colored noise. Therefore, the four
quantities needed for the synthesis are $T_{z_{i}u}$, $T_{yu}$,
$T_{z_{i}\tilde{y}}$ and $W_{y}$. They can all be obtained in a realistic
setup using input-output data and will be the ones used for identification
(see §5.1) and controller synthesis (see §5.2). For the sake of clarity and to
simplify notations, the quantity $||W_{y}\ T_{z_{i}\tilde{y}}^{c}||_{2}$ will
be replaced in the rest of the paper by $||T_{z_{i}w}^{c}||_{2}$.

Maintaining closed-loop performance in spite of modelling errors or inflow
conditions variations around the nominal case requires first and foremost the
stability robustness of the control law. From a control design point of view,
this implies considering uncertainties $\Delta$ representing a model error on
$T_{yu}$ that can lead to the instability of the feedback loop. For example,
for the block $\Delta$ represented in figure 4, if no upstream noise is
considered, we have $y=\frac{T_{yu}}{1-\Delta}u$, so that $\Delta$ represents
an inverse multiplicative uncertainty on $T_{yu}$ such that
$\Delta=\frac{T_{yu}^{real}-T_{yu}}{T_{yu}^{real}}$. This type of uncertainty
has the advantage of representing a relative error, which facilitates its
interpretation. Since $-T_{yu}^{real}K$ does not exhibit any unstable pole
($T^{real}_{yu}$ is stable because the boundary layer flow is globally stable
while $K$ is stable by design), the closed loop system is stable if and only
if the Nyquist plot of $-T_{yu}^{real}K$ does not encircle the critical point
(-1, 0), which is equivalent to $|1-T_{yu}K|>|\Delta|$ (Skogestad &
Postlethwaite, 2005). Therefore, the stability of the closed loop can be
guaranteed by working on the sensitivity function

$S=(1-T_{yu}K)^{-1}.$ (15)

Defining the $H_{\infty}$ norm of a stable SISO transfer function $G$ as

$||G(s)||_{\infty}=\underset{\omega\in\mathbb{R}}{\sup}\ |G(i\omega)|,$ (16)

we request to maintain the $H_{\infty}$ norm of the sensitivity function $S$
below a threshold, which allows to keep adequate stability margins. By
directly measuring the minimal distance between the Nyquist plot and the
critical point (-1, 0) after which the closed loop becomes unstable for a
negative feedback loop, the modulus margin $||S||_{\infty}^{-1}$ appears to be
the most generic measure for quantifying the available stability margin
(Skogestad & Postlethwaite, 2005).

Finally, maintaining optimal performance despite uncertainties on certain
frequency range of the measurement $y$ means minimizing the $H_{\infty}$ norm
of the transfer function

$\frac{u}{n}=KS.$ (17)

Desensitizing the control output $u$ on certain frequency ranges allows to be
robust to noise $n$ on the estimation sensor $y$. Even if these frequencies
are attenuated far downstream of the actuator (if they are convectively
stable, resulting in low $|T_{z_{i}\tilde{y}}|$), strong injection of energy
may occur in the direct vicinity of the actuator, which may in turn provoke
transition to turbulence in a 3D setup.

In summary, the fluidic specifications for noise-amplifier flows may be
reformulated from a control point of view as an optimization problem based on
$H_{2}$ and $H_{\infty}$ norms, in order to guarantee both performance and
robustness. The constrained minimization problem for our specific study will
be formulated in §5.2.

###  4.3 Selecting actuator and sensors

For a given external perturbation, the choice of appropriate actuator and
sensors is essential to ensure effective flow control. The input perturbation,
representing an external disturbance (acoustic noise, roughness, freestream
turbulence, etc.) is modelled by a volume forcing $w(t)\mathbf{B}_{w}(x,y)$ in
the right-hand-side of the momentum equations (2)(b), where the noise $w(t)$
is chosen white (with a variance sufficiently small for the perturbation to
remain in the linear regime) and $\mathbf{B}_{w}(x,y)$ is divergence-free and
compact in space (Bagheri et al., 2009; Semeraro et al., 2011; Belson et al.,
2013):

$\mathbf{B}_{w}=\mathbf{h}(\frac{10.66}{\delta_{0}^{*^{2}}},4.1\delta_{0}^{*},\delta_{0}^{*},1.5\delta_{0}^{*},0.15\delta_{0}^{*}),$
(18)

with

$\mathbf{h}(A_{h},x_{0},y_{0},\sigma_{x},\sigma_{y})=A_{h}\begin{pmatrix}(y-y_{0})\sigma_{x}/\sigma_{y}\\\
-(x-x_{0})\sigma_{y}/\sigma_{x}\\\
\end{pmatrix}\mathrm{exp}^{-\left(\frac{x-x_{0}}{\sigma_{x}}\right)^{2}-\left(\frac{y-y_{0}}{\sigma_{y}}\right)^{2}}.$
(19)

It is centred around the generalised inflection point in the wall-normal
direction in order to maximize the receptivity process by exciting the optimal
mechanisms of the second Mack mode, which is the most amplified, as shown by
the resolvent analysis results in section 4.1. The position of
$\mathbf{B}_{w}$ in the streamwise direction is upstream of branch I (locally
stable regions) for all frequencies according to the LLST.

For the sensors, in order to have strong observability of the disturbances, we
choose $y$ and $z(x)$ to be wall-pressure fluctuation sensors. This choice is
supported by the fact that second Mack modes exhibit strong pressure
fluctuations close to the wall, as shown by the optimal response profiles in
figure 3g. Also, that kind of sensors is commonly used in supersonic
experimental studies (Lugrin et al., 2022).

In figure 5a, we represent the quantity $F|T_{z(x)w}|^{2}$ as a function of
$\ln{F}$, where $F$ is the frequency, such that the integral represents the
$H_{2}$ norm of $T_{z(x)w}$. The module of $|T_{z(x)w}|$ is obtained by
Fourier transform of the signals from an impulse response. At each abscissa
$x$ of the plate, the energy contribution to the sensor $z(x)$ is only due to
a certain frequency bandwidth. Indeed, after reaching a peak, the magnitude
associated with a frequency rapidly decreases, as can be seen in figure 5b.
Therefore, for control, we will need to use several performance sensors
$z_{i}$ to obtain a suitable frequential representation at different
streamwise positions and capture the entire amplified bandwidth. As the
spectrum of $F|T_{z(x)w}|^{2}$ is narrow (especially downstream of the
domain), reducing $||T_{z_{i}w}^{c}||_{2}$ should also lead to a significant
reduction in $||T_{z_{i}w}^{c}||_{\infty}$.

(a)  (b)

(c)  (d)

Figure 5: a) Evolution of $F|T_{z(x)w}|^{2}$. b) Variation of the frequency
magnitude as a function of the plate abscissa. For each frequency, the green
(resp. blue) dots represent branch I and branch II of the first (resp. second)
Mack modes according to LLST. c) Evolution of the $H_{2}$ norm. The vertical
dashed line in (b) and (c) shows the streamwise location of the actuator
$\mathbf{B}_{u}$, denoted $x_{u}$. d) Evolution of the ratio
$|T_{z(x)u}|/|T_{z(x)w}|$ as a function of frequency $F$ for several plate
abscissa.

Sufficiently far downstream from $\mathbf{B}_{w}$, the most amplified
frequency at each abscissa of the domain (red line in figure 5b) is similar to
the one that could be found with the $N$-factors (see figure 2b). As the
magnitude of the perturbations increases for all frequencies in spatially
stable regions upstream of branch I (see first dot symbols in figure 5b), the
perturbations seem to be subject first to a growth due to the non-modal Orr
mechanism, before being dominated by the ”modal” growth of the unstable Mack
mode. The highest value of $|T_{z(x)w}|$ is found at the end of the domain of
interest, at a frequency of $F=0.223$, close to the frequency leading to the
highest gain in the global resolvent analysis ($F=0.237$). Therefore, the
optimal response mechanisms already observed in §4.1 are well triggered by the
chosen disturbance $\mathbf{B}_{w}$, which is therefore representative of a
more general transition scenario due to the second Mack mode.

The control goal is to create a destructive interference by generating a
second wave of appropriate amplitude and phase, which will oppose the one
generated by the upstream noise $w(t)$ (Hervé et al., 2012; Sasaki et al.,
2018a). Thus, in order to maximize the impact of the control action, the
perturbations generated by the actuator must match those induced by the
upstream noise. The incoming disturbance being mainly due to second Mack mode
instabilities, an efficient actuator can be obtained with a volume forcing
around the generalised inflection point in the wall-normal direction, in order
to maximize the receptivity process. We therefore consider
$\mathbf{B}_{u}u(t)$ in the right-hand-side of equations (2)(b) to model the
actuator, with the same divergence free spatial support as for the disturbance
$\mathbf{B}_{w}$:

$\mathbf{B}_{u}=\mathbf{h}(\frac{10.66}{\delta_{0}^{*^{2}}},867.2\delta_{0}^{*},7.79\delta_{0}^{*},1.5\delta_{0}^{*},0.5\delta_{0}^{*}).$
(20)

The actuator is placed sufficiently far downstream of $\mathbf{B}_{w}$
($x_{u}=867.2\delta_{0}^{*}$) for two reasons. The first one is to allow
disturbances to strengthen sufficiently (see figure 5c) to be easily detected
by the estimation sensor $y$ (which is close to the actuator), which in an
experimental configuration would mean placing the actuator a little upstream
of the beginning of the transition process. The second reason is to limit the
bandwidth of the frequencies to be controlled (see figure 5b) in order to keep
the complexity of the control problem reasonable. Hence, for the chosen
streamwise position of the actuator, the frequency range to be controlled is
around $F\in[0.225,0.324]$; a more upstream actuator should have controlled a
wider bandwidth. The streamwise position of the actuator remains sufficiently
upstream so that incoming perturbations are controlled over a sufficiently
long domain ($\sim 0.34\ \mathrm{m}$) representative of an experimental
configuration (the plate of the experimental tests of Kendall (1975) measured
$0.35\ \mathrm{m}$).

A comparison of $|T_{z(x)w}|$ and $|T_{z(x)u}|$ is shown in figure 5d. It can
be noted that in the vicinity of the actuator, the ratio
$|T_{z(x)w}|/|T_{z(x)u}|$ evolves with the $x$ abscissa. As this phenomenon no
longer appears for abscissas further away from the actuator and the ratio
becomes constant, it could be attributed to a non-modal transient behaviour.
Indeed, we have:

$\frac{|T_{z(x)u}|}{|T_{z(x)w}|}\propto\frac{e^{\int^{x}_{x_{u}}-(\widetilde{\alpha}_{i})_{u}dx}}{e^{\int^{x}_{x_{u}}-(\widetilde{\alpha}_{i})_{w}dx}},$
(21)

where $-(\widetilde{\alpha_{i}})_{u}$ and $-(\widetilde{\alpha_{i}})_{w}$
represent the slope of $\ln{|T_{z(x)u}|}$ and $\ln{|T_{z(x)w}|}$,
respectively. Therefore, a constant ratio implies having the same slope from a
certain distance $x$. This distance $x$ represents the non-modal distance due
to the receptivity of multiple modes to the volume forcing of the actuator on
the flow.

The impact of the position of the estimation sensor $y$ has been already
extensively studied in the noise-amplifier flow control literature (Barbagallo
et al., 2012; Belson et al., 2013; Juillet et al., 2013; Freire et al., 2020),
hence the detailed analysis for the case of the supersonic boundary layer is
left to appendix A. It is just reminded that, for a feedback design, the
estimation sensor $y$ has to be close enough to the actuator to avoid sending
outdated information and limit the effective delay impacting the maximum
achievable performance. For a feedforward design where the impact of the
actuator on the estimation sensor $y$ is assumed to be negligible in the
synthesis step ($T_{yu}=0$), the estimation sensor has to be located
sufficiently upstream of the actuator for the hypothesis to be valid.

Regarding the number of performance sensors $z_{i}$ used in the
identification/synthesis step, it was found by numerical simulations that six
probes are required to achieve nearly uniform performance along the domain
because of the need to capture the entire amplified bandwidth and the non-
modal effects due to the actuator (see appendix B). The streamwise positions
of the input perturbation, the actuator and the sensors used for the
identification and synthesis steps are summarized in table 1.

| $\mathbf{B}_{w}$ | $\mathbf{B}_{u}$ | $y$ | $z_{i}$
---|---|---|---|---
Streamwise position | $x_{w}=4.1\delta_{0}^{*}$ | $x_{u}=867.2\delta_{0}^{*}$ |  $x_{\textit{ff}}=801.2\delta_{0}^{*}$ $x_{\textit{fb}}=885.7\delta_{0}^{*}$ |  $x_{1}=933.2\delta_{0}^{*}$ $x_{2}=1029.4\delta_{0}^{*}$ $x_{3}=1125.6\delta_{0}^{*}$ $x_{4}=1317.9\delta_{0}^{*}$ $x_{5}=1510.2\delta_{0}^{*}$ $x_{6}=1766.7\delta_{0}^{*}$
Table 1: Streamwise positions of the input perturbation, the actuator and the
sensors used for the identification and synthesis steps. The position of the
estimation sensor for feedforward and feedback configurations are denoted
$x_{\textit{ff}}$ and $x_{\textit{fb}}$, respectively.

##  5 Identification and synthesis methods

###  5.1 Identification of a state-space model

Most synthesis methods require the use of state-space ROMs corresponding to
the transfers involved in the controller synthesis. For the model reduction
step, some of the input/output delays linked to the convective nature of the
flow may be discarded due to the fact that the $H_{2}$ norm is not modified by
dead-time delays. In a feedback configuration $(u,y,z_{i})$, the dead-time
delays verify $\tau_{z_{i}u}=\tau_{z_{i}\tilde{y}}+\tau_{yu}$, so that

$\begin{split}||T_{z_{i}w}^{c}||_{2}&=||\
e^{-\tau_{z_{i}\tilde{y}}s}W_{y}(T^{\prime}_{z_{i}\tilde{y}}+e^{-\tau_{yu}s}T^{\prime}_{z_{i}u}KS)\
||_{2}\\\ &=||\
W_{y}(T^{\prime}_{z_{i}\tilde{y}}+e^{-\tau_{yu}s}T^{\prime}_{z_{i}u}KS)\
||_{2},\end{split}$ (22)

where $T^{\prime}(s)$ designates the “dead-time-free” transfer function
associated to $T(s)$. The same idea can be applied also to a feedforward
design $(y,u,z_{i})$ with the result below:

$\begin{split}||T_{z_{i}w}^{c}||_{2}=||\
W_{y}(e^{-\tau_{uy}s}T^{\prime}_{z_{i}\tilde{y}}+T^{\prime}_{z_{i}u}K)\
||_{2}.\end{split}$ (23)

Thus, the only remaining delay is the one between the actuator and the
estimation sensor, $\tau_{yu}$ or $\tau_{uy}$, which is reasonably small
(compared to the delays involving $z_{i}$.) Removing unnecessary delays (for
example $\tau_{z_{i}\tilde{y}}$ in the feedback case) leads to a significant
reduction in the size of the ROMs when the dead time scale is important
compared to the time scale of the physical phenomenon to be captured (the
period of the second Mack mode). This reduction in the order of the ROMs is
beneficial both for the identification and the synthesis step: the higher the
order, the more difficult the identification and the larger the cost of the
controller synthesis.

The quantities required for the synthesis are obtained by impulse responses of
$w$ and $u$. The state-space ROMs associated to the transfer functions
$T_{z_{i}u}$, $T_{yu}$ and $T_{z_{i}\tilde{y}}$ are obtained by the subspace
identification method ERA, which requires impulse responses for each of the
inputs and involves performing a singular value decomposition to compress the
state (Juang & Pappa, 1985). This method has been used several times for the
control of 2D (Belson et al., 2013) or 3D (Morra et al., 2020; Sasaki et al.,
2018a) incompressible boundary layers. The ERA algorithm is applied after
removing (just by shifting the time axis) either $\tau_{z_{i}\tilde{y}}$ (in
the feedback case) or $\tau_{z_{i}u}$ (in the feedforward case) within the
impulses from $y$ and $u$ to $z_{i}$. The impulse responses from $y$ to
$z_{i}$ are obtained by inverse Fourier transform of $T_{z_{i}w}T_{yw}^{-1}$,
each individual transfer function being obtained by Fourier transform of an
impulse from $w$. The sampling time for ERA is $5\times dt$; the discrete time
models obtained are then converted to continuous time models by first-order
hold method (Franklin et al., 1997). As shown in figures
6(LABEL:sub@fig:ERA_1,LABEL:sub@fig:ERA_2,LABEL:sub@fig:ERA_3) for the
performance sensor $z_{6}$ and for the feedback estimation sensor
$y_{\textit{fb}}$, the constructed ROMs capture most of the dynamics.

(a)  (b)

(c)  (d)

Figure 6: (a,b,c) Comparison between impulse responses (blue lines) and the
ROMs (red circles) for the performance sensor $z_{6}$ and for the feedback
estimation sensor $y_{\textit{fb}}$. Note that for the ROMs of $T_{z_{6}u}$
and $T_{z_{6}\tilde{y}_{\textit{fb}}}$, the time axis of the impulse responses
are shifted by $\frac{t_{shift}U_{\infty}}{\delta_{0}^{*}}\simeq 897$ (black
dashed lines) which corresponds to the suppression of unnecessary dead times.
d) Comparison between the quantity $W_{y_{\textit{fb}}}$ from the linear
simulation (blue line) and the ROM (red circles).

The identification of the quantity $W_{y}$ is obtained by a vector-fitting
method (Matlab function tfest) designed to fit frequency response measurements
(Drmac et al., 2015). For this quantity, there is no uniqueness of the
identified model as the phase can vary from one model to another without
impacting the results of the synthesis (see (14)); the ROM just needs to be
stable and causal. Hence, we simply choose to define $W_{y}$ as
$W_{y}=\sqrt{PSD_{y}(\omega)}$ where $y$ is the response from an impulse in
$w$. A good agreement is achieved between $W_{y_{\textit{fb}}}$ and the ROM in
the case of the feedback estimation sensor $y_{\textit{fb}}$ (see figure 6d).

For the current application and with the six performance sensors $z_{i}$, the
sum of the orders of each ROM is $130$ for the case of the feedback
configuration and $115$ for the feedforward one. By comparison, identifying
the single transfer function $T_{z_{6}u}$ (corresponding to the farthest
performance sensor downstream) without suppressing the dead time leads to a
ROM of order $220$, which is already greater than the sum of the orders of
each ROM without their unnecessary dead-times.

In the control result section §6, because the models are of excellent quality
(see figure 6), the distinction between ROMs and real transfer functions is
not deemed necessary and the depicted results are those on the complete system
after implementation of the controllers in elsA.

###  5.2 Multi-objective structured $H_{2}$/$H_{\infty}$ synthesis

In this study, control laws are designed following a structured mixed
$H_{2}$/$H_{\infty}$ synthesis implemented in the Matlab function systune
(Apkarian et al., 2014).

Figure 7: Multiple requirements $H_{2}$/$H_{\infty}$ synthesis.

The general framework of this modern synthesis is illustrated in figure 7. In
this figure, $w_{H_{\infty}}$ (resp. $w_{H_{2}}$) and $z_{H_{\infty}}$ (resp.
$z_{H_{2}}$) represent the set of inputs and outputs whose associated
transfers are subject to $H_{\infty}$ (resp. $H_{2}$) norms. This synthesis
thus allows to minimize different $H_{2}/H_{\infty}$ norms under closed-loop
stability constraints despite model uncertainties $\Delta$. The structure of
the controller $K$ is defined by the user independently from the order of the
state-space model to be controlled, which makes it a particularly powerful and
flexible synthesis method. The set of transfer functions subject to an
$H_{2}/H_{\infty}$ norm minimization or constraints constitutes the augmented
plant; these transfer functions are composed of the transfers of the
controlled system allowing to respect the specifications, along with weighting
functions (Skogestad & Postlethwaite, 2005). Weighting functions act as
frequency domain constraints in order to shape adequately the transfer
functions to achieve specific design goals. Furthermore, weighting functions
allow to normalize the different requirements to be able to balance them
during the constrained minimization problem.

In our specific study, the structure of the controller $K$ is imposed
beforehand in the following way: (1) the controller $K$ is searched in a
state-space representation form; (2) the controller $K$ must be stable; (3) we
limit the controller order to $5$ as high-order controllers are less easily
implemented in practice (Goddard & Glover, 1995); (4) we impose a tridiagonal
state matrix which has significantly fewer parameters to determine than the
full matrix given that any real square matrix is similar to a real tridiagonal
form (McKelvey & Helmersson, 1996); (5) we impose a strictly proper controller
involving a natural roll-off of the high frequencies of $-20\ \mathrm{dB}$ per
decade in order to neglect dynamics in high frequencies and to be robust to
high frequency noise on the estimation sensor $y$ naturally present in every
experimental setup. For the controller structure imposed above, the algorithm
then solves the following constrained minimization problem:

$\begin{split}\text{minimize}&\underset{i=1,...,6}{\mathrm{max}}(||T_{z_{i}w}^{c}||_{2})\\\
&\text{subject to}\ ||W_{S}S||_{\infty}<1\text{ and
}||W_{KS}KS||_{\infty}<1.\end{split}$ (24)

This constrained minimization problem is the transcription of the fluidic
specifications established throughout §4.

Firstly, the minimization of $H_{2}$ norms of $T_{z_{i}w}^{c}$ directly allows
the reduction of the expected power for the six performance sensors $z_{i}$
used in the synthesis when they are excited by white-noise perturbations $w$
and sensed by the estimation sensor $y$. A multi-objective synthesis approach
is necessary for our problem by minimizing the expected power of sensors at
different abscissa of the flat plate instead of minimizing an overall energy.
Indeed, the disturbance energy growing as it is convected downstream, an
overall energy would then essentially account for the fluctuating energy
downstream of the domain, leaving aside the structures further upstream in the
case of a very large computational domain. Transition to turbulence appearing
locally above a certain perturbation energy threshold (see §4.1), we advocate
the need for minimizing the largest $H_{2}$ norm of the controlled system over
the set of performance sensors $z_{i}$ used to assess the local character of
transition to turbulence.

Secondly, the $H_{\infty}$ constraint on $W_{S}S$ maintains adequate stability
margins. To prevent the closed loop from being unstable in a feedback design,
a frequent choice is to ensure that $||S||_{\infty}<2$ (Skogestad &
Postlethwaite, 2005; Belson et al., 2013). Thus, the weighting function
$W_{S}$ has a constant frequency template such as $W_{S}(s)=0.5$ because the
$H_{\infty}$ constraint on $W_{S}S$ is equivalent to $|S|<1/|W_{S}|$ $\forall\
\omega\in\mathbb{R}$. This means that the system will be guaranteed stable up
to 50% of relative model errors $\Delta$ on $T_{yu}$ (see §4.2). In the case
of a feedforward design, $S(s)=1$ (because $T_{yu}=0$) and this $H_{\infty}$
constraint is always respected, which explains the unconditional stability of
the feedforward configuration.

Finally, the $H_{\infty}$ constraint on $W_{KS}KS$ is here to desensitize the
controller to new noise sources on a certain bandwidth. Our controller being
already robust to high frequency uncertainties due to the strictly proper
structure imposed, $W_{KS}$ is just designed to limit low frequency actuator
activity in case, for example, of low frequency noise on the estimation sensor
$y$.

By minimizing the maximum value between several transfer functions and using
$H_{\infty}$ norm constraints, a non-smooth optimization is performed; as non-
smooth optimization is computationally intensive (compared to LQG), it is all
the more important to obtain ROMs with the least possible states (see §5.1),
giving in our case computations of several tens of minutes.

##  6 Feedforward versus Feedback control

###  6.1 Performance on the nominal case

The results of both feedforward (denoted ’Ff’) and feedback (denoted ’Fb’)
controllers resulting from the constraint minimization problem (24) are
evaluated by implementing the controllers in the DNS solver elsA.

(a)  (b)

(c)

Figure 8: Blue, red and dashed lines represent the feedback case, the
feedforward case and the constraints (weighting functions) imposed for the
control design, respectively. a) Magnitude of $S$. b) Magnitude of $KS$. c)
Evolution of the local $H_{2}$ norm of the transfer $T_{z(x)w}$ as obtained
from the DNS simulation (those obtained with the ROMs are actually identical
at $x=x_{i}$ since the ROMs are very accurate). The vertical magenta and
orange dotted lines represent, respectively, the position of the actuator
(with the sensors $y_{\textit{fb}}$ and $y_{\textit{ff}}$ nearby) and the six
performance sensors $z_{i}$ used for synthesis. The values are normalized by
$||T_{z_{1}w}||_{2}$. The horizontal black dotted line depicts the energy
threshold $||T_{z_{1}w}||_{2}$ respected until $x_{6}$ following the
minimization of the cost functional
$\underset{i=1,...,6}{\mathrm{max}}(||T_{z_{i}w}^{c}||_{2})$.

Figure 8a shows the sensitivity function $S$ for the feedback design which
respects the $H_{\infty}$ constraint on the sensitivity function (i.e.
$|S|<1/|W_{S}|=6\ \mathrm{dB}$) imposed in the minimization problem (24)
(represented by the black dashed line). As previously explained, for the
feedforward design, $|S|=1$ (red line) and the constraint is automatically
satisfied. Figure 8b represents $|KS|$ for both the feedforward and feedback
cases. The weighting function $W_{KS}$, which allows to limit actuator
activity in case of low frequency disturbances, is also shown and we verify
that $|KS|<1/|W_{KS}|$ $\forall\ \omega\in\mathbb{R}$. For the feedback
design, $|KS|$ is close to $1/|W_{KS}|$ at low frequencies, meaning that there
is a trade-off between minimizing $H_{2}$ norms and desensitizing the
controller in the low-frequency range. We notice the natural roll-off of the
controllers of $-20\ \mathrm{dB}$ per decade at high frequencies related to
the strictly proper structure imposed in the synthesis.

The control action results in a significant reduction in the local $H_{2}$
norm of the transfers $T_{z(x)w}$ at each abscissa of the plate (see figure
8c) for both the feedforward and feedback configurations. As expected from the
literature (Belson et al., 2013; Juillet et al., 2013; Semeraro et al., 2013b;
Tol et al., 2019), the feedforward design minimizes even more the local
$H_{2}$ norm than the feedback one. Nevertheless, for both configurations, the
minimization of the cost functional
$\underset{i=1,...,6}{\mathrm{max}}(||T_{z_{i}w}^{c}||_{2})$ allowed the local
$H_{2}$ norm of $T_{z(x)w}$ not to exceed, before $x=x_{6}$, a threshold given
by the $H_{2}$ norm at $x=x_{1}$. Thus both configurations successfully
achieve the control strategy set forth in figure 2. The use of an $H_{2}$
performance criterion alongside the $H_{\infty}$ criterion on stability margin
allows to address both performance in terms of disturbance rejection and
stability robustness in the design of the feedback loop.

In addition to the reduction of the local $H_{2}$ norm along the plate, the
local $H_{\infty}$ norm $||T_{z(x)w}||_{\infty}$ has also decreased for both
the feedforward and feedback designs; this variation is directly related to
the $N$-factor envelope $\widetilde{N}$ by:

$\underset{x_{1}<x<x_{6}}{max}\ln{||T_{z(x)w}||_{\infty}}-\underset{x_{1}<x<x_{6}}{max}\ln{||T^{c}_{z(x)w}||_{\infty}}=\underset{x_{1}<x<x_{6}}{max}\widetilde{N}_{x}-\underset{x_{1}<x<x_{6}}{max}\widetilde{N}^{c}_{x}.$
(25)

More precisely, feedforward and feedback designs respectively “save” $1.13$
and $0.89$ points of $N$-factor. One might ask which is the most effective
setup for delaying transition, between minimizing
$\underset{i}{\mathrm{max}}(||T_{z_{i}w}^{c}||_{2})$ or minimizing
$\underset{i}{\mathrm{max}}(||T_{z_{i}w}^{c}||_{\infty})$, but answering the
question is beyond the scope of this study.

(a)  (b)

Figure 9: Evolution of the local $H_{\infty}$ norm of the transfer $T_{z(x)w}$
as a function of the plate abscissa for the uncontrolled (black lines),
feedback (a) and feedforward (b) cases. The evolution of $|T_{z(x)w}^{c}(F)|$
for some frequencies is also shown for the controlled cases. For vertical
lines, it is the same legend as in figure 8c.

(a)

(b)

(c)

(d)

Figure 10: Contours of $T^{\prime}_{rms}$ (a,b) and $u^{\prime}_{rms}$ (c,d)
for the uncontrolled (a,c) and feedback (b,d) cases. The white solid lines and
dashed lines represent the boundary layer thickness $\delta$ and the
generalised inflection point position $y_{g}$, respectively.

In addition to these results on wall-pressure fluctuation sensors coming from
impulse responses, we consider the global root-mean-square (r.m.s.)
temperature field (denoted $T^{\prime}_{rms}$) and streamwise velocity field
(denoted $u^{\prime}_{rms}$). For $T^{\prime}_{rms}$, whose high values are
located around the generalized inflection point (white dashed line) in the
uncontrolled case (see figure 10a), the control action reduces the amplitude
of the perturbations (see figure 10b for the feedback case). For the field
$u^{\prime}_{rms}$ in the uncontrolled case (see figure 10c), high level
regions are localized close to the wall. These levels are drastically
decreased when the control action is present (see figure 10d for the feedback
case). Regarding the feedforward design (not shown here), it further reduces
the amplitude of disturbances (as in figure 8c). By drastically reducing the
amplitude of velocity disturbances in both feedforward and feedback
configurations, while the controllers were built from wall pressure
fluctuation performance sensors, one may hope to strongly delay transition to
turbulence due to the second Mack mode in a 3D setup. Stability and
performance robustness are further addressed next.

###  6.2 Stability robustness

In the case of the feedback design, the configuration can be unstable and it
is necessary to quantify the evolution of the stability margins following
inflow condition variations or uncertainties. The closed loop system is stable
if and only if the Nyquist plot of the loop gain $-T_{yu}^{real}K$ (which is
stable) does not encircle the critical point (-1, 0). As already discussed in
§4.2, the Nyquist plot of $-T_{yu}K$ therefore allows to quantify the
available stability margins related to the distance to the critical point by
visualizing the maximum amount of error $|\Delta|$ admissible before
instability sets in. The gain and phase margins (denoted $GM$ and $PM$)
respectively represent the minimum amount of gain and phase variations
required to lose stability. In our case, the gain and phase margins
respectively stand for an estimation error in the instability’s growth rate
and convection speed which can lead to an instability of the feedback loop
(Sipp & Schmid, 2016). Inlet velocity variation is considered here to be the
most problematic variation (compared to other primitive variable variations)
as it involves multiple changes: (i) variation in time delays due to change in
convection velocity; (ii) modification of the Reynolds number $Re_{x}$
implying that for a given abscissa on the domain, the dominant frequencies are
higher (respectively lower) after an increase (respectively decrease) of
$Re_{x}$; (iii) variation of the Mach number $M_{\infty}$ implying a
modification of the neutral curves and by extension a modification of the
growth rates.

(a)  (b)

(c)  (d)

Figure 11: Evolution of the module (a) and phase (b) of $T_{yu}$ after a
variation of $\pm 5\%$ of the inlet velocity. Global view (c) and zoom near
the critical point (-1,0) (d) of the Nyquist plot of the loop gain
$-T_{yu}K_{Fb}$ (solid blue line) and $-T_{yu_{\pm 5\%}}K_{Fb}$ (dashed and
dotted red lines). The black dotted line represents the modulus margin
$||S||_{\infty}^{-1}$ (the minimal distance to instability). The black dashed
line represents the gain difference before instability and is linked to the
gain margin $GM$.

For a variation of the upstream velocity at the entry of the domain
$U_{\infty}$ of $\pm 5\%$, which induces $M_{\infty}\in[4.275,4.725]$, the new
transfer functions $T_{yu_{\pm 5\%}}$ are compared with the reference one
$T_{yu}$ in figures 11a and 11b. The greatest variations for the module appear
to be around $F=0.423$; we notice that a $5\%$ increase of the upstream
velocity implies a greater maximum value for the module at a slightly lower
frequency whereas a $5\%$ decrease in velocity implies a smaller maximum value
for the module at a slightly higher frequency (see figure 11a). The variation
of $\pm 5\%$ of the inlet velocity leads to the modification of the delays,
represented by the slope of the phase versus frequency plot (see figure 11b):
for the $5\%$ increase of the upstream velocity, the absolute value of the
slope is less and the delay is therefore shorter (with a relative variation
for the delay of $3.4\%$ compared to the reference case), whereas the opposite
is obtained in the case $-5\%$ (with a relative variation of $3.9\%$ for the
delay). Figures 11c and 11d show the Nyquist plot of the loop gains
$-T_{yu}K_{Fb}$ and $-T_{yu_{\pm 5\%}}K_{Fb}$. The variations of the upstream
velocity slightly alter the stability margins compared to those obtained in
the reference case: the phase margin stays infinite, while the gain margin
$GM$ (black dashed lines) and the modulus margin $||S||_{\infty}^{-1}$ (black
dotted lines) fluctuate respectively by a maximum of $3.6\%$ and $5.1\%$,
while remaining far from the critical point. Given the small impact of the
inflow velocity variations of $\pm 5\%$ on all margins, the feedback design
may be stable for even greater velocity variation. Therefore, unlike previous
feedback studies using LQG (Barbagallo et al., 2012; Tol et al., 2019), the
robustness to stability for a feedback design obtained with a robust synthesis
method is not a problem. Next, we examine performance robustness, which is a
different issue.

###  6.3 Performance robustness

Robustness to performance is evaluated by checking that the control laws
remain efficient in terms of expected power reduction of the different
performance sensors $z_{i}$ despite new noise sources or differences between
on-design and off-design operating conditions.

  * •

Noisy sensors
textNoisy estimation sensors are modelled by adding white Gaussian noise on
both $y_{\textit{fb}}$ and $y_{\textit{ff}}$ (see figure 12a). Both estimation
sensors are corrupted by the same amount of noise ($50\%$ of the r.m.s. value
without control action of $y_{\textit{fb}}$), which models an intrinsic defect
of the sensor, such as electronic noise, that does not depend on its position
along the domain. Nevertheless, the streamwise position of $y_{\textit{ff}}$
being quite close to that of $y_{\textit{fb}}$, the ideal signal-to-noise
ratio remains very similar for both configurations and only varies by a few
percents.

|
---|---
|

(a)  (b)

(c)  (d)

Figure 12: In all subplots, feedback and feedforward designs are in blue and
red lines, respectively. Controlled systems with ideal and noisy estimation
sensors are represented by solid and dashed lines, respectively. a) Short
sequence of $y_{\textit{fb}}$ corrupted by $50\%$ of the r.m.s. value without
control action of $y_{\textit{fb}}$. Comparison of the evolution of $PSD_{y}$
(b), $PSD_{u}$ (c) and $\mathrm{max}_{y}\ u^{\prime}_{rms}$ (d) for the
controlled systems with ideal and noisy estimation sensors. The legend in (d)
is the same as in figure 8c.

The PSD of the corrupted estimation sensors remain unchanged in the frequency
band of the second Mack mode but exhibit much larger values in low and high
frequencies (see figure 12b). This is because the PSD of white noise being
constant, the ideal signal-to-noise ratio is particularly low for frequencies
where the ideal signal energy is low. The signal $y$ is given to the
controller $K$, which generates the actuator signal $u$; the control signal
PSD for corrupted signals $y$ becomes stronger on the previously mentioned low
and high frequency bands, compared to the PSD of $u$ for ideal signals $y$
(see figure 12c). Nevertheless, thanks to the strictly proper structure and
the filter $W_{KS}$ imposed in the synthesis step, $|KS|$ have been
constrained in these frequency bands. Thus, the actuator activity remains
limited in these regions despite the important added noise and, if we look at
the evolution of the maximum along the wall-normal direction of
$u^{\prime}_{rms}$ (denoted $\mathrm{max}_{y}\ u^{\prime}_{rms}$), we keep a
performance close to the ideal case (see figure 12d). Both feedback and
feedforward configurations stay below the velocity energy threshold until
$x_{6}$ and these two designs are robust to noise on the estimation sensors.
If even noisier sensors were used, it would suffice to decrease the amplitude
of the weighting function $1/W_{KS}$ to recover performance robustness
(especially in low frequencies for the feedback configuration). For the case
illustrated in figure 12, a higher $1/W_{KS}$ (involving a less constrained
controller) could lead to an excessive injection of energy in the vicinity of
the actuator (see appendix C).

  * •

Off-design operating conditions
textPerformance robustness to off-design operating conditions is assessed by
considering the evolution of the local $H_{2}$ norm of $T_{z(x)w}$ after a
variation of free-stream density $\rho_{\infty}$ and velocity $U_{\infty}$ of
$\pm 5\%$, for both feedback and feedforward cases. The density variation may
correspond in practice to a change in altitude whereas the velocity variation
may correspond to a change in cruise speed. When $\rho_{\infty}$ is modified,
the temperature and velocity inlet values are kept constant, which means that
$M_{\infty}$ and hydrodynamic delays related to the convective behaviour are
maintained (see green dashed line figure 13a) while only $Re_{x}$ is modified
(which implies a change in the dominant frequencies at a given abscissa as
seen in figure 13b). A modification of $U_{\infty}$ on the other hand has a
much more dramatic effect since it implies variations of time delays (see
purple dashed line figure 13a) which will ultimately impact the only important
residual delay which is the one between the actuator and the estimation
sensors. It will also impact the values of $Re_{x}$ and $M_{\infty}$, which
modify the base flow profiles. Changing the base flow impacts the stability
characteristics of the boundary layer, and, in turn, the dominant frequencies
along the plate, as seen figure 13b.
textWith density variations of $\pm 5\%$ (see figure 13c), despite degraded
off-design performance, both feedback and feedforward controllers manage to
reduce the local $H_{2}$ norm compared to the case without control over a
fairly large distance on the flat plate. However, while the feedforward design
minimized the local $H_{2}$ norm more than the feedback one for the nominal
case (solid lines), it seems that this is no longer necessarily the case in
off-design situations (dotted and dashed lines). The variation in performance
between the nominal and off-design cases in the feedback configuration appears
less pronounced than in the feedforward setup, which is allowed by the
sensitivity function $S$. Although this transfer function, because of the
delay due to the actuator/estimation sensor distance, limits the achievable
performance on the nominal case for a feedback setup (see appendix A), it
allows to desensitize the system to modelling errors or to variations in
system characteristics over a certain bandwidth. Even if both designs exceed
the $H_{2}$ norm threshold at some point, they have some robustness to
performance with respect to density variations by staying below the
uncontrolled system $H_{2}$ norm all along the domain.

(a)  (b)

(c)  (d)

Figure 13: Comparison of uncontrolled pressure wavepackets generated by an
impulse of $w$ (a) and their PSD (b) at $x_{6}$ after a variation of
$\rho_{\infty}$ and $U_{\infty}$ of $-5\%$. Evolution of the local $H_{2}$
norm of $T_{z(x)w}$ after a variation of $\rho_{\infty}$ (c) and $U_{\infty}$
(d) of $\pm 5\%$ (dotted and dashed lines). The nominal cases are in solid
lines. Same legend as in figure 8c.

textThe real strength and superiority of the feedback design over the
feedforward one lies in its ability to maintain correct performance during
velocity variations (see figure 13d). While the feedback setup manages to
maintain some performance in off-design conditions by staying below the local
$H_{2}$ norm of the uncontrolled system over a fairly long distance along the
plate, the feedforward design fails to maintain the performance requirement by
amplifying the local $H_{2}$ norm. This increase in the feedforward setup may
then lead to a faster transition to turbulence, which is the opposite of the
desired objective. The velocity variations, regardless of changes in Reynolds
and Mach numbers, are indeed particularly problematic as they modify the
residual delay $\tau_{yu}$ which directly impacts (22) and (23) and therefore
may cause the controllers to activate out of phase. Thus, in the case of
noise-amplifier flows, we underline the importance to assess the robustness to
performance with respect to velocity variations, as in Fabbiane et al. (2015).
One could be tempted to robustify the feedforward setup by using an adaptive
controller structure (Fabbiane et al., 2014), but this type of approach is
only robust at long times (subject to convergence of the method) and the
problem of robustness following abrupt velocity variations would remain.
Therefore, as soon as variations or uncertainties on the inflow velocity are
present, the best trade-off between performance and robustness is a feedback
configuration.

##  7 Conclusions

A robust reactive control method has been developed in order to control the
linear growth of the second Mack mode in a 2D boundary layer over a flat plate
at Mach 4.5. The control tools (for identification and synthesis) being
mathematically well-established in a linear framework, they are perfectly
suited for this precise scenario of transition to turbulence where we seek to
control the linear growth of small perturbations.

The choice of the type and position of the actuator and sensors are based on
the study of the noise-amplifier behaviour of our flow, in order to trigger
the optimal growth mechanisms and ensure efficient flow control. During the
identification step, some unnecessary dead times related to the convective
nature of the flow are removed, allowing a significant reduction in the size
of the ROMs, which is beneficial both for the identification and synthesis
steps. Moreover, we strive to identify only quantities that could be obtained
in an experimental setup.

After identifying these useful transfers through data-driven methods, the
synthesis of the controllers is achieved with a structured mixed
$H_{2}$/$H_{\infty}$ synthesis. This robust synthesis method allows to limit
the order of the controller, to impose its structure upfront and to constrain
simultaneously several transfer functions to obtain at the same time
performance and robustness. Instead of simply minimizing a global energy, the
constraint minimization problem is posed in such a way that a shaping of the
spatial evolution of different local energy measures is realized, which seems
a more suitable approach to delay transition to turbulence. Multiple
performance sensors in the streamwise direction are therefore needed in this
study to cover the entire spectrum of amplified frequencies along the domain
and to capture the non-modal transient growth effect generated by the
actuator.

After implementing the control laws in the elsA solver, we find that
feedforward and feedback designs both manage not to exceed a certain energy
threshold on the nominal case. Moreover, the stability robustness for the
feedback design is not a problem thanks to the robust synthesis and the
constraints imposed. Regarding performance robustness, both feedforward and
feedback designs manage to reduce the amplitude of disturbances compared to
the uncontrolled case despite noisy estimation sensors or inflow density
variations. Nevertheless, for noise-amplifier flows, we stress the importance
to assess robustness to performance by changing the inflow velocity. Indeed,
this type of variation may cause the controller to activate out of phase. It
appears that the feedforward setup is completely unable to follow inflow
condition variations while the feedback setup keeps reasonable performance
over a large velocity variation of $\pm 5\%$. Therefore, the best trade-off
between performance and robustness requires a feedback configuration (in the
case of a linear time-invariant controller). This result looks contradictory
with conventional wisdom which favors a feedforward setup for noise-amplifier
flows. The widespread use of a feedforward structure is likely rooted in the
massive use of LQG synthesis for the control of noise-amplifiers. Indeed, LQG
comes with no guaranteed stability margin, which hinders its practical
application to a feedback setup. Belson et al. (2013) were among the first to
recognize the superiority of a feedback design for performance robustness in
noise-amplifier flows. The authors used loop-shaping on a simple PI
controller, but much richer feedback laws may be designed in a systematic way
using the modern robust synthesis tools of the present paper. Such tools are
already commonly used for the control of oscillator flows, where feedback is
mandatory to stabilize the unstable base flow (Flinois & Morgans, 2016;
Leclercq et al., 2019; Shaqarin et al., 2021). We expect the methodology of
the present paper, based on data-driven identification and robust synthesis on
a feedback setup to be relevant to other convectively unstable flows.

We are currently extending this study to a three-dimensional case with the
goal of delaying transition to turbulence. This implies placing multiple
estimation/performance sensors and actuators in the transverse direction and
controlling oblique waves of the first Mack mode as well as non-linearities.

Acknowledgements. This work is partially funded by the French Agency for
Innovation and Defence (AID). Their support is gratefully acknowledged. We are
also grateful to Xavier Chanteux who made his python LLST code available to
us. This study expands the work presented at the 55${}^{\text{th}}$ 3AF
International Conference on Applied Aerodynamics 12 — 14 April 2021, Poitiers
– France.

Declaration of interests. The authors report no conflict of interest.

##  Appendix A Position of the estimation sensor

To obtain the quantitative position of the sensor $y$ in our supersonic
boundary layer study for both feedforward and feedback configurations, a quick
analysis is carried out; it consists in looking at the impact of the
actuator/measurement sensor distance on the maximum achievable performance in
terms of $H_{2}$ norm reduction on the performance sensor $z_{6}$ regardless
of the desensitization to low frequency disturbances. We only look at the
performance sensor $z_{6}$ because it is the one furthest downstream from the
domain; the further downstream we are, the more we have to reduce the local
$H_{2}$ norm in order not to exceed a given threshold (see the principle
diagram in figure 2d). This performance sensor therefore plays a central role
and the position of the estimation sensor $y$ must allow a consequent
reduction of the energy of the sensor $z_{6}$. Since this analysis is only
done off-line on the ROMs and the resulting controller is not implemented on
the real complete system, the $||W_{KS}KS||_{\infty}$ constraint which was
only useful in case of new noise sources (as noisy estimation sensor) is
disabled.

For the controller structure developed §5.2, the constraint minimization
problem (24) is therefore written as

$\begin{split}&\text{minimize}\ ||T_{z_{6}w}^{c}||_{2}\\\ &\text{subject to}\
||W_{S}S||_{\infty}<1.\end{split}$ (26)

Figure 14 shows the evolution of the maximum performance achievable on the ROM
of the performance sensor $z_{6}$ as a function of the actuator/measurement
sensor distance. The $H_{2}$ norm reduction represents the quantity
$(||T_{z_{6}w}^{c}||_{2}-||T_{z_{6}w}||_{2})/||T_{z_{6}w}||_{2}$. On the one
hand, the actuator/measurement sensor distance influences very strongly the
maximum performance achievable for feedback designs (to the right of the
dotted line). On the other hand, feedforward designs (to the left of the
dotted line) are relatively unaffected by this distance over a certain range
and they perform better than feedback ones, which is consistent with the
results of the incompressible literature (Belson et al., 2013; Juillet et al.,
2013; Freire et al., 2020). The rapid drop in performance in the feedback
cases is largely due to the delay in $T_{yu}$ (Skogestad & Postlethwaite,
2005; Skogestad, 2009; Belson et al., 2013), which is the time it takes for
the wave generated by the actuator to arrive at the estimation sensor.
Intuitively, to counteract efficiently disturbances of a wavelength
$2\pi/\widetilde{\alpha}_{r}$, the actuator/measurement sensor distance must
be less than $2\pi/\widetilde{\alpha}_{r}$. Consecutively, the frequency
spectrum of the performance sensor $z_{6}$ containing a significant amount of
energy up to $F\approx 0.282$, this requires actuator/measurement sensor
distance of less than $2\pi\delta_{0}^{*}/F\sim 20\delta_{0}^{*}$ in this
case. Then, to obtain significant performance in terms of amplitude reduction,
it is decided to place the sensor $y_{\textit{fb}}$ at
$x_{\textit{fb}}=885.7\delta_{0}^{*}$ for the feedback configuration (at a
distance of $18.5\delta_{0}^{*}$ from the streamwise position of the
actuator). With regard to the feedforward design, the sensor $y_{\textit{ff}}$
is placed at $x_{\textit{ff}}=801.2\delta_{0}^{*}$ (at a distance of
$66\delta_{0}^{*}$ from the streamwise position of the actuator), in such a
way as to ensure that it is possible to disregard $T_{yu}$ in the synthesis
while having an optimal performance.

Figure 14: Evolution of the maximum performance achievable on the ROM of the
sensor $z_{6}$ as a function of the position of the measurement sensor $y$.
The dotted line represents the actuator position; feedforward and feedback
designs are respectively to the left and right of this dotted line.

##  Appendix B Evolution of performance as a function of the number of sensors
$z_{i}$ used in the synthesis

For the two sensor positions $x_{\textit{fb}}$ and $x_{\textit{ff}}$
determined previously, the following minimization problem is solved:

$\begin{split}\text{minimize}&\underset{z_{i}\in
z_{used}}{\mathrm{max}}(||T_{z_{i}w}^{c}||_{2})\\\ &\text{subject to}\
||W_{S}S||_{\infty}<1\text{ and }||W_{KS}KS||_{\infty}<1,\end{split}$ (27)

with the controller structure developed in §5.2. Not all the six performance
sensors are necessarily used for the minimisation problem and the evolution of
performance as a function of the number of sensors $z_{i}$ (and by extension
their positions) employed in the synthesis is assessed. The set of sensors
$z_{i}$ used in the synthesis is denoted $z_{used}$. In table 2, the different
configurations tested are listed: the cases labelled ’Fb$k$$z$’ (respectively
’Ff$k$$z$’) stand for feedback designs (respectively feedforward designs) with
$k$ performance sensors used in the synthesis; the performance sensors used
for each case are also given. Assuming that the transition to turbulence
process begins shortly after the streamwise position of the actuator, it is
chosen to scale the results by the local $H_{2}$ norm of the uncontrolled
system at the performance sensor $z_{1}$, which is the closest performance
sensor to the actuator. The maximum local $H_{2}$ norm between the position of
the sensors $z_{1}$ and $z_{6}$ (respectively the most upstream and the most
downstream performance sensors used in some syntheses) for the controlled
system is denoted $\underset{x_{1}<x<x_{6}}{max}||T_{z(x)w}^{c}||_{2}$.

Case |  Sensors z used for synthesis | $\underset{z_{i}\in z_{used}}{max}\left(\frac{||T_{z_{i}w}^{c}||_{2}}{||T_{z_{1}w}||_{2}}\right)$ | $\underset{x_{1}<x<x_{6}}{max}\left(\frac{||T_{z(x)w}^{c}||_{2}}{||T_{z_{1}w}||_{2}}\right)$
---|---|---|---
Without control | - | 1.96 | 1.96
Fb1$z$ | $z_{used}=\\{z_{6}\\}$ | 0.41 | 9.03
Fb3$z$ | $z_{used}=\\{z_{1},z_{4},z_{6}\\}$ | 0.67 | 1.98
Fb4$z$ | $z_{used}=\\{z_{1},z_{2},z_{3},z_{6}\\}$ | 0.90 | 1.06
Fb6$z$ | $z_{used}=\\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}\\}$ | 0.92 | 0.92
Ff1$z$ | $z_{used}=\\{z_{6}\\}$ | 0.14 | 2.48
Ff6$z$ | $z_{used}=\\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}\\}$ | 0.67 | 0.73
Table 2: Evolution of the performance after the controllers are implemented in
elsA as a function of the number of sensors $z_{i}$ used in the synthesis
step. Cases labelled ’Fb$k$$z$’ (respectively ’Ff$k$$z$’) stand for feedback
designs (respectively feedforward designs) with $k$ performance sensors used
in the synthesis. The results are normalized by the local $H_{2}$ norm of the
uncontrolled system at the position $x_{1}$.

(a)  (b)

(c)

Figure 15: a) Evolution of the local $H_{2}$ norm of the transfer $T_{z(x)w}$
from upstream noise $w$ to wall-pressure fluctuation probes $z(x)$. The
vertical magenta and orange dotted lines represent, respectively, the position
of the actuator (with the sensors $y_{\textit{fb}}$ and $y_{\textit{ff}}$
nearby) and the performance sensors $z_{i}$ that can be used for synthesis.
The values are normalized by $||T_{z_{1}w}||_{2}$. b) Comparison of
$T_{z_{6}w}$ (solid lines) and $T_{z_{1}w}$ (dashed lines) for the
uncontrolled (black lines), Fb1$z$ (blue lines) and Ff1$z$ (red lines) cases.
c) Comparison of the disturbance to performance attenuation criterion
$S_{z_{6}w}$ for different control cases. Disturbance rejection is improved
(respectively degraded) below (respectively above) the dotted line. Grey
shaded area represents the frequency bandwidth to be controlled from the
actuator to the end of the domain.

The resulting controllers are then implemented in elsA and we focus on the
evolution of the local $H_{2}$ norm of the transfers $T_{z(x)w}$ at each
abscissa of the plate. The evolution of the local $H_{2}$ norm of the
transfers $T_{z(x)w}$ for the case without control and the different feedback
cases is depicted in figure 15a (for feedforward cases, these results are
summarised in table 2). For the Fb1$z$ and Ff1$z$ cases, where the controller
is designed to minimise the energy of the performance sensor $z_{6}$, this
results in a strong reduction of the local $H_{2}$ norm at the end of the
domain; in the feedback (respectively feedforward) configuration,
$||T_{z_{6}w}^{c}||_{2}$ is even about $4.78$ (respectively $14.$) times lower
than $||T_{z_{6}w}||_{2}$. However, this significant decrease in energy
downstream of the domain was accompanied by a strong increase in the local
$H_{2}$ norm upstream in the domain (blue solid line in figure 15a for the
feedback case). The quantity
$\underset{x_{1}<x<x_{6}}{max}||T_{z(x)w}^{c}||_{2}$ for both feedforward and
feedback configurations appears greater than the uncontrolled case; this
increase of the local $H_{2}$ norm may then lead to a faster transition to
turbulence in a 3D setup, which is the opposite of the desired objective.

This increase of the local $H_{2}$ norm can be explained from figure 15b,
which represents the module of $T_{z_{6}w}$ (solid lines) and $T_{z_{1}w}$
(dashed lines) for the uncontrolled (black lines), Fb1$z$ (blue lines) and
Ff1$z$ (red lines) cases. On the one hand, the amplitudes of the dominant
frequencies of the uncontrolled system for the sensor $z_{6}$ (which is the
only one used in the synthesis for these cases) are significantly reduced both
in feedback and feedforward cases, which partly explains the significant
reduction in the $H_{2}$ norm for this transfer. On the other hand, the
amplitudes of the dominant frequencies for the sensor $z_{1}$ are amplified by
both feedback and feedforward designs, leading to an increase of the $H_{2}$
norm for this transfer and thus an amplification upstream of the domain.

Indeed, reducing the amplitude of disturbances in one part of the frequency
spectrum can lead to increasing it in the other part, which could predominate
in other abscissas of the domain. Figure 15c shows the frequency spectrum of
the disturbance to performance attenuation criterion, defined as
$S_{z_{i}w}=\frac{T_{z_{i}w}^{c}}{T_{z_{i}w}}$. For a sensor $z_{i}$,
disturbance rejection is achieved at frequencies where $|S_{z_{i}w}|<1$. We
can see from this figure that an effect similar to the waterbed effect
(Skogestad & Postlethwaite, 2005) appears: for the Fb1$z$ and Ff1$z$ cases,
the significant disturbance rejection at frequencies around $F=0.225$ is
accompanied by an amplification for higher and lower frequencies. The
frequency bandwidth to be controlled being around $F\in[0.225,0.324]$ (see
figure 5b), amplifying lower frequencies is not a problem in our case as these
will be found further downstream of $z_{6}$ and therefore not taken into
account in the computational domain. However, amplifying frequencies around
$F=0.324$ will directly impact performance on the sensor $z_{1}$ which is
dominated by these frequencies. This translates into the need to use several
sensors $z_{i}$ in the synthesis to obtain a suitable frequency representation
in different abscissas of the domain to avoid an unwanted waterbed effect.
Both Fb6$z$ and Ff6$z$ cases have lower disturbance rejection at frequencies
around $F=0.225$ but the waterbed effect on high frequencies is mitigated
compared to Fb1$z$ and Ff1$z$ cases (see figure 15c). By taking more and more
performance sensors along the plate for the synthesis, we cover a wider
spectrum of amplified frequencies. The larger the frequency bandwidth to be
rejected, the more complicated it is to obtain very high attenuation on the
spectrum. This is why the quantity $\underset{z_{i}\in
z_{used}}{max}||T_{z_{i}w}^{c}||_{2}$ increases with the number of performance
sensors used in the synthesis (see table 2). Nevertheless, due to the better
coverage of amplified frequencies by increasing the number of $z_{i}$ used in
the synthesis, a more uniform performance along the plate is obtained (see
table 2 and figure 15a).

By taking three performance sensors (one near the actuator, one near the end
of the domain and an other in between) and thus covering a wider frequency
spectrum, the Fb3$z$ case (green line in figure 15a) allows to significantly
reduce the local $H_{2}$ norm increase near the actuator compared to the
Fb1$z$. However, immediately after the position of the sensor $z_{1}$ (first
vertical orange line), yet taken into account in this synthesis, the local
$H_{2}$ norm increases and a slight bump appears in $x\approx
1020\delta_{0}^{*}$. It is associated with strong non-modal effects in the
vicinity of the actuator (see §4.3). For frequencies around $F=0.296$, those
dominant in the vicinity of the actuator, the modal behaviour is only found
for $x\gtrapprox 1136.4\delta_{0}^{*}$ (see figure 5d). Therefore, we need to
discretise the area from the actuator to the end of the transient non-modal
region with several performance sensors as in the Fb4$z$ and Fb6$z$ cases.
Because $\underset{x_{1}<x<x_{6}}{max}||T_{z(x)w}^{c}||_{2}$ is lower in the
Fb6$z$ case than in the Fb4$z$ one due to a better coverage of the amplified
frequency spectrum along the plate, six performance sensors are therefore used
in the syntheses of §6.

##  Appendix C Impact of $W_{KS}$ on the performance with noisy estimation
sensors

To illustrate the impact of the weighting function $W_{KS}$ on the
performance, the constraint minimization problem (24) is solved but with an
higher $|1/W_{KS}|$ compared to the one use all along §6. The feedback
controller resulting from this synthesis (red lines) is shown in figure 16a
and is compared to the previous one used in §6 (blue lines). The two
controllers have globally the same behaviour in the frequency bandwidth of the
second Mack mode, but the new controller has higher gain in low frequency
bandwidth. In the case where the estimation sensors are corrupted by the same
amount of white Gaussian noise as in §6.3 ($50\%$ of the r.m.s. value without
control action of $y_{\textit{fb}}$), it follows that the $u$-PSD for a
corrupted signal $y$ becomes more important in low frequencies for the
controller resulting from the synthesis with an higher $|1/W_{KS}|$ than for
the previous controller (see figure 16b). For the noisy estimation sensor case
and contrary to the controller used all along §6, the new controller leads to
a strong energy injection in the vicinity of the actuator (see red dashed line
in figure 16c). As these injected low frequencies are convectively stable,
they attenuate very quickly but the maximum along the wall-normal direction of
$u^{\prime}_{rms}$ clearly exceeds the energy threshold before the last
performance sensor $z_{i}$ used in the synthesis, which could trigger the
transition to turbulence in a 3D configuration. It should be noted that in the
case of ideal estimation sensors, the controller resulting from the synthesis
with an higher $|1/W_{KS}|$ minimizes slightly more the velocity fluctuations
compared to the previous feedback controller used in §6 (see solid lines in
figure 16c) because the constraint on $W_{KS}$ is less important. There is
therefore a trade-off between minimizing $H_{2}$ norms and desensitize the
controller in low frequency range during the synthesis.

(a)  (b)

(c)

Figure 16: Comparison of $KS$ (a), $PSD_{u}$ (b) and $\mathrm{max}_{y}\
u^{\prime}_{rms}$ (c) for a feedback controller resulting from a synthesis
with an higher $|1/W_{KS}|$ (red lines) than for the one used in §6 (blue
lines). Solid lines and dashed lines in (b) and (c) represent the cases with
ideal and noisy estimation sensors, respectively. The legend in (c) is the
same as in figure 8c.

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