Output Feedback Control of Blasius Flow withLeading Edge Using Plasma Actuator

Reza Dadfar∗ and Onofrio Semeraro∗

Royal Institute of Technology, 100 44 Stockholm, SwedenArdeshir Hanifi†

Swedish Defence Research Agency (FOI), 164 90 Stockholm, Swedenand

Dan S. Henningson‡

Royal Institute of Technology, 100 44 Stockholm, Sweden

DOI: 10.2514/1.J052141

The evolution and control of a two-dimensional wave packet developing on a flat plate with a leading edge is

investigated bymeans of direct numerical simulation. The aim is to identify and suppress the wave packets generated

by freestreamperturbations. A sensor is placed close to thewall to detect the upcomingwave packet,while an actuator

is placed further downstream to control it. A plasma actuator is modeled as an external forcing on the flow using a

model based and validated on experimental investigations. A linear quadraticGaussian controller is designed, and an

output projection is used to build the objective function.Moreover, by appropriate selection of the proper orthogonal

decompositionmodes, we identify the disturbances to be damped. A reduced-ordermodel of the input–output system

is constructed by using system identification via the eigensystem realization algorithm. A limitation of the plasma

actuators is the unidirectional forcing of the generated wall jet, which is predetermined by the electrodes’ location. In

this paper, we address this limitation by proposing and comparing two different solutions: 1) introducing an offset in

the control signal such that the resulting total forcing is oriented along one direction, and 2) using two plasma

actuators acting in opposite directions. The results are compared with the ideal case where constraints are not

accounted for the control design. We show that the resulting controllers based on plasma actuators can successfully

attenuate the amplitude of the wave packet developing inside the boundary layer.

Nomenclature

A, Ar = system matrix, reduced-order system matrixa = semimajor axis of ellipse, mB, Br = input matrix, reduced-order input matrixb = semiminor axis of ellipse, mC, Cr = output matrix, reduced-order output matrixE = power spectrum density of velocity fieldEi = energy of the ith proper orthogonal decomposition

modeG,Gr

= system transfer function and reduced-order systemtransfer function

K = control gain matrixL = estimation gain matrixM = spatial weight matrixRe = Reynolds numberr = order of the reduced modelt = time, sU = base-flow velocity, m∕s~U = modified base-flow velocity, m∕sU∞ = freestream velocity, m∕su = perturbation velocity, m∕su = estimated perturbation velocity, m∕su = perturbation velocity induced by actuator fed by

constant forcing, m∕s

~u = perturbation velocity on a modified base flow, m∕sν = kinematic viscosity, m∕sx = streamwise coordinatey = wall-normal coordinatez, v = output signalsΔt = sampling period, sδ⋆ = displacement thicknessθ� = momentum thicknessϕ, w = input signalsω = radial frequency, 1∕s

I. Introduction

T HE reduction of aerodynamic drag can positively influence theoperational costs of vehicles and aircraft; moreover, limiting

fuel consumption can lead to a reduction of the pollution. For thisreason, significant efforts have been devoted to control wall-boundedtransitional and turbulent flows; indeed, a reduction of the total dragcan be obtained by decreasing the skin friction on the aerodynamicparts of the vehicles by delaying the transition to turbulence [1]. Tothis end, passive and active control strategies are of great interest.Passive control strategies do not add external energy to the flow.

They can, for instance, be implemented by modifying the base flowvia geometric modification. Because of their simplicity and effi-ciency, these devices are an attractive approach, even though theirtime independent design precludes an influence of unsteadystructures of the flow. Thus, they can delay the transition to someextent, but they might not be able to prevent the onset of instabilities.On the other hand, active control adds energy to the system in the

form of predetermined actuation (open-loop control) or usingfeedback information from measurement sensors to determine theactuation law (closed-loop control); in this case, they can be useful forweakening and suppressing the perturbations arising in the flow.Indeed, it is well established that, if the amplitudes of the initialdisturbances are sufficiently small, the initial stage of the transition toturbulence in the wall-bounded shear flows is mostly governed bylinear mechanisms [2]. More specifically, boundary-layer flows are

Received 21 June 2012; revision received 7 November 2012; accepted forpublication 28 November 2012; published online 15 July 2013. Copyright ©2013 by Reza Dadfar, Onofrio Semeraro, Ardeshir Hanifi, Dan. S.Henningson. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 1533-385X/13 and $10.00 in correspondencewith the CCC.

*Graduate Student, Department of Mechanics, Linné Flow Centre.†Adjunct Professor, Department of Mechanics, Linné Flow Centre.‡Professor, Department of Mechanics, Linné Flow Centre.

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convectively unstable and can be regarded as noise amplifiers fromthe dynamical point of view [3]. In environments characterized bylow turbulence level, two-dimensional (2-D) perturbations aretriggered inside the boundary layer, and the Tollmien–Schlichting(TS) wave packets grow exponentially in amplitude as they traveldownstream, where finally decay or leave the observation windows[4]. This scenario is referred to as classical transition [5]. Becauseof the large sensitivity of such flows to external excitations [2],we can conveniently influence the TS waves by applying tiny localperturbations in a small region of the flow via proper devicesrequiring minute energy. Thus, mitigating the amplitude of theperturbations arising in a boundary layer can be a key factor fordelaying the transition to turbulence by using the robust and efficienttools provided by control theory.The early work regarding the combination of fluid dynamics and

control theory dates back to the papers by Joshi et al. [6], Cortelezziet al. [7], and Bewley and Liu [8].We refer to Kim and Bewley [1] fora recent review. Additional efforts have been made in the analysis ofthe stochastically forced Navier–Stokes equations by Farrell andIoannou [9] as well as Bamieh and Dahleh [10]. The amplificationdue to the transient growth has been analyzed by Jovanovic andBamieh [11]; in the latter contribution, it is shown that the streamwisevelocity component shows the largest overall response when astochastic excitation is considered.More recently, model reduction has been introduced for the design

of controllers. Indeed, because of the dimensions of the dynamicalsystem arising from the discretization of the Navier–Stokes system,characterized by 105 to 108 degrees of freedom (DOF), it is notpossible to apply the methodology for the control design in anefficient manner by using the standard control tools, usually feasiblefor systems with less than 104 DOF. This restriction can be addressedby designing a low-dimensional model that preserves the essentialdynamics of the original dynamical system [12].A classical way to obtain the reduced-order model (ROM) is the

Galerkin projection of theNavier–Stokes system onto a set of modes.The choice of these modes has a great influence on the propertyof the resulting reduced-order model; see, for instance, [12,13].When balanced truncation is considered, a method first introducedby Moore [14], the basis consists of modes representing the mostrelevant structures in the energy transfer processes from inputs tothe state (the most controllable modes) and from the state to theoutputs (the most observable modes), thus resulting in a basis fittedfor reconstructing the input–output dynamics of the system. Anapproximation of this method was proposed by Rowley [15], and it isreferred to as the balanced proper orthogonal decomposition method(BPOD). A limitation of the method is represented by the adjointsolution that needs to be availablewhen forming the bi-orthogonal setof modes. System-identification techniques allow to circumvent thislimitation; a reduced-order model of the system can be computed bysamplingmeasurements extracted directly from the flow.An exampleis represented by the eigensystem realization algorithm (ERA), firstintroduced by Juang and Pappa [16] and implemented in flowproblems byMa et al. [17]. This algorithm is theoretically equivalentto BPOD, but it uses only the information extracted from the sensorsto construct the ROM and is thus applicable to experimental data andsimulations. Once a low-order model of the system is obtained, thecontroller can be easily built by using the standard tools of controltheory.The actuator choice represents a crucial aspect of an active control

system. We can classify the different typologies of actuatorsaccordingly to their utility, by considering fluidic, moving object/surface, and plasma actuators. Much successful experimental workhas been performed by implementing different choice of actuators;for example, slot actuator and membrane were implemented asactuator by Sturzebecher and Nitsche [18] for TS wave cancellation,while speakers were used by Li andGaster [19]. Blowing and suctionactuators were employed for delaying bypass transition byMonokrousos et al. [20]. In the works by Williams et al. [21] andHeinz et al. [22], pulsed blowing jets were employed. For a broaderoverview, the interested reader can refer to the review by Cattafestaand Sheplak [23]. More recently, plasma actuators have gained

interest for their simplicity, low power consumption, high frequencyresponse, and lack of any moving parts. These characteristics makethemconvenient for being implemented in experimental environment[24]; for a review, we refer to Corke et al. [25]. One of thewell knownvariants of plasma actuator is the single dielectric barrier discharge(SDBD), which has been implemented successfully in differentapplications ranging from control of boundary-layer separation to jet-mixing enhancement and transition delay; for example, see [26,27].In this paper, we move toward a more realistic configuration, by

considering the flow past a flat plate with an elliptic leading edge,previously used by Schrader et al. [28]; although this work representsthe natural continuation of the studies previously performed in thisgroup by Bagheri et al. [29] and Semeraro et al. [30], where balancedtruncation is used in combination with a linear quadratic Gaussian(LQG) controller, limitations related to this more realistic setup areaddressed here. Indeed, the perturbation is located outside theboundary layer and is able to trigger the TS wave packets and otherstructures via the receptivity processes.Moreover, we introduce a model that reproduces the force

distribution of a plasma actuator already used in previous experi-mental setups [31]; whenmodeling it, an important feature that needsto be accounted for is the direction of the net forcing, related tothe placement of the electrodes that dictates it a priori. Often, thisconstraint is not taken into account when numerical analysis ofcontrolled flow-systems are performed; indeed, given a forcedistribution, it is often implicitly assumed that the control signalfeeding the actuator allows to switch along different diretions the netforcing, according to the sign of the signal. As already mentioned, ingeneral, this is not possible. In this work, we introduce two differentstrategies for attacking this limitation: 1) a modified control law thatallows to provide a net force oriented in only one direction byintroducing an offset, meanwhile preserving the ability of cancelingthe incoming TS waves, and 2) a different configuration includingtwo actuators, operating along two different, opposite, directions.We show that both the strategies allow to damp successfullythe perturbations generated in the boundary layer by the externaldisturbances.The paper is organized as follows. Section II deals with the flow

configuration. A detailed description of the disturbances, the plasmaactuators, and the sensors is included in Sec. III; the model reductionscheme is briefly outlined in Sec. IV. Section V provides anintroduction to the LQG control framework. The performance of thedevice are analyzed in Sec. VI, where the limitations related to theapplication of plasma actuators are addressed by using a differentmethodology. The paper finalizes with a summary of the mainconclusions (Sec. VII).

II. Flow Configuration and Governing Equations

A 2-D incompressible flow developing over a flat plate past aleading edge is considered (Figs. 1 and 2). The leading edge shape is amodified super ellipse (see Fig. 1) that provides a zero curvature at the

0 1 2 3 4 5 6−2

−1

0

1

2

3

4

Fig. 1 Modified super ellipse for the leading edgewithAR � ab � 6. The

semiminor ellipse axis b is selected as the reference length.

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junction between the leading edge and flat plate [32]. This smoothtransition reduces the effect of junction receptivity. All of the spatialdimensions in the geometry are normalizedwith the short semi-axisbof leading edge. In this study, the same aspect ratio as in Schraderet al. [28] (AR � 6) is considered.The dynamics of small-amplitude perturbations in a viscous

incompressible flow are governed by the Navier–Stokes equationlinearized around a base flow:

∂u∂t� −�U · ∇�u − �u · ∇�U − ∇p� Re−1∇2u (1a)

∇ · u � 0 (1b)

u � u0 at t � t0 (1c)

where the disturbance velocity and pressure fields are denoted byu�x; y; t� and p�x; y; t�, respectively, while x denotes the streamwisedirection and y the normal one. U�x; y� and P�x; y� represent thebase-flow velocity and pressure, respectively.The discretized linearizedNavier–Stokes equationswith boundary

conditions can be written in state space form as the following initial-value problem [12]:

du

dt� Au (2a)

u � u0 at t � 0 (2b)

whereA is the discretized linearizedNavier–Stokes operator, and u isthe discretized velocity field. Because the current flow configurationis globally stable, the eigenvalues of operator A have negative realparts [33]. However, due to the nonnormality of operator A, the flowis convectively unstable, which means that initial perturbations mayexperience a transient amplification as they propagate downstream.

A. Numerical Simulations

Direct numerical simulations (DNSs) of the flow on a flat plate pastan elliptic leading edge are performed using the spectral elementmethod [34]. This technique provides both the geometrical flexibilityof the finite element and the accuracy of spectral method. Thesimulation code is Nek5000, developed by Fischer et al. [35]. Thecode allows to simulate both fully nonlinear and linearized Navier–Stokes equations. The spatial domain is decomposed into finiteelements that, in turn, are divided into arrays of Gauss–Lobatto–Legendre (GLL) nodes. The solution of the Navier–Stokes equation

in each element is defined as a linear combination of Lagrangeinterpolant defined by an orthogonal Legendre polynomial as a basisof degreeN. A staggered grid of lower order,N − 2, for the pressureis adopted. In this method, calledPN − PN−2 [36], it is not required todefine explicitly the boundary condition for this variable. Thefollowing results are computed forN � 9 andN � 7 for the velocityand pressure grids, respectively. For the time integration, a second-order Adam–Bashforth scheme is employed.

B. Base Flow

The base flow is computed by marching in time the full Navier–Stokes system until the solution is steady. The reference speed is thefreestream velocity U∞, and the Reynolds number is defined asReb � U∞b∕ν � 1000, which is equivalent to a Reynolds numberof ReL � U∞L∕ν � 1.15 × 106, at the outflow location x � L. No-slip conditions are prescribed along the flat plate, while symmetryboundary conditions are imposed in the lower part of the domain,upstream of the leading edge. Neumann conditions for the outfloware imposed, which ensure a smooth pressure field in the end of thecomputational domain.The far-field boundary (inflowplane and freestream) is ofDirichlet

type, computed by solving the potential flow developing over anequivalent body thickened by the displacement thickness δ�, whichaccounts for the viscosity effects. The displacement thickness δ� iscomputed by combining the potential flow solution with a boundary-layer solver. This method allows the use of a far-field boundary closeto the wall and maintains zero pressure gradient on the flat-plateregion of the domain [28]. Note that the inflow boundary is 20b farfrom the leading edge, where the flow is relatively uniform and notinfluenced by the presence of the leading edge.Figure 3 displays the computational mesh for the numerical

simulation. The geometry is discretized into 6000 spectral finiteelements, formed by 100GLL points each. In Fig. 3a, the distributionof the spectral finite elements used in the region close to the leadingedge is shown. The elements are clustered near the wall where theboundary-layer effects are crucial and the actuators and sensors arelocated. In Fig. 3b, the finite elements together with theGLLpoints inthe nose region of the leading edge are depicted.Figure 4 compares the streamwise velocity obtained from direct

numerical simulation versus the Blasius boundary-layer solution onthe flat-plate part of the domain. In the region where the sensors andactuators are located (600 ≤ x∕b ≤ 1130), the DNS results are inexcellent agreement with the Blasius boundary-layer solutions. Thedisplacement and momentum loss thickness δ�∕b and θ�∕bcalculated from DNS are compared with the Blasius solution shownin Fig. 4a. The same agreement has been previously shown also bySchrader et al. [28]. Finally, Fig. 4b reports the comparison between

−20 0 20 40 60 80 100 1200

10

20

30

−1 0 1 2 30

1

2

a) b)Fig. 3 Computational grid used forDNS.The samegrid is used for computing the base flowand the perturbed flow.Thedomain extends formx∕b � −20to x∕b � 1150: a) mesh distribution of the spectral elements; along the streamwise direction for x∕b > 120 the mesh is uniform (not shown here); andb) close-up view of the leading edge region, displaying the finite elements and the GLL points.

C2 B2 C1x

yB1

20 1150

31

Fig. 2 The initial perturbation B1 is located at �x∕b;y∕b� � �−5;0.3�, upstream of the leading edge. The control action is provided by an actuator B2

located at (732,2); the estimation sensor C2 is placed at (632,2); the controller is designed based on the measurements extracted by the output C1,constituted by 10 proper orthogonal decomposition (POD) modes and spanning a region that extends approximately from 800 to 1130.

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the streamwise velocity profile at locations x∕b � 700 and x∕b �900 and the Blasius boundary layer.

C. Perturbation

The perturbation evolution is governed by the linearized Navier–Stokes equations (LNSEs); see Eq. (1). The computational domainand the geometry are the same already introduced for the base-flowcomputations. The boundary conditions along the walls and for theoutflows are of the same type already introduced; meanwhile,vanishing velocity is imposed in the far field (inflow and freestream)by enforcing null Dirichlet boundary conditions. These boundaryconditions are applied far enough from the boundary layer to havenegligible effects on the perturbation evolution.

III. Input–Output System

A schematic representation of the input–output configuration isdepicted in Fig. 2, where a sketch of the flow case is depicted togetherwith the inputs and the outputs. From the formal point of view, thesystem is described by the following linear system where the LNSEsare forced by the inputs and the measurements are extracted by theoutputs:

_u�t� � Au�t� � B1w�t� � B2ϕ�t� (3a)

v�t� � C2u�t� � Iαg�t� (3b)

z�t� � C1u�t� � Ilϕ�t� (3c)

The first input is located in front of the leading edge. The spatialdistribution is represented by the matrix B1 ∈ Rn, while thecorresponding temporal part of the input is given by the time signalw�t�. The second input B2 ∈ Rn×m represents m actuators locatedinside the boundary layer developing on the flat plate; this input is fedby the control signal ϕ�t� ∈ Rm. The output measurement v�t� ∈ Rp

in Eq. (3b) provides information about the traveling wave packet andrepresents the measurements detected by p sensors C2 ∈ Rp×n.The signalg�t� ∈ R ismodeled aswhite noisewith unit covariance

and can be considered as an input of the system; it models the noisecorrupting the measurements. By using the term Iαg�t�, whoseentries are given by α, we can represent different level of noise. Inparticular, high levels of noise corruption are represented by highvalues of α, whereas low values characterize more reliable data.Last, the second output z�t� ∈ Rk in Eq. (3c) extracts information

via the k output matrix C1i ∈ Rk×n placed far downstream in thecomputational domain; as shown later, it is used for the control designand can be regarded as the objective function of our control design.The matrix Il ∈ Rk×m contains the control penalty l in each entries;by tuning l, the control effort can be modified.

A. Initial Perturbation B1

The upstream perturbationB1 is a localized initial condition placedin front of the leading edge and outside the boundary layer. It ischosen as a Gaussian distribution, defined as

h�x; y� ��

σxγy−σyγx

�exp�−γ2x − γ2y� (4)

where

γx �x − x0σx

; γy �y − y0σy

(5)

and �x0; y0� is the center of the Gaussian distribution. The scalarquantities are σx � 4, σy � 1∕4, x0∕b � −5, and y0∕b � 0.3.The evolution of the disturbance is depicted in Fig. 5, where the

impulse response of the system to the initial condition is shown forfour different instants. During the initial stages of the evolution, thedisturbance penetrates inside the boundary layer. At time t � 700,the perturbation is mostly localized into two different regions, insideand outside of the boundary layer. The outer perturbation movesfaster, with a velocity close to the freestream velocity (U∞); theamplitude of this perturbation decreases while it is advected away.Inside the boundary layer, the presence of a TS wave packet, whichdevelopes together with longer structures stretched along thestreamwise direction (see Fig. 5), can be observed. The TS wavepacket moves slower (∼0.36U∞), but it is the unstable perturbation;indeed, it growswhile evolving along the streamwise direction. Thus,this perturbation is the most interesting for the analysis of thetransition process and control and the one we aim to control.

B. Actuators

Two different actuators are implemented in this work. AGaussian-shaped actuator, already used in previous proof-of-concept workby Bagheri et al. [12] is introduced in the new setup as referencecase. The mathematical representation is given by Eq. (4), whilethe corresponding force distribution is shown in Fig. 6a. Theperformance achieved by this idealized actuator is compared inSec. VI with the one obtained when a model for the plasma actuatorsis introduced in the control design. Among all of the differentversions of the plasma actuators, SDBD plasma actuators areconsidered in this study.Figure 7 indicates a schematic diagram of this actuator. It consists

of two electrodes, one exposed and one encapsulated in the flat-platesurface connected to an ac voltage source (order of kilovolts) andseparated by a dielectric material. If the voltage increases enough, theair above the dielectric material is ionized, and consequently ionsaccelerate in the presence of the electrical field and collide with theneutral particles of the air. The overall effect of this process is theproduction of a body force that creates a wall jet. A comprehensiveapproach to model the effects on the flow of the plasma actuators

0 200 400 600 800 1000

10−3

10−2

10−1

100

101

a)

−0.2 0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

b)Fig. 4 Baseflow: a) comparison of displacement andmomentum loss thickness obtained fromDNS (lines) andBlasius boundary layer solution (dots). Thesolid line is themomentum thickness, the dashed line is the displacement thickness, and b) comparison of streamwise velocity profile fromDNS at location

x∕b � 900 (solid line) and x∕b � 900 (dashed line) with the solution of Blasius boundary layer (dots).

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would demand a detailed analysis of complex chemistry and speciestransport to accurately describe the plasma discharge process in eachstep of the entire cycle. However, for fluid-flow problems, this levelof sophistication is unnecessary because the time and spatial scales inthe operating conditions of the plasma actuators (order of kilohertzfor time and micrometer for length) are quite different compared tothe current fluid-flow problems. Furthermore, the computational costof such methodology would be prohibitive.

An alternative approach is to focus only on the bulk characteristicsof the air and dielectric layer; in this case, the effect of the plasmaactuators can be reproduced by a body force. Although this approachis found to be simple and accurate enough [24], some limitations needto be accounted when integrating the plasma actuator model with theLQG control design for linear time-invariant control systems.First, the force distribution is not constant, and it varies as a

function of voltage amplitude and frequency [23]. A detailed

Fig. 5 Impulse response of the system to an initial perturbation B1; the streamwise velocity of the disturbance is shown at different instants of timet � �0;700;1500;2000�; the amplitude of the velocity at time t � 0 is scaled by 5 × 10−4.

AC

induced flowexposed electrode encapsulated electrode

dielectric material

Fig. 7 Schematic view of the SDBD plasma actuator.

Fig. 6 Spatial distribution of the actuators used in this paper; a) force distribution for Gaussian actuator; b) force distribution for plasma actuator. Theprofiles indicate the streamwise force distribution while the arrows display the net force field.

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quantification of the dielectric barrier discharged plasma actuatorused in this study can be found in Kriegseis [31]. This violates theassumption of time invariance of the system. In this study, we assumethat the force distribution is not varying as a function of voltage andfrequency. We use the force distribution obtained with a supplysource at 10 kV voltage. The resulting force distribution for theplasma actuator is shown in Fig. 6b.However, to test the possibility ofmanipulating the flow efficiently when a different distribution offorcing is introduced, we test different spatial distributions of theoriginal plasma actuator force distribution (10 kV) by scaling it andstudying the consequences on control action.Moreover, with predetermined electrode locations, it is impossible

to alter the force direction on demands; the wall jet is always createdin one direction. This introduces a constraint on the control design.Indeed, when applying the control law obtained by using the LQG,the sign of the control signal dictates the force direction of theactuation; for example, see [12,30]. Conversely, the geometricconstraint characterizing the plasma actuator does not allow to forcein all of the possible directions the flow; for instance, if the controlleris based on actuator forcing only along the positive direction, anegative control signal will not reverse the direction of the forcingalong the negative streamwise direction.The different strategies are described more in details in Sec. V,

while the performance are discussed in Sec. VI.The location for the actuators is chosen by considering the nature

of the analyzed flow. For convectively unstable flows, the disturbancegrows in a region marked off by branch I and branch II. The analysiscarried out by Brandt et al. [37] shows that, in between these twolocations, the sensitivity to localized perturbations results to bepractically constant. Thus, the placement of a localized actuator,whose effect is limited in a small region, is mostly dependant on theamplitudes of the growing wave packet; indeed, it is not desirable tolocate it far upstream, close to branch I, because the disturbance cansoon start to grow again. On the other hand, if the actuator is placedfar downstream, the disturbance results already amplified; moreover,in a transitional case, this would lead to an increasing of thenonlinearities affecting the flow. By following these considerations,we place the actuator halfway of the computational box, at �732; 2�.

C. Sensors and Objective Function

The measurements from the flow are extracted by the k sensorsC1

and p sensors C2. The output signal v�t� in Eq. (3b) is used for theestimation. The measurements in C2 are extracted by averagingthe velocity field using the Gaussian function [Eq. (4)] as weight; therelative positions of the sensor C2 and the actuator B2 affect thedynamics of the closed-loop system. Indeed, the input–outputconfiguration of the system is dominated by strong time delays, dueto the highly convective nature of this flow. Thus, a sensor upstreamof the actuator can measure the propagating disturbance but isnot capable of measuring the effects of the actuation; in this case,the resulting controller is more properly defined as disturbancefeedforward controller, a special case of output feedback control; see

Zhou et al. [38]. The sensor can measure the effect of the actuationonly when placed downstream or a short distance upstream ofthe actuator; however, these configurations show a quick decay ofthe performance, in particular when locations far downstreamof the actuators are investigated. For this reason, a feedforwardconfiguration is deemed more interesting for the performanceanalysis carried out in this work, and the estimation sensor C2 hasbeen placed upstream of the actuator, at (632,2).The miminization of the output signal detected in C1i can be

considered the objective function of our input–output system. Indeed,the purpose of the controller is to find a control signal ϕ�t� able toattenuate the amplitude of the disturbances detected by C1i. Theoptimal control signal is based on the information obtained by thenoisymeasurement v�t� in Eq. (3b). For this reason, theminimizationof the signal in Eq. (3c) can be regarded as the objective function ofthe controller. The objective function reads

kzk2L2�0;∞��Z

∞

0

kC1uk22 � l2kϕk22 dt (6)

where l is the control penalty and represents the expense of thecontrol. This parameter is introduced as a regularization termaccounting for physical restrictions. High values of the controlpenalty result in weak actuation and create a low-amplitude controlsignal, whereas low values of the control penalty lead to strongactuation. The output C1 is represented by a basis of POD modes.Because of the properties of the POD, the states are decomposed intothe spatial and temporal parts and are ranked according to theirenergy content. Selecting the most energetic POD modes as basis ofthe objective function C1i allows us to identify the most energeticstructures and obtain a low-order approximation of the originalsystem. This is the so-called output projection [15]. The PODmodesare mostly located far downstream, where the amplitude of the TScomponents is higher.The energy content of each mode is represented in Fig. 8. The

energy of each mode Ei corresponds to its eigenvalue obtained fromPOD. Because almost 99% of the total energy of the dynamics iscontained in the first 13 modes, these modes are enough forreconstructing the flow field properly.From an inspection of the spectrum, reported in Fig. 8a, it can be

clearly seen that, except for modes 1, 6, and 11, all of the other modesresult to be coupled two-by-two. This is mainly related to thetraveling structure of the TS waves represented by these modes.Indeed, if the covariance of a real dataset is considered, the computedPOD modes are real valued functions; thus, because waves aremathematically represented by complex functions, two POD modesare necessary to represent awavy structure traveling as awave packet;for example, see Rempfer and Fasel [39].The choice of the basis is crucial for the control design; becausewe

are interested in quenching the TSwaves only, we can use the basis asa filter by selecting only themodes that are representation of thewavytraveling structures. To do this, we select the frequency interval

1 2 3 4 5 6 7 8 9 10 11 12 13

10−3

10−2

10−1

a) b)

1 2 3 4 5 6 7 8 9 10 11 12 130

0.2

0.4

0.6

0.8

1

Fig. 8 Energy of thePODmodes. (a) Energy contentEi for each of the first 13PODmode is obtainedby their associated eigenvalue. (b) The energy ~Ei�ω�contained in the range 0.03 ≤ ω ≤ 0.1 is compared to the total energy of each mode; the ratio is reported as a dark-black bar.

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0.03 ≤ ω ≤ 0.1, characterizing the TS waves [40], and analyze theamount of energy related to those frequencies versus the wholeenergy of eachmode. As shown in Fig. 8b, themodes coming in pairsare characterized by being mostly connected to those frequencies.The spatial distribution of the low-frequency modes 1, 6, and 11 iselongated along the streamwise direction; these modes are dampedwhile traveling downstream, with a velocity higher than TS wavesand close to the freestream U∞. A more detailed analysis of theenergy distribution is shown in Fig. 9 that depicts the power spectrumdensityEi�ω� of each POD as a function of the frequency.Most of theenergy of the modes 2 and 3 is concentrated in the middle of thefrequency interval, while the distribution is spread out when higher-order modes are considered. Finally, the spatial structures of modes 1and 3 are displayed in Fig. 10.

IV. Model Reduction

Areduced-ordermodel of the systemcan be obtained by projectingthe full system onto a low-dimensional subspace spanned by r basisfunction. We can rewrite the full input–output problem [Eq. (3)] as

du

dt� Au� Bf (7a)

y � Cu�Df (7b)

where B � �B1; 0; B2� ∈ Rn×�2�m� is the input matrix;C � �C1; C2�T ∈ R�k�p�×n is the outputmatrix; f�t� � �w; g;ϕ�T ∈

R�2�m� is a vector containing the control signal ϕ�t�, the disturbanceexcitation w�t�, and the noise contamination g�t�; and y�t� ��v; z�T ∈ R�k�p� is a vector storing the signal v�t� extracted from thesensorC2 and the signal z�t�, detected by the objective function. Thefeedthrough term reads

D ��0 0 Il0 Iα 0

�∈ R�k�p�×�2�m�

The feedthrough matrix D can be neglected without affecting thegenerality of the application. The impulse response of the systemEq. (7) is given by

y � CeAtB (8)

where t is the time. At each time instant t�k� � kΔt, where k ∈ N isan integer number and Δt is the sampling period, the output of thesystem ~y�k� can be written as

~y�kΔt� � CTkB (9)

where Tk � eAkΔt is the time propagator. By introducing the innerproduct

< u; u >U � uTMu (10)

with M ∈ Rn×n representing the spatial weight matrix andu ∈ U ⊂ Rn, it is possible to derive the adjoint of Eq. (7) byapplying the definition

< Au; p >�< u; A�p > (11)

Hereafter, unless otherwise denoted, the superscript� stands for theadjoint quantities, while the superscript T represent the transpose(hermitian). As reported in Bagheri et al. [12], the resulting adjoint ofthe system in Eq. (7) reads

−dp

dt� A�p� C�f� (12a)

y� � B�p (12b)

wherep is the adjoint state, f�t�� is the adjoint input forcing, y�t�� isthe adjoint outputs of the system, and

A� � M−1ATM (13a)

B� � BTM (13b)

0.01

0.01

0.01

0.01

0.01

0.03

0.010.03

0.08

0.2

0.08

0.2

0.080.01

0.010.08

0.03

0.20.03

0.01

10.90.6

1 2 3 4 5 6 7 8 9 10 11 12 130.03

0.04

0.05

0.06

0.07

0.08

0.09

Fig. 9 Normalized energy distribution of the POD modes as a functionof frequency between 0.03 ≤ ω ≤ 0.1. Modes 1, 6, and 11 do not show

considerable energy contribution in the frequency window.

800 850 900 950 1000 1050 1100 11501

2

3

4

5

6

7

8

9

10

800 850 900 950 1000 1050 1100 11501

2

3

4

5

6

7

8

9

10

Fig. 10 PODmodes generated byusingdata collected from the impulse response of the system to each inputs: a) the isocontours of the streamwise velocitycomponent of the first POD mode i � 1, and b) the third POD mode i � 3. The solid and dotted contours indicate positive and negative velocities,respectively.

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C� � M−1CT (13c)

Note that, in the adjoint system, the adjoint inputs play the role of theoutput of the system,while the inputs of the system are represented bythe adjoint of the outputs of the original system (see [12]).

A. Empirical Balanced Truncation

Our aim is to find a basis to capture the input–output dynamics ofthe system.Among all velocity fields, some can be easily triggered bythe inputs (B) and some contribute more to the energy of the outputs(C). These states are called controllable and observable states,respectively, and can be identified by diagonalizing the corre-sponding controllability Gramian P and observability Gramian Q.The balanced modes and the corresponding adjoint set allow todiagonalize both the Gramians and to rank the states according to theobservability and controllability. Thus, it is possible to discard theunobservable/uncontrollable states that result to be redundant withrespect of the input–output behavior of the system. The computationof the controllability and observability Gramians can be performedby solving Lyapunov equation. Because the computational costfor solving large Lyapunov equations is unfeasible for a high-dimensional system, at least order O�n3�, an approximation ofthe balanced-modes basis can be used by computing empiricalGramians:

P �Z

∞

0

eAτBB�eA�τ dτ ≈ XXTM (14a)

Q �Z

∞

0

eA�τC�CeAτ dτ ≈ YYTM (14b)

The matrix Xn×mc is formed by collecting mc snapshots at discretetimes t�k� obtained from the impulses response of the system whenthe actuators are introduced as initial condition. The quadrature timecoefficient

������Δtp

is introduced for approximating the integral form ofthe Gramian:

X � �eAt1B; eAt2B; · · · eAtmc B�������Δtp

� �B; : : : ; TmcB�������Δtp

(15)

When the observability Gramian is considered, the matrix Yn×mo isformed by gathering the sequence ofmo snapshots from the impulseresponse of the adjoint system as

Y � �eA�t1C�; eA�t2C�; · · · eA�tmo C��������Δtp

� �M−1CT; : : : ;M−1TmoCT �������Δtp

(16)

The aim is to find two bi-orthogonal sets of modesΦ andΨ, such thatthe empirical Gramians

P � ΨTMPΨ; Q � ΦTMQΦ

are balanced and diagonalized, such the identity P � Q � Σ isfulfilled with Σ containing along the matrix diagonal the Hankelsingular values of the input–output system. By following theprocedure outlined in [15], the bi-orthogonal transformation isobtained by computing the singular-value decomposition (SVD) ofthe Hankel matrix defined as

H � Y�X � YTMX (17)

By factoring the Hankel matrix using SVD, we obtain

H � UΣVT � �U1 U2 ��Σ1 0

0 0

��VT1VT2

�� U1Σ1V

T1 (18)

whereΣ1 is ad × dmatrix, andd is the rank of theHankelmatrix. Thediagonal matrix Σ can be partitioned into

Σ1 ��Σr 0

0 Σd−r

�(19)

where r is the order of themodel.U1 andV1 can also be partitioned as

U1 � �Ur Ud−r �; V1 � �Vr Vd−r � (20)

Finally, the balanced modes are computed as

Φr � XVrΣ−12r ; Ψr � YUrΣ

−12r (21)

and the system parameters are

Ar �< Ψr; AΦr >M; ~E (22a)

Br �< Ψr; B >M (22b)

Cr � CΦr (22c)

B. Eigensystem Realization Algorithm

The snapshot method relies on the approximation of the Hankelmatrix by using snapshots taken from the direct and adjointsimulations. However, it can be shown that the Hankel matrix can beformed directly by collecting the signals extracted from all of theoutputs obtained from the inpulse response of the system to eachinputs (see [16,17]). Based on this observation, it can be shown thatthe ERA allows to produce a reduced-order model equivalent to abalanced realization.In the following, the algorithm is outlined by deriving it directly for

the continuous form. We start with the Hankel matrix; because thespatial matrixM is symmetric, we obtain

H � YTMX �

0BBB@

CCT...

CTmoB

1CCCAM−1M�B TB : : : TmcB �Δt (23a)

�

0@ CB · · · CTmcB

..

. ... ..

.

CTmoB · · · CTmc�moB

1AΔt (23b)

where the elements are the output of the system multiplied by timequadratureΔt. The reduced-order model is computed as follows. Theinput matrix Br is obtained by combining Eqs. (21) and (22b):

Br � ΨTr MB � Σ−12r UTr �YTMB� � Σ−1

2r UTr

0BBB@

CBCT2B

..

.

CTmoB

1CCCA

������Δtp

(24)

where ~y � YTMB is the output of the systemup to timemomultipliedby

������Δtp

. To obtain Cr, we can consider the definition of directbalanced modes in Eq. (21):

Cr � CΦr � �CX�VrΣ−12r (25)

and observe that this is equivalent to the following identity:

CX � C�BTB : : : TmcB�������Δtp

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CX � C�B TB : : : TmcB �������Δtp

� ~y�k � 0; · · · ; mc�������Δtp

(26)

Ar is calculated in the original balanced-mode method as

Ar � ΨTr MAΦr (27)

However, it is possible to observe that

Tr � eArΔt � ΨTr MeAΔt Φr � ΨTr MTΦr

� Σ−12r UTr �YTMTX�VrΣ

−12r (28)

where

YTMTX � H1 �

0B@ CTB · · · CTmc�1B

..

. . .. ..

.

CTmc�1B · · · CTmc�mo�1B

1CAΔt (29)

The entries of thematrixH1 are the outputs of the impulse response ofthe system multiplied by the time quadrature [Eq. (9)]. Finally, usingEq. (28), Ar can be calculated as

Ar �log�Tr�Δt

(30)

C. Performance of the Reduced-Order Model

In this section, the input–output behavior of the reduced-ordermodel is compared to the full linearized Navier–Stokes system.Figure 11 indicates the impulse response from B2 → C1;1, B1 → C2,and B1 → C1;1. The solid line indicates the impulse response of thefull system obtained from the Navier–Stokes system using a timestepper (Nek5000) with n ≈ 6 × 105 grid points, while the red dotspresent the results of the reduced-order model r � 120. A goodagreement between the full system and the ROM is observed bycomparing the input–output dynamics among all of the inputs and allof the outputs. The same conclusion can be carried out by comparingthe frequency response of the full system and that of the ROM. In thelinear time-invariant system, a sinusoidal input signal eiwt generates asinusoidal output signal with different magnitude kG�iω�k and phasearg�G�iw��. The frequency response is usually defined by the largestsingular value of the transfer function matrix G�iω�; see [41]. Thesystem is excited by each frequency, and when the transient behavior

vanishes and a periodic solution appears, the amplification of thedisturbances at the outputs C1 and C2 is analyzed. The largestsingular value of the output is computed and compared with thefrequency response G�iω� obtained by analyzing the ROM.Figure 12 reveals a good agreement between the full system and theROM, implying that the ROM can successfully capture the I/Ocharacteristics of the full system, especially in the frequency interval0.03 ≤ ω ≤ 0.1.

V. Control Design

The aim of this section is to introduce the controller based on thereduced-order model described in the previous section. The steps fordesigning the closed loop are the same already undertaken byBagheriet al. [29] for the 2-D boundary layer. However, the introduction ofthe constraints imposed by the plasma actuator makes it morecomplex. Indeed, as already mentioned, plasma actuators provide anet forcing only in one direction with respect to the flow direction.The forcing termB2ϕ�t� that, in the idealized case, can be oriented inany direction in the x-y domain, is now constrained to act only in onedirection, for instance the positive direction. Two different solutionsare tested to address this design restriction.

0.02 0.04 0.06 0.08 0.1 0.12 0.1410

−3

10−2

10−1

100

101

Fig. 12 The envelope of the multi-input/multi-output transfer functionmatrix G�iω� from all inputs to all outputs. The solid line represents thefrequency response of the ROM while dots indicate the full systemresponse.

0 500 1000 1500 2000 2500 3000−2

0

2x 10

−4

a)

b)

c)

0 500 1000 1500 2000 2500 3000−2

0

2x 10

−3

0 500 1000 1500 2000 2500 3000−5

0

5x 10

−3

Fig. 11 Performance of the full systemandROM: a) impulse response from inputB2 to the outputC1;1, b) impulse response fromB1 toC2, and c) impulseresponse from B1 to C1;1. The solid line is obtained from the DNS, while the dots are indicating the signals obtained from the corresponding ROM.

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1) The control design is performed by considering a model with abase flowmodified by a constant forcing; from the theoretical point ofview, the linearization is performed on a modified equilibrium state.Once the control design is completed, the forcing appears in thecontrol term as an offset of the control law. By introducing thisprocedure, the control law provides an action around a mean valueand results to be only positive (or negative). Because of thislimitation, this strategy is hereafter referred to as the constrainedcontroller (see Sec. V.B).2) Two actuators are employed, designed for acting in opposite

directions (see Sec. V.C).In the reminder of the section, these approaches are shown in

details, together with a brief introduction of the LQG controller. Theresults are discussed and compared (see Sec. VI).

A. Linear Quadratic Gaussian Design

The main idea of the linear feedback control is to determine thecontroller that minimizes the energy of disturbances in the regiondefined byC1. A classical approach to determine such control signalsis the LQG. The control signal ϕ�t� is provided in the presence of anexternal disturbance w�t�, assumed to be white noise with unitvariance. The control is designed for the actuator B2 such that themean of the output energy, z�t�,

E�z� � kzkL2�0;∞��Z

∞

0

uTCT1C1u� l2ϕTϕ dt (31)

is minimized. The design of an LQG controller involves a two-stepprocess. First the full state, represented in this case by the velocityfield, is reconstructed from the noisy measurement v�t� via anestimator. The estimated state u is computed by marching in time thedynamical system,

_u�t� � Aru�t� � B2rϕ�t� � L�C2ru − v�t�� (32)

fed by signals extracted from the system. The term L ∈ Rr×p is theestimator gain and can be computed by solving a Riccati equation[41], such that the error �u − u� is minimized. Once the state isproperly estimated, the control signal can be computed by thefollowing linear relationship:

ϕ�t� � Ku�t� (33)

where K ∈ Rm×r is referred to as the control gain. The minimizationof the cost function in Eq. (31) results in an optimization problem thatcan be solved by introducing a Riccati equation.According to the separation principle, the two steps (estimation

and full-information control) can be performed independently.Furthermore, if both problems are optimal and stable, the resultingclosed loop is optimal and stable [38]. In particular, the combinationof Eqs. (32) and (33) yields a reduced-order controller (also calledcompensator) of size r

_u�t� � �Ar � B2rK � LC2r�u�t� − Lv�t� (34a)

ϕ�t� � Ku�t� (34b)

Integrating the compensator with the full Navier–Stokes equationsyields the following closed-loop system:

_u

_u

!�

A B2K

−LC2 Ar � B2rK� LC2r

!�u

u

�

��B1 0

0 −L

��w

Iαg

�(35)

The evolution of the perturbations is simulated by marching in timethe full DNS, while the controller runs online simultaneously.

Equation (34a) is based on the reduced-order model and is solved byusing a standard Crank–Nicholson scheme.

B. Constrained Controller

The controller design introduced in the previous section does notaccount for constraints related to physical limits imposed by theactuator design. This approach is quite idealized, because a morerealistic control design usually accounts for constraints due tononlinearities, for instancemodeledwith proper saturation functions,or inherent limitations related to the control action as for the plasmaactuator considered here. As already mentioned in Sec. III.B, aplasma actuator is characterized by a predetermined direction offorcing related to the electrodes location. Thus, the direction of theforcing cannot change according to the sign of the signal. In thatconcern, the optimal controller design does not account forlimitations in the classical framework. To account for this limitation,we introduce a constant forcing �ϕ�t�, and on the top of that we add anoptimal controller signal ~ϕ�t�, such that the sum of two signals ispositive ϕ�t� � �ϕ�t� � ~ϕ�t� > 0 in the entire interval t ∈ �0; T�.From the mathematical point of view, this procedure corresponds tomodifying the base flow and building the control signal on a modellinearized around a new equilibrium point. To prove this, we beginwith the total velocity field, which can be written as

u � U� � �u� ~u� (36)

where U is the base flow, u is the steady solution obtained bymarching the perturbation equation in time with the constant forcingf � B2ϕ�t�, and ~u is the disturbance term, eventually forced by theoptimal controller ~f � B2

~ϕ�t�. The term u can be incorporated in thebase flow:

u � �U� �u� � ~u � ~U� ~u (37)

such that a modified base flow ~U is defined. By assumingU ≫ �u, themodified base flow ~U is simply obtained by summing the former baseflow, and the steady solution u is given as the solution of thelinearized perturbation equation forced with f � B2ϕ�t�:

∂ �u∂t� −�U · ∇� �u − � �u · ∇�U − ∇ �p� Re−1∇2 �u� �f (38a)

∇ · �u � 0 (38b)

We outline the procedure as follows.1) The steady state u is obtained as solution of the linearized

Navier–Stokes equation forced by a constant forcing f.2) The control design is performed on the modified base flow ~U,

obtained by summing the steady state u and the base flow U.3) The resulting controller runs on the original base flow; the

control signal obtained from the LQG solution is added to theconstant forcing f. In such a way, the deviation from the base flowused for designing the controller is incorporated in the controlforcing; the oscillating part of the control signal attenuates theamplitude of the perturbations stemming from B1.This procedure is analogous to an optimal controller. Because of

the linearity of the system, we can decompose the perturbation u intoa steady solution u related to the constant forcing and the disturbanceterm ~u, and substitute it into the LQG cost function in Eq. (31). Byexpanding the terms in the LQG cost function, the objective functionis split into three parts: a term related to the constant actuation, a termrelated to the disturbances, and a cross term. By assuming the integralof the control signal acting on the wavy part, having approximately anull mean, we can neglect the cross term. With this assumption, theminimum of the objective function can be achieved by minimizingthe two remaining terms independently. The first term is minimizedby providing the minimum possible actuation that satisfies theconstraint (i.e., a positive control signal),while the second is achievedby computing an LQG problem on the modified base flow.

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C. Two Adjacent Plasma Actuators

As a second strategy,we analyze amore expensive implementationin terms of hardware requirement, based on two actuators (see thesketch in Fig. 13). The two actuators are placed beside each other;consequently, we need to feed them with two different controlsignals. By considering the convective nature of the flow, the designcan be performed iteratively. First, an LQG optimal controller can bedesigned for the actuator B3; by assuming that this actuator is able toforce the flow only along the negative direction, we apply a propersaturation function such that the positive control signal is truncated,and a control signal ϕ�t� < 0 in the interval t ∈ �0; T� is fed into theactuator. Once the first closed loop is designed, we design a secondLQG controller for the actuator B2; we base the design on the firstclosed loop, such that we can attenuate the amplitude of the wavepacket once that the first controller has been already activated. Also,in this case, a saturation function is applied, such that the forcing isapplied in the opposite direction only (i.e., the positive one).

VI. Results

In the following section, the closed-loop performances arecompared for several cases, by introducing the LQG controller incombination with the plasma actuator. The plasma actuator action iscompared with the performance achieved by introducing theidealized Gaussian-shaped forcing already used by Bagheri et al.[12]. The sensitivity of the performance is analyzed when a differentdistribution of the plasma forcing is introduced. Finally, due to therestriction dictated by the actuator design, the strategies alreadydiscussed in the previous sections are now used for controlling the TSwave packet and compared.

A. Controller Parameters for Plasma Actuator

In Table 1, plasma actuators are considered in combination withLQGwithout any restriction related to the actuator design. The aim ofthis preliminary analysis is to compare different combinations ofcontrol penalties l and noise contamination α as reference.Objective function 6 is composed of two terms; the control penalty

l is aimed to balance the minimization, such that reduction of theamplitude of the wave packet, accounted by the term kC1uk, isaccomplished by a limited effort of the controller. The optimal valueof the control penalty usually is not known before applying thecontroller to the full DNS and involves a trial-and-error procedure. Ingeneral, small values of the control penalty correspond to a reductionof the perturbation amplitude; see cases A–C in Table 1. However,lower values of control penalties result in unfavorable behavior,such as a spurious control signal. Finally, the effects of noisecontamination are compared in cases A and D; as expected, highervalues of noise lead to a reduction of the controller performance.

B. Input–Output Analysis of the Closed-Loop System for PlasmaActuator

In this section, the closed loop is investigated from the input–output behavior perspective; in particular, the closed-loop labeled“case A” in Table 1, characterized by a control penalty l � 2 × 10−3

and a level of noise corruption α � 10−4, is considered. The analysisis carried out by considering both the signals in the time domain andthe disturbance amplitude in the frequency domain. The controlledcase A is assumed as the reference case also in the successivesections. Figure 14 reports an example of the spatial distribution ofthe perturbation velocity without control (Fig. 14a) and with control(Fig. 14b) at t � 2192. It can be clearly observed that the high-frequency, wavy structures are damped in the controlled case, whilethe low-frequency structures are revealed; by considering themaximum streamwise velocity, a reduction of 68% in amplitude canbe observed. In the following, the control action is analyzed from theinput–output dynamics point of view.

1. Input–Output Analysis in the Time Domain

In Fig. 15, the input–output behavior for case A is shown (refer toTable 1 for the parameters). In Fig. 15a, the measurement detected bythe sensor C2 is reported. This sensor is placed close to the wall, andso it can only register the perturbation evolving inside the boundarylayer. The signals related to the perturbations located close to the edgeof the boundary layer, moving at freestream velocity U∞, aredetected. The disturbances penetrate inside the boundary layer andtrigger TS wave packets, which move slower at ≈0.36U∞. The firstindication of this traveling wave packet (TS) is revealed later on, att � 1400, as easily recognizable from the oscillatory behavior. Afterthewave packet is convected past the location of the sensor, the signaltends to zero. In Fig. 15b, the control signal feeding the actuatorB2 isshown. The signal ϕ�t� in the interval t ∈ �600; 1400� reveals aninsensitive behavior to the structures moving with the freestreamvelocity. Indeed, the selected PODs are chosen to attenuate theamplitude of the disturbances in the frequency interval 0.03 ≤ω ≤ 0.1. Comparing the actuator signal ϕ�t� and the measurementsignal v�t�, a time delay of Δt � 270 is found, associated with thetime required for the wave packet to travel from the sensor to theactuator location. In Figs. 15c and 15d, the output signals detected byC1;1 and C1;3 are compared for the controlled (dashed) and theuncontrolled case (solid).

2. Disturbance Energy in the Frequency Domain

A second quantitative analysis can be performed by investigatingthe energy of the perturbations in frequency domain. In particular, thepower spectrum density of the velocity field E�ω� is evaluated in theregion x ∈ �800; 1150� by applying discrete Fourier transformperformed on a set of snapshots of the system collected in the timeinterval t ∈ �1200; 3200� with a sampling frequency 0.1 Hz.

Table 1 Cases A–D correspond to four different combinations of control penalties and noise level

Case Control penalty, l Noise corruption, α Wave-packet amplitude, kC1uk22 System norm, kGk22 Relative norm, kC1uk22�cont�∕kC1uk22�nc� × 100

NC — — — — 5.96 × 10−3 — — 100A 2 × 10−3 10−4 4.25 × 10−4 1:23 × 10−3 7.1B 1 × 10−2 10−4 8.77 × 10−4 6:35 × 10−3 14.7C 1 × 10−1 10−4 4.99 × 10−3 7:97 × 10−3 83.7D 2 × 10−3 10−2 1.71 × 10−3 2:33 × 10−3 28.6

C2 B2 C1x

yB1

B3

20 1150

31

Fig. 13 Schematic configuration for the case of two adjacent actuatorsB2 andB3.B3 forces the perturbation field along the streamwise coordinate x inthe negative direction, while B2 acts in the opposite, positive, direction.

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Fig. 14 Streamwise perturbation velocity fields; a) uncontrolled and b) controlled case at time t � 2192.

1000 1500 2000 2500 3000−2

0

2x 10

−3

a)

b)

c)

d)

1000 1500 2000 2500 3000−1

0

1

1000 1500 2000 2500 3000−5

0

5x 10

−3

1000 1500 2000 2500 3000−5

0

5x 10

−3

Fig. 15 Impulse response of the system: a) signal from input B1 to sensor C2, b) control signal feeding the actuator B2, c) measurements extracted bysensor C1;1, and d) sensor C1;3. The solid lines indicate the reference case without control while the dashed lines represent the controlled case.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

−7

10−6

10−5

10−4

10−3

Fig. 16 Power spectrum density. Solid line indicates the uncontrolled case while the dashed line depicts the controlled case when the plasma actuator isactive.

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In Fig. 16, the dashed line indicates the controlled case, while thesolid line denotes the uncontrolled cases. In the uncontrolled case,two types of disturbances characterized by low- and high-frequencystructures are clearly distinguishable. However, because of the choiceof the objective function, the controller reduces only the amplitudeof the disturbances related to the TS waves. At ω � 0.065,corresponding to the maximum energy of the uncontrolled case, thepower spectrum density is reduced up to 99%, while at ω � 0.055,corresponding to the maximum energy of the controlled case, anattenuation of 76% is found.

C. Plasma Versus Gaussian Actuators

The first performance analysis carried out is performed bycomparing the plasma actuator and the Gaussian-shaped actuatordefined by relation 4. The spatial distribution of these actuators isdepicted in Fig. 6. Different ROMs are built by using the samenumber of modes; hence, the controllers are designed by usingthe ROM specifically designed for each setup. For the plasmaactuator, the control parameter is similar to case A in Table 1, whilefor the Gaussian actuator, α � 10−4 and l � 2 × 10−2 (case G inTable 2).In Fig. 17, the power spectrum densities for both actuators are

presented and compared in the region corresponding to the basis C1.The two actuators can attenuate the disturbance amplitude in a similarway; the reduction of the wave packet amplitude is 7.9% for theplasma actuator (caseG inTable 2), while it is 13.1% for theGaussianactuators (case G in Table 1). More in detail, at ω � 0.055, theGaussian actuator gives an improvement in the energy attenuation of6.7%, with respect to the plasma actuator; on the contrary, the plasmaactuator reveals an improved performance at higher frequencies,ω � 0.065, estimated around 4.6%. In conclusion, in the desired

frequency range, the plasma actuator can attenuate the disturbanceswith the same efficiency as a Gaussian actuator.

D. Plasma Actuators with Different Sizes

The distribution of the body force for a plasma actuator is afunction of the input voltage; thus, it is not constant. Although, in thiswork, the force distribution is frozen and the reduced-order modelsare defined ad hoc starting from the given distribution, the sensitivityto different actuator sizes is analyzed to address the influence on theperformance. Three different actuators are studied: the originalactuator (case A); a larger one, where the force distribution spans aregion increased by a factor s � 150% with respect of the nominalactuator (case E); and a smaller one, where the distribution isdecreased by a factor s � 60%, labeled case F. For all cases, the noisecovariances and the control penalties are kept the same. Figure 18shows the energy of the disturbances in the frequency domain. Atω � 0.055, the large actuator s � 150% can attenuate thedisturbances 2.5% more than the original size (s � 100%), whilethe small one is characterized by a worsening of 3.2% of theperformance. At ω � 0.065, the original actuator s � 100% andsmall s � 60% actuator behave similarly (only 0.26% difference),while the larger one is 1.6% worse than case A. In conclusion,except these small differences, the discrepancies in performanceobtained by using different actuator sizes are negligible Table 3; thus,from the performance point of view, a good behavior is expected alsoin a more sophisticated framework, where the voltage influence isaccounted in the modeling.

E. Constrained Controller

The constrained controller is introduced to overcome the limitationof the plasma actuator, which can deliver forces in only one direction

Table 3 Comparing the efficiency of plasma actuators with different sizes

Case Size s Frequency ω Energy E�ω� Attenuation E�ω� − E�ω�s�100%∕Enc × 100

Plasma actuator (case E) 150% 0.055 2.86 × 10−5 −2.50%Plasma actuator (case E) 150% 0.065 6.97 × 10−6 1.64%Plasma actuator (case F) 60% 0.055 3.64 × 10−5 3.22%Plasma actuator (case F) 60% 0.065 3.32 × 10−6 0.26%

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

−7

10−6

10−5

10−4

10−3

Fig. 17 Power spectrum density E�ω� as function of the frequency ω. The solid line represents the uncontrolled case, the dashed line indicates thecontrolled case A (plasma actuator), while the line with circle symbols indicates the case with Gaussian actuator (case G).

Table 2 Comparison of different choice of actuators: plasma actuator with a size s � 150% (E); plasma actuator with asize s � 60% (F); Gaussian actuator (G); plasma actuator fed by a restricted controller (H) and two adjacent actuators (I)

Case Control penalty, l Noise level, α Wave-packet amplitude, kC1uk22 System norm, kGk22 Relative norm, kCuk22�cont�∕kCuk22�nc� × 100

NC — — — — 5.96 × 10−3 — — 100E 2 × 10−3 10−4 4.76 × 10−4 1.51 × 10−3 7.9F 2 × 10−3 10−4 4.78 × 10−4 1.83 × 10−3 8.0G 2 × 10−2 10−4 7.85 × 10−4 1.52 × 10−3 13.1H 2 × 10−3 10−4 4.25 × 10−4 4.86 × 10−2 7.1I 2 × 10−3 10−4 6.16 × 10−4 1.36 × 10−3 10.3

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and have only a positive control signal. A constant control signal ( �ϕ)is applied, and on the top of that, the optimal control signal ~ϕ isintroduced, such that the sum of the two signals is positive. In Fig. 19,the energy of the perturbation E�ω� is compared for two differentcases: the constrained LQG controller and the reference controller(caseA). The energy is evaluated in the region identified by the outputC1. The constrained controller can mitigate the disturbanceamplitude similar to the LQG controller. The similarity stems fromthe fact that a perturbation with zero frequency is added to the baseflow, and the resultant modification is quite small. Hence, th eenergyof the perturbation on the original and modified base flow actspractically in the same way in the frequency range 0.03 ≤ ω ≤ 0.1.

F. Two Adjacent Plasma Actuators

Figure 20 indicates the control signals for the two adjacentactuators (case I). The dashed line represents the control signalfeeding the actuator B3, where the positive part is truncated, whilethe solid line shows the control signal for the actuator B2, where thenegative part is truncated. Because the two actuators are not at thesame location, a small phase shift in the control signal can beobserved. Figure 19 depicts the same cases in frequency domain. Thepower spectrum densityE�ω� is depicted in Fig. 19. The results showthat the two constrained actuators B2 and B3 are able to attenuate the

perturbation in the frequency interval 0.045 ≤ ω ≤ 0.1. At thefrequency ω � 0.055, the latter could attenuate the amplitude up to65%, while the LQG controller mitigates the amplitude up to 76%. Inthe region 0.03 ≤ ω ≤ 0.045, two adjacent controllers cannotattenuate the amplitude of the perturbation even though, in this case,the energy of the disturbances in this frequency range are small.

VII. Conclusions

The active control of a flow developing on a flat plate past a leadingedge in combinationwith plasma actuators is investigated. The initialperturbation is obtained by introducing a Gaussian function locatedupstream of the leading edge; the impulse response of the system ischaracterized by freestream perturbations quickly advected down-stream in the outer region and the perturbations evolving inside theboundary layer including unstable Tollmien–Schlichting (TS). Theaim is to attenuate the disturbances by using an output feedbackcontrol. The goal is the design of a control system based on theinformation filtered via localized estimation sensors from the flow.To accomplish the task, the controller is based on a low-dimensionalmodel of the linearized Navier–Stokes equations obtained byusing system identification via an eigensystem realization algorithm.It is shown that this method can accurately capture the input–output dynamics of the system. To attenuate the amplitude of the

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

−7

10−6

10−5

10−4

10−3

Fig. 18 Power spectrumdensityE�ω� as function of the frequencyω, when plasma actuatorswith different sizes are used. The reference caseA is plotted

with a dashed line; case E (larger plasma actuator, s � 150%) is represented by a line with circle symbols. Case F (smaller plasma actuator, s � 60%) isindicated by a line with square symbols (see Sec. VI.D).

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

−7

10−6

10−5

10−4

10−3

Fig. 19 Power spectrumdensityE�ω� as function of the frequencyω. The solid line represents the uncontrolled case; caseA is depictedwith adashed line,while the restricted control (case H, see Sec. VI.E) is denoted with circles. The performance achieved by using a controller based on two actuators (seeSec. VI.F) are indicated with a line with square symbols.

0 500 1000 1500 2000 2500 3000−1

0

1

Fig. 20 Control signals for case I (two adjacent controllers, Sec.VI.F); the signal depictedwith adashed line represents the control signal feeding actuatorB3, while the solid line corresponds to actuator B2.

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perturbation, a body force representing the net force generated by asingle dielectric barrier discharge plasma actuator is introduced.The model is based on experimental investigations. The objectivefunction of the system is composed of a set of proper orthogonalmodes modes; because of the necessity of discriminating the TSwaves from the freestream disturbances, the basis of modes isselected such that only the TSwave frequency range is represented byit. The resulting controller is able to quench only the disturbanceincluded in the selected frequency range.Some limitations of the plasma actuators are addressed by

modifying the control design and carrying out several parametricanalyses for investigating the performance of the device.A first restriction of the plasma actuators is represented by the

orientation of the forcing, limited by the geometry of the device. Toaddress this limitation, two methods are introduced. First, thecontroller is constrained to act in one direction only, while preservingthe ability to cancel the wavy perturbation. In this case, a constantforcing is introduced, and on the top of that, the optimal control signalis added, based on amodified base flow. The two signals are designedin such a way that the resulting control is characterized by beingalways positive. By following this strategy, the resultant forcingcan be oriented along the original design direction of the actuator.A second alternative, more expensive in hardware terms, is todesign a controller based on two adjacent actuators, each of themcharacterized by a specific direction; in this case, two linear quadraticGaussian (LQG) controllers are designed iteratively. Both of theprocedures result in a successful attenuation of the disturbanceamplitudes, with an efficiency comparable to the standard LQGcontroller without constraints.The force distribution generated by the plasma actuators is not

constant, but it varies with the magnitude of the voltage feeding thedevice. This, in general, requires the design of a nonlinear controller.In this investigation, different actuators, characterized by differentdistributions in space, were tested. The results reveal that, apart forsmall differences, the size of the actuator does not have a hugeinfluence on the authority of the controllers.

Acknowledgments

The authors wish to thank Antonios Monokrousos for fruitfuldiscussions and Sven Grundmann for the information regardingthe plasma actuator. Computer time provided by the SwedishNational Infrastructure for Computing is gratefully acknowledged.The authors also acknowledge financial support from the SwedishResearch Council.

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