2.1 Governing equations

This paper considers a 2-D non-periodic flow between two infinite flat plates. The flow is non-dimensionalised using the maximum centreline velocity $U_{0}$ and half-height $h$ with corresponding Reynolds number $Re=(U_{0}\unicode[STIX]{x1D70C}h)/\unicode[STIX]{x1D707}$ where $\unicode[STIX]{x1D70C}$ is the density and $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid. For flow simulations a total non-dimensional length $L_{sim}=16\unicode[STIX]{x03C0}$ is considered. This section focuses in particular on the flow model that is used for control design. For control design purposes a localised region with a length of $L_{c}=8\unicode[STIX]{x03C0}$ is considered. External sources of disturbances are assumed to enter the control domain though the inflow. The geometry of the flow is shown in figure 1. A supercritical case is studied at $Re=7000$ for which the flow field is convectively unstable. However, the non-periodic flow configuration is globally stable since any initial perturbation eventually leaves the computational domain. The control objective is to stabilise convective perturbations around the steady-state parabolic velocity profile $\boldsymbol{U}(y)=[1-y^{2},~0]^{\text{T}}$ . The dynamics of small-amplitude perturbations in a viscous incompressible flow is governed by the Navier–Stokes equations linearised around the base flow and the continuity equation

where $\boldsymbol{u}(\boldsymbol{x},t)=[u(\boldsymbol{x},t),v(\boldsymbol{x},t)]$ and $p(\boldsymbol{x},t)$ denote the velocity and pressure perturbation field, $\boldsymbol{x}=(x,y)$ is the spatial coordinate and $\boldsymbol{f}(\boldsymbol{x},t)$ is an in-domain body force field per unit mass typically used for applying control. The system is closed by the boundary conditions (2.1c )–(2.1d ) where $\unicode[STIX]{x1D6E4}_{D}=\unicode[STIX]{x1D6E4}_{in}\cup \unicode[STIX]{x1D6E4}_{r}$ is the Dirichlet part of the boundary, $\unicode[STIX]{x1D6E4}_{in}$ the inflow part of the boundary, $\unicode[STIX]{x1D6E4}_{r}$ are the rigid walls and $\unicode[STIX]{x1D6E4}_{out}$ the Neumann outflow part of the boundary. $\boldsymbol{u}_{b}(\boldsymbol{x},t)$ is a prescribed velocity input profile used for boundary control at the wall boundary $\unicode[STIX]{x1D6E4}_{r}$ and for the external disturbances at the inflow boundary $\unicode[STIX]{x1D6E4}_{in}$ . The outflow boundary condition (2.1d ) is known as a no-stress condition and has proven to be well suited for unidirectional outflows (Rannacher, Turek & Heywood Reference Rannacher, Turek and Heywood1996). It is naturally satisfied by the variational formulation used in the numerical method (see appendix A). The artificial non-physical effect of this boundary condition near the outflow is investigated in § 3. In this study only boundary feedback control is considered, therefore the in-domain body force is set to zero ( $\boldsymbol{f}=0$ ) in the remainder of this section. However, in-domain disturbances are considered to evaluate the controller in § 4.

Figure 1. Channel flow geometry and control layout including the shear sensor locations $\unicode[STIX]{x1D708}_{i}$ , boundary actuator distributions $g_{i}(x)$ and controlled output distribution $q_{i}(x)$ .

2.2 Inputs and outputs

The chosen control objective is to suppress the effect of inflow disturbances on the fluctuating wall shear stress. The control actuation is achieved by means of unsteady blowing and suction at the wall and boundary shear sensors are used to extract the measurements. A feedforward actuator/sensor configuration (Belson et al. Reference Belson, Semeraro, Rowley and Henningson2013) is considered in which two point shear sensors at the walls are placed upstream of the control actuators. A schematic representation of the control layout is shown in figure 1. It is shown in Belson et al. (Reference Belson, Semeraro, Rowley and Henningson2013) that feedforward configurations achieve the best disturbance attenuation, but can be less robust to additional disturbances not seen by the sensor. The shear sensors $\unicode[STIX]{x1D742}_{m}$ are therefore placed close to the control actuators $\boldsymbol{g}_{c}$ . In addition a controlled shear output $\boldsymbol{q}$ is defined which will be used to define the performance objective of the controller. The specifications will be discussed next. The boundary actuation is modelled through the boundary conditions (2.1c ) and is decomposed into an external disturbance and a control

with $\boldsymbol{u}_{c}(\boldsymbol{x},t)$ the actuation imposed at the rigid walls and $\boldsymbol{u}_{d}(\boldsymbol{x},t)$ the external disturbance imposed at the inflow. This disturbance model is discussed in detail in the next section. To manipulate the flow, localised wall-normal blowing and suction with zero net mass flux is considered. It is assumed that the spatio-temporal actuator model is described by the following state-space description

with $\boldsymbol{\unicode[STIX]{x1D702}}_{c}(t)\in \mathbb{R}^{2}$ the actuator state that describes the magnitude of the blowing and suction, $\unicode[STIX]{x1D753}(t)\in \mathbb{R}^{2}$ the control input and $\boldsymbol{u}_{c}(\boldsymbol{x},t)$ is the actuator velocity output at the wall. The temporal dynamics is described by a first-order low-pass filter defined by ${\mathcal{A}}_{c}=-\unicode[STIX]{x1D70F}^{-1}I$ , ${\mathcal{B}}_{c}=\unicode[STIX]{x1D70F}^{-1}I$ with $\unicode[STIX]{x1D70F}$ the time constant of the filter. A fast actuator is assumed with $\unicode[STIX]{x1D70F}=0.1$ , that is a stable approximation of a pure integrator typically used for boundary control in shear flows, see e.g. Högberg & Henningson (Reference Högberg and Henningson2002), Högberg et al. (Reference Högberg, Bewley and Henningson2003a ). The actuator output at the wall is defined by ${\mathcal{C}}_{c}=\unicode[STIX]{x1D642}_{c}(\boldsymbol{x})=[\boldsymbol{g}_{c_{1}}(\boldsymbol{x}),\boldsymbol{g}_{c_{2}}(\boldsymbol{x})]$ with $\boldsymbol{g}_{c_{i}}\in L^{2}(\unicode[STIX]{x1D6E4}_{r_{i}})^{2}$ the spatial distribution function that describes how $\unicode[STIX]{x1D702}_{c_{i}}(t)$ is distributed on the rigid boundary. A localised sinusoidal spatial distribution function is considered

Such a set-up is frequently considered in a fully distributed setting to control single wavenumber pairs, see e.g. Bewley & Liu (Reference Bewley and Liu1998), Aamo & Krstic (Reference Aamo and Krstic2002), Jones et al. (Reference Jones, Heins, Kerrigan, Morrison and Sharma2015). Here a localised distribution is considered with a spatial length of $L_{g}=3\approx 0.95\unicode[STIX]{x03C0}$ and origin at $x_{g}=9\approx 2.86\unicode[STIX]{x03C0}$ . The length $L_{g}$ is less than half the wavelength of the dominant spatial perturbation mode which is $2\unicode[STIX]{x03C0}$ (see next section).

Information about the perturbation field is given by two wall-normal shear stress point measurements

where $\boldsymbol{n}$ is the inward unit normal on $\unicode[STIX]{x1D6E4}_{r}$ and $\boldsymbol{t}$ the corresponding unit tangential vector and the Dirac function $\unicode[STIX]{x1D6FF}$ indicates a point measurement. The term wall shear stress is used loosely here as the shear stress at the wall $\unicode[STIX]{x1D70F}_{xy}|_{\unicode[STIX]{x1D6E4}_{r}}=(1/Re)\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y|_{\unicode[STIX]{x1D6E4}_{r}}$ also depends on the Reynolds number. It is assumed that the Reynolds number is known, so that $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y|_{\unicode[STIX]{x1D6E4}_{r}}$ may easily be determined from measurements of $\unicode[STIX]{x1D70F}_{xy}|_{\unicode[STIX]{x1D6E4}_{r}}$ . The measurement noise $\boldsymbol{w}_{n}(t)$ is assumed to be a Gaussian stochastic process with zero means and covariances

with $\unicode[STIX]{x1D70E}_{n}^{2}$ the variance of both sensors. A feedforward configuration is chosen where the sensor is placed upstream of the control actuators at $x_{m}=7.5\approx 2.39\unicode[STIX]{x03C0}$ . In addition to the measured output, also two controlled outputs are defined

where $h(\boldsymbol{x})$ is determined by the desired performance specifications in the domain. In this study we wish to stabilise the perturbations by minimising the wall shear stress downstream of the control actuators integrated over a localised region over the boundary. To this end $h(x)$ is chosen as a Gaussian distribution function

with $x_{q}=17.5\approx 5.57\unicode[STIX]{x03C0}$ the centre of the distribution and $\unicode[STIX]{x1D70E}_{x}=1$ the radius. The controlled output is used to define the control objective in the ${\mathcal{H}}_{2}$ control framework later in this section.

2.3 Inflow disturbance model

The 2-D flow perturbations are characterised by unsteady fluctuations over a broad range of length scales and time scales. This makes the problem of estimating and controlling these perturbations inherently difficult. In particular the performance of the state estimation relies on the construction of a proper model for the external flow disturbances (Hœpffner et al. Reference Hœpffner, Chevalier, Bewley and Henningson2005). In this section a new inflow disturbance model is introduced which allows for an efficient estimation of the flow perturbations within the localised control domain. To generate the external disturbances a superposition of eigenmodes from the spectrum of the Orr–Sommerfeld (OS) operator is used. These modes are calculated from the OS equation at the desired temporal frequencies. With this approach, specific modes of the flow perturbations can thus be selected and are included in the control design. In this way the most dominant modes that contribute to transition can be precisely targeted by the controller. These modes are included in the state-space model by imposing them at the inflow boundary of the control domain. Such a boundary condition has been used to introduce disturbances in direct numerical simulations, e.g. for evaluating controllers (Baramov et al. Reference Baramov, Tutty and Rogers2004; Kotsonis et al. Reference Kotsonis, Giepman, Hulshoff and Veldhuis2013). However, the use of such boundary conditions as a disturbance model that is included in the design of the controller has so far not been reported. We consider ${\mathcal{H}}_{2}$ optimal control which is a design methodology in which the external sources of excitation are stochastic. First the disturbance model is presented for the case of stochastic excitation of the modes in § 2.3.1 and in § 2.3.2 the specific modes are selected that are included in the control design.

2.4 State-space formulation

In this section the linearised Navier–Stokes equations (LNSE) including the inputs, outputs and the inflow disturbance model are written as a boundary control system in the standard state-space format ( $\dot{\boldsymbol{u}}={\mathcal{A}}\boldsymbol{u}+{\mathcal{B}}\unicode[STIX]{x1D753}$ , $\unicode[STIX]{x1D742}={\mathcal{C}}\boldsymbol{u}$ ). This is required for defining the control objective and applying control theoretic tools. Boundary control systems do not fit directly into the standard form. However, we can extract the boundary controlled part of the dynamical model and rewrite the system on an extended state space in standard form. This method originates from Fattorini (Reference Fattorini1968) and has been applied for boundary control of wall-bounded shear flows (Högberg & Henningson Reference Högberg and Henningson2002; Högberg et al. Reference Högberg, Bewley and Henningson2003a ; Chevalier et al. Reference Chevalier, Hœpffner, Åkervik and Henningson2007). We also refer to Curtain & Zwart (Reference Curtain and Zwart1995, § 3.3) for more information on this formulation. Let ${\mathcal{X}}(\unicode[STIX]{x1D6FA})$ be the space of $n$ -dimensional divergence free functions defined on $\unicode[STIX]{x1D6FA}$ with inner product $(\boldsymbol{u}_{1},\boldsymbol{u}_{2})=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}_{1}\boldsymbol{\cdot }\boldsymbol{u}_{2}\,\text{d}\boldsymbol{x}$ and norm $\Vert \boldsymbol{u}_{1}\Vert _{2}=(\boldsymbol{u}_{1},\boldsymbol{u}_{1})^{1/2}$ where $\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in {\mathcal{X}}$ . Furthermore, let the trajectory segment $\boldsymbol{u}(\cdot ,t)=\{\boldsymbol{u}(\boldsymbol{x},t),\boldsymbol{x}\in \unicode[STIX]{x1D6FA}\}$ be the state and $\boldsymbol{u}(t)|_{\unicode[STIX]{x1D6E4}}\in {\mathcal{U}}$ the value of $\boldsymbol{u}(t)$ on the boundary defined in a separable Hilbert space ${\mathcal{U}}$ . The LNSE (2.1) in ${\mathcal{X}}(\unicode[STIX]{x1D6FA})$ , including the boundary inputs (2.2), the measurements (2.5) and the controlled output (2.7), can be written as

The operator $\mathscr{A}:D(\mathscr{A})\subset {\mathcal{X}}\mapsto {\mathcal{X}}$ corresponds to evaluating the linear differential operator of the LNSE. The pressure can be eliminated from the equations by using a space of velocity fields which are divergence free (Bewley et al. Reference Bewley, Temam and Ziane2000), which is also done here (see also appendix A for the variational formulation). $\mathscr{B}:{\mathcal{X}}\mapsto {\mathcal{U}}$ is a boundary operator which maps the flow field to its values on the boundary and ${\mathcal{C}}$ , ${\mathcal{Q}}$ are output operators, respectively defined as

To extract the boundary controlled part the first step is to construct two operators ${\mathcal{Z}}_{c}$ , ${\mathcal{Z}}_{d}$ such that

The boundary condition can then be removed by decomposing the state into

The dynamics of the new state $\boldsymbol{u}_{h}$ is governed by the following evolution equation with homogeneous boundary conditions (Curtain & Zwart Reference Curtain and Zwart1995)

where the operator ${\mathcal{A}}:D({\mathcal{A}})\mapsto {\mathcal{X}}$ is defined as

If $\boldsymbol{u}_{h}$ is a solution of the homogeneous system (2.20), then $\boldsymbol{u}$ defined by (2.19) is a solution of the original system (2.16) (Fattorini Reference Fattorini1968; Curtain & Zwart Reference Curtain and Zwart1995). Equation (2.20) contains both the temporal inputs and their time derivatives which is undesirable since they are not independent inputs. This can be eliminated by reformulating (2.20) on the extended state space ${\mathcal{X}}^{e}={\mathcal{X}}\oplus {\mathcal{U}}$

The inflow perturbation velocity and the wall actuation velocity have become a state of the system in this formulation. The external input is actually the time derivative of the boundary velocity. From the actuator model (2.3) it follows that $\boldsymbol{u}_{c}={\mathcal{C}}_{c}\boldsymbol{\unicode[STIX]{x1D702}}_{c}$ , $\dot{\boldsymbol{u}}_{c}={\mathcal{C}}_{c}{\mathcal{A}}_{c}\boldsymbol{\unicode[STIX]{x1D702}}_{c}+{\mathcal{C}}_{c}{\mathcal{B}}_{c}\unicode[STIX]{x1D753}$ and from the disturbance model (2.14) it follows that $\boldsymbol{u}_{d}={\mathcal{C}}_{d}\boldsymbol{\unicode[STIX]{x1D702}}_{d}$ , $\dot{\boldsymbol{u}}_{d}={\mathcal{C}}_{d}{\mathcal{A}}_{d}\bar{\boldsymbol{\unicode[STIX]{x1D702}}}_{d}+{\mathcal{C}}_{d}{\mathcal{B}}_{d}w_{d}$ . Substituting these expressions in (2.22), and combining this system with the actuator dynamics (2.3) and the disturbance dynamics (2.14), gives the following augmented system

where also included are the resulting output equations from the state transformation (2.19). Equation (2.23) can be compactly written as

with $\boldsymbol{u}^{e}$ the extended state. A final remark is given about the controllability of the system. By formulating the system on the extended state space (2.20) pure integrators have been added at the system external inputs. This results in additional system poles at the origin. As a result, the system in the form (2.20) is not stabilisable, which means that not all uncontrollable modes are asymptotically stable. This is a direct result of the fact that both the control and disturbance are defined at the boundary and both appear as a state in the system. It is not possible to influence the additional poles of the disturbance dynamics by means of control and vice versa (assumption (i) is violated, and assumptions (iii) and (iv) are violated for $\unicode[STIX]{x1D714}=0$ , see appendix B). By including the actuator dynamics and disturbance dynamics, the uncontrollable poles at the origin are moved to the stable left half-plane to the location of the eigenvalues of ${\mathcal{A}}_{c}$ and ${\mathcal{A}}_{d}$ . The state-space formulation (2.23) is thus stabilisable which allows the synthesis of ${\mathcal{H}}_{2}$ optimal controllers.

2.5 Finite-dimensional system

Equation (2.24) represents the continuous formulation of the flow control problem. For simulation and control design a finite-dimensional representation of (2.24) is required. In Tol et al. (Reference Tol, de Visser and Kotsonis2016) a framework is presented for deriving state-space descriptions for a general class of linear parabolic PDEs to which standard control theoretic tools can be applied. This method is also used in this work and uses multivariate B-splines of arbitrary degree and smoothness defined on triangulations (Farin Reference Farin1986; de Boor Reference de Boor and Farin1987; Lai & Schumaker Reference Lai and Schumaker2007) to find matrix representations of all operators in (2.23). This method has the flexibility of the finite element method to use local refinements and to cope with irregular domains and the high approximation power of spectral methods. The triangulations used to construct the simulation model, and the model that is used as a starting point for model reduction and control design are shown in figure 6. The use of spline spaces provides a convenient way for stating the degree and smoothness of the spline model. In addition, the approximation properties of such spline spaces have been extensively studied in the literature (Lai & Schumaker Reference Lai and Schumaker2007). Let ${\mathcal{T}}$ be the triangulation of $\unicode[STIX]{x1D6FA}$ . The spline space is the space of all smooth piecewise polynomial functions of arbitrary degree $d$ and arbitrary smoothness $r$ over ${\mathcal{T}}$ with $0\leqslant r<d$

With ${\mathcal{P}}_{d}$ the space of all polynomials of total degree $d$ and $t$ denotes a triangle. We construct a basis for the smooth divergence free spline subspace ${\mathcal{S}}$ such that ${\mathcal{S}}\subset {\mathcal{X}}$ in conjunction with a Galerkin scheme to obtain a finite-dimensional representation of the governing equations. The pressure is eliminated from the equations by using a space of velocity fields which are divergence free and a suitable choice of the variational formulation. This will also avoid singularities in the numerical method. The Galerkin-type variational formulation through which the spline approximation is determined and the corresponding numerical method are described in detail in appendix A.

Figure 6. Triangulations used for the simulation model and the model that is used for model reduction and control design. (a) Triangulation with 960 triangles used for the control model. (b) Triangulation with 1920 triangles used for the simulation model.

To derive the full-order control model a structured triangulation is used, refined near the walls to properly resolve the shear features of the flow consisting of $n_{t}=960$ triangles, and the ${\mathcal{S}}_{4}^{0}({\mathcal{T}}_{960})^{2}$ spline space is chosen as approximating space for the velocity field. $C^{0}$ continuous spline elements are chosen which allows an accurate interpolation of the actuator distribution function at the boundary. Degree $d=4$ elements are chosen which allows the construction of an exactly divergence free basis and to obtain better approximation properties (Awanou, Lai & Wenston Reference Awanou, Lai, Wenston, Chen and Lai2005; Lai & Schumaker Reference Lai and Schumaker2007). With this degree each element $t$ has a total of $N_{t}=15$ degrees of freedom. The complete basis for $L^{2}(\unicode[STIX]{x1D6FA})^{2}$ has a total of $N=n_{t}\times N_{t}\times 2=28\,800$ degrees of freedom. This basis is used to spatially discretise the system. The resulting discrete system is transformed to state-space format using a null-space projection method (Tol et al. Reference Tol, de Visser and Kotsonis2016). This projection employs a similar state transformation as in (2.19), but in a discrete setting, and results in a reduced number of states that have a minimal non-zero support for the smooth divergence free spline space ${\mathcal{S}}\subset {\mathcal{X}}$ . The reduction is equal to the total rank $R^{\ast }$ of the discrete divergence, boundary and smoothness operators. The order of the state-space model resulting from the null-space projection is $N-R^{\ast }=5569$ . The large reduction can be contributed to the fact that the constrained smooth divergence free subspace is much smaller than the unconstrained space. The order of the model is sufficiently small to allow a direct application of balanced truncation for model reduction. For the case of spatially periodic boundary conditions the accuracy of the model can be assessed via comparison of the model spectra with the temporal spectra of the Orr–Sommerfeld equation (2.15). This comparison is demonstrated in § A.2. The numerical accuracy of the first 22 dominant eigenvalues varies between $2\times 10^{-8}\leqslant |\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{k}^{OS}|\leqslant 2\times 10^{-3}$ . This is considered accurate for the purpose of control design and demonstration. A more physical validation of the model for the non-periodic case considered in this study is conducted in § 3. A different model is used for simulating the response of system. The simulation model is defined on a longer domain with a total length of $L_{sim}=16\unicode[STIX]{x03C0}$ . A similar triangulation consisting of 1920 triangles is used and the simulation model has approximately the same accuracy as the control model. In the next sections we focus on the control model and use the notation ( ${\mathcal{A}}$ , ${\mathcal{B}}$ , ${\mathcal{C}}$ , ${\mathcal{D}}$ ) to represent the full-order finite-dimensional system and use the notation ( $\unicode[STIX]{x1D63C}$ , $\unicode[STIX]{x1D63D}$ , $\unicode[STIX]{x1D63E}$ , $\unicode[STIX]{x1D63F}$ ) to represent a reduced-order system resulting from balanced truncation.

2.6 Formulation of the ${\mathcal{H}}_{2}$ control problem

Figure 7. The general control configuration. System $\unicode[STIX]{x1D642}$ , controller $\unicode[STIX]{x1D646}$ , output measurement $\unicode[STIX]{x1D742}$ , control input $\unicode[STIX]{x1D753}$ , performance objective $\boldsymbol{z}$ and exogenous disturbances  $\boldsymbol{w}$ .

In this section the feedback design problem for the state-space representation of the flow (2.24) is cast as an ${\mathcal{H}}_{2}$ optimisation problem. The state-space formulas for the optimal solution are given in appendix B. We refer to Doyle et al. (Reference Doyle, Glover, Khargonekar and Francis1989), Zhou et al. (Reference Zhou, Doyle and Glover1996), Skogestad & Postlethwaite (Reference Skogestad and Postlethwaite2005) for more detail on this control theory. The main objective of the feedback control design is to find a control input $\unicode[STIX]{x1D753}$ based on the output measurement $\unicode[STIX]{x1D742}_{m}$ that minimises the wall shear stress defined by $\boldsymbol{q}$ in the presence of the disturbances $w_{d}$ and $\boldsymbol{w}_{n}$ . First the standard control formulation that is considered by ${\mathcal{H}}_{2}$ control is presented. The application of ${\mathcal{H}}_{2}$ control to the state-space representation of the flow (2.24) will follow thereafter. Let $\boldsymbol{w}$ be the vector of exogenous disturbances and $\boldsymbol{z}$ the vector of performance measures to be minimised. The ${\mathcal{H}}_{2}$ control problem is a disturbance rejection problem and considers the standard control configuration shown in figure 7 which is described by

with $\unicode[STIX]{x1D646}(s)$ the controller to be synthesised and $\unicode[STIX]{x1D642}(s)$ the open-loop transfer function matrix of the generalised plant defined by

with the state-space realisation

To account for the state disturbances $w_{d}$ and the measurement noise $\boldsymbol{w}_{n}$ in a ${\mathcal{H}}_{2}$ control framework the state-space system (2.24) is formulated as a generalised plant (2.28) and scaled in terms of two parameters which may be individually adjusted to achieve the desired closed-loop performance. A similar scaling was also presented in Bewley & Liu (Reference Bewley and Liu1998). The control objective is to counteract the influence of the state disturbance $w_{d}$ on the controlled output defined by $\boldsymbol{q}=\bar{{\mathcal{Q}}}\boldsymbol{u}^{e}$ . Therefore the controlled output is used to define the performance measure  $\boldsymbol{z}$

which also includes a penalty on the control defined by the parameter $l$ . The parameter $l$ determines the trade-off between a low control effort ( $\unicode[STIX]{x1D753}^{\text{T}}\unicode[STIX]{x1D753}$ ) and a low controlled output energy ( $\boldsymbol{q}^{\text{T}}\boldsymbol{q}$ ). For the design of the controller, decisions must be made about the expected state disturbances and measurement noise. The temporal magnitude of these disturbances in the state-space system is defined by the expected covariances of the temporal state disturbance (2.13) and the measurement noise (2.6). In this study it is assumed that nothing is known a priori about the expected covariances. To make a parametric study for the controller design tractable, a relative magnitude of the measurement noise is defined

which is the ratio between the root mean square of the expected variance of respectively the sensor noise and the state disturbance. The state disturbance and measurement noise are respectively modelled as $w_{d}=\unicode[STIX]{x1D70E}_{d}w_{1}$ and $\boldsymbol{w}_{n}=\unicode[STIX]{x1D70E}_{n}\boldsymbol{w}_{2}$ with $w_{1}$ and $\boldsymbol{w}_{2}$ defined as white noise with unit intensity. The system is parameterised in terms of $\unicode[STIX]{x1D6FE}$ by defining a new scaled observation that is used for feedback

and the system is normalised such that $\unicode[STIX]{x1D70E}_{d}=1$ . Using this normalisation it follows from (2.30) that $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D70E}_{n}$ and the observation (2.31) is obtained by a simple change of variables. For the control design $\unicode[STIX]{x1D6FE}$ does not represent a physical root mean square value of the measurement noise, but a relative measure with respect to the state disturbance, used to tune the controller. Defining the vector of disturbances as $\boldsymbol{w}=[w_{1},~\boldsymbol{w}_{2}^{\text{T}}]^{\text{T}}$ and the following system matrices

the system (2.24) can be written as a generalised plant with the state-space formulation (2.28), that is

The ${\mathcal{H}}_{2}$ control design problem for this system is to find a controller $\unicode[STIX]{x1D646}(s)$ that, based on the measurement information $\unicode[STIX]{x1D742}$ , generates a control input $\unicode[STIX]{x1D753}$ which stabilises the system (2.33) internally and minimises

Equation (2.34) is referred to as the ${\mathcal{H}}_{2}$ -norm of the closed-loop transfer function matrix $\unicode[STIX]{x1D64F}_{zw}$ from the external disturbances $\boldsymbol{w}$ to the control objectives $\boldsymbol{z}$ and $|\unicode[STIX]{x1D61B}_{zw}^{i,j}|$ denotes the magnitude of the closed-loop transfer function from the $j$ th disturbance to the $i$ th objective. $\unicode[STIX]{x1D64F}_{zw}$ is given by

which follows from (2.26). Physically, the ${\mathcal{H}}_{2}$ norm in (2.34) can be interpreted as the amplification of the system from $\boldsymbol{w}$ to $\boldsymbol{z}$ integrated over all frequencies. In the time domain, this is equivalent to the variance amplification of stochastic disturbances (Jovanovic & Bamieh Reference Jovanovic and Bamieh2005). By minimising the ${\mathcal{H}}_{2}$ norm, the controlled output power $E[\boldsymbol{z}^{\text{T}}\boldsymbol{z}]$ of the system, due to unit white Gaussian disturbances $\boldsymbol{w}$ , is minimised. The state-space formulas for the optimal controller $\unicode[STIX]{x1D646}(s)$ that minimise (2.34) are given in appendix B. It combines a state estimator (Kalman filter) for the flow field and a state feedback, and has a state-space description of the form

with $\boldsymbol{u}_{K}^{e}$ the estimated state and ${\mathcal{A}}_{K}={\mathcal{A}}+{\mathcal{B}}_{2}{\mathcal{C}}_{K}-{\mathcal{B}}_{K}{\mathcal{C}}_{2}$ . The controller input matrix ${\mathcal{B}}_{K}$ represents the estimator gain and the output matrix ${\mathcal{C}}_{K}$ represents the state feedback gain. The controller (2.36) can be structured using the separation principle, which means that the estimator and state feedback can be tuned independently. Thus the control penalty $l$ and the estimation parameter $\unicode[STIX]{x1D6FE}$ may be individually adjusted to achieve the desired characteristics for the closed-loop system $\unicode[STIX]{x1D64F}_{zw}$ . A low value for the control penalty $l$ results in higher gain state feedback ${\mathcal{C}}_{K}$ . Similarly, when $\unicode[STIX]{x1D6FE}$ is small (high signal to noise ratio) the observation is fed back more aggressively (high observer gain ${\mathcal{B}}_{K}$ ) than when $\unicode[STIX]{x1D6FE}$ is high. The controller $\unicode[STIX]{x1D646}(s)$ in (2.36) represents the full-order controller. Such a high-order controller is usually not real time implementable for practical flow configurations. To synthesise a reduced-order controller $\unicode[STIX]{x1D646}_{r}(s)$ for the high-order plant the so-called reduce-then-design approach (Anderson & Liu Reference Anderson and Liu1989) is used, which is discussed in detail in the next section. This section also includes a parametric study for the parameters $\unicode[STIX]{x1D6FE}$ and  $l$ .

3.1 Analysis of the uncontrolled system

Figure 8. The magnitude of the spatio-temporal frequency response from the inflow disturbance $w_{1}$ to the shear along the lower wall $\unicode[STIX]{x1D708}_{1}(x,y=-1)$ . The 10 contour levels lie within $|G_{\unicode[STIX]{x1D708}_{1}w_{1}}|\in [6.9,~69.4]$ .

In this section the uncontrolled system from the disturbance input $\boldsymbol{w}$ to the shear output $\unicode[STIX]{x1D742}$ , that is $\unicode[STIX]{x1D642}_{\unicode[STIX]{x1D708}w}=[\unicode[STIX]{x1D642}_{\unicode[STIX]{x1D708}w_{1}},\unicode[STIX]{x1D642}_{\unicode[STIX]{x1D708}w_{2}}]$ in (2.26), is analysed in the frequency domain. In particular the effect of the inflow disturbance $w_{1}$ on $\unicode[STIX]{x1D742}$ is investigated from an input–output viewpoint. The disturbance input $w_{1}$ excites the Orr–Sommerfeld eigenfunction calculated for the most amplified frequency ( $\unicode[STIX]{x1D714}=0.253$ ) at the inflow, see also § 2.3 and figure 5. The perturbation shear stress created by the disturbance along the complete lower wall, $\unicode[STIX]{x1D708}_{1}(x)=(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)(x,-1)$ , is considered as output in the analysis. In this way the spatial transients created by the inflow disturbance can be evaluated and the perturbation modes that are excited can be identified. The same results hold for the upper wall due to the symmetry of the geometry. If a linear system is forced by a sinusoidal input at a particular frequency, once the initial temporal transients have died out asymptotically, the output will also be sinusoidal, at the same frequency, but with a change in amplitude and a phase shift. The magnitude amplification and phase shift of the output are equal to the magnitude and phase of the frequency response of the system. The frequency response is obtained by evaluating the transfer function on the imaginary axis, that is $s=\text{i}\unicode[STIX]{x1D714}$ . The asymptotic response for the shear output along the lower wall in the spatio-temporal frequency domain is given by

where $G_{\unicode[STIX]{x1D708}_{1}w_{1}}(\text{i}\unicode[STIX]{x1D714},x)$ is obtained from the $(1,1)$ element of

$G_{\unicode[STIX]{x1D708}_{1}w_{1}}(\text{i}\unicode[STIX]{x1D714},x)$ is the spatio-temporal frequency response function (Jovanovic & Bamieh Reference Jovanovic and Bamieh2005) from the inflow disturbance $w_{1}$ to the shear stress along the lower wall. It is a function of temporal frequency and streamwise direction. $G_{\unicode[STIX]{x1D708}_{1}w_{1}}(\text{i}\unicode[STIX]{x1D714},x)$ is visualised using the magnitude bode plot $|G_{\unicode[STIX]{x1D708}_{1}w_{1}}(\text{i}\unicode[STIX]{x1D714},x)|$ which is shown in figure 8. To support the interpretation of the magnitude the fully developed open-loop response for $\unicode[STIX]{x1D714}=0.25$ and $\unicode[STIX]{x1D714}=0.35$ is shown in figure 9. The effect of the low-pass filter (2.12) on the magnitude at the inflow and the amplification at the design frequency $\unicode[STIX]{x1D714}=0.253$ can clearly be observed. After initial spatial transients near the inflow boundary, the modal perturbations are revealed and the magnitude linearly increases or decreases depending on the frequency of $w_{1}$ . At the design frequency an insignificant transient is involved for the mode to develop in the domain. Larger transients can be observed near the inflow at other frequencies than the design point. These non-modal transients do not cause a problem for control design as they have died out in the control region ( $x>2\unicode[STIX]{x03C0}$ ). The outflow boundary condition (2.1d ) gives rise to an artificial gain near the outflow $x>6\unicode[STIX]{x03C0}$ . This does not result in reflections (wiggles) in the control domain. No special attention needs to be taken for the non-physical region as long as no measurement sensors are placed in this region. For validation purposes the exponential growth for the perturbation shear output is compared with predictions from linear stability theory. The exponential growth can be calculated using

The location of the shear sensor $x_{0}=x_{m}=2.39\unicode[STIX]{x03C0}$ and the location of the controlled shear output $x_{1}=x_{q}=5.57\unicode[STIX]{x03C0}$ are chosen to compute the growth rate. Within this region the magnitude varies linearly over a wide range of frequencies. Figure 10 shows the magnitude of $G_{\unicode[STIX]{x1D708}_{1}w_{1}}(\text{i}\unicode[STIX]{x1D714},x)$ at the two spatial locations and the exponential growth rate of the magnitude compared with the growth rates from LST. Good agreement with LST predictions can be observed. Both the model and the OS equation predict instability within the range $0.216\leqslant \unicode[STIX]{x1D714}\leqslant 0.286$ . The real part of the wavenumber $\unicode[STIX]{x1D6FC}_{r}$ and the corresponding wavelength $\unicode[STIX]{x1D706}_{x}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}_{r}$ of the perturbation can be evaluated using the phase response of the system. Let $\angle G_{\unicode[STIX]{x1D708}_{1}w_{1}}(\text{i}\unicode[STIX]{x1D714},x)$ be the phase in degrees for the shear output along the lower wall. The real part of the wavenumber can be calculated using

Figure 11 shows the phase at the two spatial locations and the resulting wavenumbers compared with the predictions from LST. It can be observed that also the wavelengths are in good agreement with LST. At $\unicode[STIX]{x1D714}=0.25$ LST predicts a wavelength of $\unicode[STIX]{x1D706}_{x}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}_{r}\approx 2\unicode[STIX]{x03C0}$ and at $\unicode[STIX]{x1D714}=0.35$ a wavelength of $\unicode[STIX]{x1D706}_{x}\approx 1.6\unicode[STIX]{x03C0}$ . These wavelengths can also be observed in figure 9.

Figure 9. Fully developed open-loop response of the streamwise perturbation velocity for two inflow disturbance frequencies. (a) Shows 10 levels in the range $u\in [-1.61,1.61]$ . (b) Shows 10 levels in the range $u\in [-1.00,1.00]$ .

These results verify that the single mode inflow disturbance model accurately captures the wavelengths and growth rates in a wider frequency band in the actuator/sensor region. Also at other frequencies than the design frequency, the disturbance will quickly develop in-domain to a travelling wave with a spatial wavelength and growth rate as predicted by the Orr–Sommerfeld equation. It provides confidence that the followed modelling procedure allows for an efficient estimation of the dominant flow perturbations in the localised control domain using wall shear sensors. In the next section the controller is designed to reduce the magnitude of the shear downstream of the control actuators.

Figure 10. (a) Magnitude of the shear output at two $x$ -locations and (b) resulting exponential growth (3.3) compared with the solutions of the Orr–Sommerfeld equation.

Figure 11. (a) Phase of the shear output at two $x$ -locations and (b) resulting wavelength (3.4) compared with the solutions of the Orr–Sommerfeld equation.

3.2 Reduced-order controller

The reduce-then-design approach (Anderson & Liu Reference Anderson and Liu1989) is used to construct a reduced-order controller for the high-order plant. First, exact balanced truncation (Moore Reference Moore1981) is applied to construct a reduced-order model of the full-order system after which the ROM is used to synthesise the optimal controller. Exact balanced truncation requires dense matrix factorisations and generally results in a computational complexity of $O(N^{3})$ and a storage requirement of $O(N^{2})$ . Exact balanced truncation is not computationally tractable for very large systems and approximate methods, such as proposed by Rowley (Reference Rowley2005), could be used in this case. However, the modelling approach in this paper avoids very large systems through localised computations allowing to apply exact balanced truncation ( $N=5569$ for the control model). Since the current flow configuration is globally stable, balanced truncation can directly be applied without the need of separating the stable and unstable subspaces. Only the application of balanced truncation for model reduction and control design is discussed in this section. We refer to Moore (Reference Moore1981) for more detail and to Rowley (Reference Rowley2005), Kim & Bewley (Reference Kim and Bewley2007) for more background in the context of flow control.

Balanced truncation extracts the most controllable and observable modes of the system and first involves creating a balanced realisation of the system such that each state has an equal measure for both controllability and observability. Let $\unicode[STIX]{x1D642}_{b}(s)=({\mathcal{A}}_{b},{\mathcal{B}}_{b},{\mathcal{C}}_{b},{\mathcal{D}}_{b})$ be a balanced realisation of the generalised plant $\unicode[STIX]{x1D642}(s)=({\mathcal{A}}_{p},{\mathcal{B}}_{p},{\mathcal{C}}_{p},{\mathcal{D}}_{p})$ given by (2.28) such that the controllability Gramian and observability Gramian respectively defined as

are given by $\unicode[STIX]{x1D64B}=\unicode[STIX]{x1D64C}=\text{diag}(\unicode[STIX]{x1D70E}_{1}^{H},\unicode[STIX]{x1D70E}_{2}^{H},\ldots ,\unicode[STIX]{x1D70E}_{N}^{H})=:\unicode[STIX]{x1D72E}$ where $\unicode[STIX]{x1D70E}_{1}^{H}\geqslant \unicode[STIX]{x1D70E}_{2}^{H}\geqslant \cdots \geqslant \unicode[STIX]{x1D70E}_{N}^{H}\geqslant 0$ are the Hankel singular values of the system. An efficient algorithm for creating balanced realisations is available in Matlab (balreal). This algorithm computes the similarity transformation $\boldsymbol{u}_{b}^{e}\mapsto \unicode[STIX]{x1D64E}\boldsymbol{u}^{e}$ , which balances the plant matrices through ${\mathcal{A}}_{b}=\unicode[STIX]{x1D64E}^{-1}{\mathcal{A}}_{p}\unicode[STIX]{x1D64E}$ , ${\mathcal{B}}_{b}=\unicode[STIX]{x1D64E}^{-1}{\mathcal{B}}_{p}$ , ${\mathcal{C}}_{b}={\mathcal{C}}_{p}\unicode[STIX]{x1D64E}$ and ${\mathcal{D}}_{b}={\mathcal{D}}_{p}$ . The similarity transformation $\unicode[STIX]{x1D64E}$ is obtained from the Cholesky factorisation of the Gramians (Laub et al. Reference Laub, Heath, Paige and Ward1987). The Gramians are computed by solving a set of Lyapunov equations (Moore Reference Moore1981). This method is also stable if the system contains nearly uncontrollable/unobservable modes which are present in the linearised Navier–Stokes operator (Bewley & Liu Reference Bewley and Liu1998; Kim & Bewley Reference Kim and Bewley2007).

The balanced realisation and corresponding singular values can be partitioned as

where $\unicode[STIX]{x1D72E}_{1}=\text{diag}(\unicode[STIX]{x1D70E}_{1}^{H},\unicode[STIX]{x1D70E}_{2}^{H},\ldots ,\unicode[STIX]{x1D70E}_{r}^{H})$ and $\unicode[STIX]{x1D72E}_{2}=\text{diag}(\unicode[STIX]{x1D70E}_{r+1}^{H},\unicode[STIX]{x1D70E}_{r+2}^{H},\ldots ,\unicode[STIX]{x1D70E}_{N}^{H})$ . The reduced-order model of order $r$ is obtained by truncating the least observable/controllable modes, that is truncating the $r+k,k=1,\ldots ,N-r$ modes: $G_{r}(s)=({\mathcal{A}}_{11},{\mathcal{B}}_{1},{\mathcal{C}}_{1},{\mathcal{D}}_{b}):=(\unicode[STIX]{x1D63C},\unicode[STIX]{x1D63D},\unicode[STIX]{x1D63E},\unicode[STIX]{x1D63F})$ . Note that balanced truncation does not depend on ${\mathcal{D}}_{b}$ and it follows that ${\mathcal{D}}_{b}={\mathcal{D}}_{p}=\unicode[STIX]{x1D63F}$ . A feature of balanced truncation is the existence of upper and lower bounds for the maximum error of the reduced-order model

with $\unicode[STIX]{x1D70E}_{r+1}^{H}$ the first neglected Hankel singular value. Figure 12 shows the first 150 Hankel singular values of the system and the upper and lower bounds for the maximum error. The steep initial drop indicates that the input–output behaviour can be captured using low-order models. However, no guarantees are available about the stability and performance of a controller designed for $\unicode[STIX]{x1D642}_{r}$ on the original system $\unicode[STIX]{x1D642}$ and the truncated dynamics should be taken into account in the performance analysis. Therefore, instead of evaluating the performance of the ROM, the performance of the reduced-order controller in combination with the original system is evaluated for increasing order  $r$ .

Figure 12. The first 150 Hankel singular values (‘●’ markers), the theoretical upper bound (dashed line) and theoretical lower bound (solid line) for the maximum error of the reduced-order model.

The reduced-order model $\unicode[STIX]{x1D642}_{r}$ is used to synthesise the ${\mathcal{H}}_{2}$ optimal reduced-order controller $\unicode[STIX]{x1D646}_{r}(s)$ that minimises (2.34) (see appendix B), and takes the form

with $\boldsymbol{u}_{K}^{e}\in \mathbb{R}^{r}$ the controller state. The resulting closed-loop system from the disturbance $\boldsymbol{w}$ to the control objective $\boldsymbol{z}$ is obtained by combining the controller (3.9) with the original system (2.33) and is given by

For the design of the controller, the performance of the closed-loop system (3.10) is characterised for different combinations of control penalties $l$ and estimation penalties $\unicode[STIX]{x1D6FE}$ . As in Bewley & Liu (Reference Bewley and Liu1998) a parametric study is conducted for the ${\mathcal{H}}_{2}$ norms of the following two closed-loop transfer functions

which are the closed-loop transfer function matrices from the disturbance to respectively the controlled output $\boldsymbol{q}$ and the control input $\unicode[STIX]{x1D753}$ . The definitions of the closed-loop system matrices $({\mathcal{A}}_{cl},{\mathcal{B}}_{cl},{\mathcal{C}}_{cl})$ follow from (3.10). The ${\mathcal{H}}_{2}$ norms of these transfer functions are related by

with $\unicode[STIX]{x1D64F}_{zw}={\mathcal{C}}_{cl}(sI-{\mathcal{A}}_{cl})^{-1}{\mathcal{B}}_{cl}$ the transfer function from the disturbance to the combined performance objective  $\boldsymbol{z}$ . A low value for $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}$ indicates a good controller performance while a low value for $\Vert \unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D719}w}\Vert _{2}$ indicates a low control effort. A finite value for these norms means an exponentially stable closed-loop system. Figure 13 shows the norms as function of the order $r$ of the controller for the combination $\unicode[STIX]{x1D6FE}=1$ , $l=1$ . The norm of the full-order controller ( $r=N$ ) is indicated by the asymptotes. It can be observed that the performance of the closed-loop system converges quickly to the case of a full-order controller. Similar results were obtained for other combinations. We select $r=50$ to design and implement the controller. With this order the performance has converged and there is no loss in performance due to the truncated dynamics. The input–output behaviour of the ROM with $r=50$ is compared to full system in figure 14. Shown is the magnitude frequency response of the transfer function $G_{\unicode[STIX]{x1D708}_{1}w_{1}}$ from the inflow disturbance $w_{1}$ to the measured output $\unicode[STIX]{x1D708}_{1}$  (a) and the transfer function $G_{q_{1}\unicode[STIX]{x1D719}_{1}}$ from the control input $\unicode[STIX]{x1D719}_{1}$ to the controlled output $q_{1}$ at the lower wall (b). There is a good agreement and the ROM accurately captures the input–output (disturbance and control) behaviour.

Figure 13. Convergence of the closed-loop system norms $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}$ (a) and $\Vert \unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D719}w}\Vert _{2}$ (b) versus the order $r$ of the reduced-order controller with $l=1$ , $\unicode[STIX]{x1D6FE}=1$ . The two dashed lines indicate the norms of the full-order ( $r=N=5569$ ) controller.

Figure 14. Performance of the ROM ( $r=50$ ). Magnitude frequency response from the state disturbance $w_{1}$ to the measured output $\unicode[STIX]{x1D708}_{1}$ (a) and from the control input $\unicode[STIX]{x1D719}_{1}$ to the controlled shear output $q_{1}$ (b) at the lower wall.

Figure 15. Contours of the closed-loop system norms $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}$  (a),  $\Vert \unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D719}w}\Vert _{2}$  (b) and the relative energy norm $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}^{2}/\Vert \unicode[STIX]{x1D642}_{qw}\Vert _{2}^{2}$  (c) with a $r=50$ reduced-order controller for different combinations of control parameter $l$ and estimation parameter $\unicode[STIX]{x1D6FE}$ ( $\Vert \unicode[STIX]{x1D642}_{qw}\Vert _{2}=16.90$ ). Controllers (I) $l=10$ , $\unicode[STIX]{x1D6FE}=1.5$ , (II) $l=20$ , $\unicode[STIX]{x1D6FE}=5$ and (III) $l=40$ , $\unicode[STIX]{x1D6FE}=15$ are considered for evaluating the closed-loop response of the system.

Figure 15 shows the contours of the ${\mathcal{H}}_{2}$ norms and the relative energy norm $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}^{2}/\Vert \unicode[STIX]{x1D642}_{qw}\Vert _{2}^{2}$ for the order $r=50$ controller. It can be observed that an energy reduction between 90 %–99 % can easily be achieved by a proper choice of the design parameters. The performance for the case $l\rightarrow \infty ,\unicode[STIX]{x1D6FE}\rightarrow \infty$ converges to the uncontrolled case. The control penalty $l$ can be used to tune the feedback gain $\unicode[STIX]{x1D63E}_{K}$ in (3.9) and determines the trade-off between control effort and magnitude of the shear perturbation $\boldsymbol{q}$ . Lower values lead to an increased controller performance (low $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}$ ) at the cost of a higher control effort. It is found that choosing $l<10$ does lead to a significantly increase in performance. The parameter $\unicode[STIX]{x1D6FE}$ can be used to tune the estimator, that is the output injection gain $\unicode[STIX]{x1D63D}_{K}$ in (3.9). Low values for $\unicode[STIX]{x1D6FE}$ (high to noise ratio) lead to a higher magnitude of estimator feedback and an increased performance. However, choosing a lower value for $\unicode[STIX]{x1D6FE}$ leads to a reduced robustness. The role of $\unicode[STIX]{x1D6FE}$ is to account for uncertainties in the estimated output which also arise in the case of unmodelled dynamics and unmodelled disturbances. High estimator gain feedback can in this case result in larger overshoots which should be avoided since they can aggravate the initial stage to transition. From the contour of $\Vert \unicode[STIX]{x1D64F}_{qw}\Vert _{2}$ it can be observed that for a given control penalty $l$ , the estimation penalty $\unicode[STIX]{x1D6FE}$ , and thus the robustness, can be increased up to the curvature of the contour level without significant loss of performance. Thus choices for $l$ and $\unicode[STIX]{x1D6FE}$ on the curvature of a desired performance level can be considered as an optimal trade-off between robustness and the desired performance. In this study robustness is valued more than control effort in determining the trade-off. Three controllers will be investigated in the next sections for evaluating the performance in the frequency domain and through numerical simulation. The design parameters for the controllers are marked in figure 15. The first (I) is a high gain controller with $l=10$ , $\unicode[STIX]{x1D6FE}=1.5$ corresponding to approximately a 99.9 % energy reduction. The second (II) is an intermediate controller with $l=20$ , $\unicode[STIX]{x1D6FE}=5$ corresponding to a 99 % energy reduction and the third (III) is a lower gain controller with $l=40$ , $\unicode[STIX]{x1D6FE}=15$ corresponding to a 90 % energy reduction.

3.3 Closed-loop frequency response

In this section the three selected controllers are evaluated in the frequency domain. The magnitude frequency response from $w_{1}$ to the controlled output $q_{1}$ (2.7) is shown in figure 16. The magnitude of the closed-loop system $T_{q_{1}w_{1}}$ is compared with the magnitude of the open-loop system $G_{q_{1}w_{1}}$ . The frequency domain performance for the three controllers is in accordance with the results in figure 15. Controller (III) limits the control effort and takes higher levels of sensor inaccuracies into account. It is more conservative also with respect to higher frequencies. The three controllers significantly suppress the most amplified frequencies close to the design frequency $\unicode[STIX]{x1D714}=0.253$ as well as the off-design frequencies. The peak magnitude is equal to the ${\mathcal{H}}_{\infty }$ norm of $T_{q_{1}w_{1}}$ which is reduced between approximately 80 %–99 % for the three controllers.

The perturbation shear reduction along the complete walls, as well as spatial transients introduced by the control can be evaluated using the spatio-temporal frequency response. Figure 17 shows the magnitude for the shear along the lower wall for the open-loop system (a) and closed-loop (b) system with controller (II). Compared to the open-loop magnitude it can be observed that the controller significantly reduces the shear in the entire downstream region of the control actuators. The magnitude at the most dominant frequencies $0.1\leqslant \unicode[STIX]{x1D714}\leqslant 0.4$ is significantly suppressed and only small amplifications are present in the region of the control actuator.

Figure 16. Closed-loop frequency response from the inflow disturbance $w_{1}$ to the controlled output $q_{1}$ along the lower wall.

Figure 17. Open-loop (a) and closed-loop (b) magnitude frequency response from the inflow disturbance $w_{1}$ to the shear stress $\unicode[STIX]{x1D708}_{1}$ along the lower wall. The 10 contour levels lie within $|G_{\unicode[STIX]{x1D708}_{1}w_{1}}|,|T_{\unicode[STIX]{x1D708}_{1}w_{1}}|\in [6.9,~69.4]$ ( $l=20$ , $\unicode[STIX]{x1D6FE}=5$ ). The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

4.1 Case A: single-frequency disturbance

In the first case a single-frequency modal disturbance, of the form (2.9), is considered with $\unicode[STIX]{x1D714}=0.253$ which has the maximum growth rate for the investigated conditions. An animation of the controlled single-frequency disturbance is provided as supplementary movie 1, available at https://doi.org/10.1017/jfm.2017.355. This disturbance is generated at the inlet $x=-4\unicode[STIX]{x03C0}$ of the simulation domain using the disturbance model presented in § 2.2. The shape of the disturbance corresponds to the eigenfunction calculated from the Orr–Sommerfeld equation at $\unicode[STIX]{x1D714}=0.253$ (see figure 5). To mimic the transitional regime in the simulations the amplitude of the perturbation is set to $A_{0}=0.01$ . First the performance of controller (II) with $l=20$ , $\unicode[STIX]{x1D6FE}=5$ and a low sensor noise $\unicode[STIX]{x1D70E}_{n}=0.01$ is investigated. Figure 18 shows the temporal evolution of the shear measurements $\unicode[STIX]{x1D742}_{m}$ that are used for feedback, the control input $\unicode[STIX]{x1D753}$ (amplitude of the blowing and suction), the perturbation energy ( $E=\Vert \boldsymbol{u}\Vert _{L^{2}}^{2}$ ) and the norm of the controlled output $\Vert \boldsymbol{q}\Vert _{2}$ . $\boldsymbol{q}$ reflects the controller performance as it is used within the control objective that is minimised by the controller, see (2.29). As the perturbation convects downstream towards the control region, the amplitude of blowing and suction increases to cancel the perturbation. The effect of the noise on the shear measurements can be observed and the resulting control input confirms the filtering and feedback of these measurements. Both control actuators at the upper and lower wall act in phase which is to be expected due to the symmetry of the geometry and the control layout. A snapshot at $t=200$ of the flow perturbation field in the control domain $x\in [0,~8\unicode[STIX]{x03C0}]$ is shown in figure 19. The performance of the state estimation is best visualised without control applied. Figure 19(a) shows the estimated flow field without control, figure 19(b) shows the real flow field without control and figure 19(c) shows the real controlled flow. The estimated flow field is computed from the controller state $\boldsymbol{u}_{K}^{e}$ through $\boldsymbol{u}_{K}^{e}\mapsto \unicode[STIX]{x1D64E}_{r}^{-1}\boldsymbol{u}_{K}^{e}$ where $\unicode[STIX]{x1D64E}_{r}^{-1}$ are the first $r$ columns of the inverse of the similarity transformation as discussed in § 3.2. It can be seen that the flow perturbations are well reconstructed in the control region where the measurements are taken, actuation is applied and where the performance objective $\boldsymbol{q}$ is defined. As a result the controller is effective in cancelling the perturbations by minimising the effect of the perturbation on $\boldsymbol{q}$ . Only low-amplitude oscillations remain. The required amplitude of the blowing and suction is of the same order as the magnitude of the perturbation as can be seen in the snapshot for the wall-normal velocity component in figure 19(c).

To compare the performance of the three controllers, the spatial evolution of the perturbation is evaluated. We define the amplitude of the streamwise velocity perturbation as

Figure 20 shows the amplitude for the three controllers with both low ( $\unicode[STIX]{x1D70E}_{n}=0.01$ ) and high ( $\unicode[STIX]{x1D70E}_{n}=0.2$ ) measurement noise. The amplitude reduction for the three controllers is in accordance with the frequency domain results in figure 16. The controllers are also robust to higher levels of sensor noise. Controller (III) takes higher sensor inaccuracies into account and the performance is preserved in the case of high sensor noise, see also table 1. Controllers (I) and (II) do not take such high measurement noise into account and the performance is less preserved. However, no severe deterioration can be observed. This can also be contributed to the simple structure of the perturbation.

Figure 19. Snapshot of the perturbation velocity within the control domain $x\in [0,~8\unicode[STIX]{x03C0}]$ at $t=200$ for the uncontrolled and controlled single frequency ( $\unicode[STIX]{x1D714}=0.253$ ) disturbance. Controller (II) with low sensor noise is considered. (a) Estimated velocity without control. (b) True velocity without control. (c) True velocity with control. The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

Figure 20. The maximum amplitude of the streamwise perturbation velocity (4.1) of the single-frequency disturbance for three controllers. (a) Feedback with low measurement noise $\unicode[STIX]{x1D70E}_{n}=0.01$ . (b) Feedback with high measurement noise $\unicode[STIX]{x1D70E}_{n}=0.2$ . The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

4.2 Case B: multiple-frequency disturbance

In the second test case a multiple-frequency disturbance is considered. An animation of the controlled multiple-frequency disturbance is provided as supplementary movie 2. The total disturbance consists of a linear combination of Orr–Sommerfeld modes. In total 16 modes in the frequency range $\unicode[STIX]{x1D714}\in [0.1,~0.4]$ are excited. Thus the disturbance is generated using 16 eigenfunctions whose shape corresponds to the eigenfunction calculated from the Orr–Sommerfeld equation at the selected frequencies. The temporal frequencies, the spatial wavelengths and spatial growth rates of these modes are listed in table 2. The spectrum includes 3 convectively unstable modes and 13 stable modes. Each mode is given the same amplitude $A_{0}=0.002$ such that the total disturbance is in the form of a wave train that is modulated as it propagates downstream. First the performance of controller (II) with a low sensor noise $\unicode[STIX]{x1D70E}_{n}=0.01$ is again investigated. The input–output signals and the closed-loop performance are shown in figure 21 and a snapshot at $t=200$ of the perturbation field in the control domain is shown in figure 22. The modulation of the perturbation can clearly be observed and the perturbation presents a richer structure as compared to the single-frequency case. With respect to the closed-loop performance the same observations can be made. The measurements are successfully filtered and the real flow is reconstructed well in the control domain as can be seen in figure 22. The controller is again able to cancel the perturbations and to suppress the perturbation wall shear stress. Although the unstable modes are dominant in the simulations, the (nearly) stable modes have not damped out and are still present as can be seen in figure 22. Nevertheless, the controller achieves nearly a full cancellation of the perturbations. This corroborates the findings of the input–output analysis presented in § 3.1 showing that the single mode disturbance model accurately captures the spatial wavelength and spatial growth of perturbations in a wider frequency band in the actuator/sensor region. As such the controller is able to effectively estimate and control a broader frequency spectrum of modes. To compare the performance of the three controllers, the spatial evolution of the perturbation is again evaluated. Since the amplitude of the perturbation also varies in time a measure for the time-averaged amplitude is defined

which is the wall-normal maximum amplitude of the root mean square (r.m.s.) streamwise velocity perturbation (Andersson, Berggren & Henningson Reference Andersson, Berggren and Henningson1999). Figure 23 shows the time-averaged amplitude for the three controllers with both low and high measurements noise. It can be observed that the amplitude reduction in case of high measurement noise for controller (I) is reduced more significantly. This is to be expected since the controller does not take high measurement inaccuracies into account. However, it still achieves a robust performance. Actually controller (I) and (II) have a comparable performance, see also table 1. This indicates that there is no large sensitivity in the choice of design parameters in case of high sensor noise. Again, the performance of controller (III) is preserved and is in accordance with its design.

Figure 21. Closed-loop performance for the multiple-frequency disturbance case. Controller (II) with low sensor noise is considered. (a) Shear measurements $\unicode[STIX]{x1D742}_{m}$ used for feedback. (b) Control input $\unicode[STIX]{x1D753}$ . (c) Perturbation energy $E=\Vert \boldsymbol{u}\Vert _{L^{2}}^{2}$ . (d) Norm of the controlled perturbation shear output $\Vert \boldsymbol{q}\Vert _{2}$ .

Figure 22. Snapshot of the perturbation velocity within the control domain $x\in [0,~8\unicode[STIX]{x03C0}]$ at $t=200$ for the uncontrolled and controlled multiple-frequency disturbance. Controller (II) with low sensor noise is considered. (a) Estimated velocity without control. (b) True velocity without control. (c) True velocity with control. The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

Figure 23. The wall-normal maximum amplitude of the root mean square streamwise perturbation velocity (4.2) of the multiple-frequency disturbance for three controllers. (a) Feedback with low measurement noise $\unicode[STIX]{x1D70E}_{n}=0.01$ . (b) Feedback with high measurement noise $\unicode[STIX]{x1D70E}_{n}=0.2$ . The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

Figure 24. Contours of the spatial distribution $\boldsymbol{F}=[F_{x},~F_{y}]^{\text{T}}$ of the in-domain disturbance used for case C.

4.3 Case C: stochastic in-domain forcing

In the third most challenging test case a stochastic in-domain forcing is considered which is generated at the upper wall near the inflow. An animation of the controlled stochastic in-domain forcing is provided as supplementary movie 3. In this case the momentum equation is forced with

where $w(t)$ is zero mean white noise with a normal distribution at unit intensity. The spatial distribution of the ‘vibrating ribbon’ at the upper wall $(y=1)$ corresponds to that of Bertolotti, Herbert & Spalart (Reference Bertolotti, Herbert and Spalart1992) and has the form $F_{x}=\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}y$ , $F_{y}=-\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}x$ with

where $\unicode[STIX]{x1D716}=0.5$ is the amplitude of the force, $\unicode[STIX]{x1D70E}_{x}=1$ , $\unicode[STIX]{x1D70E}_{y}=0.1$ the spatial lengths, $x_{r}=-3\unicode[STIX]{x03C0}$ the $x$ -position of the ribbon. The spatial distribution of this force is shown in figure 24. The body force ( $F_{x},F_{y}$ ) is both divergence free and satisfies the no-slip boundary conditions. First the performance of controller (II) with a low sensor noise is investigated. The input–output signals and the closed-loop performance are shown in figure 25 and a snapshot at $t=350$ of the perturbation field in the control domain is shown in figure 26. In addition, to better visualise the evolution of the perturbation and the controller performance, the temporal evolution for the shear stress along the lower wall for the uncontrolled case and the controlled case is shown in figure 27.

Figure 25. Closed-loop performance for the stochastic disturbance case. Controller (II) with low sensor noise is considered. (a) Shear measurements $\unicode[STIX]{x1D742}_{m}$ used for feedback. (b) Control input $\unicode[STIX]{x1D753}$ . (c) Perturbation energy $E=\Vert \boldsymbol{u}\Vert _{L^{2}}^{2}$ . (d) Norm of the controlled perturbation shear output $\Vert \boldsymbol{q}\Vert _{2}$ .

Figure 26. Snapshot of the perturbation velocity within the control domain $x\in [0,~8\unicode[STIX]{x03C0}]$ at $t=350$ for the uncontrolled and controlled stochastic disturbance. Controller (II) with low sensor noise is considered. (a) Estimated velocity without control. (b) True velocity without control. (c) True velocity with control. The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

Figure 27. Temporal evolution of the wall shear stress $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$ along the bottom wall of the channel for the stochastic forced disturbance. The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

Figure 28. The wall-normal maximum amplitude of the r.m.s. streamwise perturbation velocity (4.2) of the stochastic forced disturbance for three controllers. (a) Feedback with low measurement noise $\unicode[STIX]{x1D70E}_{n}=0.01$ . (b) Feedback with high measurement noise $\unicode[STIX]{x1D70E}_{n}=0.2$ . The triangles indicate respectively the position of the measurement sensors (▿), the actuators (▵) and the controlled outputs (▹).

The stochastic disturbance excites a spectrum of frequencies which results in large initial transients after which the disturbance develops in the form of wavepackets as can be seen in the energy plot in figure 25. The transients can also observed in the temporal evolution of the wall shear stress in figure 27 and are also present in the control region. It can be observed that the controller is still able to properly estimate the flow field and is effective in both minimising the wall shear stress and reducing the perturbation energy in the domain. Although a complete cancellation of the disturbance is not possible, the controller manages to achieve a reduction of 97 % in the controlled shear output power, see also table 1. Note that the disturbance is completely independent of the disturbance model used to design the controller. It is defined in-domain and creates initially asymmetric developing perturbations while the complete input–output layout is symmetric. Furthermore the transients are not accounted for in the control design and the perturbations are not fully developed in the control domain. Nevertheless, the controller achieves a high level of robustness to unmodelled disturbances; no overshoots can be observed in the perturbation energy and the controlled output, and the controller does not aggravate the flow. This can be contributed to the fact that the controller is able to estimate and stabilise the underlying modes that are present in the disturbance as can be seen from figure 26.

Figure 28 shows the time-averaged amplitude for the three controllers with both low and high measurements noise. It can be observed that controllers (I) and (II) have comparable performance, also for the low sensor noise case. This can be contributed to the fact that uncertainties in output measurements also arise due to the unmodelled disturbances. This indicates that there is also no large sensitivity in the choice of design parameters in case of unmodelled disturbances.