2.1. Problem set-up

In this section, we discuss the general problem of controlling an incompressible viscous fluid in a bounded domain ${\it\Omega}\subset \mathbb{R}^{n}$ ( $n=2$ or 3) by forcing on the boundary $\partial {\it\Omega}={\it\Lambda}$ . The system is modelled by the Navier–Stokes equations with Dirichlet boundary control. Let $\boldsymbol{w}(t,x)$ , $p(t,x)$ and ${\it\nu}=1/\mathit{Re}>0$ denote the velocity field, pressure field and kinematic viscosity with the Reynolds number ( $\mathit{Re}$ ) respectively. We consider the controlled Navier–Stokes equations given by

with initial data

and controlled boundary condition

with appropriate boundary condition $\partial {\it\Omega}\setminus {\it\Omega}_{c}\rightarrow \pm \infty$ . Conservation of mass requires that the boundary control $\boldsymbol{u}(t,x)$ satisfy

where $\boldsymbol{n}(x)$ is the unit outward normal to the boundary ${\it\Lambda}=\partial {\it\Omega}$ . In the problem considered here the control has the form

where for each $i=1,2,\ldots ,P$ ,

implying the assumption of a divergence-free condition, and hence the control constraint in (2.5) is satisfied automatically. Here, ${\it\Gamma}_{i}(x)$ is the spatial distribution of the control modes considered for flow control. Choice of control modes is critical and is discussed in detail in § 4.

A typical strategy for computing feedback control laws for fluids is to linearize the equations about an equilibrium point. For this study, we linearize about a time-averaged flow solution, and the objective of the control is to drive the state to this solution. Let $\boldsymbol{W}(x)$ and $P(x)$ denote velocity and pressure respectively, corresponding to a time-averaged solution to an uncontrolled problem over the interval $[0,T]$ . Then, the pair $(\boldsymbol{W}(x),P(x))$ satisfy

where, in general,

and $\otimes$ represents the Kronecker product. The first two terms vanish for $T$ -periodic flows (as in this study), and $\boldsymbol{f}$ would vanish altogether for steady state $(\boldsymbol{W}(x),P(x))$ . We define the velocity and pressure perturbation fields by $\boldsymbol{v}(t,x)=\boldsymbol{w}(t,x)-\boldsymbol{W}(x)$ and $q(t,x)=p(t,x)-P(x)$ respectively. The controlled perturbation equation becomes

where

and

We consider only continuous linear state feedback controllers, so by the Riesz representation theorem each control has the form

The structure of the functional gain provides considerable insight into the spatial nature of the controller. For example, if $\boldsymbol{h}_{j}(x)\equiv \mathbf{0}$ in ${\it\Omega}$ , then the $j$ th controller plays no role in the CL system. More typical is the case where the support of the functional gain is localized or nearly localized. If

denotes the support of the functional gain $\boldsymbol{h}_{i}(\cdot )$ , then

Therefore, the only spatial points that contribute to the feedback control law $u_{i}(t)$ are points $x\in {\it\Omega}_{i}$ , and the only place where one needs sensed or estimated information about the flow is in the region ${\it\Omega}_{i}$ . Consequently, once the control law in (2.16) is determined, the corresponding functional gain $\boldsymbol{h}_{i}(\cdot )$ provides valuable information about the optimal location of sensors and insight into developing reduced-order controllers based on state estimation techniques such as Kalman filters. We apply this basic idea to the cylinder control problem below. However, we begin with the development of the distributed parameter model so that the rough idea above can be placed on a solid theoretical basis. In addition, this formulation provides the framework to develop rigorous computational algorithms and corresponding reduced-order models for control.

In the present study, we focus on constructing numerical approximations of the PDE feedback control law and the corresponding functional gains. In particular, we use a reduced-order modelling technique developed in Akhtar, Nayfeh & Ribbens (Reference Akhtar, Nayfeh and Ribbens2008) to compute the functional gains. We then analyse these gains to provide insight into the physical behaviour of the feedback control mechanism. The control problem is defined by a flow past a circular cylinder at $\mathit{Re}=200$ , and we use suction on the cylinder surface to suppress vortex shedding. It is important to note that the objective of this study is not to design the best controller for the given problem. The main idea presented here is to use simple control laws, compute the associated functional gains, and use them to provide insight into the physical behaviour of the feedback control mechanism.

2.2. A state space formulation

Here, we provide a summary of how one formulates a rigorous distributed parameter state space model for the system of PDEs (2.12)–(2.15) (ignoring $\boldsymbol{f}$ for now). Since we are applying a Dirichlet boundary control, a detailed description of the mathematical framework requires the introduction of so-called very weak solutions to the Navier–Stokes equations and lies outside the scope of this paper. However, details may be found in Temam (Reference Temam1997), Sritharan (Reference Sritharan1998), Barbu & Triggiani (Reference Barbu and Triggiani2004), Fursikov (Reference Fursikov2004), Amrouche & Razafison (Reference Amrouche and Razafison2005), Farwig, Galdi & Sohr (Reference Farwig, Galdi and Sohr2005, Reference Farwig, Galdi and Sohr2006a ,Reference Farwig, Galdi and Sohr b ), Burns (Reference Burns, Gad el Hak and Tsai2006), Barbu, Lasiecka & Triggiani (Reference Barbu, Lasiecka and Triggiani2006), Raymond (Reference Raymond2007a ,Reference Raymond b ), Amrouche & Rodríguez-Bellido (Reference Amrouche and Rodríguez-Bellido2010) and Raymond & Thevenet (Reference Raymond and Thevenet2010).

Let

and

be the typical function spaces of divergent free vector fields, where

is the usual Lebesgue space of square integrable vector functions. For $r>0$ , $H^{r}({\it\Omega};\mathbb{R}^{n})$ is the Sobolev space of vector functions whose distributional derivatives of order up to $r$ belong to $L^{2}({\it\Omega};\mathbb{R}^{n})$ . If

with $C_{0}^{\infty }$ the space of infinitely differentiable functions with compact support, then one can show (Sritharan Reference Sritharan1991; Temam Reference Temam2001) that $\boldsymbol{H}$ is the closure of $\unicode[STIX]{x1D651}$ in $L^{2}({\it\Omega};\mathbb{R}^{n})$ and $\boldsymbol{V}$ is the closure of $\unicode[STIX]{x1D651}$ in $H_{0}^{1}({\it\Omega};\mathbb{R}^{n})$ . Let $P_{H}:L^{2}({\it\Omega};\mathbb{R}^{n})\longrightarrow \boldsymbol{H}$ be the orthogonal projection onto $\boldsymbol{H}$ (the Leray projection) and define the Stokes operator $A_{S}:D(A_{S})\subset \boldsymbol{H}\longrightarrow \boldsymbol{H}$ on the domain $D(A_{S})=H^{2}({\it\Omega};\mathbb{R}^{n})\cap \boldsymbol{V}$ by

It follows that the dense and continuous embeddings (Sritharan Reference Sritharan1991)

are compact, where the prime denotes the dual space. Let $A_{0}\boldsymbol{ : }\boldsymbol{V}\longrightarrow \boldsymbol{V}^{\prime }$ be the lifting of $A_{S}:D(A_{S})\subset \boldsymbol{H}\longrightarrow \boldsymbol{H}$ to $\boldsymbol{V}$ and define the linear operator $\mathscr{R}\boldsymbol{ : }\boldsymbol{V}\longrightarrow \boldsymbol{V}^{\prime }$ by

Hence, the Oseen operator $A({\it\nu})\boldsymbol{ : }\boldsymbol{V}\longrightarrow \boldsymbol{V}^{\prime }$ given by

is the linearized operator corresponding to the velocity–pressure pair $(\boldsymbol{W},P)$ given by (2.9). The nonlinear operator $F\boldsymbol{ : }\boldsymbol{V}\longrightarrow \boldsymbol{V}^{\prime }$ is defined by

for each ${\it\phi}\in \boldsymbol{V}$ .

In order to complete the model one must define the control input operator. The basic idea is to use the Lions structure (Lions Reference Lions1971). The reader may also refer to Bensoussan et al. (Reference Bensoussan, Da Prato, Delfour and Mitter1992a ,Reference Bensoussan, Da Prato, Delfour and Mitter b ) and Barbu et al. (Reference Barbu, Lasiecka and Triggiani2006). In particular, one ‘lifts’ $A\boldsymbol{ : }\boldsymbol{V}\longrightarrow \boldsymbol{V}^{\prime }$ to $A_{1}\boldsymbol{ : }\boldsymbol{H}\longrightarrow [D(A^{\ast })]^{\prime }$ by

where $\langle \cdot ,\cdot \rangle _{\boldsymbol{H}}$ is the inner product of the space $\boldsymbol{H}$ , and is the $L_{2}$ inner product, $\langle f,g\rangle _{H}=\int _{{\it\Omega}}\sum _{i=1}^{n}f_{i}(x)g_{i}(x)\,\text{d}x$ , and defines $B:L^{2}({\it\Lambda};\mathbb{R}^{n})\longrightarrow [D(A^{\ast })]^{\prime }$ by

where $D$ is the Dirichlet map $D:L^{2}({\it\Lambda};\mathbb{R}^{n})\longrightarrow H^{1/2}({\it\Omega};\mathbb{R}^{n})\cap \boldsymbol{H}$ . The state space model for the boundary control problem (2.12)–(2.15) now has the (very) weak formulation

where we identify $\boldsymbol{z}(t)\in \boldsymbol{H}$ with the vector field $\boldsymbol{v}(t,\cdot )$ by

The system in (2.30) has some important features that are typical of many flow control problems. It should be noted that although the Stokes operator ${\it\nu}A_{0}$ is self-adjoint, the Oseen operator $A={\it\nu}A_{0}+\mathscr{R}$ can be highly non-normal if $\mathscr{R}\neq 0$ . Therefore, the linearized control system

can be highly sensitive to parameter variations, initial data and inputs $\boldsymbol{u}(\cdot )$ . Computational algorithms for linear control problems with a non-normal $A$ operator require special effort. In addition, the nonlinear operator is conservative in the sense that

where $\langle \cdot ,\cdot \rangle _{\boldsymbol{V}}$ is the inner product of the space $\boldsymbol{V}$ ; this is the $\boldsymbol{H}^{\mathbf{1}}$ inner product $\langle f,g\rangle _{\boldsymbol{V}}=\int _{{\it\Omega}}\sum _{i=1}^{n}\sum _{j=1}^{n}((\partial f_{i}(x)/\partial x_{j})(\partial g_{i}(x)/\partial x_{j}))\,\text{d}x$ for all $\boldsymbol{v}\in \boldsymbol{V}$ , and this special structure tends to be highly sensitive to disturbances (Burns Reference Burns, Smith and Demetriou2003). Sensitivity and non-normality of the linear system lead to many difficulties in the corresponding control problem.

2.3. A feedback control problem

We now consider the feedback control problems for the Navier–Stokes system

and the corresponding linearized system

where again ${\it\nu}=1/\mathit{Re}$ and $A={\it\nu}A_{0}+\mathscr{R}$ .

New theoretical results on boundary feedback control for the Navier–Stokes equations in 2D and 3D have set the stage for implementation of practical controllers and computational approximations. The LQR approach was applied to tangential boundary control in Barbu & Triggiani (Reference Barbu and Triggiani2004) and Barbu et al. (Reference Barbu, Lasiecka and Triggiani2006) and for more general boundary control in Raymond (Reference Raymond2007a ,Reference Raymond b ) and Raymond & Thevenet (Reference Raymond and Thevenet2010). These papers provide conditions for stabilizability of the linearized and nonlinear equations.

It is important to note that there are many ‘feedback control problems’ (linear feedback, nonlinear feedback, LQR, LQG, $H^{\infty }$ , MinMax, pole placement, etc.) and several technical approaches to each of these problems. The LQR, LQG and MinMax approaches have been applied in other flow control problems (Burns et al. Reference Burns, King and Rubio1998; Fursikov et al. Reference Fursikov, Gunzburger and Hou1998a ; Burns Reference Burns, Smith and Demetriou2003, Reference Burns, Gad el Hak and Tsai2006; Hogberg, Bewley & Henningson Reference Hogberg, Bewley and Henningson2003; Fursikov, Gunzburger & Hou Reference Fursikov, Gunzburger and Hou2005).

We focus on pole placement as another example to illustrate the point that the particular control design method is not the critical issue in the sensor placement problem. What we do here could have been done using any of the previous methods. The problem of interest is to find a feedback controller that stabilizes the flow and then use the structure of the controller to find good sensor/actuator locations. The only assumption we make here is that the linearized Navier–Stokes system (2.35) is stabilizable so that there is a continuous linear feedback gain operator

such that the CL operator

generates a semigroup $S_{CL}(t,{\it\nu})=\text{e}^{A_{CL}({\it\nu})t}$ satisfying

for some $Q\geqslant 1$ and ${\it\beta}>0$ . It is important to note that if $Q>1$ , then the corresponding CL nonlinear Navier–Stokes equations

may not be exponentially asymptotically stable. However, if $Q=1$ , then one can show that the CL nonlinear Navier–Stokes system is exponentially asymptotically stable. Achievement of a CL asymptotically stable system is possible in 3D nonlinear flows with linear feedback even when the Reynolds number is greater than the critical Reynolds number. Fursikov (Reference Fursikov2004) and Burns (Reference Burns, Gad el Hak and Tsai2006) have results along this line. Moreover, in the case with control along the entire boundary, Barbu et al. (Reference Barbu, Lasiecka and Triggiani2006) have proven the existence of a linear feedback operator $\mathscr{K}:H^{1/2+{\it\varepsilon}}({\it\Omega};\mathbb{R}^{n})\cap \boldsymbol{H}\longrightarrow \boldsymbol{U}=L^{2}({\it\Lambda};\mathbb{R}^{n})$ for any small ${\it\varepsilon}>0$ , such that the nonlinear CL Navier–Stokes equation (2.39) is exponentially asymptotically stable. Furthermore, this feedback gain operator can be computed by solving an abstract Riccati operator equation.

For the problem considered in this paper with a finite number of controllers, the assumption that the system (2.35) is stabilizable with $\mathscr{K}:D(\mathscr{K})\subseteq \boldsymbol{H}\longrightarrow \mathbb{R}^{m}$ continuous implies (Riesz representation theorem) that there exist divergent free vector fields $\boldsymbol{h}_{i}(\cdot )\in \boldsymbol{H},i=1,2,\ldots ,P$ , so that

where the functions $\boldsymbol{h}_{i}(\cdot )$ are called the feedback functional gains. As noted above, we shall use the spatial information contained in the support of these functional gains to address the sensor placement problem (Burns et al. Reference Burns, King and Rubio1998).

In this study, we will use pole placement as a design methodology, thus it is important to know that the spectrum of the CL operator determines the stability of the CL system. When ${\it\Omega}$ is smooth and bounded, much is known about the spectrum of the linearized Oseen operator $A={\it\nu}A_{0}+\mathscr{R}$ and the stability of the linearized problem. The following result follows from Theorem 3.6 in Sritharan (Reference Sritharan1991).

Theorem 1 implies that in order to stabilize the linearized flow one needs only to ‘move’ a finite number of eigenvalues to the left half of the complex plane. This fact is the basis of much of the work in the paper by Barbu & Triggiani (Reference Barbu and Triggiani2004) and has been extended to boundary control in Barbu et al. (Reference Barbu, Lasiecka and Triggiani2006).

It should be noted that when linearizing about a mean flow, the term $-\boldsymbol{f}$ in (2.35) can be treated by finding a solution to the Oseen equations $({\it\nu}A_{0}+\mathscr{R})\boldsymbol{g}=\boldsymbol{f}$ . A substitution $\boldsymbol{y}=\boldsymbol{z}-\boldsymbol{g}$ produces the standard homogeneous control problem. Applying the feedback control $-\mathscr{K}\boldsymbol{y}$ to the homogeneous problem is equivalent to applying $\boldsymbol{u}(t)=-\mathscr{K}\boldsymbol{z}(t)+\mathscr{K}\boldsymbol{g}$ , where the last term applies a DC shift to the feedback control.

3.1. Proper orthogonal decomposition

Here, we present a general procedure to compute the POD modes used in this study. The velocity data $\boldsymbol{v}=(v_{1},v_{2})$ are then recorded over a shedding cycle, commonly referred to as snapshots in the POD community. The data are assembled in a matrix $\unicode[STIX]{x1D652}$ of the form

where $N$ is the number of grid points and $S$ is the number of snapshots taken at constant intervals. We note that the mean velocity field is subtracted from the $\unicode[STIX]{x1D652}$ matrix prior to computing the POD modes.

We perform a singular-value decomposition (SVD) of the ensemble data to compute the POD eigenfunctions, that is, $\unicode[STIX]{x1D652}={\it\Phi}{\it\Sigma}\unicode[STIX]{x1D651}^{\text{T}}$ , where ${\it\Phi}$ represents the POD eigenfunctions, ${\it\Sigma}$ contains the singular values ${\it\sigma}_{i}$ and $\unicode[STIX]{x1D651}$ is the matrix of right singular vectors. We compute the first $S$ POD modes using standard LAPACK routines. An important property of these eigenfunctions is orthogonality; that is, ${\it\Phi}_{i}\cdot {\it\Phi}_{j}={\it\delta}_{ij}$ , where ${\it\delta}_{ij}$ is the Kronecker delta. The optimality of the POD eigenfunction lies in capturing the greatest possible fraction of the total kinetic energy for a projection of the snapshot data onto a fixed number of basis functions. The eigenvalue ${\it\lambda}_{i}$ of $\unicode[STIX]{x1D652}^{\text{T}}\unicode[STIX]{x1D652}$ is a measure of the energy represented by the $i$ th eigenfunction and is equal to the square of the corresponding singular value ${\it\sigma}_{i}$ . Since the exterior flow problem is solved in a large domain, the standard $L_{2}$ inner product would place a significant emphasis on the outer region of the domain. To accurately capture the physics near the cylinder, we use a weighted $L_{2}$ inner product for both creating the basis functions and computing the reduced-order model. The weight function we use is motivated by the radial clustering of points towards the cylinder and is given as

3.2. Simulation cases

We simulate the flow past the cylinder at $\mathit{Re}=200$ for long enough to reach the periodic state. We note that, although the flow field at this Reynolds number is three-dimensional after the mode-A instability (Williamson Reference Williamson1996) at $\mathit{Re}\approx 180$ , we consider a two-dimensional flow at this Reynolds number to demonstrate the flow control strategy and sensor placement technique developed in this paper. We consider two simulation cases, with and without boundary control.

4.1. The control function approach

A conventional velocity expansion in POD-based reduced-order models is given by

where the $z_{i}^{r}(t)$ is the temporal coefficient of the $i$ th POD mode for the reduced-order model. However, the velocity expansion in (4.1) is not a valid candidate in the Galerkin projection, as it does not automatically satisfy the boundary conditions for arbitrary values of $z_{i}^{r}$ .

The control function approach is motivated by requiring homogeneous boundary conditions on the basis functions used in the Galerkin projection. The homogeneous boundary condition is maintained by introducing a suitable function in the velocity expansion, known as the ‘control function’.

Following Graham et al. (Reference Graham, Peraire and Tang1999), we remove the inhomogeneous boundary condition on the cylinder surface by subtracting a suitable ‘control function’ from each snapshot before generating the POD basis functions. We generate the control function by subtracting the mean velocity obtained in case I from that of case II as

Thus, the modified snapshot matrix $\unicode[STIX]{x1D652}^{mod}$ is now given as

where $u$ corresponds to the variable control input, suction in the present study, kept constant in case II specifically for this purpose. The mean flow and basis functions ( ${\it\Phi}_{i}$ ) are obtained from $\unicode[STIX]{x1D652}^{mod}$ as described in § 3.1. The ${\it\Phi}_{i}$ maintain their properties with zero velocity on the cylinder surface.

4.2. The model

The velocity expansion now includes the control function and is modified as

For the general procedure with $P$ control inputs, the reduced-order model is developed in appendix B. However, in the main text of this paper, we present the reduced-order model for a single control input as expressed in (4.4). We substitute (4.4) into (2.12), identifying $\boldsymbol{W}$ with $\bar{\boldsymbol{v}}$ , project these equations along the ${\it\Phi}_{k}$ , eliminating the pressure perturbation $q$ , and obtain

where the operators are represented by

where $k,m,n=1,2,\ldots ,M$ and where $\langle \cdot ,\cdot \rangle$ is the Euclidean inner product.

To validate the reduced-order model developed here, we consider the unactuated case, i.e. case I. Only for this section, we use the $\unicode[STIX]{x1D652}^{una}$ matrix to compute the mean velocity field (see figure 1) and the first 12 POD modes after mean subtraction. We integrate (4.5), with $u=0$ , to compute the coefficients $z_{i}^{r}$ and plot them in figure 2. The first pair of time coefficients corresponds to the shedding frequency while the second pair is its harmonic. We also observe a phase shift between the states within a pair. We then project each snapshot onto the POD modes to compute the velocity coefficient $\tilde{z}_{i}(t)$ as follows:

For the 12-dimensional Galerkin model, we investigate the two-dimensional projection of the phase portrait onto the plane ( $z_{1}^{r},z_{2}^{r}$ ) and compare it with the two-dimensional projection onto the plane ( $\tilde{z}_{1},\tilde{z}_{2}$ ) obtained using (4.13). We observe that the two projections compare well, as shown in figure 3.

Figure 2.

The velocity coefficients $z_{i}^{r},i=1,2,3,4$ , at $\mathit{Re}=200$ : (a) $z_{1}^{r}$ (solid) and $z_{2}^{r}$ (dashed); (b) $z_{3}^{r}$ (solid) and $z_{4}^{r}$ (dashed).

Figure 3.

Projection of the POD phase portrait onto the ( $z_{1}^{r},z_{2}^{r}$ )-plane (solid) and the snapshot portrait onto the ( $\tilde{z}_{1}^{r},\tilde{z}_{2}^{r}$ )-plane (triangle).

4.3. POD-based control

There are many ways to construct the control function. We compute the control function ${\it\Gamma}(x)$ from (4.2). We plot the streamwise and the normal components of the control mode in figure 4. The key advantage of this method is that the flow perturbations in the control mode are not restricted to the local regions close to the actuators. Rather, the effect of the difference between the two average fields is also present in the wake. Appropriate application of the control input, using this function, can favourably alter the flow field over a wider part of the domain. In other words, the global aspect of this method makes it a better candidate for the construction of the control mode. In the present setting, we use $\unicode[STIX]{x1D652}^{mod}$ and compute first 12 POD modes to develop a 12-dimensional reduced-order model. The first four dominant streamwise and cross-flow velocity POD modes are plotted in figures 5 and 6 respectively.

Figure 4.

Control mode at $\mathit{Re}=200$ : (a) streamwise and (b) cross-flow components.

Figure 5.

The streamwise velocity modes: (a) ${\it\Phi}_{1}^{u}$ , (b) ${\it\Phi}_{2}^{u}$ , (c) ${\it\Phi}_{3}^{u}$ , (d)  ${\it\Phi}_{4}^{u}$ .

Figure 6.

The cross-flow velocity modes: (a) ${\it\Phi}_{1}^{v}$ , (b) ${\it\Phi}_{2}^{v}$ , (c) ${\it\Phi}_{3}^{v}$ , (d)  ${\it\Phi}_{4}^{v}$ .

It is important to emphasize that the objective of this study is to compute the functional gains obtained from a simple feedback control and analyse these functions to suggest effective location of the sensors. These sensors may be used later to develop estimators to develop better controllers. The reduced-order model, developed in (4.5), is a 12-dimensional system of nonlinear ordinary differential equations (ODEs). If we can stabilize the system linearly, we can in fact stabilize the complete dynamics of the full system. According to the Hartman–Grobman theorem (Arnold Reference Arnold1988), (a) the fixed point, say $\boldsymbol{x}=\boldsymbol{x}_{0}$ , of the nonlinear system of the form $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ is stable when the fixed point $\boldsymbol{x}^{\ast }=0$ of the linear system is asymptotically stable and (b) the fixed point, $\boldsymbol{x}=\boldsymbol{x}_{0}$ , of the nonlinear system of the above form is unstable when the fixed point $\boldsymbol{x}^{\ast }=0$ of the linear system is unstable.

We linearize the system about its mean component to reduce the model to a standard linear time invariant (LTI) control problem and obtain

Equation (4.14) is an $M+1$ system of linear ODEs. Following Rediniotis et al. (Reference Rediniotis, Ko, Yue and Kurdila1999), we use $u_{c}=\dot{u}$ as the control command for finite-dimensional control, which is a convenient form for better understanding and is consistent with the control theory. Thus, $u_{c}=\dot{u}$ is a new control input to the system, $\mathbf{0}$ is a zero vector of size $M$ and $T$ denotes the transpose. Thus, we have reduced the model to a standard LTI control problem. The reduced-order model developed thus far is an open-loop (OL) control system where the control input is not a function of the states of the system. In other words, any change in the states does not alter the control input. One approach to alter the dynamics of the system is to change the poles (eigenvalues) of the state matrix $\unicode[STIX]{x1D63C}$ in the complex plane. This also allows us to design a feedback controller such that states decay to zero.

In the present framework, we choose $u_{c}$ as a linear function of the states of the system to obtain

where $\unicode[STIX]{x1D646}$ is a $1\times (M+1)$ matrix with constant gains. Substitution of (4.15) into (4.14) yields a CL system of the form

As the purpose of the control law is to stabilize the system, it is desired to have the poles of the system in the left half of the complex plane. In other words, we would like the eigenvalues of $\mathbb{A}_{c}$ to have negative real parts. This will guarantee exponential stability of the system.

To analyse the system, we move the unstable complex conjugate pair of the OL system from the right half of the complex plane to the left half of the complex plane as shown in figure 7. We note that the imaginary part of the OL pole corresponds to the vortex shedding frequency and is kept the same for the new pole locations. The pole placement technique is one method to obtain a feedback gain that favourably alters the dynamics of the system. Other techniques include the LQR technique, $H^{2}$ control and $H^{\infty }$ control. We consider three CL systems with pole locations as listed in table 1. These systems are referred to as CL1, CL2, and CL3 respectively. The choice of pole location is often driven by design specifications, such as settling time, rise time, overshoot, etc.

Figure 7.

Unstable poles moved to the left half of the complex plane.

Table 1.

Pole placement chart.

We compute appropriate gains for each CL system with the control calculated using the standard MATLAB function place which uses Ackerman’s algorithm (Ackermann Reference Ackermann1977). We then integrate the CL systems and observe their responses. In figure 8, we plot the response of the first two states $z_{1}^{r}$ and $z_{2}^{r}$ . Since the pole location is close to the imaginary axis, the system response of CL1 is slow and takes over 1000 time units to decay, with a settling time of approximately 2 %. On the other hand, the settling times of the CL2 and CL3 responses are approximately 300 and 120 time units respectively. The control input is linearly related to the states, thus $u_{c}$ also approaches zero in a similar fashion, as shown in figure 9. However, it is important to note that the movement of the poles is reflected in the control effort required to drive the states to zero. In other words, the maximum control input is close to 0.06, 0.21 and 0.8 for cases CL1, CL2 and CL3 respectively. Based on the control $\boldsymbol{u}(t)=-\mathscr{K}\boldsymbol{z}(t)+\mathscr{K}\boldsymbol{g}$ , the first component refers to $u_{c}$ , which makes the homogenized states go to zero, and the second component is the control input referring to constant suction input. For CL3, the constant suction is approximately 52 % of the free stream velocity (Akhtar & Nayfeh Reference Akhtar and Nayfeh2009).

Figure 8.

Time history of $z_{1}^{r}$ (a,c,e) and $z_{2}^{r}$ (b,d,f) with control actuation: (a,b) CL1, (c,d) CL2 and (e,f) CL3.

Figure 9.

Time history of $u_{c}$ with control actuation: (a) CL1, (b) CL2 and (c) CL3.

Using a similar model-based approach to control the vortex shedding past a cylinder, Akhtar et al. (Reference Akhtar, Naqvi, Borggaard and Burns2013) used an LQR technique to compute the appropriate gains and observed the system response. The objective was to reduce the fluctuating lift force to

where $C_{L}$ is the lift coefficient, $R$ represents a control cost and the dynamics is described by the low-dimensional model derived above. The first two states correspond to the shedding frequency observed in the lift response. If the states $\boldsymbol{z}^{r}$ are stabilized, the lift coefficient is also reduced. Thus, the optimization problem was redefined as

where $\unicode[STIX]{x1D64C}_{M\times M}$ and $\unicode[STIX]{x1D64D}_{M_{c}\times M_{c}}$ are the state weighting matrix and the control effort weighting matrix respectively, with $M=12$ , $M_{c}=1$ , $\unicode[STIX]{x1D64C}=\unicode[STIX]{x1D644}$ and $\unicode[STIX]{x1D64D}=10$ . On feeding back the optimal gains, all of the states decay to zero. It is important to note that the gains obtained through LQR and pole placement are close, leading to a similar control response to that in CL3. Akhtar et al. (Reference Akhtar, Naqvi, Borggaard and Burns2013) also demonstrated that feeding back only the dominant state ( $z_{1}^{r}$ ) stabilizes the system, while feeding back only those states corresponding to higher modes fails to stabilize the system.

6.1. Sensor placement

In order to validate our model and illustrate the idea, we consider the CL3 case to suggest the sensor locations in the wake of the cylinder. We simulate two scenarios. In each scenario, we use four locations to approximate the feedback gain given in (5.2). The choice of these locations is motivated by the magnitudes of the functional gain in the wake, in the $X=1.3$ and $X=3.0$ planes. In the first scenario, the four locations are chosen at the points where the functional gain is high (the good choice). On the contrary, we choose four locations with low functional gain magnitudes (the bad choice). The coordinates of these points along with the corresponding functional gains are provided in table 2. These locations are identified in figure 15 with good choice and bad choice scenarios represented by squares and circles respectively.

Table 2.

Sensor locations.

Figure 15.

Functional gain approximated at four locations: good (square) and bad (circle) locations.

We approximate the feedback control input using a ‘quadrature’ as follows:

where $\boldsymbol{h}_{j}$ (in table 2) represents the mean value of $\boldsymbol{h}$ over the region $\text{d}A_{j}=\text{d}X\cdot \text{d}Y$ and $\text{d}X=\text{d}Y=0.2$ . We use the mean values of $\boldsymbol{h}_{j}$ over the region $\text{d}A_{j}$ to check the convergence of the functional gains. Using $M=12$ POD modes, the mean values of $\boldsymbol{h}_{j}$ converge to the magnitudes presented in table 2. Thus, we use these mean values of $\boldsymbol{h}_{j}$ to compute the feedback control input. In figure 16, we simulate the reduced-order model with approximate gains computed from good and bad locations in the wake and compare it with a CL CFD simulation obtained from full-state feedback control.

Figure 16.

Projection of a CL simulation ( $z_{1}^{r}$ ) with gains approximated at (a) good and (b) bad locations. It should be noted that the approximate gain (blue, dashed) is computed with only four sensors and the CL simulation gain (black, solid) is obtained with full-state feedback.

The approximation of the feedback gains provided by the good data set stabilizes the system, as shown in figure 16. Although the response is slow compared with the full-order feedback control, it is important to note that the gains were approximated from only four locations rather than using the complete domain. On the other hand, if the sensors are placed at bad locations, the feedback gain approximation is poor and fails to stabilize the CL system. Based on the current choice of sensor locations, figure 16(a,b) shows the two extremes of the system response to effective and ineffective controllers respectively. An obvious option to obtain a better response is to increase the number of sensors, but this is not feasible in most practical settings. Thus, it is important to choose the correct locations for the available sensors to measure or estimate the flow field.

6.2. Functional gains versus POD modes

We emphasize that the functional gains provide an effective choice for sensor placement compared with other options, such as POD modes, wavemaker regions (Luchini, Giannetti & Pralits Reference Luchini, Giannetti and Pralits2008) or regions of absolute instability. To demonstrate the effectiveness of functional gains as the choice of sensor placement, we compare the streamwise POD modes with the streamwise functional gains, as shown in figure 17. It is important to note that the first two POD modes occur as a pair and contribute equally to capture over 95 % of the energy in the dynamical system. For direct comparison, we plot the first two streamwise POD modes in the upper half with the streamwise functional gain in the bottom half. Effective (good) and ineffective (bad) sensor locations, already determined based on the functional gains, are reproduced to compare their corresponding locations within the POD modes. A quick comparison reveals that the effective sensor locations can also be determined using the POD modes, and it may seem that placing sensors at large amplitudes of the POD modes would yield similar results to those obtained using the functional gains. However, this is not the case in general and may be misleading for effective sensor placement. To validate this statement, we choose sensor locations (white squares) where the magnitudes of the POD modes are relatively high while the corresponding functional gains are low. Arbitrary placement of a sensor at a location based on the POD mode can be ineffective, as seen from the results shown in figure 16(b). Conversely, there are locations (white circles) where there is no coherent structure present in the POD mode, but which correspond to a relatively high functional gain value and lead to a stable controlled system, as depicted in figure 16(a).

Figure 17.

Comparison of first two streamwise POD modes (upper half) with the streamwise functional gain (bottom half). The functional gain is approximated at high (white square) and low (white circle) magnitudes of the POD modes. It should be noted that the black squares and black circles from figure 15 are plotted again for reference. (a) Plot of ${\it\Phi}_{1}^{u}$ versus $h^{u}$ ; (b) plot of ${\it\Phi}_{2}^{u}$ versus $h^{u}$ .

In table 3, we note the precise magnitudes of the POD modes and functional gains at the proposed sensor locations. For the first POD mode in figure 17(a), we identify two points, $(1.74,0.21)$ and $(2.85,0.67)$ , depicted as white squares, where the magnitude of the POD mode is of the order of 0.1 while the corresponding functional gains are one order of magnitude smaller. We term these locations as good points from the POD perspective but bad choices for control purposes since the functional gains are low. Similarly, sensor placement at the locations $(2.37,0.38)$ and $(4.09,0.59)$ (white circles) indicates bad choices from the POD mode perspective, but these locations have large functional gains, thus suggesting good sensor locations. Similar conclusions can be drawn from the analysis of the second POD mode. Based on figure 17, we emphasize again that there are locations in the POD modes where their high magnitudes overlap with high values of functional gains. However, our study above shows that solely identifying the locations where the dominant POD modes have large magnitudes cannot be used to determine effective sensor locations without additional considerations. The same concept can be used to explain the significance of functional gains over other techniques such as using the wavemaker region.

Table 3.

Comparison of sensor locations using POD modes and functional gains.