Feedback control of transitional shear flows: Sensor selection for performance recovery

We consider plane Poiseuille flow at sub-critical Reynolds number of $Re=\boldsymbol{\bar{u}_{c}}h/\nu=3000$ defined by the centerline velocity of the base flow $\boldsymbol{\bar{u}_{c}}$, half height of the channel $h$, and kinematic viscosity $\nu$. As shown in figure 1, the velocity profile of the base flow is $[\boldsymbol{\bar{u}},\boldsymbol{\bar{v}},\boldsymbol{\bar{w}}]=[1-(y/h)^{2},0,0]$ with coordinate origin at the center line between walls, where $\boldsymbol{\bar{u}},\boldsymbol{\bar{v}},\boldsymbol{\bar{w}}$ represent velocity components in streamwise $x$, wall-normal $y$, and spanwise $z$ directions, respectively. The parabolic profile of the velocity is the laminar equilibrium solution of the flow. In this study, length-scale variables are non-dimensionalized by the channel half-height $h$, and velocities are non-dimensionalized by the centerline velocity of base flow $\boldsymbol{\bar{u}_{c}}$. Time is denoted by $t$.

The flow at this condition is linearly stable, since there are no unstable modes from the linear stability analysis. However, a large transient energy growth of small perturbations is observed at this flow condition, and a laminar-to-turbulent transition emerges as a result of certain flow perturbations. Hence, our objective is to design feedback control strategies to suppress the transient energy growth and further prevent the emergence of laminar-to-turbulent transition.

The flow control configuration is illustrated in figure 1. We introduce actuation in the form of blowing and suction in the wall-normal direction on the upper and lower channel walls. The velocity profile of the actuation is spatially periodic in the streamwise and spanwise directions, in accordance with the channel flow model discussed below. In this study, we consider velocity sensors along the channel interior, but the methods we introduce are valid equally with wall-based sensing as well.

We decompose the flow state $\boldsymbol{q}$ into base state $\bar{\boldsymbol{q}}$ and small perturbation $\boldsymbol{q}^{\prime}$ ($\boldsymbol{q}=\bar{\boldsymbol{q}}+\boldsymbol{q}^{\prime}$), where $\boldsymbol{q}=[\boldsymbol{u},\boldsymbol{v},\boldsymbol{w},\boldsymbol{p}]^{\mathsf{T}}$ ($\boldsymbol{p}$ is pressure), and $(\cdot){{}^{\mathsf{T}}}$ represents transpose. The kinetic energy density of a perturbation is defined as,

where $V$ is the volume of the computational domain.

By substituting the expression of $\boldsymbol{q}=\bar{\boldsymbol{q}}+\boldsymbol{q}^{\prime}$ into the Navier–Stokes equations, and assuming that the perturbation is much smaller than the base state in magnitude ($|\boldsymbol{q}^{\prime}|\ll|\bar{\boldsymbol{q}}|$), we linearize the equations by retaining linear terms and neglecting higher-order nonlinear terms as follows,

The linearzied Navier–Stokes equations are further manipulated to be expressed in terms of wall-normal velocity $\boldsymbol{v}^{\prime}$ and wall-normal vorticity $\boldsymbol{\eta}^{\prime}$ as described in (Schmid & Henningson, 2001) as,

Next, the real-valued three-dimensional perturbation of wall-normal velocity and wall-normal vorticity are expressed using Fourier expansions in the homogeneous $x$- and $z$-directions,

where $\hat{\boldsymbol{v}}(y,t)$ and $\hat{\boldsymbol{\eta}}(y,t)$ are amplitude functions of the perturbation associated with streamwise wavenumber $\alpha$ and spanwise wavenumber $\beta$. By plugging (4) into the linearized Navier–Stokes equations (2), we obtain the Orr-Sommerfeld and Squire equations as follows,

where $\mathcal{D}$ represents differentiation with respect to the wall-normal direction $y$, $\boldsymbol{{\bar{u}}}_{y}$ and $\boldsymbol{\bar{u}}_{yy}$ denote the first and second derivatives of $\boldsymbol{\bar{u}}$ with respect to $y$, and $k^{2}:=\alpha^{2}+\beta^{2}$. The governing equations (5) are in a state-space form

where ${X_{u}}=[\hat{\boldsymbol{v}},\hat{\boldsymbol{\eta}}]^{\mathsf{T}}$ is the state, and subscript $(\cdot)_{u}$ indicates uncontrolled system.

In the $y$-direction, flow variables are represented by Chebyshev polynomials with $N=101$ discrete collocation points, and a no-slip boundary condition is prescribed at the upper and lower walls for the uncontrolled baseline flow: i.e., $\hat{\boldsymbol{v}}(\pm h)={\hat{\boldsymbol{v}}}_{y}(\pm h)=\hat{\boldsymbol{\eta}}(\pm h)=0$.

In the control design, we introduce actuation in the form of wall-normal blowing and suction at the upper and lower channel walls (see figure 1). This actuation modifies the uncontrolled dynamic system (6) to form a controlled dynamical system

where $A$ is the system matrix, $X$ is the state, $U$ is the input, and ${B}$ is the input matrix that maps the influence of control inputs to the state evolution. Here the control input is selected to be ${U}=\frac{\partial}{\partial t}[\hat{\boldsymbol{v}}|_{+h},\hat{\boldsymbol{v}}|_{-h}]^{T}$, representing the rate of change of wall-normal velocity on the upper and lower walls. Since the control input $U$ represents the change of Fourier coefficients of wall-normal velocity $\hat{\boldsymbol{v}}$, in figure 1 it is shown in the form of a sinusoidal wave. The no-flow-through boundary conditions ($\hat{\boldsymbol{v}}|_{\pm h}=0$) are excluded from $X_{u}$ in (6). In the controlled case, $\hat{\boldsymbol{v}}|_{\pm h}$ are nonzero, so we append these two state variables to the flow state $X_{u}$ to form a new state $X=[X_{u},\hat{\boldsymbol{v}}|_{+h},\hat{\boldsymbol{v}}|_{-h}]^{\mathsf{T}}$. Analogously, the dynamics matrix $A_{u}$ is modified and denoted by $A$ to account for the new state.

In all that follows, we transform all quantities to their equivalent real-valued representations so that $A\in\mathbb{R}^{n\times n}$, $X\in\mathbb{R}^{n}$, ${B}\in\mathbb{R}^{n\times m}$, and $U\in\mathbb{R}^{m}$. Further, all measured sensor outputs $Y\in\mathbb{R}^{p}$ to be considered in this study will be represented by the output equation

Further details about the model formulation can be found in (McKernan et al., 2006).

In this study, we aim to use feedback control to reduce the transient energy growth (TEG). We consider TEG due to some initial flow perturbation $X(t_{0})=X_{0}$.

The associated system response is given by $X(t)=\mathrm{e}^{A(t-t_{0})}X_{0}$, and the associated perturbation kinetic energy is given as,

where $Q=Q^{\mathsf{T}}>0$. Further, the maximum TEG is defined as,

which results from a so-called worst-case or optimal perturbation (Butler & Farrell, 1992b). The system can exhibit TEG whenever $G>1$. In this study, we always evaluate the maximum TEG for a given system. The worst-case perturbations are calculated using the algorithm proposed in (Whidborne & Amar, 2011). We emphasize that, in general, perturbation for the uncontrolled flow will be different from the one for the controlled flow, and so these perturbations must be determined independently.

To use feedback control to achieve TEG reduction, the control input vector $U$ at a given instant is determined from available system information. For a full-state feedback control, the state $X$ is assumed to be known and available for feedback, i.e., full-information control:

where the design variable $K\in\mathbb{R}^{m\times n}$ is called the state feedback gain matrix.

LQR synthesis is based on solving,

subject to the linear dynamic constraint given in (7), where $R>0$. The resulting full-state feedback LQR controller gain matrix $K$ is determined from the solution of an algebraic Riccati equation (Brogan, 1991). Although LQR controllers will not necessarily minimize TEG, they have been shown to reduce TEG in shear flows and to exhibit robustness to parametric uncertainties (Ilak & Rowley, 2008; Martinelli et al., 2011; Sun & Hemati, 2019).

Outside of numerical simulations, full-state feedback controllers are typically not practically viable for flow control; such controllers require knowledge of the full state of the flow, which is usually not directly available for feedback in practice. In order to achieve feedback control with the measured information from sensors, in this study, we propose to use a static output feedback (SOF) control structure. For SOF control, the control input is determined directly from the measured output $Y$ in an analogous manner to full-state feedback, with

where the design variable $F\in\mathbb{R}^{m\times p}$ is called the SOF feedback gain matrix. The SOF form of LQR control was proposed for TEG control in (Yao & Hemati, 2018, 2019).The SOF-LQR solves the same minimization problem as shown in (12), but with an SOF constraint on the feedback law. This is equivalently written as,

The SOF-LQR can be solved using iterative Anderson-Moore methods (Rautert & Sachs, 1997; Syrmos et al., 1997). In this study, we use the Anderson-Moore algorithm with Armijo-type adaptation proposed in (Yao & Hemati, 2018). The details are given in Appendix A.

The first approach we propose stems from the fact that a controller gain matrix contains important information regarding the relative contribution of individual sensors and actuators to the controlled closed-loop dynamics in (15). If we design a controller gain $F_{n}$ based on a sufficiently rich library of candidate sensors, then each column in $F_{n}$ corresponds to a specific sensor in the candidate library, represented as a row of $C_{n}$. Thus, by evaluating the relative norm of each column in $F_{n}$, we can determine the relative contribution of a particular sensor to the control action and the closed-loop dynamics in (15). Here, we assess the relative importance of sensors by evaluating the relative $L_{2}$-norm of each column in the gain matrix $F_{n}$. Let $F_{n}(:,i)$ indicate the $i^{th}$ column of $F_{n}$, then the $L_{2}$-norm is calculated as

where $F_{n}(j,i)$ is the $j^{th}$ element of the $i^{th}$ column of $F_{n}$. Consider, for example, that a column in $F_{n}$ of all zeros and the associated row in $C_{n}$ could be removed completely without altering the closed-loop system response in (15). In general, columns of $F_{n}$ with large norm contribute more to the closed-loop response than the columns of $F_{n}$ with small norm. Thus, the rows in $C_{n}$ associated with the dominant columns in $F_{n}$ indicate sensors that are “more important” for the controlled system dynamics. Therefore, we identify the subset of $r$ columns with the largest relative norm, denoted by their indices ${j_{1},\cdots,j_{r}}$. Then, a new output matrix $C_{r}$ containing a reduced set of $r$ sensors can be constructed from the rows ${j_{1},\cdots,j_{r}}$ of $C_{n}$. In principle, a gain matrix $F_{r}$ for this reduced set of sensors can be determined directly from the columns ${j_{1},\cdots,j_{r}}$ of $F_{n}$, but it is actually beneficial to compute an optimal gain $F$ by re-designing the SOF gain for the system $(A,B,C_{r})$. In the present work, we consider linear quadratic control objectives, so the re-design is performed based on SOF-LQR synthesis via an iterative Anderson-Moore algorithm (see Appendix A).

The method proposed here for sensor selection by column-norm evaluation (CE) is summarized in Algorithm 1. Note that it can be useful to scale the outputs of the candidate sensor library so that each row of $C_{n}$ has unit norm. This results from the fact that the scaling of the sensor output is inversely associated with the scale of the associated column norm in $F_{n}$. Performing this scaling to unit norm output tends to be important when a heterogeneous set of sensors is used to construct the candidate library (e.g., velocity, pressure, and shear-stress); otherwise, the importance of a particular type of sensor can be artificially inflated or deflated by the nature of observable quantity itself.

Lastly, note that a similar procedure to Algorithm 1 can be formulated for actuator selection by considering the relative norm of rows in $F_{n}$, each corresponding to an actuator represented as a column of the input matrix $B$ in (15). These ideas are closely related to sparse controller synthesis techniques based on convex optimization, which can be used to explicitly promote sparsity and design controller gains with many columns or rows of all zeros (Lin et al., 2013; Polyak et al., 2014).

Another sensor selection procedure can be devised using the notion of balanced truncation (Moore, 1981; Antoulas, 2005; Laub et al., 1987). Balanced truncation is a model reduction method for linear systems that works by truncating states with a lesser contribution to a system’s input-output dynamics. To do so, the state-space is first transformed into balanced coordinates $\bar{X}_{b}=TX$, so that the controllability Gramian $W_{c}$ and the observability Gramian $W_{o}$ are equal and diagonal. That is, in balanced coordinates we have $\bar{W}_{c}=\bar{W}_{o}=diag(\sigma_{1},\cdots,\sigma_{n})$, where $\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{n}$ are the system’s Hankel Singular Values (HSVs). Larger HSVs indicate directions of state-space with greater contribution to the input-output dynamics, whereas smaller HSVs indicate directions of state-space with lesser contribution to the input-output dynamics. As such, truncation in balanced coordinates of the $n-r$ states with the smallest HSVs will allow a reduction to an $r$-dimensional state-space, while preserving information that is most important for capturing the input-output dynamics. This idea can be extended for sensor selection from a sufficiently rich library of candidate sensors $C_{n}$, since a reduction to an $r$-dimensional state-space will necessarily lead to an output matrix with redundant rows. Upon eliminating these redundant rows in $C_{n}$—e.g., by means of a pivoted QR decomposition—we will be left with $r$ linearly independent rows (sensors) that are relevant for feedback control.

The balancing transformation can be determined by first computing the system Gramians from the associated Lyapunov equations,

Then, using the lower triangular Cholesky factorizations $L_{o}$ and $L_{c}$ of $W_{o}$ and $W_{c}$, respectively, and the singular value decomposition of their product $L_{o}^{T}L_{c}=\Phi\Sigma\Psi^{T}$, the balancing transformation can be computed as

By retaining the $r$ states with the largest HSVs and truncating the rest, the reduced-order system resulting from balanced truncation can be expressed as

where $A_{b}\in\mathbb{R}^{r\times r}$, $B_{b}\in\mathbb{R}^{r\times m}$, $C_{b}\in\mathbb{R}^{p\times r}$, the reduced-order state $X_{b}\in\mathbb{R}^{r}$, input vector $U\in\mathbb{R}^{m}$, and output vector $Y_{b}\in\mathbb{R}^{p}$. Since $r<n$, the output matrix $C_{b}$ will necessarily have redundant rows. Thus, only $r$ linearly independent rows (sensors) are needed to achieve the same feedback control performance as the reduced-order system with the full sensor library. To see this, consider that the static output feedback control determines the control action directly from the measured output as $U=F_{n}Y_{n}$. This can be approximately achieved based on the reduced-order model attained via balanced truncation as $U=F_{n}Y_{n}\approx F_{n}Y_{b}$ (see figure 2). Now, since $Y_{b}=C_{b}X_{b}$ and $C_{b}$ has redundant rows, it is possible to exactly reproduce this control signal from a reduced set of $r$ outputs $Y_{r}=C_{s}X_{b}$ with $C_{s}\in\mathbb{R}^{r\times r}$ as $U=F_{n}Y_{b}=F_{r}Y_{r}$. This final output signal can be represented in the original basis for the full $n$-dimensional state-space, which determines the specific sensors that are needed to recover full-information feedback control performance using output feedback control.

In order to down-select to a set of $r$ linearly independent rows from the output matrix $C_{b}$, we make use of the column-pivoted QR decomposition as

where $\mathbf{Q}\in\mathbb{R}^{r\times r}$ is an orthogonal matrix and $\mathbf{R}\in\mathbb{R}^{r\times p}$ is an upper triangular matrix with diagonal elements $r_{ii}$, with $|r_{11}|\geq|r_{22}|\geq\dots\geq|r_{nn}|$. The independent rows $j_{1},\cdots,j_{r}$ of $C_{b}$ can be identified by entries in the column permutation matrix $\mathbf{E}\in\mathbb{R}^{p\times p}$. Thus, only these rows of $C_{b}$ are retained in order to down-select the number of sensors and obtain the associated output matrix $C_{s}$ shown graphically in figure 2c. Similarly, these same indices can be used to construct the output matrix $C_{r}$ in the original $n$-dimensional basis for the state-space: simply construct $C_{r}$ from rows $j_{1},\cdots,j_{r}$ of $C_{n}$. All that remains at this point is to re-design an SOF controller based on the system $(A,B,C_{r})$ that uses this reduced set of sensors. A summary of this balanced truncation (BT) sensor selection approach is summarized as Algorithm 2 below. Note that, as with the CE sensor selection approach, this BT approach can be formulated analogously for actuator selection as well. Similar ideas for using balanced model reduction methods for sensor and actuator selection have been investigated in (Manohar et al., 2018b).

The choice of $r$ in the carrying out the above steps to is important for ensuring that the final sensor configuration is able to achieve the desired control performance. One way to make this determination is to evaluate the difference between the input-output dynamics of the $r$-dimensional reduced-order balanced truncation model relative to the $n$-dimensional full-order model. We can evaluate this error by first defining an error system $\Sigma_{e}$ that maps any input $U$ to the associated error at the output $e=Y_{n}-Y_{b}$:

Then, an appropriate $r$ can be determined by evaluating an associated system norm of $\Sigma_{e}$ and ensuring it falls below some threshold $err$. A small system norm When the error system holds small system norms, the ROM captures the original system’s input-output dynamics well. In this study, we can consider both the $\mathcal{H}_{2}$- and $\mathcal{H}_{\infty}$-norms of $\Sigma_{e}$. The $\mathcal{H}_{2}$-norm of $\Sigma_{e}$ is denoted as $\|\Sigma_{e}\|_{\mathcal{H}_{2}}$ and corresponds to the root-mean-square of the impulse response of $\Sigma_{e}$. The $\mathcal{H}_{\infty}$-norm of $\Sigma_{e}$ is denoted by $\|\Sigma_{e}\|_{\mathcal{H}_{\infty}}$ and corresponds to the peak value of the largest singular value of $\Sigma_{e}$.

The sensor configurations and the associated worst-case SOF-LQR control performance for streamwise disturbance with $(\alpha,\beta)=(1,0)$ are presented in figure 5. The maximum TEG of the full-information LQR controller (blue dotted line), the maximum TEG ($G$) for the SOF-LQR controlled flow are reported as a function of the number of sensors in figure 5a, with CE configurations denoted as red crosses and BT configurations as green dots. From this analysis, we see that SOF-LQR control recovers full-information LQR TEG performance when $r_{BT}\geq 6$ sensors are used based on the BT approach. Using the CE approach, SOF-LQR TEG performance recovers to within 5% of the full-information LQR TEG when $r_{CE}\geq 14$ sensors are used. In this case, the sufficiently rich library of candidate sensors $C_{n}$ was constructed using only $\boldsymbol{\hat{v}}$ sensors. As such, the observed differences between the sensor configurations from BT and CE is purely due to the locations of these sensors. The specific sensor configurations obtained by the CE and BT sensor selection methods are illustrated in figure 5b and figure 5c respectively. By both approaches, the near-wall sensors are identified as important for the control starting with as few as $r_{CE}=r_{BT}=3$ sensors. Except for the case with $r_{CE}=9$, all the sensors selected by the CE approach are located either close-to-wall or clustered symmetrically about the channel centerline in the range $|y/h|=[0.77,0.84]$. In contrast, the BT approach tends to yield asymmetrical arrangements in the channel and distributes sensors more uniformly between the channel walls. That said, BT does tend to place a higher concentration of sensors in a similar range as that of the CE from $|y/h|=[0.79,1)$. As we will see, near-wall $\boldsymbol{\hat{v}}$ information also tends to be important for performance recovery due to oblique and spanwise disturbances as well.

We further investigate control performance for streamwise disturbances using the CE sensor configuration for $r_{CE}=14$ and the BT sensor configuration for $r_{BT}=6$. The sensor locations along with the optimal disturbance profiles for the SOF-LQR controllers leading from the CE and BT sensor arrangements are shown in figure 6. As can be seen here, the SOF-LQR controllers designed for sensor configurations from the CE and BT approaches yield similar optimal disturbance profiles as the full-information LQR controller. The sensors selected by the CE approach capture the near-wall spatial features of the $\boldsymbol{\hat{v}}$ disturbance profile. The BT approach yields sensors arrangements that tend to capture spatial features throughout the channel. Interesting, the largest qualitative differences in the SOF-LQR optimal disturbance profiles arise in $\boldsymbol{\hat{u}}$, even though the sensor configurations are restricted to only $\boldsymbol{\hat{v}}$ sensors. These differences—compared to the full-information case—are most visually prominent near the centerline for the BT configuration and near the walls for the CE configuration.

Sensor configurations and the associated worst-case SOF-LQR control performance for oblique disturbances with $(\alpha,\beta)=(1,1)$ are reported in figure 7. Figure 7a illustrates the maximum TEG ($G$) performance as a function of the number of sensors associated with the sensor configurations determined by CE (red cross) and BT (green dots) methods relative to the full-information LQR control performance (blue dotted line). The CE and BT approaches both recover full-information LQR performance when at least $r_{CE}=14$ and $r_{BT}=13$ total sensors are used, respectively. In figure 7b and figure 7c, the specific sensor configurations obtained by the CE and BT sensor selection methods are reported. Note that both approaches were applied to the same candidate library of streamwise ($\boldsymbol{\hat{u}}$) and and wall-normal ($\boldsymbol{\hat{v}}$) sensors. The BT method identifies both streamwise and wall-normal velocity sensors as important; whereas, the CE approach identifies that only wall-normal velocity sensors are important. In both approaches, the near-wall wall-normal velocity sensors are immediately identified as important for control, starting with $r_{CE}=r_{BT}=2$ sensors. As the number of sensors is increased, the CE approach identifies wall-normal velocity sensors in the range $|y/h|=[0.48,0.63]$ as important. In contrast, the BT approach does not select any $\boldsymbol{\hat{v}}$ sensors far from the walls until at least $r_{BT}\geq 11$ sensors are to be used. Instead, the BT approach tends to place $\boldsymbol{\hat{u}}$ sensors at or near the centerline of the channel. When $r_{BT}=7$ or greater, SOF-LQR control based on the BT sensor arrangement yields TEG performance comparable to the full-information LQR controller. This appears to be due to the placement of $\boldsymbol{\hat{u}}$ sensors in the range $|y/h|=[0.5,1)$. The CE approach requires more sensors for this performance recovery. For SOF-LQR controllers based on the CE approach, TEG performance is comparable to the full-information LQR control when $r_{CE}\geq 13$. As in the BT case, the CE approach finds that information in the approximate range $|y/h|=[0.5,1)$ is important for recovering this performance, except now it is $\boldsymbol{\hat{v}}$ information that is deemed important.

We will investigate control performance for oblique disturbances in the remainder using the CE sensor configuration for $r_{CE}=14$ and the BT sensor configuration for $r_{CE}=13$. The optimal disturbance profile leading to the maximum TEG for controllers based on these CE and BT sensor arrangements are reported in figure 8.

Similar to the streamwise disturbance case, the SOF-LQR optimal disturbance profiles from both the CE and BT sensor configurations are qualitatively similar to the full-information LQR optimal disturbance profiles. In both the CE and BT approaches, the placement of $\boldsymbol{\hat{v}}$ sensors indicates that near-wall wall-normal velocity information is important. In the CE case, the $\boldsymbol{\hat{v}}$ sensors capture the dominant spatial features of the $\boldsymbol{\hat{v}}$ disturbance profile. The distribution of $\boldsymbol{\hat{v}}$ sensors along the interior of the channel suggests that spatial derivatives $\boldsymbol{\hat{v}}$ are important. In contrast, the BT approach introduces only a single $\boldsymbol{\hat{v}}$ sensor away from the walls, which captures information near the spatial peak in the optimal wall-normal velocity disturbance profile. The BT approach emphasizes sensors that capture the prominent spatial features of the $\boldsymbol{\hat{u}}$ optimal disturbance profile.

Sensor configurations and the associated worst-case SOF-LQR control performance for spanwise disturbances with $(\alpha,\beta)=(0,2)$ are reported in figure 9. The maximum TEG ($G$) for the controlled flow is reported as a function of the number of sensors for sensor configurations determined by CE (red crosses) and BT (green dots) methods in figure 9a. These are compared with the maximum TEG ($G$) for the full-information LQR controller (blue dotted line). From this analysis, it is evident that the BT approach yields a controller that recovers—at least approximately—the full-information control performance when $r_{BT}\geq 10$. In contrast, the CE approach requires $r_{CE}\approx 32$ sensors to get within $1\%$ of the full-information TEG performance. Investigating the specific sensor configurations obtained by the CE and BT sensor selection methods—see figures 9b and 9c, respectively—provides some guidance on why this way be the case. The BT method identifies both streamwise and wall-normal velocity sensors as important; whereas, the CE approach identifies that only wall-normal velocity sensors are important. Even more significant, the BT approach immediately ($r_{BT}=2$) identifies that near-wall sensors are important for control. In contrast, CE does not identify this same near-wall information as important until $r_{CE}\geq 32$. Up until $r_{CE}=32$, all of the sensors from the $CE$ approach are clustered symmetrically about the channel center line in the range $|y/h|=[0.187,0.588]$. The BT approach does not always yield a symmetric sensor arrangement, but the sensors are more evenly distributed throughout the interior of the channel. Given the TEG performance achieved with these sensor configurations, it is evident that wall-normal information in the vicinity of the walls and at the channel center line are important for TEG reduction for spanwise disturbances. Wall-normal velocity information at the center line is important, as the introduction of such a sensor allows the BT approach to recover full-information control performance with $r_{BT}=10$ sensors. In some of the BT arrangements, the center line $\boldsymbol{\hat{v}}$ sensor is replaced by or augmented with a $\boldsymbol{\hat{u}}$ sensor. In these cases, a pair of $\boldsymbol{\hat{v}}$ sensors tend to appear asymmetrically about the center line.

We will investigate control performance for spanwise disturbances in the remainder using the CE sensor configuration for $r_{CE}=32$ and the BT sensor configuration for $r_{BT}=16$. Note that the choice of $r_{BT}=16$ is motivated by the fact that the direct numerical simulations in section 5 run to completion more quickly for this case than for $r_{BT}=10$. The optimal disturbance profile leading to the maximum TEG for controllers based on these CE and BT sensor arrangements are reported in figure 10. Interestingly, the optimal disturbance profiles for the SOF-LQR controllers are strikingly similar with one another and to optimal disturbance profile for the full-information LQR controller—even more so than in the previous cases considered for streamwise and oblique disturbances. In some sense, this is to be expected based on previously reported findings regarding the $(\alpha,\beta)=(0,2)$ disturbance case. This wave-number pair exhibits the highest TEG among all wave-number pairs, and is the least affected by control (Aniketh & Hemati, 2019; Martinelli et al., 2011; Sun & Hemati, 2019). In both the CE and BT approaches, we see that the $\boldsymbol{\hat{v}}$ sensors capture the dominant spatial features of the $\boldsymbol{\hat{v}}$ disturbance profile. However, only the BT sensor placements are able to capture the spatial profile associated with the $\boldsymbol{\hat{u}}$ component of the optimal disturbance. Although the $\boldsymbol{\hat{u}}$ component is smaller in magnitude compared with the $\boldsymbol{\hat{v}}$ component, this has an important consequence for TEG control performance.

Up to this point, all of the control performance analysis has been conducted for nominal conditions at $Re=3000$. In this section, we will investigate the robustness of the TEG reduction when the controllers designed for $Re=3000$ are applied at “off-design” sub-crital $Re$ values. The specific sensor arrangements we will investigate here correspond to the cases with a corresponding detailed analysis in the sections above. These are summarized in table 1.

To investigate robust TEG performance, we again consider closed-loop system responses for optimal perturbations that result in the maximum TEG ($G$) for a given closed-loop system with control applied at the off-design $Re$. Results for SOF-LQR controllers designed at $Re=3000$ then applied at $Re=[500,5500]$ are reported in figure 11. The performance of both controllers at off-design Reynolds numbers is strikingly similar to the corresponding performance at on-design conditions at Reynolds numbers higher than $1000$. These results demonstrate that SOF-LQR controllers can be designed based on BT and CE sensor arrangements robustly recover full-information LQR control performance; however, this robustness is not unconditional.

For streamwise wave disturbances at $Re\leq 500$, the SOF-LQR controllers designed for $Re=3000$—using either CE or BT sensor arrangements—fail to reduce TEG relative to the uncontrolled flow. For spanwise wave disturbances at $Re\leq 1000$, the SOF-LQR designed for $Re=3000$ based on the BT sensor arrangement lacks the same robust performance recovery characteristics as the CE sensor arrangement. In fact, the BT arrangement fails fails to reduce TEG relative to the uncontrolled flow at $Re\leq 500$. However, designs based on the CE sensor arrangement maintain the TEG reduction even at these lower Reynolds numbers. As the TEG under smaller Reynolds number is relatively low, it is not surprised that the TEG reduction ability also decreases. Though there are less chances that the transition will happen compare to higher Reynolds numbers, this suggests us that we need to design the SOF-LQR controllers separately for TEG reduction at low Reynolds numbers. Indeed, SOF-LQR controllers designed using either BT and CE sensor arrangements at “on-design” conditions would recover the full-information LQR performance and reduce TEG relative to the uncontrolled flow.

Direct numerical simulations are performed with various optimal disturbance amplitudes for each wavenumber pair. As shown in figure 12, the transient energy growth density for the smallest disturbance amplitude considered overlaps the linear result. Moreover, we find that the amplification of initial disturbance is suppressed as the amplitude of disturbance increases and nonlinear effects become prominent. Although smaller transient energy growth is observed in the cases with larger initial disturbances, the laminar-to-turbulent transition still emerges as indicated by dashed lines in figure 12. Indeed, large values of the absolute transient energy $E$, rather than the amplification $E/E_{0}$, is more likely to lead to transition. This also indicates that for the same initial disturbance, an ability to reduce the transient amplification can serve to suppress the laminar-to-turbulent transition.

In the controlled flow, the sensor selection strategy evaluated in the direct numerical simulations are listed in table 1. These strategies require the least number of sensors in output feedback control to recover the full-state LQR control performance in terms of reducing transient energy growth. For the streamwise disturbance with $(\alpha,\beta)=(1,0)$, the threshold of minimal seed that leads to a laminar-to-turbulent transition increases from $E_{0}=1.0\times 10^{-4}$ to $E_{0}=1.0\times 10^{-3}$. For the oblique disturbance, the threshold increases from $E_{0}=5.0\times 10^{-5}$ to $E_{0}=5.0\times 10^{-4}$. However, for the spanwise disturbance, the controllers do not suppress transition, and actually lead to an earlier emergence of transition. This occurs for the full-information controller as well (Sun & Hemati, 2019). Although TEG performance is recovered, this case shows that TEG reduction is not the only objective to be considered when it comes to transition control.