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

Abstract

The choice and placement of sensors and actuators is an essential factor determining the performance that can be realized using feedback control. This determination is especially important, but difficult, in the context of controlling transitional flows. The highly non-normal nature of the linearized Navier–Stokes equations makes the flow sensitive to small perturbations, with potentially drastic performance consequences on closed-loop flow control. Full-information controllers, such as the linear quadratic regulator (LQR), have demonstrated some success in reducing transient energy growth and suppressing transition; however, sensor-based output feedback controllers with comparable performance have been difficult to realize. In this study, we propose two methods for sensor selection that enable sensor-based output feedback controllers to recover full-information control performance: one based on a sparse controller synthesis approach, and one based on a balanced truncation procedure for model reduction. Both approaches are investigated within linear and nonlinear simulations of a sub-critical channel flow with blowing and suction actuation at the walls. We find that sensor configurations identified by both approaches allow sensor-based static output feedback LQR controllers to recover full-information LQR control performance, both in reducing transient energy growth and suppressing transition. Further, our results indicate that both the sensor selection methods and the resulting controllers exhibit robustness to Reynolds number variations.

Graphic abstract

Appendices

Appendix A Anderson-Moore algorithm with Armijo-type adaptation

Algorithm 3 presents the Anderson-Moore algorithm with Armijo-type adaptation that can be used to solve an SOF-LQR controller [16, 24]. The SOF-LQR problem consists of a coupled set of nonlinear matrix equations whose solution may not yield a global optimal solution. The Anderson-Moore algorithm is an iterative method for computing a locally optimal solution. The algorithm requires an initial stabilizing SOF control, which can influence the resulting SOF-LQR controller. In this work, Algorithm 3 is initialized using a SOF control gain determined using an iterative linear matrix inequality approach [21, 24]. Subsequently, the complexity of a single iteration of the Anderson-Moore algorithm with Armijo type adaptations is of ${\mathcal {O}}(n^3)$. Additional details regarding the SOF-LQR synthesis methods used here can be found in [24].

Appendix B A comparison of column-norm evaluation with convex-optimization-based sparse control synthesis

The column-norm evaluation (CE) method proposed in Sect. 3.1 is a simple heuristic that facilitates sensor selection for large-scale problems. The CE heuristic was inspired by the convex-optimization-based methods proposed in [41]. Here we will provide justification for the CE method by comparing with a convex-optimization-based design on a system for which convex synthesis is computationally tractable. Although the CE method is still heuristic in nature, this example will establish that it is at least a reasonable heuristic to consider when computationally intensive convex-optimization-based methods are not tractable.

Consider the linear state-space system,

$$\begin{aligned} {\dot{x}}&= Ax + Bu \end{aligned}$$

(B1)

$$\begin{aligned} y&=Cx, \end{aligned}$$

(B2)

where

$$\begin{aligned} \begin{aligned} A&= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -0.0046 &{} -0.1978 &{} 0.0039 &{} 0.0133 &{} 0.0127 &{} -0.0285 &{} 0 &{} 0 \\ 0.0380 &{} -0.5667 &{} -0.0029 &{} -0.0014 &{} -0.0100 &{} -0.0232 &{} 0 &{} 0 \\ 0.3259 &{} 0.3570 &{} -0.2947 &{} -0.4076 &{} -0.8152 &{} 0.1064 &{} 1.0000 &{} 0 \\ -0.0045 &{} -0.0378 &{} 0.0070 &{} -0.0654 &{} -0.0397 &{} 0.0709 &{} 0 &{} 0 \\ -0.4020 &{} -0.2149 &{} 0.2266 &{} -0.4093 &{} -0.8210 &{} -0.2786 &{} 0 &{} 1.0000 \\ -0.0730 &{} 0.5683 &{} 0.0148 &{} 0.2674 &{} 0.1442 &{} -0.7396 &{} 0 &{} 0 \\ -9.8100 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 9.8100 &{} 0 &{} 0 &{} 0 &{} 0 \end{array} \right] , \\ B&= I_{8\times 8},\\ C&= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0.0676 &{} -1.1151 &{} 0.0062 &{} -0.0170 &{} -0.0129 &{} 0.1390 &{} 0 &{} 0 \\ 0.1221 &{} 0.1055 &{} -0.0682 &{} 0.0049 &{} 0.0106 &{} 0.0059 &{} 0 &{} 0 \\ -0.0001 &{} 0.0039 &{} 0.0010 &{} 0.1067 &{} 0.2227 &{} 0.0326 &{} 0 &{} 0 \\ -0.0016 &{} 0.0035 &{} -0.0035 &{} 0.1692 &{} 0.1430 &{} -0.4070 &{} 0 &{} 0 \end{array} \right] . \end{aligned} \end{aligned}$$

The dual of this system was used to demonstrate row-sparse controller synthesis in [41] and was reported as benchmark problem HE3 [49] corresponding to a linearized 8-state model of the Bell 201A-1 Helicopter. We work with this modified version of the problem here so as to study column-sparse controller synthesis, corresponding to sensor selection and the CE method.

Our first step will be to compute an optimal controller according to [41]. Since $B=I$, this turns out to be a convex problem that can be solved via (B3) in the sparse LQR step of Algorithm B, to be described momentarily. The initial condition is set to $x_0 = [1,1,...,1]^T$, following the choice in [41]. Solving for the exact optimal control gain $F^{exact}$, we obtain

$$\begin{aligned} F^\text {exact} = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} -0.0181 &{} -0.9803 &{} 0.1325 &{} 0.0028 \\ 0.7672 &{} -0.1281 &{} -0.0674 &{} 0.0205 \\ -1.8853 &{} 19.6594 &{} -0.6169 &{} 1.5326 \\ 0.0896 &{} -0.1614 &{} -0.7120 &{} -0.4899 \\ 0.2274 &{} -0.1401 &{} -6.6221 &{} -3.0483 \\ 0.0154 &{} 0.0517 &{} -0.8353 &{} 0.1169 \\ -1.4943 &{} 21.1077 &{} -4.2401 &{} 0.0429 \\ 0.9987 &{} -4.3681 &{} -11.9121 &{} -6.6448 \end{array} \right] . \end{aligned}$$

The optimal cost associated with this solution is $J^\text {exact}=1.32 \times 10^3$.

Table 3 Column norm values of the SOF gain matrix F

In order to apply the CE heuristic, we evaluate the column norms of the gain matrix $F^\text {exact}$ (see Table 3). According to the CE heuristic, the first column of $F^\text {exact}$ (i.e., the sensor associated with the first row of C) is least important to the closed-loop performance, and so can be removed. By the CE approach, the next candidate for removal would be the second column of $F^\text {exact}$, or the second row of C. Finally, the last candidate for removal would be column 3 of $F^\text {exact}$, or the sensor associated with the third row of C. In our example, we opt to remove two sensors. Following the CE sequence identified above, we have

$$\begin{aligned} C^\text {CE} = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0.1221 &{} 0.1055 &{} -0.0682 &{} 0.0049 &{} 0.0106 &{} 0.0059 &{} 0 &{} 0 \\ -0.0001 &{} 0.0039 &{} 0.0010 &{} 0.1067 &{} 0.2227 &{} 0.0326 &{} 0 &{} 0 \end{array} \right] . \end{aligned}$$

We now re-solve for the optimal control via (B3), which yields

$$\begin{aligned} F^\text {CE} = \left[ \begin{array}{c@{\quad }c} -0.9823 &{} 0.1772 \\ -0.0393 &{} 0.0313 \\ 19.8284 &{} -0.3111 \\ -0.1582 &{} -0.7959 \\ -0.2287 &{} -7.7153 \\ 0.1156 &{} -0.8048 \\ 21.3852 &{} -4.6557 \\ -4.5979 &{} -13.9553 \end{array} \right] . \end{aligned}$$

The cost associated with this solution will be $J^\text {CE} = 1.39 \times 10^3 \approx 1.05J^\text {exact}$. Note that to achieve a sparse sensor solution, a small (5%) price was paid on the performance.

Now to compare with a more rigorous (i.e., non-heuristic) solution approach to establish some justification for the CE heuristic. A column-sparse controller can be obtained by working with Algorithm 4, listed at the end of this appendix. This is a convex-optimization-based controller synthesis that seeks a column-sparse design. The row-sparse control synthesis version of this algorithm was proposed in [41]. The basic idea is to relax the original objective function J to allow for some sparsity in the resulting control design. This is reflected in the condition (B4), where $\xi$ is a user-selected design parameter. A larger $\xi$ will allow for a sparser controller, while $\xi \rightarrow 1$ will tend toward the original optimal control solution. For additional details on the derivation of this algorithm, we refer the reader to [41].

In applying Algorithm 4, we set $\xi = 1.25$ in (B4). This was the same value used in the example reported in [41]. In step 3, we use $10^{-8}$ for the zeroing threshold. The resulting controller gain is

$$\begin{aligned} F^\text {convex} = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} -0.9662 &{} 0.1954 &{} 0 \\ 0 &{} -0.0314 &{} 0.0404 &{} 0 \\ 0 &{} 19.5898 &{} -0.5827 &{} 0 \\ 0 &{} -0.1424 &{} -0.7770 &{} 0 \\ 0 &{} -0.1352 &{} -7.6064 &{} 0 \\ 0 &{} 0.1383 &{} -0.7835 &{} 0 \\ 0 &{} 21.2576 &{} -4.8017 &{} 0 \\ 0 &{} -4.3871 &{} -13.7127 &{} 0 \end{array} \right] . \end{aligned}$$

The associated cost of this column sparse controller is $J^\text {convex} = 1.39 \times 10^3 \approx 1.05J^\text {exact}$.

Two observations regarding these solutions provide justification for the CE method: (1) the same set of sensors is removed/retained between the CE method and the convex-optimization-based synthesis, and (2) the associated cost function for both the CE and convex-optimization-based methods are comparable ($1.05J^\text {exact}$). We note that the specific gain values are different between the two methods. This is an artifact of the manner in which the gains were computed. In the convex-optimization-based control synthesis, the zero rows in the C matrix were not removed, whereas in the CE method they were handling the C matrix in the same manner.

Since the same sparse sensor solution is obtained, a re-design of $F^\text {convex}$ with those zero rows removed does yield an identical control solution $F^\text {CE}$ as obtained by the CE method. The converse is also true. We note that the correspondence between the CE method and the convex-optimization-based method is dependent on the user-specified design variable $\xi$. As such, this correspondence between solutions is not guaranteed. Nonetheless, as we saw in this example, it is possible for the CE method to achieve a comparable solution as the convex-optimization-based approach for particular values of the design parameter $\xi$.

While we have established some justification for the CE method, this justification is by no means universal. The example problem selected here for demonstration was chosen to be convex. This was done to facilitate comparisons with a unique solution—removing the complications arising from iterative methods and multiple local minima in the context of non-convex problems. Even in this simple context of convex problems, the CE method can have tremendous computational advantages over the convex-optimization-based for larger-scale problems such as fluid flows. The semi-definite programming problem underlying Algorithm 4 will scale with ${\mathcal {O}}(n^6)$ in general. Without a computationally tractable alternative design algorithm, one must begin to consider model reduction (see e.g., [15]). When the problem is non-convex—which is the more realistic case in flow control because the actuation will also be sparse—these computational challenges compound. As such, an efficient heuristic method—such as the CE approach—is most welcome.

Fig. 18

Grid resolution study of transient kinetic energy density E for a streamwise-wave and b spanwise-wave disturbance cases

Appendix C Grid resolution study

Grid resolution studies have been performed for streamwise- and spanwise-wave disturbances, as shown in Fig. 18. The coarse and refined meshes contain $64\times 101 \times 64$ and $128\times 101 \times 64$ grid points in the x-, y- and z-direction, respectively. The grids have been tested using different optimal disturbance amplitudes. At the transient energy growth and laminar-to-turbulent transition stages, the results from the coarse and refined meshes overlap before the flows have completely become turbulent. The results in Fig. 18 indicate that the grid resolution is sufficient for the flow condition considered in the present work, which does not require fully resolving the turbulent flow. Because we further examine the transition mechanism in the channel flow, the refined mesh is adopted in the direct numerical simulation to better capture relatively small-scale structures present during the transition process. Moreover, because the flow with oblique-wave disturbance has similar characteristics to the one with streamwise-wave disturbance, the grid selected for streamwise-wave disturbance is sufficient to resolve the flow response to the oblique-wave disturbance.