Actuator and sensor placement for closed-loop control of convective instabilities

Abstract

This work deals with the characterization of the closed-loop control performance aiming at the delay of transition. We focus on convective wavepackets, typical of the initial stages of transition to turbulence, starting with the linearized Kuramoto–Sivashinsky equation as a model problem representative of the transitional 2D boundary layer; its simplified structure and reduced order provide a manageable framework for the study of fundamental concepts involving the control of linear wavepackets. The characterization is then extended to the 2D Blasius boundary layer. The objective of this study is to explore how the sensor–actuator placement affects the optimal control problem, formulated using linear quadratic Gaussian (LQG) regulators. This is carried out by evaluating errors of the optimal estimator at positions where control gains are significant, through a proposed metric, labelled as $\gamma$. Results show, in quantitative manner, why some choices of sensor–actuator placement are more effective than others for flow control: good (respectively, bad) closed-loop performance is obtained when estimation errors are low (respectively, high) in the regions with significant gains in the full-state-feedback problem. Unsatisfactory performance is further understood as dominant estimation error modes that overlap spatially with control gains, which shows directions for improvement of a given set-up by moving sensors or actuators. The proposed metric and analysis explain most trends in closed-loop performance as a function of sensor and actuator position, obtained for the model problem and for the 2D Blasius boundary layer. The spatial characterization of the $\gamma$-metric provides thus a valuable and intuitive tool for the problem of sensor–actuator placement, targeting here transition delay but possibly extending to other amplifier-type flows.

Appendices

Appendix A: $\mathcal {H}_{2}$ norm

Given a stable transfer function G(s), the $\mathcal {H}_{2}$ norm is defined as

$$\begin{aligned} \begin{array}{c} \parallel G(s)\parallel _{2}^{2}:=\frac{1}{2\pi }Tr\bigg (\int _{0}^{+\infty } G(j\omega )^{H}G(j\omega )\mathrm{d}\omega \bigg ) = Tr\bigg (\int _{0}^{+\infty }g(t)^\mathrm{T}g(t)\mathrm{d}t\bigg ) \end{array} \end{aligned}$$

(30)

where g(t) is the impulse response in the time domain. The equivalence between both expressions is given by Parseval’s theorem.

Writing G(s) in the state-space form

$$\begin{aligned} \left\{ \begin{array}{ll} {\dot{q}}(t) = Aq(t) + Bu(t)\\ y(t) = Cq(t) \end{array} \right. \end{aligned}$$

(31)

which has m inputs, l outputs and state of order n. The output $g_{j}(t)$, which corresponds to the jth column of g(t), is the output for the impulsive input $u_{j}(t)=[\dots \; \delta (t)\; \ldots ]^\mathrm{T}$, which is nonzero only in the jth position, where it has an impulsive forcing. Considering $q(0)=0$,

$$\begin{aligned} g_{j}(t) = C\int _{0}^{\infty }e^{A(t-\tau )}Bu_{j}(\tau )\mathrm{d}\tau =Ce^{At}B_{j} \end{aligned}$$

(32)

where $B_{j}$ is the jth column of matrix B. Then, Eq. (30) can be written as

$$\begin{aligned}&\parallel G(s)\parallel _{2}^{2}:=Tr\bigg (\int _{0}^{+\infty }g(t)^\mathrm{T}g(t)\mathrm{d}t\bigg ) = Tr\bigg (\int _{0}^{+\infty }B^\mathrm{T}e^{A^\mathrm{T}t}C^\mathrm{T}Ce^{At}B\mathrm{d}t \bigg )\nonumber \\&\quad =Tr\big (B^\mathrm{T}P_{O}B\big )\nonumber \\&\parallel G(s)\parallel _{2}^{2}:=Tr\bigg (\int _{0}^{+\infty }g(t)g(t)^\mathrm{T}\mathrm{d}t\bigg ) = Tr\bigg (\int _{0}^{+\infty }Ce^{At}BB^\mathrm{T}e^{A^\mathrm{T}t}C^\mathrm{T}\mathrm{d}t \bigg )\nonumber \\&\quad =Tr\big (CP_{C}C^\mathrm{T}\big ) \end{aligned}$$

(33)

where $P_{O}=\int _{0}^{+\infty }e^{A^\mathrm{T}t}C^\mathrm{T}Ce^{At}\mathrm{d}t$ is the observability Gramian and $P_{C}=\int _{0}^{+\infty }e^{At}BB^\mathrm{T}e^{A^\mathrm{T}t}\mathrm{d}t$ is the controllability Gramian.

The squared $\mathcal {H}_{2}$ norm of the system is given by the integral of its output energy in time when it is forced by an impulsive input (30) or by the expected value of the output energy density in the steady state when it is forced by a unit spectral power white-noise forcing.

Given the generalized approach to consider the control of a flow excited by a unit spectral density white-noise forcing, the impulsive response simulation is preferred over the usual attempts to obtain a stochastic forcing. As the latter is not strictly possible, a precise result for the $\mathcal {H}_{2}$ norm can be obtained only through averaging over several realizations, while (32) can be obtained in a single simulation for a single-input single-output (SISO) system and provide a precise result (Fig. 24). For a multiple-input multiple-output (MIMO) system, m impulsive simulations are needed for the $\mathcal {H}_{2}$ norm calculation. The expressions (33) can also be a workaround to stochastic simulations.

Fig. 24

Output energy power or the squared $\mathcal {H}_{2}$ norm for the uncontrolled system in Fig. 4 with input $B_{d}$ and output $C_{z}$

To show this equivalence, the result in [18] is developed.

The output for an input u(t) in (31) is

$$\begin{aligned} y_{l\times 1}(t) = \int _{0}^{t}C_{l\times n}\varPhi _{A}(t,s)_{n\times n}B_{n\times m}u(s)_{m\times 1}\mathrm{d}s \end{aligned}$$

(34)

where $\varPhi _{A}(t,s) = e^{A(t-s)}$. The ith output component is

$$\begin{aligned} y_{i}(t) = \int _{0}^{t}C_{i*}\varPhi _{A}(t,s)Bu(s)\mathrm{d}s \end{aligned}$$

(35)

where $C_{i*}$ is the ith line of matrix C. The energy E(t) of the measurement of the flow subjected to the actuation u(t) is given by

$$\begin{aligned} E(t) = \sum _{i}{\overline{y}}_{i}(t)y_{i}(t) \end{aligned}$$

(36)

where

$$\begin{aligned} {\overline{y}}_{i}(t)y_{i}(t)= & {} \int _{0}^{t}{\overline{C}}_{i*}{\overline{\varPhi }}_{A}(t,s')\\&{\overline{B}}{\overline{u}}(s')\mathrm{d}s'\int _{0}^{t}C_{i*}\varPhi _{A}(t,s)Bu(s)\mathrm{d}s=\\= & {} \int _{0}^{t}\int _{0}^{t}{\overline{C}}_{i*}{\overline{\varPhi }}_{A}(t,s') {\overline{B}}{\overline{u}}(s')C_{i*}\varPhi _{A}(t,s)Bu(s)\mathrm{d}s'\mathrm{d}s \end{aligned}$$

In tensor notation,

$$\begin{aligned} \langle E(t)\rangle =\int _{0}^{t}\int _{0}^{t}{\overline{C}}_{ij} {\overline{\varPhi }}_{Ajk}(t,s'){\overline{B}}_{kp}C_{ir} \varPhi _{Arv}(t,s)B_{vw}\langle {\overline{u}}_{p}(s') u_{w}(s)\rangle \mathrm{d}s'\mathrm{d}s \end{aligned}$$

(37)

where $\langle \, \rangle$ denotes ensemble averaging. As u is a unit spectral density white-noise forcing,

$$\begin{aligned} \langle {\overline{u}}_{p}(s')u_{w}(s)\rangle= & {} \delta _{pw}\delta (s-s')\\ \langle E(t)\rangle= & {} \int _{0}^{t}{\overline{C}}_{ij}{\overline{\varPhi }}_{Ajk}(t,s) {\overline{B}}_{kp}C_{ir}\varPhi _{Arv}(t,s)B_{vw}\delta _{pw}\mathrm{d}s\nonumber \\ \langle E(t)\rangle= & {} \int _{0}^{t}{\overline{C}}_{ij}{\overline{\varPhi }}_{Ajk}(t,s) {\overline{B}}_{kp}C_{ir}\varPhi _{Arv}(t,s)B_{vp}\mathrm{d}s \nonumber \\ \langle E(t)\rangle= & {} B^\mathrm{T}_{pk}\Bigg (\int _{0}^{t}\varPhi ^\mathrm{T}_{Akj}(t,s)C^\mathrm{T}_{ji}C_{ir} \varPhi _{Arv}(t,s)\mathrm{d}s\Bigg )B_{vp}.\nonumber \end{aligned}$$

(38)

Let $P_{O}(t) = \int _{0}^{t}\varPhi _{A}^\mathrm{T}(t,s)C^\mathrm{T}C\varPhi _{A}(t,s)\mathrm{d}s$, $P_{O} = \lim _{t \rightarrow \infty } P_{O}(t)$ and $\langle E^{\infty }\rangle = \lim _{t \rightarrow \infty }\langle E(t)\rangle$. Then,

$$\begin{aligned} \langle E^{\infty }\rangle = B^\mathrm{T}_{pk}P_{O_{kv}}B_{vp} \end{aligned}$$

(39)

In matrix notation,

$$\begin{aligned} \langle E^{\infty }\rangle = Tr(B^\mathrm{T}P_{O}B) = \parallel G(s)\parallel _{2}^{2} \end{aligned}$$

(40)

Showing

$$\begin{aligned} \langle E^{\infty }\rangle = Tr(CP_{C}C^\mathrm{T}) = \parallel G(s)\parallel _{2}^{2} \end{aligned}$$

(41)

is analogous.

Appendix B: $\mathcal {H}_{2}$ norm as modes superposition

From (41), performing the singular value decomposition on matrix $P_{C}$, as in (19)

$$\begin{aligned} P_{C} = \left[ \begin{array}{ccc} \ldots &{} \mid \phi _{ v}\mid &{} \ldots \\ \end{array} \right] \left[ \begin{array}{ccc} \ddots &{} &{} \\ &{} \sigma _{c_v} &{} \\ &{} &{} \ddots \end{array} \right] \left[ \begin{array}{c} \vdots \\ \hline \phi ^\mathrm{T}_{ v} \\ \hline \vdots \end{array} \right] \end{aligned}$$

In tensor notation,

$$\begin{aligned} P_{C_{kr}} = \sigma _{c_v}\phi _{kv}{\phi }_{rv} \end{aligned}$$

The $\mathcal {H}_{2}$ squared norm can then be written as

$$\begin{aligned} \parallel G(s)\parallel _{2}^{2} ={C}^\mathrm{T}_{ki}P_{C_{kr}}C^\mathrm{T}_{ri}= \sigma _{c_v}\phi _{kv} {C}^\mathrm{T}_{ki}{\phi }_{rv}C^\mathrm{T}_{ri} \end{aligned}$$

In matrix notation,

$$\begin{aligned} \parallel G(s)\parallel _{2}^{2} = \sum _{v}\sigma _{c_v} \sum _{i}\langle \phi _{v},C^\mathrm{T}_{*i}\rangle ^{2} \end{aligned}$$

(42)

where $C^\mathrm{T}_{*i}$ is the ith column of matrix $C^\mathrm{T}$.

Showing

$$\begin{aligned} \parallel G(s)\parallel _{2}^{2} = \sum _{r}\sigma _{o_r}\sum _{p} \langle \psi _{r},B_{*p}\rangle ^{2} \end{aligned}$$

(43)

where $B_{*p}$ is the pth column of matrix B and $\sigma _{o_r}$ and the modes $\psi _{r}$ come from the singular value decomposition of matrix $P_{O}$, is analogous.

The expressions (42) and (43) allow the visualization of the $\mathcal {H}_{2}$ norm as the superposition of matrices C and B with the controllability and observability modes, respectively. Besides, they provide a direct formula to measure the influence of the model reduction on the $\mathcal {H}_{2}$ norm.