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Review Article

Linear Closed-Loop Control of Fluid Instabilities and Noise-Induced Perturbations: A Review of Approaches and Tools1

[+] Author and Article Information
Denis Sipp

ONERA-The French Aerospace Lab,
Meudon F-92190, France
e-mail: denis.sipp@onera.fr

Peter J. Schmid

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: pjschmid@imperial.ac.uk

Manuscript received September 20, 2015; final manuscript received March 13, 2016; published online May 3, 2016. Assoc. Editor: Jörg Schumacher.

Appl. Mech. Rev 68(2), 020801 (May 03, 2016) (26 pages) Paper No: AMR-15-1112; doi: 10.1115/1.4033345 History: Received September 20, 2015; Revised March 13, 2016

This review article is concerned with the design of linear reduced-order models and control laws for closed-loop control of instabilities in transitional flows. For oscillator flows, such as open-cavity flows, we suggest the use of optimal control techniques with Galerkin models based on unstable global modes and balanced modes. Particular attention has to be paid to stability–robustness properties of the control law. Specifically, we show that large delays and strong amplification between the control input and the estimation sensor may be detrimental both to performance and robustness. For amplifier flows, such as backward-facing step flow, the requirement to account for the upstream disturbance environment rules out Galerkin models. In this case, an upstream sensor is introduced to detect incoming perturbations, and identification methods are used to fit a model structure to available input–output data. Control laws, obtained by direct inversion of the input–output relations, are found to be robust when applied to the large-scale numerical simulation. All the concepts are presented in a step-by-step manner, and numerical codes are provided for the interested reader.

Copyright © 2016 by ASME
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References

Figures

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Fig. 1

Block diagram of a typical feedback control setup, including plant, compensator, external disturbance sources (w,g), control signal u, measurement signal y, and objective output z

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Fig. 2

Sketch of flow over an open cavity (a) and a backward-facing step (b)—two generic flow configurations representing an oscillator and noise-amplifier flow, respectively. The actuator (u), flow sensor (y), and performance sensor (z) are marked by colored symbols for each configuration. External upstream disturbance sources are indicated by w in the case of an amplifier flow. (a) Feedback configuration and (b) feed-forward configuration.

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Fig. 3

(a) Eigenvalues for flow over an open cavity for four different Reynolds numbers, displayed in the complex frequency (ω)–growth-rate (σ) plane. (b) Principal global modes for flow over an open square cavity at Re=7500, visualized by contours of streamwise velocity. (c) Corresponding adjoint global mode.

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Fig. 4

Modulus of transfer functions for different compositions of the Galerkin bases for flow over an open cavity at Re = 7500. The bases are composed of eight global modes (8 GM) and a varying number of balanced (bPOD) modes.

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Fig. 5

Block diagram of a typical feedback control setup for oscillator flows, including plant, compensator, external noise sources (w,g), control signal u, measurement signal y, and objective output z

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Fig. 6

(a) Principal balanced mode and (b) associated adjoint balanced mode, visualized by contours of the streamwise velocity component. The actuator and sensor locations are indicated by black symbols.

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Fig. 7

Sketch of model reduction procedure (Galerkin projection) for the system matrix. Unstable global modes are indicated in red (direct modes in Vu and adjoint modes in Wu), and balanced modes are indicated in green (direct modes in Vs and adjoint modes in Ws). The reduced system matrix A¯ consists of a diagonal submatrix containing the unstable eigenvalues (red symbols) and a dense submatrix (dark blue) describing the reduced stable subspace dynamics.

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Fig. 8

Modulus of the closed-loop transfer function between sensor noise g and objective signal z (blue curve) and between sensor noise g and control signal u (red curve) for open-cavity flow at Re = 7500 with (l2=∞ and G/W=∞) (SGL).

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Fig. 9

Block diagram illustrating the compensator design process (a) and the application of the reduced-order compensator to the full plant (b). The performance and robustness of the design are evaluated for the configuration on the right.

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Fig. 10

Objective sensor signal z (a) and control signal u (b) for compensated flow over an open cavity at Re = 7500 in the SGL. The simulation has been initialized with the most unstable global mode, whose amplitude has been chosen sufficiently small such that the entire simulation remains in the linear regime. No disturbance/noise sources w and g have been applied.

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Fig. 11

Objective sensor signal z (a) and control signal u (b) for compensated flow over an open cavity applying a compensator designed for Re = 7500 in the SGL to a flow at a Reynolds number of Re = 7000. The initial condition consists of the most unstable global mode, with a sufficiently low amplitude for the simulation to remain in the linear regime.

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Fig. 12

Closed-loop perturbed system. The closed-loop system is composed of the actual cavity flow Tyu and the controller Kuy. The actual transfer function Tyu is slightly different from the reduced transfer function T¯yu by a multiplicative factor 1+Δ. The perturbation Δ displays one input (the control signal zd = u) and one output (the actuator noise wd). We have also indicated an additional input to the closed-loop system, wy, which is required for well-posedness of the H∞ design framework.

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Fig. 13

Modulus of the open-loop transfer function Tyu (solid black line), including GM a+(ω) and a−(ω) (dashed lines), for flow over an open cavity at Re = 7500 stabilized by (a) an SGL-compensator and (b) an MG-compensator. Note that the open-loop transfer function for Re = 7000, included in red, is contained within the bounds over the entire frequency range only for the MG-compensator.

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Fig. 14

Contours of gain margins GM+ (a) and GM (b) and PM (c) in the  log10(G/W)−log10(ℓ2) plane for flow over an open cavity at Re = 7500. Dark contours indicate good robustness properties. The white contour levels represent the least stable eigenvalue of the flow-field at Re = 7000 coupled to a compensator designed at Re = 7500. The white bullet point indicates the coordinates (G/W=1587−ℓ2=2041) corresponding to the LTR calculations.

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Fig. 15

Time signal of objective measurement z (a) and control u (b) for cavity flow at Re = 7000 stabilized by an LTR-compensator designed for Re = 7500

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Fig. 16

Temporal evolution of the perturbation kinetic energy for increasing amplitudes of the initial condition, contrasting the performance of SGL-compensation and robustified LTR-compensation (G/W=1587−ℓ2=2041), for cavity flow at Re = 7500

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Fig. 17

Objective measurement z = y ((a) and (c)) and associated control signal u ((b) and (d)) for control of an open cavity flow at Re = 6000. (a) and (b) SGL-compensator (l2=∞ and G/W=∞). (c) and (d) LTR-compensator with (l2=200 and G/W=250).

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Fig. 18

Impulse responses between u and y. (a)–(d) for different Reynolds numbers, the shear-stress sensor being located at xs=1.05. (f)–(i) For different locations of the estimation sensor (the Reynolds number being equal to Re = 7000). Zeros of the open-loop transfer function T¯yu. (e) For different Reynolds numbers, the shear-stress sensor being located at xs=1.05. (j) For different locations of the estimation sensor (the Reynolds number being equal to Re = 7000).

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Fig. 19

Gain margin GM in the  log10(G/W)−log10(ℓ2) plane. (a)–(d) For different Reynolds numbers, the shear-stress sensor being located at xs=1.05. (e)–(h) For different locations xs of the estimation sensor y: (e) xs=0.75, (f) xs=0.5, (g) xs=0.25, and (h) xs=0.15. The actuator is located at x=−0.1, and the Reynolds number is equal to Re = 7000.

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Fig. 20

Sketch indicating the transfer of information to be modeled by the ARMAX structure. Transfer of information (1) from the control u to the performance sensor z, (2) from the upstream sensor y to the performance sensor z, (3) the observable part of the disturbance environment w measured by y, and (4) the part of w, unobservable by y but impacting the performance sensor z.

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Fig. 21

Learning data set consisting of the recorded measurements from the upstream estimation sensor y (a), input signal u (b), and downstream performance sensor z (c). The validation of the model is shown in (d) where the predicted output (solid black line), for a forcing different from the learning set and for a different disturbance environment, is compared to the true signal (red symbols) from the full system. The sensor measurements y and z and the actuator signal u were considered noise-free.

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Fig. 22

(a) Spectrum magnitude of clean signal z (red) and clean signal y (black) in simulation without control u = 0. The blue lines relate to the chosen u-signals. The solid lines correspond to the raw signals and the dotted lines to the filtered ones. The vertical solid line refers to ωc=2, the dashed-line to ωS, and the dashed–dotted line to ωDNS. (b) Autocorrelation function of the z-signal (red) and cross-correlation functions of the u (black) and y (blue) signals with the z-signal in simulation with control u.

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Fig. 23

Impulse responses (a) from u to z in open-loop (black solid line), from y to z in open-loop (red solid-line) and from y to z in closed-loop (dashed-red line). The impulse response of the control law from y to u is shown in (b).

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Fig. 24

Magnitude (a) and phase (b) of the control law between y and u. The red line refers to the (original) control law obtained for the sampling time ΔtS=0.1, while the black line corresponds to the resampled law (bilinear transform) at the DNS time step ΔtDNS=2×10−3. The vertical solid line refers to ωc=2, the dashed line to ωS, and the dashed–dotted line to ωDNS. Note that the controller is also used to filter the high frequencies present in the y signal, which justifies that the control law must be resampled with respect to the acquisition time ΔtDNS.

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Fig. 25

(a) Control signal u(t) versus time. (b) Performance sensor z(t). (c) Global perturbation energy E(t). In all the plots, the black (respectively, red) line represents the uncontrolled (respectively, controlled) case.

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