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Review Articles

Input-Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows

[+] Author and Article Information
S. Bagheri, D. S. Henningson

Department of Mechanics and Linné Flow Center, Royal Institute of Technology (KTH), S-10044 Stockholm, Sweden

J. Hœpffner

Institut de Recherche sur les Phénomènes Hors Équilibre (IRPHÉ), CNRS-Université d’Aix-Marseille, F-13384 Marseille, France

P. J. Schmid

Laboratoire d’Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France

Appl. Mech. Rev 62(2), 020803 (Feb 19, 2009) (27 pages) doi:10.1115/1.3077635 History: Received March 19, 2008; Revised March 26, 2008; Published February 19, 2009

This review presents a framework for the input-output analysis, model reduction, and control design for fluid dynamical systems using examples applied to the linear complex Ginzburg–Landau equation. Major advances in hydrodynamics stability, such as global modes in spatially inhomogeneous systems and transient growth of non-normal systems, are reviewed. Input-output analysis generalizes hydrodynamic stability analysis by considering a finite-time horizon over which energy amplification, driven by a specific input (disturbances/actuator) and measured at a specific output (sensor), is observed. In the control design the loop is closed between the output and the input through a feedback gain. Model reduction approximates the system with a low-order model, making modern control design computationally tractable for systems of large dimensions. Methods from control theory are reviewed and applied to the Ginzburg–Landau equation in a manner that is readily generalized to fluid mechanics problems, thus giving a fluid mechanics audience an accessible introduction to the subject.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Overview of the open-loop and closed-loop analyses performed in this review. The response in terms of the flow state, kinetic energy, and sensor signal to impulse, and harmonic and stochastic inputs of the parallel, nonparallel, convectively unstable, and globally unstable Ginzburg–Landau equation is investigated in Secs. 2,3. Model reduction of the system is performed in Sec. 4 followed by optimal (LQG), robust (H∞), and reduced-order control design in Sec. 5.

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Figure 2

Local stability concepts based on the linear response of the parallel Ginzburg–Landau equation to a temporally and spatially localized pulse at t=0 and x=0, displayed in the x-t-plane. (a) Stable configuration μ0≤0: The solution at t=t1>0 is damped everywhere. (b) Convectively unstable configuration 0<μ0<μt: The solution at t=t1 is amplified but is zero along the ray x/t=0. (c) Absolutely unstable configuration μt≤μ0: The state is amplified at t=t1 and nonzero along the ray x/t=0.

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Figure 3

The neutral stability curve for the parallel Ginzburg–Landau equation (with cu=0.2) in the (μ0,k)-plane

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Figure 4

The neutral absolute stability curve for the parallel Ginzburg–Landau equation (with γ=1−i) in the (μ0,Umax)-plane

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Figure 5

(a) The spatiotemporal evolution of a disturbance in a globally unstable flow. The disturbance grows exponentially until the cubic nonlinear term −|q|2q (see Refs. 30,45 for details of the nonlinear Ginzburg–Landau equation) causes the disturbance to saturate and oscillate. (b) The energy that corresponds to the evolution in (a) is shown in red, and the linear exponential growth for the linear Ginzburg–Landau equation is shown in dashed black.

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Figure 6

(a) Linear transient growth of a perturbation in space and time: An optimal initial perturbation grows as it enters the unstable domain at branch I at x=−8.2 until it reaches branch II at x=8.2. The two dashed lines depict branches I and II. (b) The corresponding optimal energy growth of the convectively unstable flow in (a).

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Figure 7

The first (a) and second (b) global (black lines) and adjoint eigenmodes (red lines) of the Ginzburg–Landau equation with the absolute value shown in solid and the real part in dashed. The gray area marks the region of instability.

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Figure 8

Shape of an optimal disturbance with the absolute value shown in black, the real and imaginary parts shown in blue and red, respectively. The gray region marks the unstable region, where disturbances grow exponentially. The maximum value of the optimal disturbance is located close to branch I.

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Figure 9

Optimal energy growth, Emax, as a function of time. S configuration μ0<0: The perturbation energy decays exponentially for all time. CU configuration 0<μ0<μc: The perturbation energy is amplified initially but decays to zero asymptotically. GU configuration μc<μ0: The perturbation energy grows exponentially asymptotically. The values of the parameters used in the computations are listed in Table 1.

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Figure 10

Global spectrum of the subcritical Ginzburg–Landau equation (see Table 1), where all the eigenvalues (blue dots) are in the stable half-plane. The unstable domain is in gray and the exact global spectrum is indicated in green. The numerically computed global eigenvalues (blue dots) exhibit a characteristic split, aligning with the resolvent contour that approximately represents machine precision. The resolvent norm contours range from 10−1 to 1015.

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Figure 11

Example of the input-output behavior of the Ginzburg–Landau equation with one input and two outputs. In (a) the evolution in space and time of the state when forced by random noise is shown. The region between the dashed lines is convectively unstable. The locations of the forcing B(x=−11), the first output C1 (at branch I), and the second output C2 (at branch II) are marked by arrows. In (b) and (c), the output signals y1=C1q and y2=C2q are shown, and in (d) the input signal u is shown. Note that in (c) the amplitude of the output signal y1 is less than 1, but further downstream in (b), the second output signal y2 has an amplitude close to 10. This illustrates the amplifying behavior of the system.

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Figure 12

Impulse response of the Ginzburg–Landau equation: (a) The state response to an impulse introduced at t=0 and xw=−11. (b) The impulse response at branch II. The convective character of the instability is evident: A wavepacket grows as it enters the unstable domain but is gradually convected away from this domain before it begins to decay.

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Figure 13

Input-output pseudospectra where the black transfer function contour levels are {100,101,103,104,105,106}. The red contour (with level 208) represents the largest contour value that crosses the imaginary axis. The blue symbols indicate the eigenvalues of A.

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Figure 14

(a) The state response to harmonic forcing located upstream of branch I (lower of the two dashed lines). The largest response is at branch II (upper dashed line) for ω=−0.65. (b) The frequency response, where the output is a Gaussian function (see Appendix), is located at branch II. In the gray area, all forcing frequencies are amplified in the unstable domain, and all other frequencies are damped illustrating a filtering effect. This response corresponds to the thick dashed line representing the imaginary axis in the pseudospectra plot in Fig. 1, and the peak value ‖G‖∞=208 corresponds to the red contour level.

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Figure 15

The response to stochastic forcing. (a) The evolution of the state energy for five different simulations (black lines), the mean state energy given by the solution of the algebraic Lyapunov equation (red solid line), and the energy averaged over 50 simulations (thick dashed line). (b) The thick red line shows the rms value of the Ginzburg–Landau equation when excited by random forcing w at the location marked with an arrow. Five representative snapshots of the response to this forcing are shown by black thin lines; the average over 50 simulations is displayed by a thick blue dashed line.

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Figure 16

The state covariance/controllability Gramian P of the Ginzburg–Landau equation. The Gramian describing how the state components are influenced by an input corresponds in a stochastic framework to the state covariance for white noise as input. The red circle signifies the forcing location (xw=−11), and the dashed box marks the region of instability. The states that are most sensitive to forcing, and thus controllable, are located downstream, at branch II.

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Figure 17

The first (a) and second (b) POD modes obtained from an eigenvalue decomposition of the controllability Gramian in Fig. 1. Note that these modes are orthogonal. The absolute value is shown in solid and the real part in dashed. The gray area marks the region of instability.

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Figure 18

The observability Gramian Q of the Ginzburg–Landau equation. The red circle marks the location of the output C at branch II. The initial states that contribute most to the output are located upstream, at branch I.

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Figure 19

The spatial support of the first 20 global (a), POD (b), and balanced modes (c). The spatial support is defined as the region where the amplitude of a particular mode is larger than 2% of its maximum amplitude. The location of the input (just upstream of branch I) and output (at branch II) is marked with red and green dashed lines, respectively. The global modes span only the region around branch II. The first POD modes (b) are located at branch II, even though the higher modes quickly recover the input. The balanced modes (c) cover the region between the input and output with only two modes. The areas marked with light gray in (a) and (c) represent the spatial support of the adjoint modes for the global and balanced modes. The spatial separation in x of the direct and adjoint modes, shown in (a) for global modes, is absent in (b) for the balanced modes.

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Figure 20

The controllability modal residuals (black line) of the first 20 global modes given by Eq. 56, which is the product of the overlap of the actuator and adjoint mode ⟨ψi,B⟩ (red) and the sensitivity defined by (⟨ψi,ϕi⟩)−1 (blue). Although the overlap of the spatial support of the actuator decreases for higher modes, the controllability still increases due to the rapid growth of the receptivity of higher modes to forcing, quantified by the inverse of ⟨ψi,ϕi⟩.

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Figure 21

Model reduction error of the POD (black), balanced (red), and global (green) modes. For the balanced modes, the error always decays by increasing the number of modes, in contrast to the error of POD modes. The error does not decay at all for the first 50 global modes due to the failure to project the input B located upstream of branch I onto the global eigenmodes located close and downstream to branch II.

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Figure 22

Comparison of the frequency response of the full Ginzburg–Landau equation with three reduced-order models. The blue dashed lines represent the full model of order n=220. The performance of reduced-order models based on r=2, 4, and 6 modes are shown in the (a), (b), and (c), respectively. The red lines represent the balanced modes, the black lines represent the POD modes, and green lines represent the global eigenmodes. We observe that the balanced modes capture the peak value of the frequency response, which represents the main characteristic of the input-output behavior. The approximation of the frequency response for the open-loop case is unsatisfactory for POD models of orders 2 and 4 and for all global-mode models.

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Figure 23

The first (a) and second (b) balanced modes. The modes are nonorthogonal and the adjoint balanced modes are shown in red. The absolute value is shown in solid and the real part in dashed. The gray area marks the region of instability.

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Figure 24

Hankel singular values of the approximate balanced truncation are marked with colored symbols and the exact balanced truncation with black symbols. The number of singular values that are correctly captured increases with the number of snapshots (red: 1000; green: 500; blue: 70 snapshots).

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Figure 25

(a) The mean of the error covariance tr(PM) (lower dashed line) obtained by solving the Riccati equation 80 is compared to the estimation error (blue line) obtained by marching the estimator in time 72. Also, the mean value of the state (top dashed line/red line) is shown and found to be nearly three orders of magnitude larger than the estimation error. It is evident that both the state and the estimation error reach a steady state. (b) The rms value of the error and the state are shown in blue and red lines, respectively. The red and green Gaussian functions represent the location of the input (stochastic disturbances) and the sensor. The error attains its minimum value just downstream of the sensor location and increases upstream and downstream of it.

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Figure 26

The controlled Ginzburg–Landau equation with stochastic excitation: (a) White noise w with zero mean and unit variance W=1 forces the system at x=−11, just upstream of unstable region with input B1 as a Gaussian function (green). Measurements y(t) of the state (red Gaussian) contaminated by white noise with zero mean and variance G=0.1 are taken at xs=0. The actuator u with control penalty R=1 is placed upstream of the sensor at xu=−3. The rms values of the uncontrolled and LQG-controlled state are given by the solid red and black lines, respectively. The absolute value of the state |q| is shown in an x-t-plane in (b), while the lower plot (c) displays the kinetic energy E=‖q‖M as a function of time. The control is only engaged for t∊[250,750]. Dashed lines in (c) indicate the mean value computed from Lyapunov equation.

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Figure 27

(a) Comparison of the frequency response of the open-loop (red), LQG-controlled (black), and H∞-controlled (blue) Ginzburg–Landau equations. For the open loop, the ∞-norm corresponding to the peak value of the response is 151, whereas the 2-norm corresponding the to integral of the response is 20.5. The H∞-controller minimizes the peak value to 18.4 and reduces the 2-norm to 8.7. The LQG/H2-controller, on the other hand, minimizes the 2-norm to 6.1 and reduces the peak value to 20.8. (b) The energy evolution of an optimal disturbance is shown for the convectively unstable Ginzburg–Landau equation (red line) and the closed-loop system computed with LQG/H2 (black) and H∞ (blue).

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Figure 28

Actuator and sensor placement for the supercritical Ginzburg–Landau equation, which yields a stabilizable and detectable system. The spatial support of the actuator (blue bar), sensor (red bar), the unstable domain (gray region), and the unstable global mode (black lines) together with its corresponding adjoint mode (red lines) are shown.

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Figure 29

The spatiotemporal response to an impulse in time induced at x=−10 for the uncontrolled system (a) and LQG-controlled system (b).

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Figure 30

(Top) The perturbation energy of an initial condition, which illustrates the asymptotic growth and the decay of the global mode of the controlled and uncontrolled systems. (Bottom) The spectrum of the uncontrolled (red) and LQG-controlled (black) Ginzburg–Landau equations. The exponential growth of the wavepacket in Fig. 2 is due to one unstable global mode of the open loop shown by the red circle in the unstable half-plane (gray region). The LQG-based closed loop is stable with no unstable eigenvalues.

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Figure 31

The frequency response of the closed feedback loop based on a LQG compensator. The blue dashed lines represent the full model of order n=220. The performance of reduced-order models based on r=2, 4, and 6 modes are shown in (a), (b), and (c), respectively. The red lines represent the balanced modes, the black lines represent the POD modes, and the green lines represent the global eigenmodes. We observe that reduced-order controller based on balanced modes outperforms the other two models. The poor performance of the reduced order based on POD and global modes is directly associated with the unsatisfactory approximation of the open-loop case in Fig. 2.

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