On Model-Based Feedback Flow Control Jonathan Epps, Miguel Palaviccini, Louis Cattafesta

1. On Model-Based Feedback Flow Control Jonathan Epps, Miguel Palaviccini, Louis Cattafesta MAE Department, University of Florida Interdisciplinary Microsystems Group Florida Center for Advanced Aero-Propulsion IFFC-2 Poitiers, France December 8-10, 2010 Supported by AFOSR, NSF, and FCAAP

2. Outline • Choices for Feedback Flow Control Methods • Desirable Model Features • POD-Galerkin Model Shortcomings • Extensions to POD-Galerkin Models • Example I – 2-D cylinder wake (Re=100) • Nonlinear model and controllers • Example II – compressible cavity oscillations • Dynamic phasor control • Limit-cycle oscillations • Lightly-damped, linear, stable oscillations? • Balanced models and the connection between theory and experiments • Outlook

3. Feedback Flow Control ChoicesRef: Pastoor et al., JFM, 2008

4. M=0.275 Example: Adaptive Black-Box ModelCavity Oscillations • Control works but lacks physical insight and basis • Ref: Kegerise et al., JSV, 2007

5. Overview • Model-based feedback flow control techniques require an analytically tractable plant model. • Motivates development of minimal-order flow models • POD models are useful for flow control but… • Exhibit inherent limitations that must be circumvented • Various extensions have been proposed • Focus here is on those most amenable to experimental implementation

6. POD – Limitations & Extensions for Control • “Standard” POD method has limitations • How to respect effects of boundary conditions? Pressure gradient? • When actuation is introduced, flow structures change. How do we account for this? • Low-energy features (e.g., acoustic feedback) can be important to the dynamics. How do we account for these? • Some recent extensions • Traveling POD: shift reference frame for traveling waves • Mode-interpolation techniques • Double POD • Shift modes: add additional mode(s) to capture transient • Phenomenological models: instead of Galerkin projection, base models on physical intuition • Balanced truncation: use adjoint simulations to weight modes according to dynamical importance

7. Model Requirements for Control Synthesis • (a) natural flow (I) as initial condition, • (b) actuated flow (II) not far from the desired controlled flow, • (c) natural transient from (II) to (I) when actuation is turned off, • (d) actuated transient from (I) to (II), • (e) suitability of the model for control design, • (f) possibility of observer design from sensor signals, • (g) implementable in experiments • Ref: Noack et al., AIAA 2004-2408 and JFM 2003. Model should describe dynamics near I and II

8. Generalized Galerkin System • Generalized Galerkin approximation • Leads to generalized system model via Galerkin projection • Objective: obtain a minimal Galerkin model suitable for control • Ref: Noack et al., AIAA 2004-2408

9. Example - Cylinder Wake Minimal Extended Galerkin POD Model Based on 3-mode Galerkin POD model with a shift mode for cylinder wake (Noack et al., 2003, Tadmor et al. 2004). Galerkin approximation is u1 u2 umean u3 us

10. Example: Cylinder Wake Resulting Dynamical Systems Model After projection, phase averaging, & transforming to cylindrical coordinates: obtain which has a limit cycle

11. Example: Cylinder Wake Control of the Cylinder Wake (Tadmor et al., 2004; King et al., 2005) Adding actuation (transverse oscillation velocity w/ appropriate phase) to the unforced system model yields

12. Example: Cylinder Wake Using: transforms the controlled system to

13. Example: Cylinder Wake Key Dynamical Aspects (Tadmor et al., 2004) Amplitude Dynamics and Inertial Manifold of the natural system provide insight (actuation amplitude g assumed constant)

14. Example: Cylinder Wake Lessons Learned • Qualitatively, the model does a good job predicting the salient features of the controlled flow, including the existence of globally stable and unstable limit cycles and a lower bound on the reduction of fluctuation energy for the given control policy. • The low order model is quantitatively accurate near the open-loop limit cycle, but diverges rapidly as the vortices are suppressed and the base flow changes. • Techniques suggested for a posteriori corrections to the model parameters via nonlinear model estimation. • Additional model extensions required.

15. Nonlinear Control Approaches • King et al. (2005) developed several nonlinear controllers based on the “Dynamic Phasor” model recast in the following form • Also described in Rowley and Juttijudata (2005)

16. Energy-Based Controller Dynamics Phasor Models: Cylinder Wake (King et al., 2005) A simple energy-based controller was developed by averaging the control influence on the oscillation amplitude over half a period when cos(Φ-θ)>0. This mean influence is inserted into the dynamics, and the amplitude A is then forced to decay with a rate of –k. In general, the control is: Note: This assumes we know a1, a2, and a3. In real-time experiments, we will need to estimate these.

17. Energy-Based Controller Dynamics Phasor Models: Cylinder Wake (King et al., 2005) Energy Based Control Results k=0.0075

18. Other Nonlinear Controllers • Damping controller that preserves the natural oscillation frequency ω • Input-output linearization • Lyapunov controller based on • Backstepping controller • State feedback control based on a linear parametrically varying model • Opposition controller Have similar effects as the energy-based controller, but the commanded input is undesirable.

19. DNS of Nonlinear Controllers • DNS of the seven controllers were performed • Simplified Galerkin model no longer accurately reflects the flow for peak suppression and performance suffers • Note: passive splitter plate reduces fluctuations by ~60% Unactuated flow - contours of u Energy-based control – max reduction

20. Application to Cavity Oscillations? • Rowley and Juttijudata (2005) postulated a similar, two-state, dynamical system model for cavity oscillations. • Parameters σ, α, ω “tuned” to match unforced oscillator • Parameter estimation methods would be used in experiments • Parameters b1 and b2 “tuned” to match observed transient in simulations w/ sinusoidal forcing at ω

21. Dynamic Phasor Control • As before, the input is chosen to be u = rc cos(θ – θc) and the model is averaged over one period (Krylov-Bogoliubov) • Assumes r is changing slowly, dθ/dt~ω, and inputs u are small • An appropriate choice of θc and rc yields the final model, which, for 0 < κ < σ, has a periodic orbit with amplitude • Idea is to reduce amplitude of oscillations but stay within the range of validity of the model (κ cannot be too large)

22. Dynamic Phasor Control • Kalman filter, assuming dr/dt = 0, is used to estimate the states • where η is a p sensor measurement and • L1 > 0, and L2 = ω – L12 / 2ω (chosen for stable, critically damped observer dynamics) • Both the model and the state observer will only work well when the oscillations are near ω, which depends on the Mach number • Could also estimate ω (as in Pastoor et al. 2008)

23. Dynamic Phasor Control • Model-based control works well for the design Mach number • Oscillations are completely eliminated for 1 ≤ κ / σ ≤ 3 • Too high a value of κ causes the system to leave the region of validity of the model, resulting in increased oscillations.

24. Dynamic Phasor Control • Performance is very sensitive to Mach number (design M=0.6) M=0.55 M=0.65 M=0.7

25. Dynamic Phasor Control • Can this approach be applied in experiments? • Issues • Need to account for strong influence of Mach number? • Adaptive parameter estimation and control? • Only applies to limit-cycle oscillation • Cavity oscillations often lightly-damped, stable, linear • What can we do in this case?

26. Balanced Truncation(Linear Systems) • Consider a linear (stable) state-space system • Idea for obtaining a reduced-order model: • Change to coordinates in which x1 is “most important” state, x2 “less important”,…, xn “least important” • Then throw out (truncate) the least important states • How to define “most important” states? • Two important concepts: controllability and observability • Most controllable states are ones easily excited by an input • Most observable states are ones that have a large effect on output • Balance these concepts: x1 is most controllable and most observable, etc. • Typically produces better control-oriented models than POD/Galerkin

27. What are you interested in capturing? States easily excited by an input States that have large influence on the output Overview of balanced truncation Hankel singular values

28. Balanced POD • Can use standard & adjoint simulations to compute approximate balanced truncation with cost similar to POD (“Balanced POD”) • Rowley, Int. J. Bifurc. Chaos, 2005 • Advantages • Explicitly incorporates effects of actuators and sensors • Considering observability effectively weights the dynamical importance of various modes: low-energy modes that affect the dynamics (e.g., acoustic waves) will be strongly observable, and will not be truncated • Guaranteed error bounds for linear systems, close to best achievable by any model • Disadvantages • Works only for linear systems • Extensions available for nonlinear systems • Computation intractable for systems with more than about 104 states • Not applicable to experiments!

29. Application to Experimental Control of Cavity Oscillations • Limit cycle or lightly damped stable oscillations? • Assuming Gaussian input disturbances, then output puff is Gaussian for a linear system M=0.45 M=0.34 M=0.34 M=0.45: lightly damped oscillations M=0.34: w/ control M=0.34: self-sustained oscillations Rowley et al. (2006)

30. Application to Cavity Oscillations -Eigensystem Realization Algorithm • Illingworth et al., J Sound Vib, (2010), doi:10.1016/j.jsv.2010.10.030 • Cattafesta et al., AIAA-97-1804 • Construct Hankel matrix H(0) from input/output data • Results in balanced model • H(0) = observability x controllability • Calculate A B C D to achieve a balanced realization • SVD and truncate at order n

31. Application to Cavity Oscillations- 2D DNS • LQG Controller Design AFTER dynamic phasor control! • Includes effects of disturbances and noise • Control feedback law • Quadratic cost function • Use Kalman filter to estimate unknown states Weight matrices used to penalize large system states and large control inputs

32. Application to Cavity Oscillations • System identification using ERA compared to spectral analysis • 140 states nearly matches spectral analysis • 8 states models frequency range of Rossiter modes

33. Application to Cavity Oscillations • Results indicate excellent suppression that is robust to Mach # Mdesign=0.6 OL CL M=0.5 M=0.7 OL OL CL CL

34. Present Experiments at UF • Bandpass filter around Rossiter modes @ M=0.3

35. Present Experiments at UF • ERA n=20 • Reasonable comparison versus conventional frequency response • Next step is estimator and then LQG

36. Outlook • Nonlinear (limit cycle) reduced-order models can be obtained via extensions to standard POD that are suitable for control design. • Dynamic phasor control (appropriate amplitude and phase) via physically motivated and formal methods can suppress the oscillations. Lower |u| is associated with energy-based control. • These reduced-order models must “respect” the range of validity of the model. • If the control is too aggressive, the model will no longer be valid and the control performance will suffer. • Suggests additional model extensions and/or adaptive parameter estimation or… • Dynamic phasor control of nonlinear oscillator produces a stable, lightly damped oscillator system and then linear control is applied. • Robust, linear control approaches possible provided a “balanced” reduced-order model can be obtained. • Balanced POD for simulations (requires adjoint) • Balanced state-space realization via ERA for experiments

37. References B. Noack, K. Afanasiev, M. Morzynski, G. Tadmore, and F. Thiele, “A hierarchy of low-dimensional models for the transient and post-transient cyclinder wake.” J. Fluid Mech., vol. 497, pp. 335-363, 2003. G. Tadmor, B. Noack, M. Morzynski, and S. Siegel, “Low-Dimensional Models For Feedback Flow Control. Part II: Control Design and Dynamic Estimation.” Proc. AIAA 2nd Flow Control Conference., pp. 2004-2409, 2004. R. King, M. Seibold, O. Lehman, B. Noack, M. Morzynski, and G. Tadmor, “Nonlinear Flow Control Based on a Low Dimensional Model of Fluid Flow.” In Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems (ed. T. Meureret al. ). Lecture Notes in Control and Information Sciences, vol. 322, pp. 369-386, 2005. C. Rowley and V. Juttijudata, “Model-based Control and Estimation of Cavity Flow Oscillations.” Proc. 44th IEEE Conference on Decision and Control., December 2005. C. Rowley, D. Williams, T. Colonus, R. Muray, and D. Macmynowski V., “Linear Models for Control of Cavity Oscillations. Journal of Fluid Mechanics, vol. 547, pp.317-330, 2006. M. Morzynski, W. Stankiewicz, B. Noack, R. King, F. Thiele, and G. Tadmor, “Continuous Mode Interpolation for Control-Oriented Models of Fluid Flow.” In Active Flow Control (ed. R. King). Notes on Numerical Fluid Mechanics and Multidisciplinary Design., vol. 95, pp. 260-278, 2007. S. Siegal, K. Cohen, J. Seidel, and T. McLaughlin, “State Estimation of Transient Flow Fields Using Double Proper Orthogonal Decomposition (DPOD).” In Active Flow Control (ed. R. King). Notes on Numerical Fluid Mechanics and Multidisciplinary Design., vol. 95, pp. 105-118, 2007. L. Henning and R. King, “Drag Reduction by Closed-Loop Control of a Separated Flow Over a Bluff Body with a Blunt Trailing Edge.” Proc. 44th IEEE Conference on Decision and Control., pp. 494-499, Dec. 2005. S. Illingworth, A. Morgans, and C. Rowley. “Feedback Control of Flow Resonances Using Balanced Reduced-Order Models.” Journal of Sound and Vibration, to appear.