1 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca Brandt, Peter Schmid Input-output analysis, model reduction and control applied to the Blasius boundary layer - using balanced modes

2 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Linearized Navier-Stokes for Blasius flow Discrete formulationContinuous formulation

3 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Input-output configuration for linearized N-S

4 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Solution to the complete input-output problem Initial value problem: flow stability Forced problem: input-output analysis

5 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Ginzburg-Landau example Entire dynamics vs. input-output time signals

6 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Input-output operators Past inputs to initial state: class of initial conditions possible to generate through chosen forcing Initial state to future outputs: possible outputs from initial condition Past inputs to future outputs:

7 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Most dangerous inputs, creating the largest outputs Eigenmodes of Hankel operator – balanced modes Observability Gramian Controllability Gramian

8 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Controllability Gramian for GL-equation Correlation of actuator impulse response in forward solution POD modes: Ranks states most easily influenced by input Provides a means to measure controllability Input

9 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Observability Gramian for GL-equation Correlation of sensor impulse response in adjoint solution Adjoint POD modes: Ranks states most easily sensed by output Provides a means to measure observability Output

10 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Controllability and Observability Gramians Correlation of actuator impulse response in forward solution POD modes: Correlation of sensor impulse response in adjoint solution Adjoint POD modes:

11 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Balanced modes: eigenvalues of the Hankel operator Combine snapshots of direct and adjoint simulation Expand modes in snapshots to obtain smaller eigenvalue problem

12 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Snapshots of direct and adjoint solution in Blasius flow Direct simulation:Adjoint simulation:

13 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Balanced modes for Blasius flow adjoint forward

14 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Properties of balanced modes Largest outputs possible to excite with chosen forcing Balanced modes diagonalize observability Gramian Adjoint balanced modes diagonalize controllability Gramian Ginzburg-Landau example revisited

15 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Model reduction Project dynamics on balanced modes using their biorthogonal adjoints Reduced representation of input-output relation, useful in control design

16 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Impulse response DNS: n=10 5 ROM: m=50 Disturbance Sensor Actuator Objective Disturbance Objective

17 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Frequency response DNS : n=10 5 ROM: m=80 m=50 m=2 From all inputs to all outputs

18 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Optimal Feedback Control – LQG controller Find an optimal control signal   (t) based on the measurements (t) such that in the presence of external disturbances w(t) and measurement noise g(t) the output z(t) is minimized.  Solution: LQG/H2 cost function K LL w z g (noise)

19 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 LQG controller formulation with DNS Apply in Navier-Stokes simulation

20 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Performance of controlled system Noise Sensor Actuator Objective controller

21 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Performance of controlled system NoiseSensorActuatorObjective Control off Cheap Control Intermediate control Expensive Control

22 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Conclusions Input-output formulation ideal for analysis and design of feedback control systems – applied in Blasius flow Balanced modes Obtained from snapshots of forward and adjoint solutions Give low order models preserving input-output relationship between sensors and actuators Feedback control of Blasius flow Reduced order models with balanced modes used in LQG control Controller based on small number of modes works well in DNS Outlook: incorporate realistic sensors and actuators in 3D problem and test controllers experimentally

23 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008

24 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Message Need only snapshots from a Navier-Stokes solver (with adjoint) to perform stability analysis and control design for complex flows Main example Blasius, others: GL-equation, jet in cross-flow

25 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Outline Introduction with input-output configuration Matrix-free methods using Navier-Stokes snapshots The initial value problem, global modes and transient growth Particular or forced solution and input-output characteristics Reduced order models preserving input-output characteristics, balanced truncation LQG feedback control based on reduced order model Conclusions

26 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Background Global modes and transient growth Ginzburg-Landau: Cossu & Chomaz (1997); Chomaz (2005) Waterfall problem: Schmid & Henningson (2002) Blasius boundary layer, Ehrenstein & Gallaire (2005); Åkervik et al. (2008) Recirculation bubble: Åkervik et al. (2007); Marquet et al. (2008) Matrix-free methods for stability properties Krylov-Arnoldi method: Edwards et al. (1994) Stability backward facing step: Barkley et al. (2002) Optimal growth for backward step and pulsatile flow: Barkley et al. (2008) Model reduction and feedback control of fluid systems Balanced truncation: Rowley (2005) Global modes for shallow cavity: Åkervik et al. (2007) Ginzburg-Landau: Bagheri et al. (2008) Invited session on Global Instability and Control of Real Flows, Wednesday 8-12, Evergreen 4

27 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 The forced problem: input-output Ginzburg-Landau example Input-output for 2D Blasius configuration Model reduction

28 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Input-output analysis Inputs: Disturbances: roughness, free-stream turbulence, acoustic waves Actuation: blowing/suction, wall motion, forcing Outputs: Measurements of pressure, skin friction etc. Aim: preserve dynamics of input-output relationship in reduced order model used for control design

29 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Feedback control LQG control design using reduced order model Blasius flow example

30 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 LQG feedback control Estimator/ Controller Reduced model of real system/flow cost function

31 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Riccati equations for control and estimation gains