Dan Henningson

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Input-output analysis, model reduction and
control applied to the Blasius boundary layer -
using balanced modes

• Dan Henningson
• collaborators
• Shervin Bagheri, Espen Åkervik
• Luca Brandt, Peter Schmid

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Linearized Navier-Stokes for Blasius flow
Discrete formulation
Continuous formulation
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Input-output configuration for linearized N-S
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Solution to the complete input-output problem

• Initial value problem flow stability
• Forced problem input-output analysis

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Ginzburg-Landau example

• Entire dynamics vs. input-output time signals

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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

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Most dangerous inputs, creating the largest
outputs

• Eigenmodes of Hankel operator balanced modes

• Controllability Gramian

• Observability Gramian

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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

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Observability Gramian for GL-equation
Output

• Correlation of sensor impulse response in adjoint
  solution
• Adjoint POD modes
• Ranks states most easily sensed by output
• Provides a means to measure observability

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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

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Balanced modes eigenvalues of the Hankel
operator

• Combine snapshots of direct and adjoint
  simulation
• Expand modes in snapshots to obtain smaller
  eigenvalue problem

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Snapshots of direct and adjoint solution in
Blasius flow
Direct simulation
Adjoint simulation
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Balanced modes for Blasius flow
adjoint
forward
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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

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Model reduction

• Project dynamics on balanced modes using their
  biorthogonal adjoints
• Reduced representation of input-output relation,
  useful in control design

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Impulse response
Disturbance Sensor
Actuator Objective
Disturbance Objective
DNS n105 ROM m50
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Frequency response
From all inputs to all outputs
DNS n105 ROM m80 m50 m2
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Optimal Feedback Control LQG
cost function
g (noise)
Ly
fKk
z
w
controller

Find an optimal control signal f (t) based
on the measurements y(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
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LQG controller formulation with DNS

• Apply in Navier-Stokes simulation

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Performance of controlled system
controller
Noise
Sensor
Actuator
Objective
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Performance of controlled system
Noise
Sensor
Actuator
Objective
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Conclusions

• Input-output formulation ideal for analysis and
  design of feedback control systems
• 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

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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

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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

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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

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The forced problem input-output

• Ginzburg-Landau example
• Input-output for 2D Blasius configuration
• Model reduction

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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

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Feedback control

• LQG control design using reduced order model
• Blasius flow example

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LQG feedback control
cost function
Reduced model of real system/flow
Estimator/ Controller
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Riccati equations for control and estimation gains