Input-output analysis, model reduction and control applied to the Blasius boundary layer - using balanced modesDan Henningson
collaborators
Shervin Bagheri, Espen Åkervik
Luca Brandt, Peter Schmid
Performance of controlled system
Noise Sensor Actuator Objective Control off
Cheap Control
Intermediate control
Expensive Control
Comparing the urms-values of the controlled and uncontrolled case, we see that the disturbances begin to decay after they reach the actuator and are very small at the location of the objective function, z. The cheap and intermediate control works very good, whereas in the expensive control the control energy is weighted too much in the objective function to allow for sufficient control effort.
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
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
Message: … This means that the only thing that is needed for stability and control design of truly complex flows is a good Navier-Stokes solver and its adjoint. No need for special stability codes nor memory demanding storage of big matrices. We will exemplify these ideas in Blasius flow, jet in cross-flow and the shallow cavity, as well as using the Ginzburg-Landau equation.
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
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
I want to briefly mention some recent work that has served as a foundation for the work I will present here. The realization that transient growth associated with global modes can describe locally convectively growing disturbances we first discussed by Cossu and Chomaz (1997) using the Ginzburg-Landau equation. Since then this has been shown in several shear flows and we will also discuss it in this talk. The use of matrix free methods, in particular Krylov-Arnoldi methods has been around for some time, e.g Edwards et al (1994), but only recently been applied to a number of more complicated shear flows. We will also discuss these methods in this talk. With Krylov methods the global modes and transients can be calculated using only snapshots from a Navier-Stokes solver. Snapshots of a Navier-Stokes solution and its adjoint, can also be used to calculate so called balances modes, which preserve important input-output characteristics for control design. In recent work Rowley (2005) introduced these methods for fluids problems and we will further apply them to spatially developing flows here.
The forced problem: input-output
Ginzburg-Landau example
Input-output for 2D Blasius configuration
Model reduction
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
We now get to the input-output analysis, i.e. the mapping between input and output signals. The figure shows a flat plate where several inputs and outputs have been indicated. We have incoming disturbances as inputs, this could be sound, free-stream vortical disturbances or roughness at the wall, for example. Another input is actuation, such as localized blowing and suction. Outputs can be various sensor signals, used for control purposes or as measure of the success of the control (objective function). The aim is to preserve the dynamics of the input-output relationship in reduced order models for control design purposes.
Feedback control
LQG control design using reduced order model
Blasius flow example
LQG feedback control
Estimator/ Controller Reduced model of real system/flow cost function
In the LQG (Linear Quadratic Gaussian) framework we asume that the forcing w is stochastic and we also add noise g to the measurement. We also add an estimator which we would like to force to have the same solution as our original system, using only information avail able from the measurements. We force the estimator with the difference in the two measurements, times a measurement gain L. From the estimated flow we calculate the control signal through a control gain K. The LQG framework gives us a way to optimally calculate the measurement gain L and the control gain K.
Riccati equations for control and estimation gains
We choose the control gain K to minimize the objective function. The optimal K is found from the solution to a Riccati equation. The measurement gain L is chosen such that the covariance of the estimation error P is minimized. The optimal L is also found from a Riccati equation (dual to that of the one for the control problem).
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