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
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
Linearized Navier-Stokes for Blasius flow
Discrete formulation Continuous formulation
I want to introduce the Blasius flow case that I will use as an example throughout most of this talk. The gray area represents the computational domain, starting at Re-based on delta* = 1000. It is 1000 units long and contains a fringe region at the end 200 units long. The governing equation for disturbances in this flow is the linearized Navier-Stokes equations, seen here. In this talk we will not use the continuous formulation, but assume that the equations have been discretized. When you see du/dt=Au it will simply mean the discretized Navier-Stokes equations. I will use this state space formulation in the rest of the talk.
Input-output configuration for linearized N-S
We start by introducing the state-space formulation for the input-output problem for the Blasius boundary layer. 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 inputs and outputs are introduced through the operators B and C, respectively. This yields the state-space formulation of the linearized Navier-Stokes equation, which we have written in discrete form. A is obtained after discretization and projection of the Navier-Stokes equations on a divergence-free space. For simplicity both input and output operators are here approximated by Gaussian functions.
Solution to the complete input-output problem
Initial value problem: flow stability
Forced problem: input-output analysis
The formal solution to the complete input-output problem can be written a matrix exponential and an integral over the forcing. The matrix exponential is just the formal solution to the initial value problem which gives us the stability properties of the flow, both asymptotic and transient. The forced solution is what is considered in the input-output analysis. We will consider the forced problem in this talk.
Ginzburg-Landau example
Entire dynamics vs. input-output time signals
As an example we consider the Ginzburg-Landau equation in state space form. Recall that the operators B and C determine the location and spatial extent of the input and outputs. The GL-equation is an one-dimensional advection-diffusion equation with an unstable domain between the dashed lines. The left figure shows the entire dynamics in an x-t-diagram. The left figures show three input/output signals. The input is a white noise signal introduced through the operator B_1, just before the unstable domain. There are two outputs, one at branch one extracted through the operator C_1 and one at branch two, extracted through the operator C_2. While the output signal at branch one shows many similarities to the input, the output at branch two show an amplified signal where most of the high frequencies have been filtered out. The latter output signal show familiar structures of wavepackets, similar to more complicated fluid instability problems. Recall that the mapping from the inputs to the outputs can be written by this convolution integral originating from the particular solution to the state space problem.
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:
Now we will introduce several input-output operators that will help us in the analysis. We introduce a controllability operator from past inputs to the initial state u_0. This amounts to limiting the initial conditions to the ones possible to generate with the chosen forcing. We also introduce an observability operator from the initial state to future outputs. This amounts to consider the possible outputs that can be generated by this class of initial conditions. Combining the two operators we can obtain a mapping from past inputs to future outputs through an operator H, called the Hankel operator. We will use these operators in the analysis of the input-output relationships.
Most dangerous inputs, creating the largest outputs
Observability Gramian Controllability Gramian Eigenmodes of Hankel operator – balanced modes
We can consider the most dangerous input. This can be written as the maximum of the inner product of Hf with itself. Using the definition of the Hankel operator and its adjoints we can move all of the operators to the left hand side of the inner product. Linear algebra then tells us that the maximum of that expression is given by the largest eigenvalue of this operator combination. In the expression for the eigenvalue problem the controllability Gramian P and the observability Gramian Q appear. The Gramians are two matrices that are frequently used in control theory to determine controllability and observability of a system. Note that the Gramians can be written as combinations of the controllability and obervability operators and their adjoints.
Controllability Gramian for GL-equation
Input Correlation of actuator impulse response in forward solution
POD modes:
Ranks states most easily influenced by input
Provides a means to measure controllability
Let us take a look at the controllability Gramian for the Ginzburg-Landau equation. It can be viewed as the correlation of the actuator impulse response in the forward solution, and be approximated by XX^T, where X is a matrix of snapshots. The figure shows the matrix and that the largest entries appears at the downstream of the domain, around the upper branch. Recall that eigenmodes of the correlation matrix are POD modes. The modes of the controllability Gramian ranks the states most easily influenced by the input and provides a means to measure controllability.
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
Let us take a look at the observability Gramian for the Ginzburg-Landau equation. It can be viewed as the correlation of the sensor impulse response in the adjoint solution, and be approximated by YY^T, where Y is a matrix of snapshots. The figure shows the matrix and that the largest entries appears at the upstream of the domain, around the lower branch. Eigenmodes of the correlation matrix are now adjoint POD modes. The modes of the observability Gramian ranks the states most easily sensed by the output and provides a means to measure observability.
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:
Balanced modes: eigenvalues of the Hankel operator
Combine snapshots of direct and adjoint simulation
Expand modes in snapshots to obtain smaller eigenvalue problem
We can now calculate the balanced modes as eigenvalues of the Hankel operator, ranking the most dangerous inputs. This can be approximated by combining snapshots of the direct and adjoint simulations, as proposed by Rowley (2005). The Hankel eigenvalue problem can be approximated as XX^TYY^Tu=sig^2u, which is a very large eigenvalue problem. It can be made much smaller by expanding in the snapshots, similar to what is done when POD modes are found using the Sirovich snapshot method.
Snapshots of direct and adjoint solution in Blasius flow
Direct simulation: Adjoint simulation:
We would like to construct the snapshot matrices for the Blasius problem. Here we show four snapshots of the forward and the adjoint solution, making up parts of the X and Y snapshot matrces. The impulse response of the input creates a wavepacket that propagates downstream and the impulse response of the sensor in the adjoint simulation create an upward moving wave packet propagating upstream. The adjoint wavepacket is tilted upward against the shear showing the presence of the Orr-mechanism.
Balanced modes for Blasius flow
adjoint forward
Here we see a number of the balanced modes and their adjoints.
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
The balanced modes have a number of interesting properties. They give the largest outputs possible to excite with the chosen forcing. The forward balanced modes diagonalize the observability Gramian, and the adjoint balanced modes diagonalize the controllability Gramian, with the same eigenvalues on the diagonal, schematically shown for the Ginzburg-Landau problem here.
Model reduction
Project dynamics on balanced modes using their biorthogonal adjoints
Reduced representation of input-output relation, useful in control design
These modes are very good to use for model reduction if one is interested in preserving the input-outout relationship in a small model. We can project the dynamics on the balanced modes by expanding in a sum of such modes, using their biorthogonal adjoints by to project out the expansions coefficients. The reduced order model can then be written in the same state space form, where the expressions for the matrices A, B and C are found by the projection.
Impulse response
DNS: n=105
ROM: m=50 Disturbance Sensor Actuator Objective Disturbance Objective
We can now evaluate the reduced order model by considering the impulse responses from the inputs to the outputs. The responses appear as wavepackets in the outputs.
The response using our linearized DNS code with 100 000 degrees of freedom is shown in black and the red dahsed line shows the response of a reduced-system with only 50 dof. The reduced order model matches perfectly with the DNS.
Frequency response
DNS: n=105
ROM: m=80
m=50
m=2
From all inputs to all outputs
Another way to evaluate the reduced order model is to force the system harmonically and consider the frequency response. Due to the linear nature of the system, the output is a signal with the same frequency as the input, but a different amplitude and a phase shift. The response using our linearized DNS code with 100 000 degrees of freedom is shown in red and the black line shows the response of a reduced-system with only 80 dof, which matches perfectly with the DNS. Using 50 dof (blue line) we capture the complete response for the growing frequencies. As we see the I/O behavior is captured very well.
Optimal Feedback Control – LQG
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 cost function f=Kk Ly w z g
(noise)
Now, that we have our reduced model, it is time to use it to design a controller.
The LQG (Linear Quadratic Gaussian) – problem is to find a control signal phi based on the measurements psi such that in the presence of external disturbances w and measurement noise g the output is minimized.
The solution of the LQG problem involves two Riccati equations based on the reduced order model. This results in a reduced-order controller which coupled to the full-order open-loop system results in closed loop feedback control system.
LQG controller formulation with DNS
Apply in Navier-Stokes simulation
Now we have the control and estimation gains and can apply the controller in the Navier-Stokes simulations. We use a small number of balanced modes in the estimator, which is run along side of the flow calculations. The estimator is driven by the measurement y and returns the control signal phi to the flow actuator in the DNS.
Performance of controlled system
Noise Sensor Actuator Objective controller
We can now evaluate the performance of the controlled system. We excite the system with white noise.
The LQG-controller is engaged for t=1500 – 4500. In the absence of any control effort the stochastic input is amplified and show up at wavepackets in the two output signals. When the control is activated, after a time delay the amplitude begins to decay . When the controller is again disengaged, the amplitude of the disturbance begins to grow.
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