Model Order Reduction for Large Scale Dynamical Systems Lecture 5: Control Theory SVD based Methods

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Model Order Reduction for Large Scale Dynamical
SystemsLecture 5 Control Theory (SVD based)
Methods
NXP Semiconductors
Tamara Bechtold
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Basic Approximation Methods Overview
Balanced truncation (BTA) Hankel norm
(HNA) Singular Perturbations (SPA) POD
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Outline

• Short Introduction (for the guests)
• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation
• Hankel Norm Approximation (HNA)
• Case Studies
• Tutorial on SLICOT Library

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Outline

• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation
• Hankel Norm Approximation (HNA)
• Case Studies
• Tutorial on SLICOT Library

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Transfer Function of the Linear Dynamical System

• Often the transfer functions are used as a metric
  for approximation

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Impulse Response

• For a given initial value x0 x(t0) and a given
  input u, the solution of the state-space
  equations is
• Therefore, the output is given by
• If u(t) d (t), y(t) h(t) is called impulse
  response and is defined as
• Transfer function G(s) is the Laplace
  transformation of the impulse response

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Controllability and Observability Gramians

• h(t) can be decomposed into an input-to-state map
  and an state-to-output map
  .
• Thus, the input d causes the state x(t), while
  the initial condition x(0) causes the output
  .
• The grammians corresponding to x and ? are

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Energy Aspects

• The significance of grammians stems from the fact
  that the minimal energy required to steer the
  state of the system from 0 to xr is given by
• The maximal energy produced by observing the
  output of the system with initial state x0 is
  given by
• This provides a way to determine states that are
  hard to reach and/or difficult to observe!
• Furthermore, it can be shown (see lecture 4) that
  those states are hard to reach that have a
  significant component in the span of those
  eigenvectors of P that correspond to small ?(P).
• Analog, those states that have a significant
  component in the span of the eigenvectors of Q
  that correspond to small ?(Q) are difficult to
  observe.

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Related Question

• Given a stable system S , does there
  exist a basis in the state-space in which states
  that are difficult to reach are also difficult to
  observe?
• If yes, we could truncate them, i. e. reduce the
  system order!
• Answer to this question is affirmative. The
  transformation that achieves this goal is called
  a balancing transformation.

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Outline

• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation
• Hankel Norm Approximation (HNA)
• Case Studies
• Tutorial on SLICOT Library

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Basic Idea of Balanced Truncation

• Find a basis transformation T, such that the
  gramians of the transformed system satisfy
• The two gramians are transformed as follows
• Note The product PQ transforms under similarity
  and hence, the eigenvalues ?i(PQ) are
  input-output invariants.
• Their square roots are called Hankel Singular
  Values (HSV) and determine how well a model can
  be approximated by a reduced order model.

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Lyapunov Equations

• The key for the comutation of the HSVs is the
  observation that the gramians P, Q are the
  unique, symetric positive definite solutions to
  the linear matrix equations, known as Lyapunov
  equations

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Clasic BTA Algorithm

• 1. Solve
  for P.
• 2. Solve
  for Q.
• 3. Compute Cholesky factors,
  .
• 4. Compute SVD of Cholesky factors
  .
• 5. Compute the balancing transformation matrices
• 6. Form the balanced realization transformations
  as
• 7. Select the reduce model order and partition
  realization conformally.
• 8. Truncate to form the reduced
  realization.

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Properties of Balanced Truncations
Slide by courtesy of Siep Weiland
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Proof (For the Interested Reader)
Slide by courtesy of Siep Weiland
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Outline

• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation (SPA)
• Hankel Norm Approximation (HNA)
• Case Studies
• Tutorial on SLICOT Library

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Truncation and Residualzation

• Given
• A reduced order model can be obtained by
  eliminating x2, i. e., truncating the state (set
  x2 0). The resulting system is
• An alternative to state truncation is state
  residualization (set ). We get a
  Singular perturbation approximation

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Outline

• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation (SPA)
• Hankel Norm Approximation (HNA)
• Case Studies
• Tutorial on SLICOT Library

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Hankel Norm Approximation (HNA)
The H8 schatten norm of G (L2 induced norm of S)
is defined as the maximum of the highest peek of
the frequency response, i. e. as the largest
singular value of the transfer function
evaluated on the imaginary axes
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Comparision of the Methods
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Outline

• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation
• Hankel Norm Approximation (HNA)
• Case Studies
• Tutorial on SLICOT Library

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Pyrotechnical Microthruster

• EU project µPYROS developed new generic
  microsystems able to deliver an impulse-bit
  thrust or pressure waves within a sub millimeter
  volume of silicon.

Real basic research program to overcome technical
and scientific difficulties and come up with a
demonstrator for a concrete application SPACE
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Microthruster Strusrture
Igniting solid-fuel (new material)
1mm
Propulsive solid-fuel (new material)

• Processes within microthruster

Heat transfer from the resistor to the
ignition substance
Ignition
Membrane rapture
Sustained combustion
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Mathematical Model

• Equation of heat conduction (no external thermal
  effects)
• Initial and boundary conditions
• Joule low

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Finite Element Model

• Initial boundary conditions
• zero initial temperature distribution
• Dirichlet boundary condition TG 0 (at the
  bottom G of the chip)
• Dimension of the problem n 1071, , 79000

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Ports

• Inputs constant heat generation of 150mW applied
  to the resistor
• This corresponds to the single input of the form
• Outputs

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Simulation Result Transient Simulation

• boundary conditions can be varied over time
• necessary for determining the duration time ?
  until ignition (Tignit 400 C)
• requires time consuming calculation

? ? 30ms
temperature C
calculation time for transient simulation gt 120
minutes for a mesh of 5300 nodes
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Tunable Optical Filter

• The DFG project AFON aimed at the development of
  an optical filter, which is tunable by thermal
  means.
• The thin-film filter is configured as a membrane
  in order to improve thermal isolation.
• Fabrication is based on silicon technology.
• Wavelength tuning is achieved through thermal
  modulation of resonator optical thickness, using
  metal resistor deposited onto the membrane.

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Modeling

• Mathematical model heat transfer PDE.
• Spatial discretization with finite elements leads
  to an ODE system.

Temperature distribution after 0.25s of heating
with 1mW.

• Initial boundary conditions
• zero initial temperature distribution
• Dirichlet boundary condition TG 0 (at the
  bottom G of the chip)
• Dimension of the problem n 1668, , 106437

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Ports

• Inputs constant heat generation of 1mW applied
  to the resistor
• This corresponds to the single input of the form
• Outputs

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Simulation Results Transient Simulation
Step response (outer plot) in the centrale
membrane node (Memb1)and step response errors
(inner plot) of the optical filter for the
constant input power of 1mW
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Where to find Models for Testing

• Different models of the microthruster can be
  found at http//www.imtek.de/simulation/index.php?
  pagehttp//www.imtek.uni-freiburg.de/simulation/b
  enchmark/
• We will test SLICOT with very small models
  (dimension 25)
• We will test MOR for ANSYS with very big models
  (dimension 100000)
• We will also combine different approaches

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Transmission Line

• An academic model (created in Pstar)
• DAE System (for MOR this is not so nice as an
  ODE system)
• Dimension of the problem 10 6002
• SISO setup

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Simulation Results
Harmonic response to the
Transient response to the unity step input
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Outline

• Recapitulation of the important items so far
• Balanced Truncation Approximation (BTA)
• Singular Perturbation Approximation
• Hankel Norm Approximation (HNA)
• Summary
• Tutorial on SLICOT Library

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SLICOT-Subroutine LIbrary in Control Theory

• Collection of Fortran 77 subroutines
• Using subroutines from BLAS and LAPACK
• Free for academic research (www.slicot.de)
• Contents
• Usage call the routines from Fortran, C, C,
  MATLAB, Mathematica

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Model Reduction Software in SLICOT

• Reduction of stable models
• Reduction of unstable models

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SLICOT Matlab Comparision
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Overall Performance

• SLICOT has a parallel version.
• Yet, the computational complexity is O(n3).
• Limited to small systems.
• Useful for further reduction of the compact model

Computational times for microthruster model in
seconds on Sun Ultra-80 with 4GB RAM and 450MHz
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Mathlink Interface to SLICOT

• Code by Dr. Rudnyi
• GPL - licensed
• Sources for Slicot.m, slicot.tm and slicot.cpp
  available at http//modelreduction.com/soft/slicot
  /

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Package Post4MOR

• Offers postprocessing of the reduced models
• Reading-in of the original/reduced model in
  MatrixMarket format
• Transient and harmonic simulation
• Plotting the results
• GPL-licensed package available at
  http//modelreduction.com/soft/Post4MOR/
• Can be used in combination with SLICOT
  (slicot.m), MOR for ANSYS (C implementation of
  Arnoldi method) or with any other MOR software!

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Following DEMO is due to the kind help of Kiril
Gordine and Peter FeuersteinThank you