Introduction to Simulation Lecture 25

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Introduction to Simulation - Lecture 25
Model-Order Reduction II
Jacob White
Thanks to Luca Daniel, Jing Li, Joel Phillips,
Michal Rewienski,
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MOR Outline

• Dynamic Linear Case
• Rational Functions
• Projection Framework
• Krylov Methods
• Hankel Reduction and TBR
• Mention a few issues

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Heat Conducting Bar
Demonstration Example
State-Space Description
Given the right scaling
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State-Space Description
Dynamic Linear case

• Original Dynamical System - Single Input/Output
• Reduced Dynamical System
• q ltlt N, but input/output behavior preserved

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An Aside on Transfer Functions Laplace
Transform
Rewrite the ODE in transformed variables
? Transfer Function
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An Aside on Transfer Functions Meaning of H(s)
For Stable Systems, H(jw) is the frequency
response
?Sinusoid
Sinusoid with shifted phase and amplitude
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An Aside on Transfer Functions EigenAnalysis
Transfer Function
Apply Eigendecomposition
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Original System Transfer Function
Reduced Model Transfer Function
Model Reduction Find a low order rational
function matching H(s)
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Degrees of Freedom
Reduced Model Dynamical System
Reduced Model Transfer Function
coefficients
coefficients
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Variable Changes Do not change transfer functions
Reduced Model Transfer Function
Similarity (x Sw) Transformed Transfer Function
Many Dynamical Systems have the same transfer
function!!
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Rational Function Fitting by point matching

• Can match 2q points
• cross multiplying generates a linear system

For i 1 to 2q
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Point Matching Matrix can be ill-conditioned

• Columns contain progressively higher powers of
  the test frequencies
• Must orthogonalize columns during construction

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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Importance of Fitting at low frequency
Correct Steady State behavior requires accurate
match at low frequencies
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Taylor Series Expansion and Moments
Original System Transfer Function Moments
Moments
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Moment Matching for accurate low frequency
behavior
Reduced Model Matches Original Systems Moments
Cross-Multiplying and Matching Terms
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Point Matching Versus Moment matching
Point matching can be very inaccurate in between
points
Moment (derivatives) matching accurate around
expansion point, but inaccurate on wide
frequency band
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Heat Conducting Bar
Dynamic Linear Case
Heat applied at one end, temperature measured at
the other
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Keeping Eigenmodes versus matching moments
Dynamic Linear Case
Heat Flow Results
q1
q1
q3
q10
Exact
Exact
Keep qth slowest modes
Matches q moments
q3
N100
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Explicit Moment Matching Problem
System of equations extremely ill-conditioned
Columns become linearly dependent for large q!
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Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Problems with explicit fitting methods

• Linear Systems for fitting ill-conditioned
• Need specialized algorithms which avoid explicit
  fitting matrix construction
• Rational function must be converted to
  state-space
• Needed by most simulation tools
• Requires root finding procedure, very sensitive
  to parameter variation

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Projection Framework
Dynamic Linear Case
Dimension Reducing Change of Variables
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Projection Framework
Dynamic Linear Case
Change of variables
Equation Testing
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Projection Framework
Dynamic Linear Case
Assumed Biorthogonal Relationship between V and U

• Original System
• Substitute
• Test by multiplying by V
• Previous Slide Assumed that V and U biorthogonal

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Projection Framework
Dynamic Linear Case
Forming the reduced system matrix
qxq

NxN

• No explicit A need, Only Matrix-vector products

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Projection Framework
Dynamic Linear Case
VU can preserve definiteness properties

• Original System
• Reduced System
• If A is ( or -) definite, so is Ar
• Preserves stability in the definite case
• Can also preserve passivity

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Projection Framework
Dynamic Linear Case
Approaches for Picking U and V

• Use Eigenvectors
• Use Time Series Data
• Compute
• Use the SVD to pick q lt k important vectors
• Use Frequency Domain Data
• Compute
• Use the SVD to pick q lt k important vectors
• Use Krylov Subspace Vectors?
• Use Singular Vectors of System Grammians?

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Aside on Krylov Subspaces - Definition
The order k Krylov subspace generated from matrix
A and vector b is defined as
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Projection Framework
Dynamic Linear Case
Moment Matching Theorem
If
And
Then
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Projection Framework
Dynamic Linear Case
Special Case Moment Matching Theorem
If U and V are such that
Then the first q moments of reduced system match
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A Projection Alternative
Dynamic Linear Case
First Invert A before applying reduction
Form reduced model by projecting inverse of A
The Projection Theorem Still Holds!!
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Projection Alternative
Model-Order Reduction
Heat Flow Results
N100
q1
q2
Exact
Matches q moments
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Noninverse Formulation
Model-Order Reduction
Heat Flow Results
N100
q1
q3
Exact
Matches q moments
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Computing U
Dynamic Linear Case
Need for Orthogonalization
Vectors will line up with dominant eigenspace!
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Computing U
Dynamic Linear Case
Need for Orthogonalization
b
orthogonalize
orthogonalize
orthogonalize

• Only requires solves with A and vector inner
  products

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Computing orthogonal U
Dynamic Linear Case
Arnoldi Algorithm
For i 1 to q For j 1 to i
Generates q1 vectors!
Orthogonalize New Vector
Normalize
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Computing U
Dynamic Linear Case
Arnoldi Identity
Rank 1 matrix
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Computing U
Dynamic Linear Case
Arnoldi Identities Continued
Multiplying U by the inverse of A yeilds
Multiplying by the transpose of U
By orthogonality
The Projection Alternative Reduced Model
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Two Existing Approaches

• Moment Matching
• Accurate over a narrow band.
• Matching function value and derivatives.
• Cheap
• O(n) if A is very sparse.

• TruncatedBalanced Realization
• Wide-band accuracy.
• Does not follow all details.
• Theoretical error bound.
• Expensive O(n3)

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Reminder about Eigenanalysis
Transfer Function
Apply Eigendecomposition
Should keep controllable and observable modes,
but should they be the eigenmodes?
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Truncated Balanced Realization Non-symmetric
Systems

• 1. Calculate controllability gramian, P, and
  observability gramian, Q, by solving two Lyapunov
  equations,

• 2. If have Cholesky factors,
• 3. Projection

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Properties of the TBR Reduction

• Globally accurate reduced model.

• Maximum frequency domain error is bounded by,

• Guaranteed stable.

• Expensive
• Lyapunov equation solve
• Singular value decomposition

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Solving Lyapunov Equations

• How to find (approximate) P, Q, or just as good,
  their factors Zb, Zc, efficiently?

• No expensive operations on A no matrix
  decompositions.

• Cheap operations matrix-vector products and
  solves.

• Low rank approximations.
• Zb, and Zc have only a few columns.
• Recent Approach Cholesky-Free ADI methods

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Reduction Based on Hankel Operators
Hankel Operator Maps Past Inputs to Future Outputs
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The Hankel Operator has an SVD

• The Singular Values of the Hankel Operator

• P, Q are Observability and Controllability
  Grammians
• P and Q are NxN matrices
• Hankel Operator has a finite set of singular
  values
• Reduction by ignoring small singular values
• Just like with any matrix

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Summary

• Dynamic Linear Case
• Rational Functions
• Projection Framework
• Krylov Methods
• TBR and Hankel Reduction
• Optimal Reduced Model
• Extremely computationally expensive