Model Reduction of Systems with Symmetries
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BART VANLUYTEN, JAN C. WILLEMS, BART DE MOOR
44 th IEEE Conference on Decision and Control
12-15 December 2005
Model Reduction of Systems with Symmetries
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Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
Outline
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Introduction Mathematical modeling Computational complexity issues
Properties of physical models:• Conservativeness• Dissipativity• Symmetries
How can we reduce a symmetric model and obtain a reduced order model that preserves the symmetry?
Model reduction
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Outline
Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
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Linear Time Invariant Systems Linear time-invariant input-output systems in
discrete time
with Equivalently
with Block Hankel matrix
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Outline
Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
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Unitarily Invariant Norms
Square matrix is unitary
The norm on is said to be unitarily invariant
Ex: Frobenius norm of
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SVD-truncation Singular Value Decomposition (SVD) of
with
and unitary
Rank k SVD-truncation
with
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Rank SVD-truncation of M is the unique optimal rankapproximation in the Frobenius norm if the gap condition holds
Theorem: Assume that has the symmetry
with and unitary, then, if the gap condition holds, the rank SVD-truncation has the same symmetry
SVD-truncation of Matrices with Symmetries
gap condition
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Assume that has the symmetry
with and unitary, then, if the gap condition holds, the rank SVD-truncation has the same symmetry
Theorem:
Proof:
Symmetric matrix is not symmetric in this sense
SVD-reduction of Matrices with Symmetries
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Outline
Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
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Matrices with equal Rows/Columns Permutation matrix:
-th and -th rows of are equal
-th and -th rows of are equalgap condition
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Matrices with Zero-Rows/-Columns Diagonal matrix:
-th row of is equal to 0
-th row of is equal to 0gap condition
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Block Circulant Matrices Block circulant matrix generated by
with
Equivalent definition
where
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is block circulant
is block circulant
The same holds for • block -circulant matrices• block skew-circulant matrices
SVD-truncation of block circulant matrix can very nicely be computed using the Discrete Fourier Transform (DFT)
Block Circulant Matrices
gap condition
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Outline
Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
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Systems with Pointwise Symmetries
• permutation
• permutation
•
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Systems with Pointwise Symmetries Proposition:
Assume is stable ( ), and it has the symmetry:
with and given unitary matrices.
Then, if , the balanced reduced system of order has the same symmetry:
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Systems with Pointwise Symmetries
• permutation
• permutation
•
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Special cases:• Even
• Odd
• Even/Odd
Periodic Impulse Response
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Periodic Impulse Response [Sznaier et al] Impulse response is periodic with period
is circulant
Find a -th order reduced model which is also periodic with period
Find such that• is circulant • is small•
Truncated SVD of • gives optimal approximation in any unitarily invariant norm• is again block circulant
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Outline
Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
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Reduction of Interconnected Systems Model Reduction while preserving interconnection
structure
Markov parameters are circulant ! Approaches
• Reduce the building block to order 1, interconnect to get order 2
• Reduce the interconnected system to order 2, and view as interconnection of two systems of order 1
S: order 4
• gives best results
• uses our theory
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Reduction of Interconnected Systems
with
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Reduction of Interconnected Systems Interconnected system is given by
After second order balanced reduction, we have
has the same symmetry as !!
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Reduction of Interconnected Systems• The 8-th order interconnected system• The second order system obtained by interconnecting two
first order approximations of the building blocks• The second order system obtained by approximating the
reduced interconnected system with an interconnection of two identical first order building blocks Input 1 to Output 1
Input 2 to Output 2
Input 1 to Output 2
Input 2 to Output 1
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Outline
Introduction
LTI systems
SVD-reduction of matrices with symmetries
Examples
Application to model reduction
Simulation example
Conclusion
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Conclusion
Model reduction of systems with• Pointwise symmetries• Periodic impulse responses
Model reduction based on SVD preserves these symmetries if the ‘gap condition’ is satisfied
Results based on the fact that SVD-truncation of matrix with unitary symmetries leads to a lower rank matrix with the same symmetries if the ‘gap condition’ holds