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

28

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