BART VANLUYTEN, JAN C. WILLEMS, BART DE MOOR 44 th IEEE Conference on Decision and Control December 2005 Model Reduction of Systems with Symmetries

2 Introduction LTI systems SVD-reduction of matrices with symmetries Examples Application to model reduction Simulation example Conclusion Outline

3 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

4 Outline Introduction LTI systems SVD-reduction of matrices with symmetries Examples Application to model reduction Simulation example Conclusion

5 Linear Time Invariant Systems Linear time-invariant input-output systems in discrete time with Equivalently with Block Hankel matrix

6 Outline Introduction LTI systems SVD-reduction of matrices with symmetries Examples Application to model reduction Simulation example Conclusion

7 Unitarily Invariant Norms Square matrix is unitary The norm on is said to be unitarily invariant Ex: Frobenius norm of

8 SVD-truncation Singular Value Decomposition (SVD) of with and unitary Rank k SVD-truncation with

9 Rank SVD-truncation of M is the unique optimal rank approximation 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

10 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

11 Outline Introduction LTI systems SVD-reduction of matrices with symmetries Examples Application to model reduction Simulation example Conclusion

12 Matrices with equal Rows/Columns Permutation matrix: -th and -th rows of are equal -th and -th rows of are equal gap condition

13 Matrices with Zero-Rows/-Columns Diagonal matrix: -th row of is equal to 0 gap condition

14 Block Circulant Matrices Block circulant matrix generated by with Equivalent definition where

15 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

16 Outline Introduction LTI systems SVD-reduction of matrices with symmetries Examples Application to model reduction Simulation example Conclusion

17 Systems with Pointwise Symmetries permutation

18 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:

19 Systems with Pointwise Symmetries permutation

20 Special cases: Even Odd Even/Odd Periodic Impulse Response

21 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

22 Outline Introduction LTI systems SVD-reduction of matrices with symmetries Examples Application to model reduction Simulation example Conclusion

23 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

24 Reduction of Interconnected Systems with

25 Reduction of Interconnected Systems Interconnected system is given by After second order balanced reduction, we have has the same symmetry as !!

26 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

27 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