Model Reduction of Systems with Symmetries

1. Model Reduction of Systems with Symmetries

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

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 • Equivalentlywith • 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. SVD-truncation of Matrices with Symmetries • Rank SVD-truncation of M is the unique optimal rankapproximation in the Frobenius norm if the gap condition holds • Theorem: gap condition Assume that has the symmetrywith and unitary, then, if the gap condition holds, the rank SVD-truncation has the same symmetry

10. SVD-reduction of Matrices with Symmetries • Theorem: • Proof: • Symmetric matrix is not symmetric in this sense Assume that has the symmetrywith and unitary, then, if the gap condition holds, the rank SVD-truncation has the same symmetry

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 -th row of is equal to 0 gap condition

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

15. Block Circulant Matrices • 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) 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 • 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 • permutation

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

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 structureMarkov 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