Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

1. Perturbation analysis of TBR model reduction in application totrajectory-piecewise linear algorithm for MEMS structures. Dmitry Vasilyev, Michał Rewieński, Jacob White Massachusetts Institute of Technology

2. Outline • Background • Trajectory-piecewise linear (TPWL) framework for model order reduction • TBR-based reduction procedure for TPWL model reduction • Numerical example: MEMS switch • Perturbation analysis of TBR-generated models • Conclusions

3. Model reduction problem • Requirements for reduced model • Want q << n (cost of simulation is q3) • Want yr(t) to be close to y(t) • Original complex model: • Reduced model:

4. Projection basis approach to reduction • Pick biorthogonal projection matrices W and V • Projection basis are columns of V and W • Yields inefficient representation for f r • Evaluating WTf(Vxr) requires order n operations: x Vxr=x x n q V xr f f r=WTf xr Vxr f(Vxr) WTf(Vxr)

5. TPWL approximation of f( ). Extraction algorithm • Compute A1 • Obtain W1 and V1using linear reduction for A1 • Simulate training input, collect and reduce linearizations Air = W1TAiV1f r (xi)=W1Tf(xi) Initial system position x1 x3 x2 … xn Training trajectory Non-reduced state space

6. Krylov-subspace methods Fast Don’t guarantee accuracy Balanced-truncation methods Expensive (~n3) Guarantee accuracy Obtaining projection basis For example, V=W= colspan(A-1B, A-2B, … , A-q B) We are using this algorithm

7. Our Approach: x1 W1TA1V1 x2 x1 W1TA2V1 … W1TA3V1 xn W1TAnV1 We used single linear reduction for obtaining projection basis. There are more options: we can perform several reductions and then aggregate bases.

8. Our Approach: • Use TPWL to handle nonlinearity • Before we used Krylov-subspace linear reduction (less accurate) • Here we use TBR for projection matrices W and V x0 x2 x1 … xn

9. Hankel operator TBR reduction u y LTI SYSTEM t t Past input Future output X (state) P (controllability) Which states are easier to reach? Q (observability) Which states produces more output? TBR algorithm includes into projection basis most controllable and most observable states

10. Micromachined device example FD model non-symmetric indefinite Jacobian

11. TPWL-TBR results– MEMS switch example Errors in transient Unstable! Odd order models unstable! Even order models beat Krylov ||yr – y||2 Why??? Order of reduced system

12. Hankel singular values, MEMS beam example This is the key to the problem. Singular values are arranged in pairs! # of the Hankel singular value

13. Outline • Background • Trajectory-piecewise linear (TPWL) framework for model order reduction • TBR-based reduction procedure for TPWL model reduction • Numerical example: MEMS switch • Perturbation analysis of TBR-generated models • Conclusions

14. Problem statement Consider two LTI systems: Perturbed: ( A, B, C ) Initial: ( ) ~ ~ ~ TBR reduction TBR reduction ~ Projection basis V Projection basis V Define our problem: How perturbation in the initial system affects TBR projection basis?

15. TBR reduction algorithm • Compute Controllability and observability gramians P and Q • Compute Cholesky factor of P: P = RTR • Compute SVD of RQRT: UΣ2UT = RQRT • Projection basis V is first q columns of the matrix T = RTU Σ-1/2 Our goal: How perturbation in the initial system affects balancing transformation T ?

16. Step 1 - Gramians 1) Compute Controllability and observability gramians P and Q Lyapunov equation for P AP + PAT = -BBT Ã=A + δA Perturbation (assumed small) AδP + δPAT = -(δAP +P(δA)T) (Keeping 1st order terms) Small δA result in small δP (same for Q)

17. Step 2 – Cholesky factors 2) Compute Cholesky factor of P: P = RTR How we compute R (SPD) P= UDUT, R = UD1/2UT P + δP => R + δR Perturbations (assumed small) (Always solvable for δR if the initial system is controllable) RδR + δRRT = δP Small δP result in small δR

18. Step 3 – balancing SVD Perturbation behavior of TBR projection is dictated by: 3) Compute SVD of RQRT: UΣ2UT = RQRT Symmetric eigenvalue problem for RQRT

19. Perturbation theory for symmetric eigenvalue problem Eigenvectors ofRQRT: Eigenvectors of RQRT+ Δ: Mixing of eigenvectors (assuming small perturbations): ciklarge when λi0 ≈ λk0

20. Results of the analysis The closer Hankel singular values lie to each other, the more corresponding eigenvectors of V tend to intermix! • Analysis implies simple recipe for using TBR • Pick reduced order to insure • Remaining Hankel singular values are small enough • The last kept and first removed Hankel Singular Values are well separated • Helps insure that all linearizations stably reduced

21. TPWL-TBR results– MEMS switch example Errors in transient Unstable! Odd order models unstable! Even order models beat Krylov ||yr – y||2 Why??? Order of reduced system

22. Hankel singular values, MEMS beam example This is the key to the problem. We violate our recipe by picking odd-order models! # of the Hankel singular value

23. Eigenvalue behavior of linearized models Eigenvalues of reduced Jacobians, q=8 Eigenvalues of reduced Jacobians, q=7 Another view on the even-odd effect: TBR is adding complex-conjugate pair

24. Conclusions • In this work we used TBR-based linear reduction procedure to generate TPWL reduced models • We performed an analysis of TBR algorithm with respect to perturbation in the system, and suggested a simple recipe for using TBR as a linear reduction algorithm in TPWL framework • Our observations shows that our derivations are correct.