1. Perturbation analysis of TBR model reduction in application to trajectory-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 q 3 ) Want y r (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 W T f(Vx r ) requires order n operations: Vx r =x x n x xrxr V q f r =W T f f xrxr Vx r f(Vx r ) W T f(Vx r )

5. 1.Compute A 1 2.Obtain W 1 and V 1 using linear reduction for A 1 3.Simulate training input, collect and reduce linearizations A i r = W 1 T A i V 1 f r (x i )=W 1 T f(x i ) TPWL approximation of f( ). Extraction algorithm Non-reduced state space Initial system position Training trajectory x1x1 x2x2 x3x3 xnxn …

6. Obtaining projection basis Krylov-subspace methods Fast Don’t guarantee accuracy Balanced-truncation methods Expensive (~n 3 ) Guarantee accuracy For example, V=W= colspan(A -1 B, A -2 B, …, A -q B) We are using this algorithm

7. Our Approach: x1x1 x1x1 x2x2 xnxn … W1TA1V1W1TA1V1 W1TA2V1W1TA2V1 W1TA3V1W1TA3V1 W1TAnV1W1TAnV1 We used single linear reduction for obtaining projection basis. There are more options: we can perform several reductions and then aggregate bases.

8. Use TPWL to handle nonlinearity Before we used Krylov-subspace linear reduction (less accurate) Here we use TBR for projection matrices W and V Our Approach: x0x0 x1x1 x2x2 xnxn …

9. TBR reduction LTI SYSTEM X (state) t u t y Hankel operator Past input Future output 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 non-symmetric indefinite Jacobian FD model

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

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

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: Initial: ( ) Perturbed: ( A, B, C ) TBR reduction Projection basis V Define our problem: How perturbation in the initial system affects TBR projection basis? ~ ~ ~ ~

15. TBR reduction algorithm Our goal: How perturbation in the initial system affects balancing transformation T ? 1)Compute Controllability and observability gramians P and Q 2)Compute Cholesky factor of P : P = R T R 3)Compute SVD of RQR T : UΣ 2 U T = RQR T 4)Projection basis V is first q columns of the matrix T = R T U Σ -1/2

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

17. Step 2 – Cholesky factors 2) Compute Cholesky factor of P : P = R T R P= UDU T, R = UD 1/2 U T How we compute R (SPD) Perturbations (assumed small) P + δP => R + δR RδR + δRR T = δP (Always solvable for δR if the initial system is controllable) Small δP result in small δR

18. Step 3 – balancing SVD 3) Compute SVD of RQR T : UΣ 2 U T = RQR T Perturbation behavior of TBR projection is dictated by: Symmetric eigenvalue problem for RQR T

19. Perturbation theory for symmetric eigenvalue problem Eigenvectors of RQR T : Eigenvectors of RQR T + Δ : Mixing of eigenvectors (assuming small perturbations): c i k large when λ i 0 ≈ λ k 0

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 Order of reduced system || y r – y || 2 Odd order models unstable! Even order models beat Krylov Why??? Unstable!

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

23. Eigenvalue behavior of linearized models Eigenvalues of reduced Jacobians, q=7 Eigenvalues of reduced Jacobians, q=8 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.