Perturbation analysis of TBR model reduction in application

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

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

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

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

TPWL approximation of f( ). Extraction algorithm x 1 Initial system position x 2 x 3 … xn Training trajectory Non-reduced state space 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 Air = W 1 TAi. V 1 f r (xi)=W 1 Tf(xi)

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

Our Approach: W 1 TA 1 V 1 W 1 TA 2 V 1 W 1 TA 3 V 1 x 1 x 2 … xn W 1 TAn. V 1 We used single linear reduction for obtaining projection basis. There are more options: we can perform several reductions and then aggregate bases.

Our Approach: x 0 n Use TPWL to handle nonlinearity x 1 x 2 … xn n n Before we used Krylov-subspace linear reduction (less accurate) Here we use TBR for projection matrices W and V

TBR reduction Hankel operator u t y LTI SYSTEM Past input t 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

Micromachined device example FD model non-symmetric indefinite Jacobian

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

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

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

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

TBR reduction algorithm 1) Compute Controllability and observability gramians P and Q 2) Compute Cholesky factor of P: P = RTR 3) Compute SVD of RQRT: UΣ 2 UT = RQRT 4) 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 ?

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

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

Step 3 – balancing SVD Perturbation behavior of TBR projection is dictated by: 3) Compute SVD of RQRT: UΣ 2 UT = RQRT Symmetric eigenvalue problem for RQRT

Perturbation theory for symmetric eigenvalue problem Eigenvectors of RQRT : Eigenvectors of RQRT + Δ : Mixing of eigenvectors (assuming small perturbations): cik large when λi 0 ≈ λk 0

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

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

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

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

Conclusions n n n 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.