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Perturbation analysis of TBR model reduction in application to trajectory-piecewise linear algorithm for MEMS structures

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### 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

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

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:

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)

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

Krylov-subspace methods

Fast

Don’t guarantee accuracy

Balanced-truncation methods

Expensive (~n3)

Guarantee accuracy

Obtaining projection basisFor example,

V=W=

colspan(A-1B, A-2B, … , A-q B)

We are using this algorithm

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.

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

Hankel operatorTBR 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

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

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

- 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

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?

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 ?

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)

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

Step 3 – balancing SVD

Perturbation behavior of TBR projection is dictated by:

3) Compute SVD of RQRT: UΣ2UT = RQRT

Symmetric eigenvalue problem

for RQRT

Perturbation theory for symmetric eigenvalue problem

Eigenvectors ofRQRT:

Eigenvectors of RQRT+ Δ:

Mixing of eigenvectors (assuming small perturbations):

ciklarge when λi0 ≈ λk0

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

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

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=8

Eigenvalues of reduced Jacobians, q=7

Another view on the even-odd effect:

TBR is adding complex-conjugate pair

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.

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