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

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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

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Perturbation analysis of TBR model reduction in application to trajectory-piecewise linear algorithm for MEMS structures. Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology. Outline. Background - PowerPoint PPT Presentation

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Page 1: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

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

Page 2: Dmitry Vasilyev, Michał Rewie ń ski, 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

Page 3: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

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:

( )( ( )) ( )

( ) ( )

dx tf x t Bu t

dty t Cx t

( )( ( )) ( )

( ) ( )

rr r r

r r r

dx tf x t B u t

dt

y t C x t

Reduced model:

( ) , :r q r q qx t f

Page 4: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

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:

Vxr=x

x

n x xrV q

f r=WTff

xr Vxr f(Vxr) WTf(Vxr)

Page 5: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

1.Compute A1

2.Obtain W1 and V1 using linear reduction for A1

3.Simulate training input, collect and reduce linearizations Ai

r = W1TAiV1

f r (xi)=W1Tf(xi)

TPWL approximation of f( ). Extraction algorithm

Non-reduced state space

Initial system position

Training trajectory

x1

x2x3

xn

Page 6: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Obtaining projection basis

Krylov-subspace methods Fast

Don’t guarantee accuracy

Balanced-truncation methods

Expensive (~n3)

Guarantee accuracy

( )( ) ( )

( ) ( )

dx tAx t Bu t

dty t Cx t

For example, V=W=colspan(A-1B, A-2B, … , A-q B)

We are using this algorithm

Page 7: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Our Approach:

x1

x1x2

xn

W1TA1V1

W1TA2V1

W1TA3V1

W1TAnV1

We used single linear reduction for obtaining projection basis.

There are more options: we can perform several reductions and then aggregate bases.

Page 8: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

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:

0

( () ( )( ( )))Ti

Ti

nr r r r TTPWL i i

i

W AVW f xf x w x x W x

x0

x1x2

xn

Page 9: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

TBR reduction

LTI SYSTEM

X (state)

tu

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

Page 10: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Micromachined device example

4 2 2

4 2 20

3

ˆ ( )

( )((1 6 ) ) 12

w

elec a

u u uEI S F p p dy

x x t

d puK u p p

dt

non-symmetric indefinite Jacobian

FD model

Page 11: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

0 5 10 15 2010-3

10-2

10-1

100

101

102

TBR TPWL modelKrylov TPWL model

TPWL-TBR results– MEMS switch example

Errors in transient

Order of reduced system

||yr –

y|| 2

Odd order models unstable!

Even order models beat Krylov

Why???

Unstable!

Page 12: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

0 5 10 15 20 25 30

10 -6

10 -5

10 -4

10 -3

Hankel singular value

Hankel singular values, MEMS beam example

# of the Hankel singular value

This is the key to the problem.

Singular values are arranged in pairs!

Page 13: Dmitry Vasilyev, Michał Rewie ń ski, 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

Page 14: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Problem statement

Consider two LTI systems:Initial:

( )Perturbed:( A, B, C )

TBR reduction

TBR reduction

Projection basis V Projection basis V

Define our problem: How perturbation in the initial system

affects TBR projection basis?

, ,A B C

~

~ ~ ~

Page 15: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

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 = RTR

3)Compute SVD of RQRT: UΣ2UT = RQRT

4)Projection basis V is first q columns of the

matrix

T = RTU Σ-1/2

Page 16: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Step 1 - Gramians

1) Compute Controllability and

observability gramians P and Q

AP + PAT = -BBT Lyapunov equation for P

Perturbation (assumed small)Ã=A + δA

AδP + δPAT = -(δAP +P(δA)T) (Keeping 1st order terms)

0

( ( ) )TA t T AtP e AP P A e dt

Small δA result in small δP

(same for Q)

Page 17: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Step 2 – Cholesky factors

2) Compute Cholesky factor of P: P =

RTRP= UDUT, R = UD1/2UT How we compute R (SPD)

Perturbations (assumed small)P + δP => R + δR

RδR + δRRT = δP (Always solvable for δR if the initial system is

controllable)

Small δP result in small δR

2 2min

1|| || || ||

2 | Re( ( )) |R P

R

Page 18: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Step 3 – balancing SVD

3) Compute SVD of RQRT: UΣ2UT = RQRT

Perturbation behavior of TBR projection is dictated by:

Symmetric eigenvalue problemfor RQRT

Page 19: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Perturbation theory for symmetric eigenvalue problem

Eigenvectors of RQRT :

Eigenvectors of RQRT + Δ :

Mixing of eigenvectors (assuming small perturbations):

cik large when λi

0 ≈ λk0

0

1

Nk

k i ii

e c e

0 0

0 0

( ),

Tk k ii

k i

e ec k i

0 0 01 2, , ..., Ne e e

Page 20: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Results of the analysis The closer Hankel singular

values lie to each other, themore 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

0 0

0 0

( ),

Tk k ii

k i

e ec k i

Page 21: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

0 5 10 15 2010-3

10-2

10-1

100

101

102

TBR TPWL modelKrylov TPWL model

TPWL-TBR results– MEMS switch example

Errors in transient

Order of reduced system

||yr –

y|| 2

Odd order models unstable!

Even order models beat Krylov

Why???

Unstable!

Page 22: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

0 5 10 15 20 25 30

10 -6

10 -5

10 -4

10 -3

Hankel singular value

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!

Page 23: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

Eigenvalue behavior of linearized models

Eigenvalues of reduced Jacobians,

q=7Eigenvalues of reduced Jacobians, q=8

Another view on the even-odd effect:TBR is adding complex-conjugate pair

-3 -2 -1 0 1x 105

-8

-6

-4

-2

0

2

4

6

8 x 106

First linearization pointSecond linearization point

-3 -2 -1 0 1x 105

-8

-6

-4

-2

0

2

4

6

8 x 106

First linearization pointSecond linearization pointFifth linearization point

Page 24: Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology

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.