Dmitry Vasilyev, Michał Rewie ń ski, Jacob White Massachusetts Institute of Technology
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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:
( )( ( )) ( )
( ) ( )
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
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)
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
…
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
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.
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
…
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
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
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!
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!
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: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
~
~ ~ ~
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
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)
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
Step 3 – balancing SVD
3) Compute SVD of RQRT: UΣ2UT = RQRT
Perturbation behavior of TBR projection is dictated by:
Symmetric eigenvalue problemfor 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 ≈ 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
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
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!
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!
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
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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