1. A TBR-based Trajectory Piecewise-Linear Algorithm for Generating Accurate Low-order Models for Nonlinear Analog Circuits and MEMS
Dmitry Vasilyev, Michał Rewieński, Jacob White
Massachusetts Institute of Technology

2. Outline
Background
Trajectory-piecewise linear (TPWL) framework for model order reduction
Choice of projection bases
TBR-based reduction procedure for TPWL model reduction
Examples and computational results
Issues in selecting order of the model
Efficiency and accuracy
Future work and conclusions

3. Differential Equation Model
Original complex model:
Model can represent:
Finite-difference spatial discretization of PDEs
Circuits with linear capacitors and inductors
Need accurate input-output behavior

4. Model reduction problem
Original complex model:
Reduced model:
Requirements for reduced model
Want q << n (cost of simulation is q3)
Want yr(t) to be close to y(t)

5. 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 n x V xr q f
f r=WTf

6. Trajectory Piecewise Linear approximation of f( ) [Rewieński, 2001]
Training trajectory x0 x2 x1 …
wi(x) is zero outside circle xn
Simulating trajectory

7. Projection and TPWL approximation yields efficient f r( )
q x 1
Air Ai WT V = q
Air q n n

8. TPWL approximation of f( ). Extraction algorithm
Compute A1
Obtain W1 and V1 using linear reduction for A1
Simulate training input, collect and reduce linearizations Air = W1TAiV1 f r (xi)=W1Tf(xi)
Initial system position x0 x2 x1 … xn
Training trajectory
Non-reduced state space

9. Linearized system has nonsymmetric, indefinite Jacobian
Example problem
RLC line
Linearized system has nonsymmetric, indefinite Jacobian

10. Numerical results – nonlinear RLC transmission line
System response for input current i(t) = (sin(2π/10)+1)/2
Input:
training
input
testing
input
Voltage at node 1 [V]
Time [s]

11. Outline
Background
Trajectory-piecewise linear (TPWL) framework for model order reduction
Choice of projection bases
TBR-based reduction procedure for TPWL model reduction
Examples and computational results
Issues in selecting order of the model
Efficiency and accuracy
Future work and conclusions

12. Key issue: choosing projection basis
Krylov-subspace methods
Fast
Don’t guarantee accuracy
Balanced-truncation methods
Expensive (~n3)
Guarantee accuracy

13. Key issue: choosing projection
Krylov-subspace methods
Balanced-truncation methods
Result:
projection matrices W and V

14. The matter of this presentation
Which method more suitable for TPWL?
Krylov-subspace methods
Balanced-truncation methods
Can we use it?
Our presentation aims to answer this question
Used in previous works

15. Reminder: TBR reduction algorithm
Given linear system (A, B, C)
Compute Controllability and observability gramians P and Q
Compute Cholesky factor of P: P = RTR
Compute SVD of RQRT: UΣ2UT = RQRT Diagonal values of Σ are called the Hankel singular values
Projection basis V contains first r columns of the balancing transformation T = RTU Σ-1/2

16. Our Approach: Use TPWL to handle nonlinearity
Use TBR for projection matrices W and V x0 x2 x1 … xn

17. Outline
Background
Trajectory-piecewise linear (TPWL) framework for model order reduction
Choice of projection bases
TBR-based reduction procedure for TPWL model reduction
Examples and computational results
Issues in selecting order of the model
Efficiency and accuracy
Future work and conclusions

18. Numerical results – RLC transmission line
TBR-based TPWL beat
Krylov-based
4-th order TBR TPWL reaches the limit of TPWL representation
Error in transient
||yr – y||2
Order of the reduced model

19. Micromachined device example
FD model
non-symmetric
indefinite Jacobian

20. 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

21. Outline
Background
Trajectory-piecewise linear (TPWL) framework for model order reduction
Choice of projection bases
TBR-based reduction procedure for TPWL model reduction
Examples and computational results
Issues in selecting order of the model
Efficiency and accuracy
Future work and conclusions

22. Illustrating even-odd behavior for MEMS beam example
Observation: First-point Jacobian has
many complex-conjugate
eigenvalues.
Just curious: How complex-conjugate
pairs are represented by
TPWL models?

23. Eigenvalue behavior of linearized models
Eigenvalues of reduced Jacobians, q=7
Eigenvalues of reduced Jacobians, q=8
TBR is adding complex-conjugate pair

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

25. Explanation of even-odd effect – Problem statement
Consider two LTI systems:
Initial:
( )
Perturbed:
( )
TBR reduction
TBR reduction ~
Projection basis V
Projection basis V
Define our problem:
How perturbation in the initial system
affects TBR projection basis?

26. 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 r columns of the matrix T = RTU Σ-1/2
Our goal:
How perturbation in the initial system
affects balancing transformation T ?

27. TBR reduction algorithm
Perturbation behavior of TBR projection is dictated by:
3) Compute SVD of RQRT: UΣ2UT = RQRT
Symmetric eigenvalue problem
for RQRT

28. Perturbation theory for symmetric eigenvalue problem
Eigenvectors of M0 :
Eigenvectors of M0 + Δ :
Mixing of eigenvectors (assuming small perturbations):
cik large when λi0 ≈ λk0

29. Explaining even-odd behavior
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

30. Outline
Background
Trajectory-piecewise linear (TPWL) framework for model order reduction
Choice of projection bases
TBR-based reduction procedure for TPWL model reduction
Examples and computational results
Issues in selecting order of the model
Efficiency and accuracy
Future work and conclusions

31. Reducing cost of TBR reduction - Combined Krylov-TBR algorithm
Krylov reduction (Wi , Vi):
Ai = WiTAVi
Bi = WiTB
Ci = CVi
Initial Model:
(A B C), n
Intermediate Model:
(Ai Bi Ci), ni
TBR reduction (Wt , Vt):
Ar = WtTAVt
Br = WtTB
Cr = CVt
Reduced Model:
(Ar Br Cr), q
Experiments showed no difference
between TBR and this algorithm

32. Performance of Krylov- TBR TPWL MOR extraction procedures*
Initial
model
size N
TBR
TPWL
q=6
Krylov-TBR
TPWL,
Krylov
q = 30
1500
1268 s
30.57 s
26.34 s
800
181.8 s
8.57 s
7.75 s
400
23.75 s
2.73 s
3.03 s
Cost of Krylov-TBR almost equals Krylov
*Matlab implementation

33. Comparing accuracies of Krylov TPWL method and TBR-based TPWL algorithm
Accuracy in transient *
Order of the reduced model needed to achieve this accuracy for our models
Krylov-based TPWL
TBR-based TPWL 5% 12 2 1% 20 4
0.5%
>30 6
5x reduction in order – 125x improvement in efficiency
*Testing input equal to training input

34. Proposed improvement W1 , V1 W2 , V2 Wagg , Vagg
To aggregate projection bases:
Biorthogonalization
W1TV1 = Ik1 × k1
W2TV2 = Ik2 × k2
make
WaggTVagg = IN_agg × N_agg
Nagg ≤ k1 + k2
W1 , V1
W2 , V2
Wagg , Vagg
Question. How to remove redundant directions?
(in case of Krylov reduction we used SVD, since
Krylov uses orthogonal projection)

35. Future work
Are TBR-based TPWL models valid for unstable linearizations?
What about systems having the following form (i.e. circuits with nonlinear capacitors):

36. Conclusions
In this work we used TBR-based linear reduction procedure to generate TPWL reduced models
Order reduced 5 times while maintaining comparable accuracy with Krylov TPWL method (efficiency improved 125 times!)
Combined Krylov-TBR reduction allows to extract TPWL models at low cost
One should take care of repeated or almost equal Hankel singular values when applying this method.