A TBR-based Trajectory Piecewise-Linear Algorithm for Generating Accurate Low-order Models for Nonlinear Analog Circui

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

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

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

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

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

WT Projection and TPWL approximation yields efficient f r( ) q x1 Air Ai q V = Air q n n

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 x0 x2 x1 … xn Training trajectory Non-reduced state space

Example problem RLC line Linearized system has nonsymmetric, indefinite Jacobian

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]

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

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

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

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 = RQRTDiagonalvalues of Σ are called the Hankel singular values • Projection basis V contains first r columns of the balancing transformationT = RTU Σ-1/2

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

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

Micromachined device example FD model non-symmetric indefinite Jacobian

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

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

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

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

Explanation of even-odd effect – Problem statement Consider two LTI systems: Perturbed: ( ) 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 r columns of the matrix T = RTU Σ-1/2 Our goal: How perturbation in the initial system affects balancing transformation T ?

TBR reduction algorithm 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 ofM0 : Eigenvectors ofM0 + Δ: Mixing of eigenvectors (assuming small perturbations): ciklarge when λi0 ≈ λk0

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

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

Performance of Krylov- TBR TPWL MOR extraction procedures* Cost of Krylov-TBR almost equals Krylov *Matlab implementation

Comparing accuracies of Krylov TPWL method and TBR-based TPWL algorithm 5x reduction in order – 125x improvement in efficiency *Testing input equal to training input

Proposed improvement 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)

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

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.

A TBR-based Trajectory Piecewise-Linear Algorithm for Generating Accurate Low-order Models for Nonlinear Analog Circui