Bradley N. Bond

Parameterized model order reduction of nonlinear dynamical systems

In this paper we present a parameterized reduction technique for non-linear systems. Our approach combines an existing non-parameterized trajectory piecewise linear method for non-linear systems, with an existing moment matching parameterized technique for linear systems. Results and comparisons are presented for two examples: an analog non-linear circuit, and a MEM switch.

A Piecewise-Linear Moment-Matching Approach to Parameterized Model-Order Reduction for Highly Nonlinear Systems

This paper presents a parameterized reduction technique for highly nonlinear systems. In our approach, we first approximate the nonlinear system with a convex combination of parameterized linear models created by linearizing the nonlinear system at points along training trajectories. Each of these linear models is then projected using a moment-matching scheme into a low-order subspace, resulting in a parameterized reduced-order nonlinear system. Several options for selecting the linear models and constructing the projection matrix are presented and analyzed. In addition, we propose a training scheme which automatically selects parameter-space training points by approximating parameter sensitivities. Results and comparisons are presented for three examples which contain distributed strong nonlinearities: a diode transmission line, a microelectromechanical switch, and a pulse-narrowing nonlinear transmission line. In most cases, we are able to accurately capture the parameter dependence over the parameter ranges of plusmn50% from the nominal values and to achieve an average simulation speedup of about 10x.

Guaranteed stable projection-based model reduction for indefinite and unstable linear systems

In this work we present a stability-preserving projection framework for model reduction of linear systems. Specifically, given one projection matrix (e.g. a right-projection matrix), we derive a set of linear constraints for the other projection matrix (e.g. the left-projection matrix) resulting in a projection framework that is guaranteed to generate a stable reduced model. Several efficient techniques for solving the proposed system of constraints are presented, including an optimization problem formulation for finding the optimal stabilizing projection, and a formulation with computational complexity independent of the size of the original system. The resulting algorithms can create accurate stable and passive models of arbitrary indefinite systems at a significantly cheaper cost than existing methods such as balanced truncation. Nevertheless, our algorithms integrate fully and effortlessly with most of the available standard model order reduction approaches for very large systems generated in VLSI applications (such as moment-matching methods, POD, or poor manpsilas TBR), which can guarantee stability and passivity only in very specialized cases. Our algorithms have been tested on a large variety of typical VLSI applications, including field-solver-extracted models of RF inductors for analog applications, power distribution grids for large VLSI digital integrated circuits, and MEMS devices for sensing and actuation applications. The results have been successfully compared to those from existing and much more expensive stabilizing reduction techniques.

Stabilizing schemes for piecewise-linear reduced order models via projection and weighting functions

In this paper we present several results concerning the stabilization of piecewise-linear reduced order models. We include proofs of internal and external stability for models whose system matrices possess special structures. We then introduce a new projection scheme, and a new set of weighting functions which allow us to extend some of these results to piecewise-linear systems comprised of arbitrary matrices, at least one of which is Hurwitz. Included are an algorithm for creating switching piecewise-linear reduced models comprised of globally exponentially stable systems, and stable simulation results for a system which produces unstable results when using the standard TPWL method.

Passive reduced order modeling of multiport interconnects via semidefinite programming

In this paper we present a passive reduced order modeling algorithm for linear multiport interconnect structures. The proposed technique uses rational fitting via semidefinite programming to identify a passive transfer matrix from given frequency domain data samples. Numerical results are presented for a power distribution grid and an array of inductors, and the proposed approach is compared to two existing rational fitting techniques.

Automated compact dynamical modeling: an enabling tool for analog designers

In this paper we summarize recent developments in compact dynamical modeling for both linear and nonlinear systems arising in analog applications. These techniques include methods based on the projection framework, rational fitting of frequency response samples, and nonlinear system identification from time domain data. By combining traditional projection and fitting methods with recently developed convex optimization techniques, it is possible to obtain guaranteed stable and passive parameterized models that are usable in time domain simulators and may serve as a valuable tool for analog designers in both top-down and bottom-up design flows.