We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.
A compact nonlinear model order-reduction method (NORM) is presented that is applicable for time-invariant and periodically time-varying weakly nonlinear systems. NORM is suitable for model order reduction of a class of weakly nonlinear systems that can be well characterized by low-order Volterra functional series. The automatically extracted macromodels capture not only the first-order (linear) system properties, but also the important second-order effects of interest that cannot be neglected for a broad range of applications. Unlike the existing projection-based reduction methods for weakly nonlinear systems, NORM begins with the general matrix-form Volterra nonlinear transfer functions to derive a set of minimum Krylov subspaces for order reduction. Moment matching of the nonlinear transfer functions by projection of the original system onto this set of minimum Krylov subspaces leads to a significant reduction of model size. As we will demonstrate as part of comparison with existing methods, the efficacy of model reduction for weakly nonlinear systems is determined by the achievable model compactness. Our results further indicate that a multipoint version of NORM can substantially improve the model compactness for nonlinear system reduction. Furthermore, we show that the structure of the nonlinear system can be exploited to simplify the reduced model in practice, which is particularly effective for circuits with sharp frequency selectivity. We demonstrate the practical utility of NORM and its extension for macromodeling weakly nonlinear RF communication circuits with periodically time-varying behavior. In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a "training input." Computational results and performance data are presented for an example of a micromachined switch and selected nonlinear circuits. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or recently developed quadratic reduction techniques. Also, we propose a procedure for a posteriori estimation of the simulation error, which may be used to determine the accuracy of the extracted trajectory piecewise-linear reduced-order models. Finally, it is shown that the proposed model order reduction technique is computationally inexpensive, and that the models can be constructed "on the fly," to accelerate simulation of the system response. The problem of automated macromodel generation is interesting from the viewpoint of system-level design because if small, accurate reduced-order models of system component blocks can be extracted, then much larger portions of a design, or more complicated systems, can be simulated or verified than if the analysis were to have to proceed at a detailed level. The prospect of generating the reduced model from a detailed analysis of component blocks is attractive because then the influence of second-order device effects or parasitic components on the overall system performance can be assessed. In this way overly conservative design specifications can be avoided. This paper reports on experiences with extending model reduction techniques to nonlinear systems of differential-algebraic equations, specifically, systems representative of RF circuit components. The discussion proceeds from linear time-varying, to weakly nonlinear, to nonlinear time-varying analysis, relying generally on perturbational techniques to handle deviations from the linear time-invariant case. The main intent is to explore which perturbational techniques work, which do not, and outline some problems that remain to be solved in developing robust, general nonlinear reduction methods.