Variability and statistical analysis flow for dynamic linear systems with large number of inputs

Fast analysis of the dynamics of large linear systems with large number of inputs, such as power grid (PG) nets, is a required component of system verification platforms. Such analysis, exhibiting a considerable memory footprint and requiring intensive computations and advanced numerical techniques, has been the framework of recent approaches. However analyzing the effect of design variability, which can have a critical impact on the power distribution across the chip, especially when considering its dynamic performance, poses a unmet challenge. Existing approaches collect information about the voltage and current fluctuations in key nodes that may lead to erroneous behavior or relevant performance changes. This is achieved through repetitive extraction and/or simulation of the large linear RC network for a very broad number of parameter settings. Unfortunately network size and the plethora of different settings that requires investigation implies that such an approach can be exceedingly time consuming, even if parallel architectures are used. In order to address such a challenge, this paper introduces an alternative analysis flow that builds a parameterized model of the time domain node voltages on the fly, using the nominal time domain simulation as starting point. Once such model is generated, the effect of variability in the time response can be efficiently evaluated for multiple settings, allowing collection of relevant variation and statistic information of the impact of a large number of parameters in the current design. The performance of the methodology is evaluated on an set of standard PG extracted netlists, showing large improvements in terms of speed with modest memory requirements while maintaining an acceptable degree of accuracy.