Addressing Computational Complexity of High Speed Distributed Circuits Using Model Order Reduction

Advanced in the fabrication technology of integrated circuits (ICs) over the last couple of years has resulted in an unparalleled expansion of the functionality of microelectronic systems. Today’s ICs feature complex deep-submicron mixed-signal designs and have found numerous applications in industry due to their lower manufacturing costs and higher performance levels. The tendency towards smaller feature sizes and increasing clock rates is placing higher demands on signal integrity design by highlighting previously negligible interconnect effects such as distortion, reflection, ringing, delay, and crosstalk. These effects if not predicted in the early stages of the design cycle can severely degrade circuit performance and reliability. The objective of this thesis is to develop new model order reduction (MOR) techniques to minimize the computational complexity of non-linear circuits and electronic systems that have delay elements. MOR techniques provide a mechanism to generate reduced order models from the detailed description of the original modified nodal analysis (MNA) formulation. The following contributions are made in this thesis: 1. The first project presents a methodology for reduction of Partial Element Equivalent Circuit (PEEC) models. PEEC method is widely used in electromagnetic compatibility and signal integrity problems in both the time and frequency domains. The PEEC model with retardation has been applied to 3-D analysis but often result in large and dense matrices, which are computationally expensive to solve. In this thesis, a new moment matching technique based on Multi-order Arnoldi is described to model PEEC networks with retardation. 2. The second project deals with developing an efficient model order reduction algorithm for simulating large interconnect networks with nonlinear elements. The proposed methodology is based on a multidimensional subspace method and uses constraint equations to link the nonlinear elements and biasing sources to the reduced order model. This approach significantly improves the simulation time of distributed nonlinear systems, since additional ports are not required to link the nonlinear elements to the reduced order model, yielding appreciable savings in the size of the reduced order model and computational time. 3. A parameterized reduction technique for nonlinear systems is presented. The proposed method uses multidimensional subspace and variational analysis to capture the variances of design parameters and approximates the weakly nonlinear functions as a Taylor series. An SVD approach is presented to address the efficiency of reduced order model. The proposed methodology significantly improves the simulation time of weakly nonlinear systems since the size of the reduced system is smaller than the original system and a new reduced model is not required each time a design parameter is changed

MODEL ORDER REDUCTION OF NONLINEAR DYNAMIC SYSTEMS USING MULTIPLE PROJECTION BASES AND OPTIMIZED STATE-SPACE SAMPLING

Model order reduction (MOR) is a very powerful technique that is used to deal with the increasing complexity of dynamic systems. It is a mature and well understood field of study that has been applied to large linear dynamic systems with great success. However, the continued scaling of integrated micro-systems, the use of new technologies, and aggressive mixed-signal design has forced designers to consider nonlinear effects for more accurate model representations. This has created the need for a methodology to generate compact models from nonlinear systems of high dimensionality, since only such a solution will give an accurate description for current and future complex systems.The goal of this research is to develop a methodology for the model order reduction of large multidimensional nonlinear systems. To address a broad range of nonlinear systems, which makes the task of generalizing a reduction technique difficult, we use the concept of transforming the nonlinear representation into a composite structure of well defined basic functions from multiple projection bases.We build upon the concept of a training phase from the trajectory piecewise-linear (TPWL) methodology as a practical strategy to reduce the state exploration required for a large nonlinear system. We improve upon this methodology in two important ways: First, with a new strategy for the use of multiple projection bases in the reduction process and their coalescence into a unified base that better captures the behavior of the overall system; and second, with a novel strategy for the optimization of the state locations chosen during training. This optimization technique is based on using the Hessian of the system as an error bound metric.Finally, in order to treat the overall linear/nonlinear reduction task, we introduce a hierarchical approach using a block projection base. These three strategies together offer us a new perspective to the problem of model order reduction of nonlinear systems and the tracking or preservation of physical parameters in the final compact model

Fast nonlinear model order reduction via associated transforms of high-order volterra transfer functions

We present a new and fast way of computing the projection matrices serving high-order Volterra transfer functions in the context of (weakly and strongly) nonlinear model order reduction. The novelty is to perform, for the first time, the association of multivariate (Laplace) variables in high-order multiple-input multiple-output (MIMO) transfer functions to generate the standard single-s transfer functions. The consequence is obvious: instead of finding projection subspaces about every s i, only that about a single s is required. This translates into drastic saving in computation and memory, and much more compact reduced-order nonlinear models, without compromising any accuracy. © 2012 ACM.published_or_final_versio

Model order reduction for neutral systems by moment matching

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On the Generation of Large Passive Macromodels for Complex Interconnect Structures

This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed technique