Stochastic macromodeling of nonlinear systems via polynomial chaos expansion and transfer function trajectories

A novel approach is presented to perform stochastic variability analysis of nonlinear systems. The versatility of the method makes it suitable for the analysis of complex nonlinear electronic systems. The proposed technique is a variation-aware extension of the Transfer Function Trajectory method by means of the Polynomial Chaos expansion. The accuracy with respect to the classical Monte Carlo analysis is verified by means of a relevant numerical example showing a simulation speedup of 1777 X

Extracting Analytical Nonlinear Models from Analog Circuits by Recursive Vector Fitting of Transfer Function Trajectories

This paper presents a technique for automatically extracting analytical behavioral models from the netlist of a nonlinear analog circuit. Subsequent snapshots of the internal circuit Jacobian are sampled during time-domain analysis and are then processed into Transfer Function Trajectories (TFT). The TFT data project the nonlinear dynamics of the system onto a hyperplane in the mixed state-space/frequency domain. Next Recursive Vector Fitting (RVF) algorithm is used to extract an analytical Hammerstein model out of the TFT data in an automated fashion. The resulting RVF model equations are implemented as an accurate nonlinear behavioral model in the time domain. The model is guaranteed stable by construction and can trade off complexity for accuracy. The technique is validated on a high-speed analog buffer circuit containing 70 linear and nonlinear components, showing a 7X speedup

Parameterized macromodeling of passive and active dynamical systems

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Fast Simulation of Analog Circuit Blocks under Nonstationary Operating Conditions

This paper proposes a black-box behavioral modeling framework for analog circuit blocks operating under small-signal conditions around non-stationary operating points. Such variations may be induced either by changes in the loading conditions or by event-driven updates of the operating point for system performance optimization, e.g., to reduce power consumption. An extension of existing data-driven parameterized reduced-order modeling techniques is proposed that considers the time-varying bias components of the port signals as non-stationary parameters. These components are extracted at runtime by a lowpass filter and used to instantaneously update the matrices of the reduced-order state-space model realized as a SPICE netlist. Our main result is a formal proof of quadratic stability of such Linear Parameter Varying (LPV) models, enabled by imposing a specific model structure and representing the transfer function in a basis of positive functions whose elements constitute a partition of unity. The proposed quadratic stability conditions are easily enforced through a finite set of small-size Linear Matrix Inequalities (LMI), used as constraints during model construction. Numerical results on various circuit blocks including voltage regulators confirm that our approach not only ensures the model stability, but also provides speedup in runtime up to 2 orders of magnitude with respect to full transistor-level circuits

Combined parametric and worst case circuit analysis via Taylor models

This paper proposes a novel paradigm to generate a parameterized model of the response of linear circuits with the inclusion of worst case bounds. The methodology leverages the so-called Taylor models and represents parameter-dependent responses in terms of a multivariate Taylor polynomial, in conjunction with an interval remainder accounting for the approximation error. The Taylor model representation is propagated from input parameters to circuit responses through a suitable redefinition of the basic operations, such as addition, multiplication or matrix inversion, that are involved in the circuit solution. Specifically, the remainder is propagated in a conservative way based on the theory of interval analysis. While the polynomial part provides an accurate, analytical and parametric representation of the response as a function of the selected design parameters, the complementary information on the remainder error yields a conservative, yet tight, estimation of the worst case bounds. Specific and novel solutions are proposed to implement complex-valued matrix operations and to overcome well-known issues in the state-of-the-art Taylor model theory, like the determination of the upper and lower bound of the multivariate polynomial part. The proposed framework is applied to the frequency-domain analysis of linear circuits. An in-depth discussion of the fundamental theory is complemented by a selection of relevant examples aimed at illustrating the technique and demonstrating its feasibility and strength

Volterra Series-Based Time-Domain Macromodeling of Nonlinear Circuits

Volterra series (VS) representation is a powerful mathematical model for nonlinear circuits. However, the difficulties in determining higher order Volterra kernels limited its broader applications. In this paper, a systematic approach that enables a convenient extraction of Volterra kernels from X-parameters is presented. A concise and general representation of the output response due to arbitrary number of input tones is given. The relationship between Volterra kernels and X-parameters is explicitly formulated. An efficient frequency sweep scheme and an output frequency indexing scheme are provided. The least square linear regression method is employed to separate different orders of Volterra kernels at the same frequency, which leads to the obtained Volterra kernels complete. The proposed VS representation based on X-parameters is further validated for time-domain verification. The proposed method is systematic and general-purpose. It paves the way for time-domain simulation with X-parameters and constitutes a powerful supplement to the existing blackbox macromodeling methods for nonlinear circuits.postprin