A mesh-free adaptive parametric macromodeling strategy with guaranteed stability

This paper proposes a fully automated procedure for the generation of behavioral time-domain macromodels of complex multiport electric, electronic or electromagnetic systems, whose response depends on several design parameters. The latter are embedded in closed form in the macromodel structure through a mesh-free radial basis function representation, which allows scalability to a possibly large number of parameters. A greedy process is proposed to iteratively select a reduced number of training frequency responses, so that the macromodel accuracy is enforced uniformly in the parameter space. Examples with up to ten independent parameters demonstrate the effectiveness of proposed algorithm

Towards fully automated high-dimensional parameterized macromodeling

This paper presents a fully automated algorithm for the extraction of parameterized macromodels from frequency responses. The reference framework is based on a frequency-domain rational approximation combined with a parameter-space expansion into Gaussian Radial Basis Functions (RBF). An iterative least-squares fitting with positivity constraints is used to optimize model coefficients, with a guarantee of uniform stability over the parameter space. The main novel contribution of this work is a set of algorithms, supported by strong theoretical arguments with associated proofs, for the automated determination of all the hyper-parameters that define model orders, placement and width of RBFs. With respect to standard approaches, which tune these parameters using time-consuming tentative model extractions following a trial-and-error strategy, the presented technique allows much faster tuning of the model structure. The numerical results show that models with up to ten independent parameters are easily extracted in few minutes

Infinite-dimensional calculus under weak spatial regularity of the processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus

A simple planning problem for COVID-19 lockdown: a dynamic programming approach

A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach

Data-driven extraction of uniformly stable and passive parameterized macromodels

A Robust algorithm for the extraction of reduced-order behavioral models from sampled frequency responses is proposed. The system under investigation can be any Linear and Time Invariant structure, although the main emphasis is on devices that are relevant for Signal and Power Integrity and RF design, such as electrical interconnects and integrated passive components. We assume that the device under modeling is parameterized by one or more design variables, which can be related to geometry or materials. Therefore, we seek for multivariate macromodels that reproduce the dynamic behavior over a predefined frequency band, with an explicit embedded dependence of the model equations on these external parameters. Such parameterized macromodels may be used to construct component libraries and prove very useful in fast system-level numerical simulations in time or frequency domain, including optimization, what-if, and sensitivity analysis. The main novel contribution is the formulation of a finite set of convex constraints that are applied during model identification, which provide sufficient conditions for uniform model stability and passivity throughout the parameter space. Such constraints are characterized by an explicit control allowing for a trade-off between model accuracy and runtime, thanks to some special properties of Bernstein polynomials. In summary, we solve the longstanding problem of multivariate stability and passivity enforcement in data-driven model order reduction, which insofar has been tackled only via either overconservative or heuristic and possibly unreliable methods

On stabilization of parameterized macromodeling

We propose an algorithm for the identification of guaranteed stable parameterized macromodels from sampled frequency responses. The proposed scheme is based on the standard Sanathanan-Koerner iteration in its parameterized form, which is regularized by adding a set of inequality constraints for enforcing the positiveness of the model denominator at suitable discrete points. We show that an ad hoc aggregation of such constraints is able to stabilize the iterative scheme by significantly improving its convergence properties, while guaranteeing uniformly stable model poles as the parameter(s) change within their design range

Enabling fast power integrity transient analysis through parameterized small-signal macromodels

In this paper, we present an automated strategy for extracting behavioral small-signal macromodels of biased nonlinear circuit blocks. We discuss in detail the case study of a Low DropOut (LDO) voltage regulator, which is an essential part of the power distribution network in electronic systems. We derive a compact yet accurate surrogate model of the LDO, which enables fast transient power integrity simulations, including all parasitics due to the specific layout of the LDO realization. The model is parameterized through its DC input voltage and its output current and is thus available as a SPICE netlist. Numerical experiments show that a speedup up to 700X is achieved when replacing the extracted post-layout netlist with the surrogate model, with practically no loss in accuracy

Efficient EM-based variability analysis of passive microwave structures through parameterized reduced-order behavioral models

In this contribution we demonstrate how reduced-order behavioral models allow for extremely accurate and computationally efficient electromagnetic-based variability analysis of microwave passive structures. In particular, we report the MonteCarlo analysis of a wideband matching network at Ka-band, designed with a commercial foundry GaN-HEMT process PDK. As sources of variation we considered the thickness of the two dielectric layers available in the PDK to implement MIM capacitors of different order of magnitude, both exploited in the network. Based on a limited set of electromagnetic simulations, a parameterized behavioral model is extracted and then translated into a parameterized circuit equivalent (SPICE netlist) straightforward to be imported into RF CAD tools. The adopted model, implementing a rational approximation of the simulated S-parameters with rational dependence on the two parameters, provides excellent agreement with electromagnetic simulations, robustness against port impedance change and good extrapolation capabilities

Structured black-box parameterized macromodels of integrated passive components

A novel black-box model representation and identification process is introduced, specifically designed to extract layout-scalable behavioral macromodels of passive integrated devices from sampled frequency-domain responses. An automated choice of structured frequency-domain basis functions enables extremely accurate approximations for responses characterized by high dynamic ranges over extended frequency bands, overcoming the main limitations of standard approaches. Numerical results confirm that the proposed structured approach provides robust and reliable scalable models, with guaranteed stability and passivity over the frequency band and parameter space of interest