Automated compact dynamical modeling: An enabling tool for analog designers The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bond, Bradley N., and Luca Daniel. “Automated Compact Dynamical Modeling.” ACM Press, 2010. 415. Web. 26 June 2012. © 2010 Association for Computing Machinery (ACM) As Published http://dx.doi.org/ 10.1145/1837274.1837379 Publisher Association for Computing Machinery (ACM) Version Final published version Citable link http://hdl.handle.net/1721.1/71214 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. 26.1 Automated Compact Dynamical Modeling: An Enabling Tool for Analog Designers Bradley N. Bond Luca Daniel Massachusetts Institute of Technology Massachusetts Institute of Technology [email protected] [email protected] ABSTRACT semidefinite matrices. For linear systems with such specific prop- In this paper we summarize recent developments in compact dy- erties, a body of methods exist (e.g. [1, 2]) that can generate au- namical modeling for both linear and nonlinear systems arising tomatically compact stable and passive dynamical models suitable in analog applications. These techniques include methods based for time domain circuit/gate level simulations. On the contrary, on the projection framework, rational fitting of frequency response when working with analog applications, one needs to handle non- samples, and nonlinear system identification from time domain data. linear circuit blocks or microelectromechanical (MEMS) devices. By combining traditional projection and fitting methods with re- Even when considering only the linear part of the analog systems cently developed convex optimization techniques, it is possible to (e.g. RF inductors, passive power combiners, transmission lines) obtain guaranteed stable and passive parameterized models that are the constraint of “constant semidefinite matrices” is always inevitably usable in time domain simulators and may serve as a valuable tool violated. This is because the electromagnetic field solvers used to for analog designers in both top-down and bottom-up design flows. analyze analog structures inevitably introduce numerical discretiza- tion errors at high frequencies, and because they are forced by the large size of the problems to use advanced formulations and ap- Categories and Subject Descriptors: J.6 [Computer Aided En- proximation techniques such as substrate Green functions and fast gineering]: Computer-aided design (CAD), G.1.6 [Optimization]: matrix-vector products. In many cases no internal description of Convex programming, I.6.0 [Simulation and Modeling]: General the systems is even available, and the only information provided by General Terms: Algorithms the field solvers, or by measured data, is a set of frequency response Keywords: Model reduction, Analog design, Semidefinite pro- values (if the system is linear), or a set of time domain input-output gramming, Parameterized modeling, Compact modeling signals (if the system is nonlinear). Another important difference between digital and analog sys- 1. INTRODUCTION tems is that in digital applications higher level system simulators (e.g. timing analyzers) or top-down design methodologies do not As feature sizes keep decreasing and operating frequencies keep require dynamical state-space models for each gate, but rather just increasing, the performance of many complex analog components a small set of simple equations or tabulated values relating per- and systems is becoming crucially dependent on what the design- formance (or behavioral) parameters. However, in order to enable ers identify as “non-idealities” or “second order effects”. These are system level simulation and fast tradeoff explorations for analog phenomena such as non-linearities, process variations, skin effects, applications, leading to robust designs insensitive to process varia- proximity effects, and substrate and magnetic coupling. Analog tions and to all high frequency second order effects, it is essential to designers would therefore greatly benefit from tools that generate have the ability to instantiate instantaneously parameterized com- automatically compact dynamical models which capture quantita- pact dynamical state-space models of circuit blocks. tively all such second order effects. Digital designers have indeed To better highlight the differences between digital and analog been already benefitting for decades from model order reduction applications, we would like to emphasize that the classical name, tools to speed-up circuit simulations within the context of a bottom- “model order reduction”, used in the digital community, is com- up library characterization step for timing analysis. However, sev- pletely inadequate to describe the variety of techniques that are be- eral significant differences between analog and digital applications ing developed to handle analog systems. We propose instead the limit severely the usability of the existing model order reduction name “automated compact dynamical modeling”. Here the term techniques. dynamical indicates state space models, emphasizing the fact that The first important difference is that, within the digital applica- for system level analog simulations it is not sufficient to generate tions, the systems to be “reduced” are mainly, if not exclusively, simple static behavioral models relating performance parameters RLC interconnect networks generated by parasitic extractors (e.g. of a circuit block as in digital applications. The term compact as resistance, capacitance and inductance extractors). Such systems opposed to “reduced order”, emphasizes the fact that model com- are linear, and are always described by constant, structured, and plexity is not strictly tied to the model order (i.e. the number of equations) as is the case in the digital interconnect linear system world. The complexity of the generated system should instead be measured in terms of how efficiently the generated model can be employed in a time domain simulator. When working with analog nonlinear systems one quickly realizes that small model orders do Permission to make digital or hard copies of all or part of this work for not necessarily imply fast simulations. Finally, although in some personal or classroom use is granted without fee provided that copies are cases one might want to “reduce” a large model already given in not made or distributed for profit or commercial advantage and that dynamical state space form, in all the other cases typical of the copies bear this notice and the full citation on the first page. To copy analog world, one is only given frequency domain data, or input- otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. DAC'10, June 13-18, 2010, Anaheim, California, USA Copyright 2010 ACM 978-1-4503-0002-5 /10/06...$10.00 415 26.1 output time domain data. Therefore a compact dynamical model For weakly nonlinear systems it is common to use a polynomial might need to be “‘identified” or “fitted” as opposed to “reduced”. expansion for the approximation. For example, for a simplified As a matter of fact, in order to address the many new challenging nonlinear system x˙ = f(x)+Bu, the vector field f(x) is approx- analog constraints, during the last few years the automated compact imated as dynamical modeling community has been forced to move beyond ˜ f(x) ≈ f(x)=A0 + A1x + A2(x ⊗ x)+.... traditional projection-based model order reduction approaches, em- (5) bracing for instance recent developments in convex optimization. k th where Ak is an N × N matrix containing the k derivatives of In some cases, some of the more traditional projection appraoches f(x). The new vector field f˜(x) can be easily projected given the or unconstrained fitting techniques have been adapted to the analog Kronecker product property (V xˆ) ⊗ (V xˆ)=(V ⊗ V )(ˆx ⊗ xˆ), world by simply adding a perturbation step that enforces proper- resulting in the reduced vector field ties such as stability and passivity as a post-processing operation. ˆ T ˜ ˆ ˆ ˆ In other cases, completely new techniques have been, and are still f(ˆx)=U f(V xˆ)=A0 + A1xˆ + A2(ˆx ⊗ xˆ)+... currently being invented. ˆ T ˆ T ˆ T The remainder of this paper is organized as follows. Section 2 where A0 = U A0, A1 = U A1V and A2 = U A(V ⊗ V ). presents projection methods for stable and passive modeling of Analysis of polynomial reduced models can be further simplified linear and nonlinear systems described in state-space form. Sec- using ideas from Volterra theory, as was done in [4, 5]. tion 3 surveys methods for compact modeling of linear systems An alternative approach, better suited for highly nonlinear sys- using rational approximation techniques for transfer matrix fitting, tems, consists of interpolating between a collection of local models combined with optimization-based stability and passivity enforce- ˜ ˜ ment. Section 4 summarizes recent developments in stable dynam- f(x) ≈ f(x)= wi(x)fi(x), (6) ical model fitting techniques using optimization to enforce stability i in nonlinear dynamical systems. Lastly, Section 5 discusses what where w(x) are weighting functions. The local models are chosen further steps need to be taken, both by model reduction researchers ˜ so that each individual model fi(x) can be reduced using projection and by the rest of the EDA community, to help compact dynamical modeling become a mainstream tool supporting analog design. ˆ T ˜ T ˜ f(ˆx)=U f(V xˆ)= wi(ˆx)U fi(V xˆ), (7) i T ˜ ˜ 2. PROJECTION-BASED METHODS such that U fi(V xˆ) has low complexity. For instance, fi could be ˜ polynomial functions, or even linear fi(x)=Aix + ki,inwhich T ˜ T T 2.1 The Traditional Projection Framework case U fi(V xˆ)=U AiV xˆ + U ki. The key idea behind the Most of the traditional model order reduction techniques can be Trajectory PieceWise Linear (TPWL) method [6] and related tech- interpreted within a projection framework.