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31
A trajectory piecewiselinear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices
 in Proc. Int. Conf. ComputerAided Design
"... Abstract—In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewiselinear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then ..."
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Cited by 144 (8 self)
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Abstract—In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewiselinear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a “training input. ” Computational results and performance data are presented for an example of a micromachined switch and selected nonlinear circuits. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or recently developed quadratic reduction techniques. Also, we propose a procedure for a posteriori estimation of the simulation error, which may be used to determine the accuracy of the extracted trajectory piecewiselinear reducedorder models. Finally, it is shown that the proposed model order reduction technique is computationally inexpensive, and that the models can be constructed “on the fly, ” to accelerate simulation of the system response. Index Terms—Microelectromechanical systems (MEMS), model order reduction, nonlinear analog circuits, nonlinear dynamical systems, piecewiselinear models. I.
Krylov Subspace Techniques for ReducedOrder Modeling of Nonlinear Dynamical Systems
 Appl. Numer. Math
, 2002
"... Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Kry ..."
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Cited by 93 (5 self)
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Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reducedorder bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the VolterraWiener representation of the bilinear system. It is shown that the twosided Krylov subspace technique matches significant more number of multimoments than the corresponding oneside technique.
Emerging simulation approaches for micromachined devices
 IEEE Trans. Comput. Aided Design of Integrated Circuits and Systems
"... Abstract—In this survey paper, we describe and contrast three different approaches for extending circuit simulation to include micromachined devices. The most commonly used method, that of using physical insight to develop parameterized macromodels, is presented first. The issues associated with fit ..."
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Cited by 30 (3 self)
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Abstract—In this survey paper, we describe and contrast three different approaches for extending circuit simulation to include micromachined devices. The most commonly used method, that of using physical insight to develop parameterized macromodels, is presented first. The issues associated with fitting the parameters to simulation data while incorporating design attribute dependencies are considered. The numerical model order reduction approach to macromodeling is presented second, and some of the issues associated with fast solvers and model reduction are summarized. Lastly, we describe the recently developed circuitbased approach for simulating micromachined devices, and describe the design hierarchy and the use of a catalog of parts. Index Terms—Extraction, macromodeling, MEMS, micromachining, microsystems, modelorder reduction, simulation.
An efficient reducedorder modeling approach for nonlinear parametrized partial differential equations
"... For general nonlinear parametrized partial differential equations (PDEs), the standard Galerkin projection is no longer efficient to generate reducedorder models. This is because the evaluation of the integrals involving the nonlinear terms has a high computational complexity and cannot be preco ..."
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Cited by 15 (1 self)
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For general nonlinear parametrized partial differential equations (PDEs), the standard Galerkin projection is no longer efficient to generate reducedorder models. This is because the evaluation of the integrals involving the nonlinear terms has a high computational complexity and cannot be precomputed. This situation also occurs for linear equations when the parametric dependence is nonaffine. In this paper, we propose an efficient approach to generate reducedorder models for largescale systems derived from PDEs, which may involve nonlinear terms and nonaffine parametric dependence. The main idea is to replace the nonlinear and nonaffine terms with a coefficientfunction approximation consisting of a linear combination of precomputed basis functions with parameterdependent coefficients. The coefficients are determined efficiently by an inexpensive and stable interpolation at some precomputed points. The efficiency and accuracy of this method are demonstrated on several test cases, which show significant computational savings relative to the standard Galerkin projection reducedorder approach. Copyright q
Model order reduction for nonlinear dynamical systems based on trajectory piecewiselinear approximations
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2005
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LYAPUNOV EQUATIONS, ENERGY FUNCTIONALS, AND MODEL ORDER REDUCTION
"... Abstract. We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous ob ..."
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Cited by 13 (8 self)
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Abstract. We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems. Key words. Lyapunov equations, Gramians, energy functionals, balanced truncation, model order reduction, bilinear systems, stochastic systems AMS subject classifications. 93A15, 93A30, 93C10 93E20, 93B40 1. Introduction. Model
ReducedOrder Modeling of Weakly Nonlinear MEMS Devices with TaylorSeries Expansion and Arnoldi Approach
 TRANSDUCERS MAGAZINE (S&T EDIGEST)
, 2004
"... In this paper, we present a new technique by combining the Taylor series expansion with the Arnoldi method to automatically develop reducedorder models for coupled energy domain nonlinear microelectromechanical devices. An electrostatically actuated fixedfixed beam structure with squeezefilm dam ..."
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Cited by 9 (0 self)
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In this paper, we present a new technique by combining the Taylor series expansion with the Arnoldi method to automatically develop reducedorder models for coupled energy domain nonlinear microelectromechanical devices. An electrostatically actuated fixedfixed beam structure with squeezefilm damping effect is examined to illustrate the modelorder reduction method. Simulation results show that the reducedorder nonlinear models can accurately capture the device dynamic behavior over a much larger range of device deformation than the conventional linearized model. Compared with the fully meshed finitedifference method, the model reduction method provides accurate models using orders of magnitude less computation. The reduced MEMS device models are represented by a small number of differential and algebraic equations and thus can be conveniently inserted into a circuit simulator for fast and efficient systemlevel simulation.
Krylov subspaces from bilinear representations of nonlinear systems
 COMPEL
"... Abstract – For efficient simulation of stateoftheart dynamical systems as arise in all aspects of engineering, the development of reducedorder models is of paramount importance. While linear reduction techniques have received considerable study, increasingly nonlinear model reduction is becoming ..."
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Cited by 5 (0 self)
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Abstract – For efficient simulation of stateoftheart dynamical systems as arise in all aspects of engineering, the development of reducedorder models is of paramount importance. While linear reduction techniques have received considerable study, increasingly nonlinear model reduction is becoming a significant field of interest. From a circuits and systems viewpoint, systems involving micromachined devices or systems involving mixed technologies necessitate the development of reducedorder nonlinear models. From a control systems viewpoint, the design of controllers for nonlinear systems is greatly facilitated by nonlinear model reduction strategies. To this end, the paper proposes two novel modelreduction strategies for nonlinear systems. The first involves the development, in a novel manner as compared to previous approaches, of a reducedorder model from a bilinear representation of the system while the second involves a reducing a polynomial approximation using subspaces derived from a related bilinear representation. Both techniques are shown to be effective through the evidence of a standard test example.
Compact Electrothermal Model of Semiconductor Device with Nonlinear Convection Coefficient
 In: Thermal, Mechanical and MultiPhysics Simulation and Experiments in MicroElectronics and MicroSystems. Proceedings of EuroSimE 2005
"... Compact thermal models for semiconductor devices are usually constructed under the assumption of constant material properties as well as the convection coefficient. However, in many cases this is the cause of undesired deviations from experimental results. In this paper, we present an extension of m ..."
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Cited by 3 (2 self)
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Compact thermal models for semiconductor devices are usually constructed under the assumption of constant material properties as well as the convection coefficient. However, in many cases this is the cause of undesired deviations from experimental results. In this paper, we present an extension of model order reduction to construct a compact thermal model for the case when a convection coefficient is nonlinear. As an example, we use transient thermal simulation of a semiconductor device with a temperaturedependent convection coefficient. We demonstrate that the model order reduction technique introduced in this paper can treat such nonlinearity very efficiently. At the same time, the reduced model is accurate enough to replace the original model. 1.