Article
Nonsmooth Modeling and Simulation for Switched Circuits
Lecture Notes in Electrical Engineering (Impact Factor: 0.11). 10/2010; 69. DOI: 10.1007/9789048196814
Source: OAI
ABSTRACT
Nonsmooth Modeling and Simulation for Switched Circuits concerns the modeling and the numerical simulation of switched circuits with the nonsmooth dynamical systems (NSDS) approach, using piecewiselinear and multivalued models of electronic devices like diodes, transistors, switches. Numerous examples (ranging from introductory academic circuits to various types of power converters) are analyzed and many simulation results obtained with the INRIA opensource SICONOS software package are presented. Comparisons with SPICE and hybrid methods demonstrate the power of the NSDS approach. Nonsmooth Modeling and Simulation for Switched Circuits is intended to researchers and engineers in the field of circuits simulation and design, but may also attract applied mathematicians interested by the numerical analysis for nonsmooth dynamical systems, as well as researchers from Systems and Control.

 "Typically this class of systems is obtained using two or more linear vector fields that are defined on different regions separated by discontinuity boundaries. In particular, a circuit having an ideal switch with state feedback can be modeled with a planar piecewise linear system where the discontinuity boundary is defined by a straight line, see Sec. 1.1.7 of [1]. Planar linear differential systems are completely understood using only linear algebra and they do not present isolated periodic orbits, so called limit cycles. "
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ABSTRACT: This paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line, Σ, and only one Σsingularity of monodromic type. The proofs are based in an extension of Rolle's Theorem for dynamical systems on the plane. 
 "The study of sweeping processes with perturbations was introduced by Castaing, Dúc, Ha and Valadier [9] and Castaing amd Monteiro Marques [10]. The interest in the theory of sweeping processes comes from the fact that it has numerous practical applications in nonsmooth mechanics, analysis of hysteresis phenomena, mathematical economics and in the modeling of switched electrical circuits (see, e.g., the monographs by Acary, Bonnefon and Brogliato [1], Drábek, Krej˘ ci and Taka˘ c [16], Monteiro Marques [34] and the references therein). "
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ABSTRACT: We study wellposedness of sweeping processes with stochastic perturbations generated by a fractional Brownian motion and convergence of associated numerical schemes. To this end, we first prove new existence, uniqueness and approximation results for deterministic sweeping processes with bounded $p$variation and next we apply them to the stochastic case. 
 "In that case, (7) is formulated as a linear complementarity problem which could be solved very efficiently using standard softwares. 2) Regulation in Power Converters: A large number of electrical circuits with nonsmooth devices (diodes, switches, etc.), such as power converters, are modeled using complementarity relations which is a special kind of variational inequality when the setvalued map S(t) = K, where K ⊆ R d K is some closed convex cone (see [1], [26] for examples). Let K * denote the dual cone to K, defined as: K * := {v ∈ R d K  v, w ≥ 0, ∀w ∈ K}. "
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ABSTRACT: We consider the problem of designing state feedback control laws for output regulation in a class of dynamical systems which are described by variational inequalities and ordinary differential equations. In our setup, these variational inequalities are used to model state trajectories constrained to evolve within timevarying, closed, and convex sets, and systems with complementarity relations. We first derive conditions to study the existence and uniqueness of solutions in such systems. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available; In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed.
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