Vincent Acary

This chapter is devoted to present the mathematical tools which are used in this book to analyze the nonsmooth circuits and their time-discretizations. This chapter does not aim at being exhaustive. The unique objective is that the book be sufficiently self-contained and that all the mathematical notions which are the foundations of the nonsmooth dynamical systems that are presented, be easily available to the readers who are not familiar with such tools. For this reason the results are given without proofs. After a brief recall of some basic tools, we come back to the circuits of Chap. 1 and rewrite their dynamics using new mathematical frameworks. Many of the tools which are presented in this chapter, will be used, or presented in an other way in Chap. 4.

In the NonSmooth Dynamical Systems (NSDS) approach, the standard description of elements by means of explicit and smooth functions is enriched by new elements described by generalized equations. The characteristics of the electronic devices can be then nonsmooth and even multivalued. These new elements are called the electrical “nonsmooth elements”. Some examples have already been studied in Chaps. 1 and 2. The description of nonsmooth components relies a lot on mathematical notions from Convex Analysis and the Mathematical Programming theory. A significant amount of informations on these aspects has already been provided in Chap. 2, starting from the very simple academic circuits examples presented in Chap. 1. In this chapter we take advantage of the material of the foregoing chapters to arrive at the general mathematical formalisms which are used in the NSDS approach to simulate the electrical circuits of Chaps. 7 and 8. The first subsections briefly recall some basic facts which are exposed in more details in Chap. 2, in particular Sects. 2.1 and 2.3. In a way similar to the foregoing chapter, the time argument is dropped from the state variables, in order to lighten the presentation of the dynamics.

This chapter is devoted to present numerical simulation results obtained with the siconos platform, on several simple circuits: the first circuit has been built to show that conventional analog simulators fail to converge; the other circuits are classical diode-bridge wave rectifiers, and the last one is a circuit that exhibits a sliding mode. In this chapter and in Chap. 8, five simulation software packages were used: This chapter is devoted to present numerical simulation results obtained with the siconos platform, on several simple circuits: the first circuit has been built to show that conventional analog simulators fail to converge; the other circuits are classical diode-bridge wave rectifiers, and the last one is a circuit that exhibits a sliding mode. In this chapter and in Chap. 8, five simulation software packages were used: siconos: the platform developed at INRIA Grenoble Rhône-Alpes (France) dealing with nonsmooth dynamical systems with dedicated time integrators and algorithms to solve sets of equations and inequalities (for instance LCP: linear complementarity problems). ngspice: an open-source version of the original SPICE3F5 software developed by Berkeley university. Even if this version may differ from existing commercial ones, it shares with them a common set of models and the solving algorithms belong also to the same class that deals with regular functions. smash: a commercial version of spice developed by Dolphin Integration (see http://www.dolphin.fr). eldo: a commercial version of spice with Newton-Raphson and OSR (one step relaxation) algorithms developed by Mentor Graphics (http://www.mentor.com). plecs: a simulink/matlab toolbox dedicated to the simulation of power electronics circuits (see http://www.plexim.com). The models and algorithms come from the hybrid approach. In our work the freely available demonstration version of plecs has been used.

... rely on a linear complementarity formulation for handling contact in a stable way, and ... unbounded second derivatives which ruin the stability of the fixed step integration schemes. ... While this approximate formulation captures both dynamic and static fric-tion properly, it introduces ...

In this paper we present an extension of Moreau’s sweeping process for higher order systems. The dynamical framework is carefully introduced, qualitative, dissipativity, stability, existence, regularity and uniqueness results are given. The time-discretization of these nonsmooth systems with a time-stepping algorithm is also presented. This differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary-slackness variables. Applications of such high-order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes.

The Siconos Platform is a scientific computing software dedicated to the modeling, simulation, control, and analysis of nonsmooth dynamical systems (NSDS). It is developed in the Bipop team-project at INRIA (The French National Institute for Research in Computer Science and Control (http://bipop.inrialpes.fr).) in Grenoble, France, and distributed under GPL GNU license. Siconos aims at providing a general and common tool for nonsmooth problems in various scientific fields like applied mathematics, mechanics, robotics, electrical circuits, and so on. However, the platform is not supposed to re-implement the existing dedicated tools already used for the modeling of specific systems, but to integrate them. The Automatic Circuit Equations Formulation (ACEF) module is the implementation of the automatic circuit equation extended to general nonsmooth components. From a SPICE netlist, possibly augmented by some nonsmooth components, the ACEF build a dynamical formulation that can be simulated by Siconos.

In this report it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an {\em event-capturing}, or {\em time-stepping} algorithm) allows for accumulation of events (Zeno phenomena) and for multiple switching surfaces (i.e., a sliding surface of codimension $\geq 2$). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied.

In this chapter, some basic facts on the circuit equation formulation and simulation which are shared by most of the analog spice-like simulators are presented. The formulation of the circuit equations is based on two basic ingredients: conservative laws given by the Kirchhoff laws in currents and voltages, constitutive equations of the electrical components, which lead to a Differential Algebraic Equation (DAE). We will give some details on the DAEs in Sects. 3.1 to 3.6 and the chapter will end in Sect. 3.7 on conventional techniques for the numerical analog simulation of circuits.

This section is dedicated to introduce simple circuits that contain electronic devices with a nonsmooth current/voltage characteristic. Examples are RLC circuits with so-called ideal diodes, ideal Zener diodes, ideal switches. The main peculiarities of their dynamics are highlighted through detailed analysis. The parallel with simple nonsmooth mechanical systems is made. Last, but not least, the numerical method that will be used in the remainder of the book is introduced.

In this chapter we present a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary slackness variables. The proposed model incorporates the state jumps via high-order distributions through the extension of Moreau’s sweeping process, which is a special type of differential inclusion. The time-discretization of these nonsmooth systems, which is non-trivial, is also presented. Applications of such high-order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes, where it may be helpful for a better understanding of the closed-loop dynamics induced by some feedback laws.

It has been seen in Chaps. 1, 2 and 3 that electrical circuits with nonsmooth multivalued electronic devices, can be recast under various mathematical formalisms (see Sect. 2.7 for a summary). For the sake of the numerical integration of those circuits, one needs a small set of general formulations which are suitable for a subsequent time-discretization. In other words, the simple examples that are analyzed in details in Chaps. 1 and 2 possess a too simple dynamics to be characteristic representatives of the general issue of nonsmooth circuits. Especially, the material of Chap. 3 teaches us that DAEs are ubiquitous in circuits (a well-known fact, indeed). It is therefore necessary to obtain mathematical formalisms that incorporate not only the nonsmooth and multivalued models of the electronic devices (ideal diodes, Zener diodes, etc.), but also the equations obtained from the MNA (see Sects. 3.5 and 3.6).

This chapter is devoted to present the mathematical tools which are used in this book to analyze the nonsmooth circuits and their time-discretizations. This chapter does not aim at being exhaustive. The unique objective is that the book be sufficiently self-contained and that all the mathematical notions which are the foundations of the nonsmooth dynamical systems that are presented, be easily available to the readers who are not familiar with such tools. For this reason the results are given without proofs. After a brief recall of some basic tools, we come back to the circuits of Chap. 1 and rewrite their dynamics using new mathematical frameworks. Many of the tools which are presented in this chapter, will be used, or presented in an other way in Chap. 4.

In the NonSmooth Dynamical Systems (NSDS) approach, the standard description of elements by means of explicit and smooth functions is enriched by new elements described by generalized equations. The characteristics of the electronic devices can be then nonsmooth and even multivalued. These new elements are called the electrical “nonsmooth elements”. Some examples have already been studied in Chaps. 1 and 2. The description of nonsmooth components relies a lot on mathematical notions from Convex Analysis and the Mathematical Programming theory. A significant amount of informations on these aspects has already been provided in Chap. 2, starting from the very simple academic circuits examples presented in Chap. 1. In this chapter we take advantage of the material of the foregoing chapters to arrive at the general mathematical formalisms which are used in the NSDS approach to simulate the electrical circuits of Chaps. 7 and 8. The first subsections briefly recall some basic facts which are exposed in more details in Chap. 2, in particular Sects. 2.1 and 2.3. In a way similar to the foregoing chapter, the time argument is dropped from the state variables, in order to lighten the presentation of the dynamics.

This chapter is devoted to present numerical simulation results obtained with the siconos platform, on several simple circuits: the first circuit has been built to show that conventional analog simulators fail to converge; the other circuits are classical diode-bridge wave rectifiers, and the last one is a circuit that exhibits a sliding mode. In this chapter and in Chap. 8, five simulation software packages were used: This chapter is devoted to present numerical simulation results obtained with the siconos platform, on several simple circuits: the first circuit has been built to show that conventional analog simulators fail to converge; the other circuits are classical diode-bridge wave rectifiers, and the last one is a circuit that exhibits a sliding mode. In this chapter and in Chap. 8, five simulation software packages were used: siconos: the platform developed at INRIA Grenoble Rhône-Alpes (France) dealing with nonsmooth dynamical systems with dedicated time integrators and algorithms to solve sets of equations and inequalities (for instance LCP: linear complementarity problems). ngspice: an open-source version of the original SPICE3F5 software developed by Berkeley university. Even if this version may differ from existing commercial ones, it shares with them a common set of models and the solving algorithms belong also to the same class that deals with regular functions. smash: a commercial version of spice developed by Dolphin Integration (see http://www.dolphin.fr). eldo: a commercial version of spice with Newton-Raphson and OSR (one step relaxation) algorithms developed by Mentor Graphics (http://www.mentor.com). plecs: a simulink/matlab toolbox dedicated to the simulation of power electronics circuits (see http://www.plexim.com). The models and algorithms come from the hybrid approach. In our work the freely available demonstration version of plecs has been used.

... rely on a linear complementarity formulation for handling contact in a stable way, and ... unbounded second derivatives which ruin the stability of the fixed step integration schemes. ... While this approximate formulation captures both dynamic and static fric-tion properly, it introduces ...

In this paper we present an extension of Moreau’s sweeping process for higher order systems. The dynamical framework is carefully introduced, qualitative, dissipativity, stability, existence, regularity and uniqueness results are given. The time-discretization of these nonsmooth systems with a time-stepping algorithm is also presented. This differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary-slackness variables. Applications of such high-order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes.

The Siconos Platform is a scientific computing software dedicated to the modeling, simulation, control, and analysis of nonsmooth dynamical systems (NSDS). It is developed in the Bipop team-project at INRIA (The French National Institute for Research in Computer Science and Control (http://bipop.inrialpes.fr).) in Grenoble, France, and distributed under GPL GNU license. Siconos aims at providing a general and common tool for nonsmooth problems in various scientific fields like applied mathematics, mechanics, robotics, electrical circuits, and so on. However, the platform is not supposed to re-implement the existing dedicated tools already used for the modeling of specific systems, but to integrate them. The Automatic Circuit Equations Formulation (ACEF) module is the implementation of the automatic circuit equation extended to general nonsmooth components. From a SPICE netlist, possibly augmented by some nonsmooth components, the ACEF build a dynamical formulation that can be simulated by Siconos.

In this report it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an {\em event-capturing}, or {\em time-stepping} algorithm) allows for accumulation of events (Zeno phenomena) and for multiple switching surfaces (i.e., a sliding surface of codimension $\geq 2$). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied.

In this chapter, some basic facts on the circuit equation formulation and simulation which are shared by most of the analog spice-like simulators are presented. The formulation of the circuit equations is based on two basic ingredients: conservative laws given by the Kirchhoff laws in currents and voltages, constitutive equations of the electrical components, which lead to a Differential Algebraic Equation (DAE). We will give some details on the DAEs in Sects. 3.1 to 3.6 and the chapter will end in Sect. 3.7 on conventional techniques for the numerical analog simulation of circuits.

This section is dedicated to introduce simple circuits that contain electronic devices with a nonsmooth current/voltage characteristic. Examples are RLC circuits with so-called ideal diodes, ideal Zener diodes, ideal switches. The main peculiarities of their dynamics are highlighted through detailed analysis. The parallel with simple nonsmooth mechanical systems is made. Last, but not least, the numerical method that will be used in the remainder of the book is introduced.

In this chapter we present a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary slackness variables. The proposed model incorporates the state jumps via high-order distributions through the extension of Moreau’s sweeping process, which is a special type of differential inclusion. The time-discretization of these nonsmooth systems, which is non-trivial, is also presented. Applications of such high-order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes, where it may be helpful for a better understanding of the closed-loop dynamics induced by some feedback laws.

It has been seen in Chaps. 1, 2 and 3 that electrical circuits with nonsmooth multivalued electronic devices, can be recast under various mathematical formalisms (see Sect. 2.7 for a summary). For the sake of the numerical integration of those circuits, one needs a small set of general formulations which are suitable for a subsequent time-discretization. In other words, the simple examples that are analyzed in details in Chaps. 1 and 2 possess a too simple dynamics to be characteristic representatives of the general issue of nonsmooth circuits. Especially, the material of Chap. 3 teaches us that DAEs are ubiquitous in circuits (a well-known fact, indeed). It is therefore necessary to obtain mathematical formalisms that incorporate not only the nonsmooth and multivalued models of the electronic devices (ideal diodes, Zener diodes, etc.), but also the equations obtained from the MNA (see Sects. 3.5 and 3.6).