Multirate methods for the transient analysis of electrical circuits

Multirate integration is an important tool to increase the speed of the transient analysis of circuits. This paper shows an approach for the “Compound-Fast” multirate algorithm how to control the errors at the coarse and the refined time-grid by means of the independent stepsizes of these grids. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Funnel Control for Linear DAEs

We study funnel control for linear differential-algebraic multi-input multi-output systems which are not necessarily regular. We show that the funnel controller (that is a static nonlinear output error feedback) achieves - for a special class of right-invertible systems with asymptotically stable zero dynamics - tracking of a reference signal by the output signal within a pre-specified performance funnel. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frequency compensation for a class of DAE's arising in electrical circuits

Structured perturbations of regular pencils of the form , are considered which model the addition of a capacitance c in an electrical circuit in order to improve the frequency response. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structural analysis of electrical circuits including magnetoquasistatic devices

Modeling electric circuits that contain magnetoquasistatic (MQS) devices leads to a coupled system of differential-algebraic equations (DAEs). In our case, the MQS device is described by the eddy current problem being already discretized in space (via edge-elements). This yields a DAE with a properly stated leading term, which has to be solved in the time domain. We are interested in structural properties of this system, which are important for numerical integration. Applying a standard projection technique, we are able to deduce topological conditions such that the tractability index of the coupled problem does not exceed two. Although index-2, we can conclude that the numerical difficulties for this problem are not severe due to a linear dependency on index-2 variables.     

Funnel control for fully actuated systems under a fragment of signal temporal logic specifications

Temporal logics have lately proven to be a valuable tool for various control applications by providing a rich specification language. Existing temporal logic-based control strategies discretize the underlying dynamical system in space and/or time. We will not use such an abstraction and consider continuous-time systems under a fragment of signal temporal logic specifications by using the associated robust semantics. In particular, this paper provides computationally-efficient funnel-based feedback control laws for a class of systems that are, in a sense, feedback equivalent to single integrator systems, but where the dynamics are partially unknown for the control design so that some degree of robustness is obtained. We first leverage the transient properties of a funnel-based feedback control strategy to maximize the robust semantics of some atomic temporal logic formulas. We then guarantee the satisfaction for specifications consisting of conjunctions of such atomic temporal logic formulas with overlapping time intervals by a suitable switched control system. The result is a framework that satisfies temporal logic specifications with a user-defined robustness when the specification is satisfiable. When the specification is not satisfiable, a least violating solution can be found. The theoretical findings are demonstrated in simulations of the nonlinear Lotka–Volterra equations for predator–prey models.     

Spectral reduction for control systems modeling passive integrated circuits

Linear control systems modeling passive integrated circuits are examined. A new algebraic method of spectral reduction equipped with efficient tools for preserving passivity is proposed and justified. For RC networks (circuits), this method is similar to and can be regarded as an extension of the well-known PACT method, which is based on congruence transforms. Up to now, such an extension seemed to be impossible, and different techniques were used for the reduction of RCL and RCLM networks. Some numerical results are presented.     

Simulating harmonic oscillator and electrical circuits: a didactical proposal

A Mathematica ^? package is described that uses simulations and animations to illustrate key concepts in harmonic oscillation and electric circuits for students not majoring in physics or mathematics. Students are not required to know the Mathematica^? environment: a user-friendly interface with buttons functionalities and on-line help allows immediate use of the application. Students who are familiar with the programming language have the opportunity to see the set up code that is used, and they can eventually customize it. The goal of the application presented is to enhance student insight, in the spirit of learning by experimentation.     

Global properties of time-delayed feedback control in electronic circuits

Global properties play an important role in real life applications of time-delayed feedback control of chaos. The success of control is not only determined by the change of the local stability of some unstable periodic state embedded in the chaotic attractor, but also related to a basin of attraction large enough to capture the chaotic trajectory within a reasonable time. We report on electronic circuit experiments where the control performance is essentially affected by the type of bifurcations limiting the control regime and their respective global properties. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analogue Implementation of the Funnel Controller

In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Ramsey numbers for circuits

Denote by r_k(n) the largest integer for which there exists some way of coloring the edges of a complete graph on r_k(n) vertices in k colors without forcing the appearance of a monochromatic simple circuit of length n. A new lower bound for r_k(5) is obtained.     

Variational integrators for electric circuits

Variational integrators are symplectic-momentum preserving integrators that are based on a discrete variational formulation of the underlying system. So far, variational integrators have been mainly developed and used for a wide variety of mechanical systems. In this work, we develop a variational integrator for the simulation of electric circuits. An appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. Finally, a corresponding time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. In this way, a variational integrator is constructed that gains several advantages. A comparison to standard integration techniques shows that even for simple LCR circuits a better long-time energy behavior and frequency preservation can be obtained. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotation numbers for unions of circuits

Let G be a simple undirected graph which has p vertices and is rooted at x. Informally, the rotation number h(G, x) of this rooted graph is the minimum number of edges in a p-vertex graph F, such that for each vertex v of F, there exists a copy of G in F with the root x at v. In this paper, we calculate a lower bound for the rotation number of the graph which is the disjoint union of circuits C_k, C_e where 4 ? k < ?, give infinite classes where this bound is exact, and obtain classes of rotation numbers for the case k = 4.     

Easily tested circuits for linear functions

It is shown that a linear Boolean function of n variables can be realized over the basis {&,V,−} by an irredundant circuit admitting a unit diagnostic test of the length ] log(n−1)[ +2 for inverse output errors of elements.     

A sufficient condition for hamiltonian circuits

A theorem is proved that is (in a sense) the best possible improvement on the following theme: If G is an undirected graph on n vertices in which $\text{|Γ(S)| ≥}\text{1}\text{3}\text{(n + | S | + 3)}$ for every non-empty subset S of the vertices of G, then G is Hamiltonian.     

Variational integrators for nonlinear electric circuits

The variational framework for linear electric circuits introduced in [1] is extended to general nonlinear circuits. Based on a constrained Lagrangian formulation that takes the basic circuit laws into account the equations of motion of a nonlinear electric circuit are derived. The resulting differential-algebraic system can be reduced by performing the variational principle on a reduced space and regularity conditions for the reduced Lagrangian are presented. A variational integrator for the structure-preserving simulation of nonlinear electric circuits is derived and demonstrated by numerical examples. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudorandom bits for constant depth circuits

For every integerd we explicitly construct a family of functions (pseudo-random bit generators) that convert a polylogarithmic number of truly random bits ton bits that appear random to any family of circuits of polynomial size and depthd. The functions we construct are computable by a uniform family of circuits of polynomial size and constant depth. This allows us to simulate randomized constant depth polynomial size circuits inDSPACE(polylog) and inDTIME(2 ^polylog ). As a corollary we show that the complexity class AM is equal to the class of languages recognizable in NP with a random oracle. Our technique may be applied in order to get pseudo random generators for other complexity classes as well; a further paper [16] explores these issues.Part of this work was done while the first author was in U. C. Berkeley, visiting the Hebrew University of Jerusalem.     

Length of diagnostic tests for Boolean circuits

Mathematical Notes -