Time‐domain properties of reactive dual circuits

The present work explores some effects of the replacement of capacitors by inductors and vice versa in state and semistate models of lumped circuits. Such a replacement, when performed together with an inversion of the capacitance and inductance matrices, yields a transformation of the form λ→λ^?1 in the system spectra. In the semistate context, this covers in particular extremal cases in which null eigenvalues or infinite ones with higher index appear in the matrix pencil associated with the model; these cases describe certain pathological circuit configurations. This approach leads to a discussion of new properties of strictly passive circuits; specifically, from the known fact that the index of strictly passive circuits does not exceed two, we derive that the index of null eigenvalues in this setting cannot exceed one. This precludes in particular Takens‐Bogdanov degeneracies, defined by an index‐two double‐zero eigenvalue, in strictly passive circuits. Although the results are addressed in a linear context, they can be extended via linearization to non‐linear problems, as it is the case in the transformation of singularity‐induced bifurcation phenomena into steady bifurcations discussed at the end of the paper. Copyright © 2006 John Wiley & Sons, Ltd.     

A note on stability of linear time‐variant electrical circuits having constant eigenvalues

The main aspect of this paper is to show that the stability of linear time‐variant systems cannot be estimated from the location of the eigenvalues. For this purpose, two simple time‐variant electrical circuits are presented, which have constant eigenvalues. As will be shown, the time‐variant circuits can be asymptotically stable although there is a positive eigenvalue and this circuit can be unstable despite negative eigenvalues only. The idea behind is a suited time‐variant state transformation of a linear time‐invariant system. An electrical interpretation of both systems and of state transformations allows for an energetic evaluation from an electrical point of view even though the analytical solution is not necessarily known. Copyright © 2012 John Wiley & Sons, Ltd.     

An efficient piecewise‐linear DC analysis method for general non‐linear circuits

The convergence problems of conventional DC analysis can be partly avoided by using piecewise‐linear analysis. This paper proposes a piecewise‐linear DC analysis method that can efficiently handle arbitrary couplings between non‐linear circuit elements. Piecewise‐linear modelling of the non‐linear circuit elements is automatically performed during simulation, using simplicial subdivisions. The number of linear regions, and thereby iterations, is considerably reduced by combining the common parts of separate simplicial subdivisions. Due to these reasons and since the method is formulated with the commonly used modified nodal approach, it has been possible to implement the method in the general‐purpose circuit simulator APLAC. The correct operation of the method is demonstrated with three examples. Copyright © 1999 John Wiley & Sons, Ltd.     

An interval algorithm for finding all solutions of non‐linear resistive circuits

In this letter, an efficient algorithm is proposed for finding all solutions of non‐linear (not piecewise‐linear) resistive circuits. This algorithm is based on interval analysis, the dual simplex method, and the contraction methods. By numerical examples, it is shown that the proposed algorithm could find all solutions of systems of 500–700 non‐linear circuit equations in acceptable computation time. Copyright © 2004 John Wiley & Sons, Ltd.     

A combined approach for the identification of continuous non‐linear systems

The identification of a non‐linear continuous output‐only system from a time series is considered for the case that the functional form of the model is not known beforehand. To estimate both functions and parameters, a combination of non‐parametric modelling based on non‐linear regression and parametric modelling based on a multiple shooting algorithm is proposed. This strategy to determine non‐linear differential equations is exemplified on experimental data from a chaotic circuit where an accurate reconstruction of the observed attractor is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Semistate models of electrical circuits including memristors

In this paper we present several semistate or differential‐algebraic models arising in nodal analysis of nonlinear circuits including memristors. The goal is to characterize the tractability index of these models under strict passivity assumptions, a key issue for the numerical simulation of circuit dynamics. We show that the main model, which combines memristors' fluxes and charges, is index two. From a technical point of view, this result is based on the use of a projector along the image of the leading matrix, in contrast to previous index analyses. For charge‐controlled memristors, the elimination of fluxes yields an index one system in topologically nondegenerate circuits, and an index two model otherwise. Analogous results are also proved to hold for flux‐controlled memristors. Our framework accommodates coupling effects among resistors, memristors, capacitors and inductors. Copyright © 2010 John Wiley & Sons, Ltd.     

On simplex‐based piecewise‐linear approximations of non‐linear mappings

Simplex‐based piecewise‐linear (PWL) approximations of non‐linear mappings are needed when the robust PWL analysis is used to directly solve non‐linear equations. This paper proposes a straightforward technique for transforming the well‐known approximations into another form. This new form is computationally more efficient, since it preserves the sparse structure of the original Jacobian matrix. Furthermore, this new form of PWL approximation explicitly relates the simplex‐based PWL analysis to the conventional formulation of the Katzenelson algorithm. The proposed transform technique is also extended to treat groupwise‐separable mappings and, finally, non‐separable but sparse mappings that arise in real‐life simulation of large electronic circuits. In this paper, all these (transformed) simplex‐based PWL approximations are compared in terms of their generality and efficiency. The computational efficiency of the PWL approximation that utilizes sparsity is validated with realistic simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Towards analog implementations of PWL two‐dimensional non‐linear functions

This paper deals with the circuit implementation of non‐linear algebraic bivariate functions. The synthesis procedure is based on a piecewise‐linear approximation technique and on a corresponding circuit architecture, whose basic element is a circuit block with the input/output function y(x) = max(0; x). Some known CMOS circuit structures that can be used to obtain such a block are considered, and their main advantages and drawbacks are pointed out. The static and dynamic features of both the single circuit block and the overall architecture for two‐dimensional PWL functions are illustrated by way of examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Second order mem‐circuits

This paper presents a comprehensive taxonomy of so‐called second‐order memory devices, which include charge‐controlled memcapacitors and flux‐controlled meminductors, among other novel circuit elements. These devices, which are classified according to their differential and state orders, are necessary to get a complete extension of the family of classical nonlinear circuit elements (resistors, capacitors, and inductors) for all possible controlling variables. Using a fully nonlinear formalism, we obtain nondegeneracy conditions for a broad class of second‐order mem‐circuits. This class of circuits is expected to yield a rich dynamic behavior; in this regard, we explore certain bifurcation phenomena exhibited by a family of circuits including a charge‐controlled memcapacitor and a flux‐controlled meminductor, providing some directions for future research. Copyright © 2014 John Wiley & Sons, Ltd.     

A method for the approximate synthesis of cellular non‐linear networks—Part 2: Circuit reduction

In this paper, we face the problem of model reduction in piecewise‐linear (PWL) approximations of non‐linear functions. The reduction procedure presented here is based on the PWL approximation method proposed in a companion paper and resorts to a strategy that exploits the orthonormality of basis functions in terms of a proper inner product. Such a procedure can be favourably applied to the synthesis of the resistive parts of cellular non‐linear networks (CNNs) to reduce the complexity of the resulting circuits. As an example, the method is applied to a case study concerning a CNN for image processing. Copyright © 2003 John Wiley & Sons, Ltd.     

Constructive approximation of non‐linear discrete‐time systems

It is known that large classes of approximately‐finite‐memory maps can be uniformly approximated arbitrarily well by the maps of certain non‐linear structures. As an application, it was proved that time‐delay networks can be used to uniformly approximate arbitrarily well the members of a large class of causal nonlinear dynamic discrete‐time input–output maps. However, the proof is non‐constructive and provides no information concerning the determination of a structure that corresponds to a prescribed bound on the approximation error. Here we give some general results concerning the problem of finding the structure. Our setting is as follows. There is a large family 𝒢 of causal time‐invariant approximately‐finite‐memory input‐output maps G from a set S of real d‐vector‐valued discrete‐time inputs (with d⩾1) to the set of ℝ‐valued discrete‐time outputs, with both the inputs and outputs defined on the non‐negative integers 𝒵₊. We show that for each ϵ>0, any Gϵ𝒢 can be uniformly approximated by a structure map H(G, ·) to within tolerance ϵ, and we give analytical results and an example to illustrate how such a H(G, ·) can be determined in principle. Copyright © 2000 John Wiley & Sons, Ltd.     

A robustification tool for adaptive non‐linear control

This paper presents a conceptually simple robustification approach for the adaptive control of a class of non‐linear systems with static and dynamic uncertainties. This approach generates a new class of robust adaptive non‐linear controllers and is based upon a combined application of the well‐known adaptive backstepping and recent non‐linear small‐gain techniques. The presented method is illustrated via a third‐ order chemical reactor with only temperature information, and under relaxed conditions. An adaptive output‐feedback stabilizer is obtained without resorting to any state observer. Copyright © 1999 John Wiley & Sons, Ltd.     

A new method for solving non‐linear carrier diffusion equation in axial direction of broad‐area lasers

The solution of non‐linear carrier diffusion equation associated with the axial direction of broad‐area laser has been achieved by a hybrid asymptotic‐numerical method that combines WKB approximation and the variation of constants method. The non‐linearity has been taken into account by using an iterative scheme: carrier lifetime has been linearized and calculated as a function of carrier density known from the previous iteration. Non‐uniform photon density has been taken into account. As the new method is based on analytical solutions, it is very fast and not vulnerable to large gradients of carrier concentration usually occurring in the vicinity of laser facets. Therefore, it can be useful in complicated self‐consistent thermal models of broad‐area lasers. The obtained results are compared with two simplified cases: solution of linear diffusion equation with constant coefficients and calculations neglecting the diffusion. Copyright © 2010 John Wiley & Sons, Ltd.     

Electrical characterization of linear and non‐linear random networks and mixtures

A material composed of a mixture of distinct homogeneous media can be considered as a homogeneous one at a sufficiently large observation scale. The problem of mixture characterization has been exactly solved in case of linear random mixtures, that is, materials for which the various components are isotropic, linear and mixed together as an ensemble of particles having random shapes and positions. In the present work the authors briefly review the linear theory and then consider mixtures of non‐linear media. In particular they give formulas for obtaining their constitutive equations for current density, electrical displacement, and magnetic induction. These relations have been derived by means of heuristic considerations on random networks and they have been verified with simulations obtaining a high degree of accuracy. Copyright © 2003 John Wiley & Sons, Ltd.     

Guaranteed recursive non‐linear state bounding using interval analysis

The problem considered here is state estimation in the presence of unknown but bounded state perturbations and measurement noise. In this context, most available results are for linear models, and the purpose of the present paper is to deal with the non‐linear case. Based on interval analysis and the notion of set inversion, a new state estimator is presented, which evaluates a set estimate guaranteed to contain all values of the state that are consistent with the available observations, given the perturbation and noise bounds and a set containing the initial value of the state. To the best of our knowledge, it is the first estimator for which this claim can be made. The precision of the set estimate can be improved, at the cost of more computation. Theoretical properties of the estimator are studied, and computer implementation receives special attention. A simple illustrative example is treated. Copyright © 2002 John Wiley & Sons, Ltd.     

Bifurcation analysis of a piecewise smooth system with non‐linear characteristics

In previous works, there are no results about the bifurcation analysis for a piecewise smooth system with non‐linear characteristics. The main purpose of this study is to calculate the bifurcation sets for a piecewise smooth system with non‐linear characteristics. We first propose a new method to track the bifurcation sets in the system. This method derives the composite discrete mapping, Poincaré mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. As an illustrated example, we then apply this general methodology to the Rayleigh‐type oscillator containing a state‐ period‐dependent switch. In the circuit, we can find many subharmonic bifurcation sets including global bifurcations. We also show the bifurcation sets for the border‐collision bifurcations. Some theoretical results are verified by laboratory experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient algorithm for finding all DC solutions of piecewise‐linear circuits

An efficient algorithm is proposed for finding all DC solutions of transistor circuits where characteristics of transistors are represented by piecewise‐linear (PWL) convex monotone functions. This algorithm is based on a simple test (termed the linear programming, LP, test) for non‐existence of a solution to a system of PWL equations in a given region. In the conventional LP test, the system of PWL equations is transformed into an LP problem by surrounding component PWL functions by rectangles. Then the dual simplex method is applied, by which the number of pivotings per region becomes very small. In this letter, we propose a new LP test using the dual simplex method and triangles. The proposed test is not only efficient but also more powerful than the conventional test using the simplex method or rectangles. Copyright © 2008 John Wiley & Sons, Ltd.     

Differential equation description and Chebyshev approximation of linear time‐invariant circuits

The approximation technology of analogue circuit functions is crucial to the computer‐aided simulation, analysis, and design automation of electronic circuits. Chebyshev polynomials and various differential equations are proposed in this paper to approximate the functions of linear time‐invariant circuits. The coefficient calculation methods of the Chebyshev expansion and the differential equation matrices are thoroughly deduced, and the construction methods employed in the functions and the actual time mapping of the linear time‐invariant circuits are presented in this paper. An example of an analogue filter verifies the effectiveness and accuracy of the proposed approximation algorithm and elaborates on the selection process of the order number and the time step length of the Chebyshev expansion according to the demanded truncation error.