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.     

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.     

Cellular non‐linear networks for minimization of functionals. Part 2: Examples

A method for the definition of cellular non‐linear networks able to find approximate minima of rather a large class of continuous functionals is illustrated through three examples. The method, based on the spatial discretization of continuous functionals and on the theory of potential functions for resistive circuits, has been presented in Part 1 of this paper. The first example (related to electromagnetic‐field theory) has the main purpose to show some aspects of the application procedure. The other two examples concern, respectively, a possible image‐processing application of the method (where a parallel processing is highly desirable) and a comparison with another method proposed in the literature on CNNs. Copyright © 2001 John Wiley & Sons, Ltd.     

Cellular non‐linear networks for minimization of functionals. Part 1: Theoretical aspects

A method for the definition of cellular non‐linear networks able to find approximate minima of rather a large class of continuous functionals is proposed and discussed from a theoretical point of view. The method is based on the spatial discretization of continuous functionals and on the theory of potential functions for resistive circuits. The discretization of the continuous functionals is obtained by resorting to the finite difference method or to the finite element method. The spatial discretization converts a functional into a function of a finite set of variables. By exploiting the theory of potential functions for resistive circuits, from such a function one can derive a lumped circuit that makes it possible to find an approximate minimum of the given functional. Copyright © 2001 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 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.     

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.     

PAC learning in non‐linear FIR models

The PAC learning theory creates a framework to assess the learning properties of static models for which the data are assumed to be independently and identically distributed (i.i.d.). The present paper first extends the idea of PAC learning to cover the learning of modelling tasks with m‐dependent data, and then applies the resulting framework to evaluate learning of non‐linear FIR models. Also, the learning properties of FIR modelling with radial basis function networks are further specified. These results include an upper bound on the size of the data set required to train an FIR radial basis function network, provided that the input data are uniformly distributed. Copyright © 2001 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.     

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.     

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.     

Hypothetical simulation of non‐linear systems with memory,using an ideal computer with infinite speed and capacity

The solution of equations describing non‐linear dynamic systems is considered. A computer with infinite speed and capacity—as a hypothetical device—is assumed. The method is based on the classical sampling theorem, Newton iteration and discrete Fourier transform. Algorithms for transient and periodic cases, working simultaneously in the time and frequency domain are introduced and demonstrated. Copyright © 2000 John Wiley & Sons, Ltd.     

Adaptive output tracking for a class of non‐linear time‐delay systems

In this paper the output tracking control problem for a class of non‐linear time delay systems with some unknown constant parameters is addressed. Such a problem is solved in the case that the non‐linear time‐delay system has full delay relative degree and stable internal dynamics. It is supposed moreover that the output and its time derivatives until n?1, where n is the length of the state vector (euclidean part), do not depend explicitly on the unknown parameters. This work is the first step towards the application of the methodologies of adaptive control for non‐linear delayless systems, based on tools of differential geometry, to non‐linear time‐delay systems too. Copyright © 2004 John Wiley & Sons, Ltd.     

A cellular non‐linear network for image fusion based on data regularization

In this paper, a synthesis method developed in the last few years is applied to derive a cellular non‐linear network (CNN) able to find an approximate solution to a variational image‐fusion problem. The functional to be minimized is based on regularization theory and takes into account two complementary principles, namely, knowledge source corroboration and belief enhancement/withdrawal, both typical of data‐fusion approaches. The obtained CNN has been tested by simulations (i.e. by numerically integrating the circuit state equations) in some case studies. The quality of the results is good, as turns out from comparisons with some standard methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Direct adaptive control for non‐linear uncertain systems with exogenous disturbances

A direct adaptive non‐linear control framework for multivariable non‐linear uncertain systems with exogenous bounded disturbances is developed. The adaptive non‐linear controller addresses adaptive stabilization, disturbance rejection and adaptive tracking. The proposed framework is Lyapunov‐based and guarantees partial asymptotic stability of the closed‐loop system; that is, asymptotic stability with respect to part of the closed‐loop system states associated with the plant. In the case of bounded energy L₂ disturbances the proposed approach guarantees a non‐expansivity constraint on the closed‐loop input–output map. Finally, several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Adaptive estimation and rejection of unknown sinusoidal disturbances through measurement feedback for a class of non‐minimum phase non‐linear MIMO systems

This paper develops an adaptive estimation method to estimate unknown disturbances in a class of non‐minimum phase non‐linear MIMO systems. The unknown disturbances are generated by an unknown linear exosystem. The frequencies, phases and amplitudes of the disturbances are unknown, the only available information of the disturbances is the number of distinctive frequencies. The system considered in this paper is a class of MIMO non‐linear systems in the output feedback form which can be non‐minimum phase. The proposed estimation algorithm provides exponentially convergent estimates of system states, unknown disturbances in the system and frequencies of the disturbances characterized by the eigenvalues of the exosystem. Moreover, based on the stabilization controller for the disturbance free system, the estimates of the disturbances are used to solve the disturbance rejection problem. The unknown disturbances are compensated completely with the stability of the whole closed‐loop system. Copyright © 2006 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.     

An approximation for the input conductivity function of the non‐linear resistive grid

The 2‐term approximations for the input conductivity functions, i_in=f_in(v_in), of a grid of similar weakly non‐linear (parabolic) conductors, and the grid's symmetric cuts, measured between two close nodes, are derived, using a semi‐empirical method; the results of a relevant PSpice simulation are presented. The functions f_in(v_in) of the grid's symmetric cuts possess a common analytical feature. Simulation results show that the error in the calculation of the non‐linear terms in the input functions is less than 1 per cent. Copyright © 2001 John Wiley & Sons, Ltd.