Mauro Marini

Necessary and sufficient condition for absolute stability of neural networks

The main result in this paper is that for a neural circuit of the Hopfield type with a symmetric connection matrix T, the negative semidefiniteness of T is a necessary and sufficient condition for Absolute Stability. The most significant theoretical implication is that the class of neural circuits with a negative semidefinite T is the largest class of circuits that can be employed for embedding and solving optimization problems without the risk of spurious responses. >

On a class of nonsymmetrical neural networks with application to ADC

For nonzero initial conditions a neural network may stop in a spurious state-that is, in an equilibrium point that does not correspond to the correct digital representation of the input signal. A method based on a particular class of nonsymmetrical neural networks is proposed for eliminating the problem of stopping in spurious states. The dynamical behavior of these structures is studied to prove that they are characterized by a unique equilibrium point which is globally attractive-that is, the system will converge toward this point for every choice of initial conditions and for every choice of (continuous) nonlinearities. The explicit expression obtained for the unique equilibrium point permits one to design the connection strengths between neurons so that the equilibrium coincides with the desired output for a given input signal. The proposed design procedure is applied to the classical example of A/D conversion, showing that this A/D converter structure has no spurious states. The A/D was simulated using SPICE, and experimental results obtained with a discrete component prototype of the converter are presented. >

A condition for global convergence of a class of symmetric neural circuits

A sufficient condition is proved guaranteeing that a class of neural circuits that includes the Hopfield model as a special case is globally convergent towards a unique stable equilibrium. The condition only requires symmetry and negative semi-definiteness of the neuron connection matrix T and is extremely simple to check and apply in practice. The consequences of the above result are discussed in the context of neural circuits for optimization of quadratic cost functions. >

Analog network testability measurement: a symbolic formulation approach

A symbolic formulation approach is applied to the problem of computing testability features of analog linear networks. The program, SAPTES, obtained by following this approach is presented. The program can be a very useful tool for designers and researchers in the field of linear analog circuits. SAPTES, which is written in LISP and runs on MS-DOS personal computers, is able to compute the testability of linear circuits of rather high complexity (composed of tens of components and nodes). Computational times range from a few tens of seconds to some tens of minutes on medium speed computers. The program is easily transportable to workstations or a mainframe, and, for the mainframe, program performance will considerably increase. >

Cellular neural network approach to a class of communication problems

In this paper we discuss the design of a cellular neural network (CNN) to solve a class of optimization problems of importance for communication networks. The CNN optimization capabilities are exploited to implement an efficient cell scheduling algorithm in a fast packet switching fabric. The neural-based switching fabric maximizes the cell throughput and, at the same time, it is able to meet a variety of quality of service (QoS) requirements by optimizing a suitable function of the switching delay and priority of the cells. We also show that the CNN approach has advantages with respect to that based on Hopfield neural networks (HNNs) to solve the considered class of optimization problems. In particular, we exploit existing techniques to design CNNs with a prescribed set of stable binary equilibrium points as a basic tool to suppress spurious responses and, hence to optimize the neural switching fabric performance.

On absolute stability of neural networks

The aim of this paper is to discuss the role of Absolute Stability (ABST) in the design of neural optimization solvers and to find necessary and sufficient conditions for ABST for some classes of neural networks of applicative interest. By ABST it is meant that there is a unique equilibrium point attracting all trajectories of motion and that this property is valid for all neuron activation functions belonging to a specified class of nonlinear mappings and for all constant neural network inputs. ABST neural networks are best suited for solving optimization problems being devoid of spurious suboptimal responses for every choice of the activation function and of the input vector. A necessary and sufficient condition for ABST has been found for symmetric neural networks of the Hopfield type. In this paper, we show that the concept of ABST can be applied also to special classes of nonsymmetric Hopfield neural networks and to neural models different from the Hopfield one. It is shown in particular that necessary and sufficient conditions for ABST can be found for two interesting classes of nonsymmetric networks, namely, cooperative Hopfield-type networks and composite neural networks with variable and constraint neurons used for solving linear and quadratic programming problems in real time.<<ETX>>

Multifrequency measurement of testability with application to large linear analog systems

With increasing electronic circuit complexity, assessing the testability features becomes a necessity during the design, implementation, and operational or maintenance phases of an analog system. A quantitative measure of testability, based on several multifrequency stimuli, is adopted which is able to handle multiple faults and may provide information also on the degree of complexity encountered in a specific test. An efficient and practical algorithm is proposed which is associated with the result of Sen and Saeks and has a well-defined interpretation even with a large number of circuit parameters liable to failure. The described technique is a basis for optimizing the number and allocation of the selected test points; furthermore, it may serve as an aid in functional partitioning of the same system to facilitate testing and/or reduce computational complexity. An application to a classical active filter is also given.

Improvements to numerical testability evaluation

In this paper, some recent theoretical developments are reviewed and associated algorithms are proposed to determine the numerical testability for large multiinput multioutput systems. In our approach, the modified nodal analysis and the usual techniques for the sensitivity computation in the frequency domain are employed. According to the theoretical basis provided by Sen and Saeks [1] the testability evaluation is related to the computation of the number of linearly independent columns in a convenient form of the sensitivity matrix with rational entries having a common denominator. Then, by extending some results already obtained in [3], it is shown that the above-mentioned number can be determined by computing the numerical rank of a matrix comprised of the coefficients obtained by expanding the numerators of the sensitivities in a suitable orthogonal polynomial series. The numerical rank computation is simplified, particularly for large systems, through an algorithm based on the estimation of the polynomial degrees, which is performed by the iterative comparison between the Chebycheff and the corresponding Stirling coefficients.

On the asymptotic behavior of the solutions of a class of second-order linear differential equations

Abstract We give necessary and sufficient conditions for the solutions of the differential equation ( p ( t ) x ′( t ))′ = q ( t ) x ( t ) to be bounded together with their first derivatives. We also study the asymptotic behavior of the solutions.

Positive decreasing solutions of quasi-linear difference equations

The second order nonlinear difference equation is considered. A full characterization of limit behavior of all positive decreasing solutions is established. The obtained results answer some open problems formulated for Sturm-Liouville discrete operator. A comparison with the continuous case jointly with similarities and discrepancies is given as well.