Finding all stable equilibria of cellular neural networks

An efficient algorithm is given for finding all stable equilibrium states of a cellular neural network. the method is based upon a new theorem concerning the network dynamics. the theorem implies the implementation of a simple sign test which, combined with a binary tree search, efficiently performs the task of searching the equilibria. Since the proposed algorithm uses information about the network architecture and dynamic behaviour, it exhibits two important advantages over other more general methods for searching the equilibria of resistive piecewise-linear circuits. First, the regions which cannot possibly include stable steady states can be eliminated even before starting to perform the algorithm. Secondly, there is no need to solve any set of linear equations, which is usually the most tedious part of equilibrium-searching methods.     

A generalized theorem of Reichert for biquadratic minimum functions

This paper is concerned with a generalized theorem of Reichert for biquadratic minimum functions, which states that any biquadratic minimum function realizable as the impedance of networks with n reactive elements and an arbitrary number of resistors can be realized with n reactive elements and two resistors. First, a series of constraints on networks realizing minimum functions are presented. Furthermore, by discussing the possible resistor edges incident with vertices of the reactive‐element graphs, it is proved that any minimum function realizable as the impedance of networks with three reactive elements and an arbitrary number of resistors can be realized with three reactive elements and two resistors, from which the validity of the case of n = 3 follows. Similarly, the validity of the case of n = 4 is proved. Together with the Bott–Duffin realizations, the generalized theorem of Reichert for biquadratic minimum functions is finally proved. The results of this paper are motivated by passive mechanical control. Copyright © 2016 John Wiley & Sons, Ltd.     

On the number of conductances required for realizing Y and K matrices

Upper bounds are established on the number of conductances required for realizing a real symmetric matrix Y as the short-circuit conductance matrix of a resistive n-port network containing no negative conductances, and for the realization of a real matrix K as the potential factor matrix of a similar network without negative conductances. These results are the consequence of the properties of the modified cut-set matrix of an n-port and a theorem in the theory of linear programming.     

Short-circuit admittance matrix synthesis with rc common-ground networks and a minimum number of grounded finite-gain phase-inverting voltage-controlled voltage sources

A new procedure for the synthesis of active RC networks when grounded finite-gain phase-inverting voltage-controlled voltage sources serve as active elements is developed. It is proved that an arbitrary n × n matrix of real rational functions of the complex frequency variable can be realized as the short-circuit admittance matrix of a grounded transformerless active RC n-port network containing (n+1) grounded finite-gain phase-inverting voltage-controlled voltage sources (VCVSs). In general all the (n+1) grounded VCVSs are necessary. The structure proposed to prove a general theorem is later simplified for the realization of a restricted but important class of real rational matrices to obtain considerable savings in the computation volume and in the number of passive components used for the realization of the network. Examples are given to illustrate presented synthesis procedures.     

Some recent results in structured robust stability: An overview

Some recent results on robust stability with structured perturbations, using the polynomial framework, are presented in this paper without proof. As background we first describe Kharitonov's theorem and give an interpretation of it as a generalization of the Hermite—Bieler interlacing theorem. The need for a generalization of this result for tackling the control problem is explained, and our new results are then presented. An important generalization of Kharitonov's theorem that solves the box problem in parameter space is described. Some efficient formulae for the l²-stability margin in parameter space are also given. The results are illustrated by examples.     

Full and reduced-order H∞ filtering of Takagi–Sugeno fuzzy time-varying delay systems: Input–output approach

This article addresses the issue of delay-dependent H_∞ filtering design for TakagiŮSugeno fuzzy time-varying delay systems using the input–output approach. A three-term approximation model has been used to transform the original system into two interconnected subsystems. Since the nonquadratic Lyapunov–Krasovskii functional requires to deal with the membership function's (MF) time derivative, upper-bound inequalities have been added to the obtained conditions. Based on the scaled small gain theorem, nonquadratic Lyapunov–Krasovskii functional approach and considering the bounds of the MF time-derivative, the H_∞ full- and reduced-order filters are designed and then formulated in terms of linear matrix inequalities. Finally, illustrative examples are presented to demonstrate the validity of the proposed methods.     

Theory on resistance of m × n cobweb network and its application

A basic theorem of equivalent resistance between two arbitrary nodes in an m × n cobweb network in both finite and infinite conditions is discovered, and two conjectures on the equivalent resistance are proved in terms of the basic theorem. We built a tridiagonal matrix equation by means of network analysis and made a diagonalization method of matrix transformation and work out its explicit expressions. The new formulae obtained here can be effectively applied in complex impedance network, especially the formulation leads to the occurrence of resonances and a series of novel results in RLC (denote resistor, inductance and capacitance) network. These curious results suggest the possibility of practical applications to resonant circuits. Copyright © 2014 John Wiley & Sons, Ltd.     

Audio source separation: solutions and problems

The problem of separating out a number of audio sources observed from an array of microphones in a real room environment has received a great deal of attention in the past decade. While there are now a number of workable methods that can even deal with relatively high reverberation (IEEE Trans Audio Speech Process, 2003; 11 :489–497), a number of interesting problems still remain. In this paper, the authors review the methods based around independent component analysis, discussing the various choices available in algorithm design. We then explore the issue of sensitivity to speaker movement which appears to impose fundamental limitations on BSS performance. Copyright © 2004 John Wiley & Sons, Ltd.     

Considerations on the equivalence between non‐uniform transmission lines

We investigate the relation of equivalence between non‐uniform lossless transmission lines (LC lines or lines, for short) with sectionally smooth distributed parameters (series inductance L(x) and shunt capacitance C(x) per unit length, x being the spatial variable). The equivalence condition is derived by heuristic and well‐grounded inference. The condition provides us with the basis on which we can generate by rote a line equivalent to a fixed original line according to a specified change of spatial variable (elasticity function). The concept of intermediary lines between a pair of equivalent lines is introduced to enhance precise understanding of the condition and the procedure to generate equivalent lines. In addition, fused parameters are defined as a generalization of distributed parameters, in terms of which the argument and formulas are developed in a brief and inclusive manner. It is shown to be possible, on generating equivalent lines, to preassign a fused parameter of the equivalent line or one of the associated intermediary lines instead of specifying the elasticity function. Equivalent lines thus obtained are called conditioned equivalent lines. We discuss how to get conditioned equivalent lines as well as equivalency tests of arbitrarily given pair of LC lines to present some comprehensible and promising methods involving a versatile, graphical technique. We also refer to such interesting topics as a normalization of LC lines, a kind of conservation law of fused parameters between equivalent lines and the scope of realizable characteristics by LC lines conditioned in an arbitrary way. Some formerly known equivalence conditions are located appropriately in the context of our theory. Further, the equivalence condition is shown to be not only sufficient but also necessary for LC lines whose distributed parameters may have a finite number of points of discontinuity by invoking a theorem for an inverse scattering problem. Copyright © 2005 John Wiley & Sons, Ltd.     

The describing function method and free vibrations in nonlinear systems

The most popular method of investigation of periodic oscillations in nonlinear systems is the describing function method. Two theorems on the existence of a free vibration in a given neighbourhood of the approximate vibration obtained by this method are presented in the paper. Moreover, for a typical transfer function K(jω) of the linear part of the system and a typical describing function of the non-linear element, the theorem on the estimation of frequency of the free vibration is given.     

Hybrid sequences for rapidT 1 imaging

This article generalizes the concept of the Look-LockerT ₁-measurement sequence to include both EPI-like and Snapshot FLASH-like elements and it provides a bridge between a number of previously demonstrated methods of quantitativeT ₁ imaging. It is shown that a segmentedk-space acquisition provides numerous advantages if sufficient time is available.     

Ideal rectifier n-ports are solvable

It is proved that an n-port composed of ideal transformers, gyrators and ideal rectifiers has a unique solution for the port variables when it is excited by generators in series with strictly positive resistances. the multiplicity of solutions for the internal variables is discussed. Connections are established with the theory of linear inequalities and with the coloured branch theorem of graph theory.     

The vector potential revisited

The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts: 1) the so-called transverse part represented by a divergenceless vector; and 2) the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when the Helmholtz theorem is used are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, the Aharonov-Bohm effect is chosen here. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. It is shown that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is determined uniquely by the magnetic field and does not alter by a gauge transformation. On the other hand, the longitudinal part depends on the choice of special gauge. It is shown that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.     

An efficient exact numerical solution for scattering by a circular cylinder

An efficient and exact spectral integral method (SIM) for the general problems of scattering by a circular cylinder is presented, both for acoustic and electromagnetic cases. Fast Fourier transform (FFT) and convolution theorem help us to execute the algorithm efficiently with computational complexity. The Fourier coefficients of the integral kernels are computed in a semianalytical way to accomplish a fast convergence rate with spectral accuracy. Compared with the method not using the semianalytical forms, without the increase of the computing and memory abundance, it saves more than 90% sampling points to achieve about the same accuracy, and with the same number of sampling points, several to more than ten orders of magnitude more accuracy can be achieved. For example, for the tested large wavenumber case with 2 × 10⁶ wavelengths on the boundary, only two sampling points per wavelength are required to achieve a relative error of less than 0.001%. More than 90% computational time is saved compared with the normal summation formula of harmonic cylinder expansions. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

On the maximal power transfer theorem for n-ports

The maximal power problem is considered for several classes of terminating networks, in particular an arbitrary resistive n-port termination, and the case of a single resistor terminating each port. A duality theorem is established that converts the problem of maximizing power to the problem of minimizing a related function. The latter problem is solved completely for arbitrary resistive terminations, and examples are given. For completeness, some previously published work is discussed.     

On the almost sure central limit theorem for ARX processes in adaptive tracking

The goal of this paper is to highlight the almost sure central limit theorem (ASCLT) for martingales to the control community. We shall establish the ASCLT for the least squares estimator of the unknown parameter of a controllable ARX(p,q) process in adaptive tracking. The usual notion of controllability for ARX(p,q) processes allows us to avoid the use of a persistent excitation in the adaptive tracking control. We shall also provide strongly consistent estimators of the even moments of the driven noise as well as two quadratic strong laws for the average costs and the estimation errors sequences. Our theoretical results are illustrated by numerical experiments.     

Separation conditions and criteria for uniform approximation of input–output maps

We consider multidimensional shift-invariant input–output maps G from a relatively compact set of functions S to a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated arbitrarily well using a certain structure consisting of a not-necessarily linear dynamic part followed by a non-linear memoryless section that may contain sigmoids or radial basis functions, etc. The dynamic part is comprised of a finite number of dynamic maps h₁,…,h_j drawn from a set H of maps that satisfy a certain continuity condition. In our results certain separation conditions, of the kind associated with the Stone–Weierstrass theorem, play a prominent role. Here they emerge as criteria for approximation, and not just sufficient conditions under which an approximation exists. In particular, one of the theorems given is a result to the effect that universal approximation can be achieved using the structure we consider if and only if the set H satisfies the separation condition that (hu₁)(0)≠(hu₂)(0) for some h∈H whenever u₁,u₂∈cl(S) and u₁≠u₂ (where cl(S) denotes the closure of S). This holds even if the elements of H are not linear. © 1998 John Wiley & Sons, Ltd.     

On state-space synthesis of reciprocal n-ports

The paper considers the state-space synthesis of reciprocal n-ports with a minimum number of reactive elements. After formulating the synthesis problem new procedures are given for the lossless and for the passive cases which use a directly determinable reciprocal minimal realization as starting point. The methods are presented in terms of admittance matrices and then restated for impedance, hybrid and scattering matrix synthesis.     

交流支路的电气剖分方法:(一) 普通支路的剖分

用传统的电力网络分析方法很难准确得到网络源流沿交流支路传输的路径剖分信息。针对此问题,从电力网络的基本原理出发,给出了交流支路3类电气剖分的定义,构造了3类电气剖分的方法,归纳出相应的电气剖分定理。由这些方法和定理可以准确地掌握网络源流沿剖分子路径传输的电气量的性质和数量,且剖分子路径与原支路具有电气上的等效性,并严格满足电力网络的基本电气规律。算例结果显示了所提出的方法和定理的正确性。最后,简述了电气剖分的优点及进一步工作的重点,并就相关问题进行了讨论。