M‐matrices and global convergence of discontinuous neural networks

Abstract

The paper considers a general class of neural networks possessing discontinuous neuron activations and neuron interconnection matrices belonging to the class of M-matrices or H-matrices. A number of results are established on global exponential convergence of the state and output solutions towards a unique equilibrium point. Moreover, by exploiting the presence of sliding modes, conditions are given under which convergence in finite time is guaranteed. In all cases, the exponential convergence rate, or the finite convergence time, can be quantitatively estimated on the basis of the parameters defining the neural network. As a by-product, it is proved that the considered neural networks, although they are described by a system of differential equations with discontinuous right-hand side, enjoy the property of uniqueness of the solution starting at a given initial condition. The results are proved by a generalized Lyapunov-like approach and by using tools from the theory of differential equations with discontinuous right-hand side. At the core of the approach is a basic lemma, which holds under the assumption of M-matrices or H-matrices, and enables to study the limiting behaviour of a suitably defined distance between any pair of solutions to the neural network

A new scheme for power factor correction and active filtering for six-pulse converters loads

“…References [13][14][15][16][17], and references therein. Indeed, such networks are ideally suited for solving optimization problems.…”

“…In Reference [13], cellular neural networks (CNNs) were used for tracking and obstacle avoidance in an autonomous robot. In References [14][15], the properties of global exponential convergence of discontinuous neural networks and delayed bidirectional associative memory neural networks (DBAMNN) were discussed. Then, in Reference [16], the convergence dynamics of general neural networks (GNNs) under almost periodic stimuli was investigated.…”

“…Hence two groups of ADALINEs are adopted in the d-axis and q-axis of the synchronous rotating frame to derive the dc components in d − q reference frame, corresponding to the fundamental components of load currents in stationary phase a − b − c frame. The drawbacks regarding to the sophisticated network structure and computational complexity in References [13][14][15][16][17] can be significantly simplified, which makes it possible to implement the algorithms in low cost digital signal processor (DSP) and facilitate the practical implementation of the proposed system.…”

“…The proportional-integral (PI) controllers are implemented in daxis and q-axis based on the decoupled state-space equations of the T-type active power filter. The load side dynamics are incorporated in the feed-forward control loop by using two groups of synchronous frame adaptive linear neural networks (SADALINEs) [13][14][15][16][17][18][19][20][21][22][23] in d-axis and q-axis, respectively, for fast estimation of the fundamental components of nonlinear load currents. Besides, a separate proportional-integral (PI) controller is adopted in the outer voltage loop for balancing the active power flow of the voltage source inverter.…”

Global Exponential Stability and Global Convergence in Finite Time of Neural Networks with Discontinuous Activations

“…Considering this, some researchers have studied the exponential stability analysis problem for neural networks since one can use exponential convergence rate to determine the speed of neural computations [9,12,13,17]. And [9,10,12,13] study convergence in finite time which is especially desirable in the design of real time neural optimization solvers, and it cannot be achieved in smooth dynamical systems, since in that case there can be only asymptotic convergence toward an equilibrium point [27].…”