Showing papers by "Anthony N. Michel published in 1994"

Dynamical Systems with Saturation Nonlinearities: Analysis and Design

Abstract: to dynamical systems with saturation nonlinearities.- Qualitative theory of control systems with control constraints and state saturation: Two fundamental issues.- Asymptotic stability of dynamical systems with state saturation.- Null controllability of discrete-time dynamical systems with control constraints and state saturation.- Stability analysis of one-dimensional and multidimesional state-space digital filters with overflow nonlinearities.- Criteria for the absence of overflow oscillations in fixed-point digital filters using generalized overflow characteristics.- Stability analysis of state-space realizations for multidimensional filters with overflow nonlinearities.- to part III.- Analysis and synthesis of a class of neural networks with piecewise linear saturation activation functions.- Sparsely interconnected neural networks for associative memories with applications to cellular neural networks.- Robustness analysis of a class sparsely interconnected neural networks with applications to design problem.

Global stability and local stability of Hopfield neural networks with delays.

Abstract: It is well known that Hopfield neural networks without delays exhibit no oscillations and possess global stability (i.e., all trajectories tend to some equilibrium). In the present paper we show that if the bound \ensuremath{\tau}\ensuremath{\beta}\ensuremath{\parallel}${\mathit{T}}_{2}$\ensuremath{\parallel}1 is satisfied, then a corresponding Hopfield neural network with delays \ensuremath{\tau}g0, interconnection matrix ${\mathit{T}}_{2}$ associated with delays, and gain of the neurons given by \ensuremath{\beta}, will exhibit similar qualitative properties as the original Hopfield neural network without delays (\ensuremath{\parallel}${\mathit{T}}_{2}$\ensuremath{\parallel} denotes the matrix norm induced by the Euclidean vector norm). Specifically, we show that if the above bound is satisfied, then a Hopfield neural network without delays and a corresponding Hopfield neural network with delays will have identical asymptotically stable equilibria, and both networks are globally stable. In addition to the above, we provide in the present paper an effective method of determining the asymptotic stability of an equilibrium of a Hopfield neural network with delays, assuming that the above bound is satisfied. Our results are consistent with the results reported by Marcus and Westervelt [Phys. Rev. A 39, 347 (1989)]. Specifically, the present results, all of which are obtained by rigorous proof, give support to these results, which are based on linearization arguments, numerical simulations, and experimental results.

Sparsely interconnected neural networks for associative memories with applications to cellular neural networks

Abstract: We first present results for the analysis and synthesis of a class of neural networks without any restrictions on the interconnecting structure. The class of neural networks which we consider have the structure of analog Hopfield nets and utilize saturation functions to model the neurons. Our analysis results make it possible to locate in a systematic manner all equilibrium points of the neural network and to determine the stability properties of the equilibrium points. The synthesis procedure makes it possible to design in a systematic manner neural networks (for associative memories) which store all desired memory patterns as reachable memory vectors. We generalize the above results to develop a design procedure for neural networks with sparse coefficient matrices. Our results guarantee that the synthesized neural networks have predetermined sparse interconnection structures and store any set of desired memory patterns as reachable memory vectors. We show that a sufficient condition for the existence of a sparse neural network design is self feedback for every neuron in the network. We apply our synthesis procedure to the design of cellular neural networks for associative memories. Our design procedure for neural networks with sparse interconnecting structure can take into account various problems encountered in VLSI realizations of such networks. For example, our procedure can be used to design neural networks with few or without any line-crossings resulting from the network interconnections. Several specific examples are included to demonstrate the applicability of the methodology advanced herein. >

Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities

Abstract: We utilize the second method of Lyapunov to establish sufficient conditions for the global asymptotic stability of the trivial solution of percent nonlinear, shift-invariant 2-D (two-dimensional) systems. We apply this result in the stability analysis of 2-D quarter plane state-space digital filters, which are endowed with a general class of overflow nonlinearities. Utilizing the l/sub /spl infin// vector norm and the p/sup th/ power of the l/sub p/ vector norm for 1/spl les/p >

Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices

Abstract: Establishes a set of new sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors use these results to establish necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors relate the above results to the existence of quadratic Lyapunov functions for linear time-invariant systems with interval-valued coefficient matrices. Using the above results, the authors develop an algorithm to determine the Hurwitz and the Schur stability properties of interval matrices. The authors demonstrate the applicability of their results by means of two specific examples. >

Necessary and sufficient conditions for the controllability and observability of a class of linear, time-invariant systems with interval plants

Abstract: We establish necessary and sufficient conditions for the controllability of single-input/multi-output linear time-invariant systems with interval plants and for the observability of multi-input/single-output linear time-invariant systems with interval plants. In arriving at our results, we determine allowable plant uncertainty bounds for controllability and observability. Two specific examples are considered to demonstrate the applicability of our results. >

Qualitative analysis of dynamical systems determined by differential inequalities with applications to robust stability

Abstract: In this paper we develop a Lyapunov stability theory for finite dimensional continuous-time dynamical systems described by a system of first-order ordinary differential inequalities. We utilize this theory to establish sufficient robust stability criteria for a large class of finite dimensional, continuous-time dynamical systems described by systems of ordinary differential equations. We demonstrate the applicability of the methodology advanced herein by means of a specific example that has been considered in the literature. In terms of computational complexity and conservatism of stability criteria, the present results frequently offer improvements over existing results. >

Stability of interconnected systems: results without quasimonotonicity conditions

Abstract: Addresses the stability analysis of a large class of nonlinear, continuous-time, finite dimensional dynamical systems which satisfy sector conditions. For such systems the authors define finite sets of extreme systems. The authors' stability results which are phrased in terms of the qualitative properties of the extreme systems provide a systematic method for the stability analysis of an important class of problems. By means of specific examples the authors demonstrate that when applicable, the present results frequently offer advantages over existing results (e.g., over existing stability results of interconnected systems). The authors show that the present results constitute robustness stability results.

Stability and stabilizability of a class of systems with state saturation and parameter uncertainties

Abstract: For linear systems with parameter uncertainties and subject to state saturation, we establish results concerning two important issues: (a) global asymptotic stability of an equilibrium, and (b) stabilizability by means of linear state feedback. Systems of the type considered herein capture two important phenomena commonly encountered in the modeling process: (i) system parameter uncertainties (which in the present case are modeled by means of interval matrices), and (ii) operation of systems over a wide range (which in the present case is accounted for by state saturation nonlinearities).

On Lyapunov stability of a family of nonlinear time-varying systems

Abstract: Investigates several types of Lyapunov stability of an equilibrium of a family of finite dimensional dynamical systems determined by ordinary differential (difference) equations. By utilising the extreme systems of the family of systems, the authors establish sufficient conditions, as well as necessary conditions (converse theorems) for several robust stability types. The authors' results enable them to realize a significant reduction in the computational complexity of the algorithm of Brayton and Tong in the construction of computer generated Lyapunov functions. Furthermore, the authors demonstrate the applicability of the present results by analyzing robust stability properties of equilibria for Hopfield neural networks and by analyzing the Hurwitz and Schur stability of interval matrices. >

Possible bursting phenomena in self-tuning control

Abstract: In this paper, in the case when there are no modelling errors, bursting phenomena in extended-least-squares- (ELS-) based self-tuning control are analysed It is shown that in the absence of external excitation and without imposing structural conditions on the system model, bursts of the tracking error and consequently drift of the parameter estimates are possible By analysing the possible occurrence of bursting, an upper bound on the convergence rate of the tracking error is established It is demonstrated that it is the self-excitation mechanism which provides global stability of the adaptive system In other words, the possible instability of the adaptive loop generates a bounded condition number of the covariance matrix This in turn implies stabilization of the adaptive system In this paper we also discuss a stochastic gradient algorithm with a convergence rate similar to that presently established for the ELS-based self-tuning controller