Paolo Nistri

Mathematical models for hysteresis

The various existing classical models for hysteresis, Preisach, Ishlinskii, Duhem–Madelung, are surveyed, as well a more modern treatments by contemporary workers. The emphasis is on a clear mathematical description of the formulation and properties of each model. In addition the authors try to make the reader aware of the many open questions in the study of hysteresis.

Global convergence of neural networks with discontinuous neuron activations

The paper introduces a general class of neural networks where the neuron activations are modeled by discontinuous functions. The neural networks have an additive interconnecting structure and they include as particular cases the Hopfield neural networks (HNNs), and the standard cellular neural networks (CNNs), in the limiting situation where the HNNs and CNNs possess neurons with infinite gain. Conditions are derived which ensure the existence of a unique equilibrium point, and a unique output equilibrium point, which are globally attractive for the state and the output trajectories of the neural network, respectively. These conditions, which are applicable to general nonsymmetric neural networks, are based on the concept of Lyapunov diagonally-stable neuron interconnection matrices, and they can be thought of as a generalization to the discontinuous case of previous results established for neural networks possessing smooth neuron activations. Moreover, by suitably exploiting the presence of sliding modes, entirely new conditions are obtained which ensure global convergence in finite time, where the convergence time can be easily estimated on the basis of the relevant neural-network parameters. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. In particular, global convergence is addressed by using a Lyapunov-like approach based on the concept of monotone trajectories of a differential inclusion.

Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain

This paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. The discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity, while the delay accounts for the finite switching speed of the neuron amplifiers, or the finite signal propagation speed. It is known that the delay in combination with high-gain nonlinearities is a particularly harmful source of potential instability. The goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay. More precisely, conditions are given under which there is a unique equilibrium point of the neural network, which is globally exponentially stable for the states, with a known convergence rate. The conditions are easily testable and independent of the delay. Moreover, global convergence in finite time of the state and output is investigated. In doing so, new interesting dynamical phenomena are highlighted with respect to the case without delay, which make the study of convergence in finite time significantly more difficult. The obtained results extend previous work on global stability of delayed neural networks with Lipschitz continuous neuron activations, and neural networks with discontinuous neuron activations but without delays.

Generalized neural network for nonsmooth nonlinear programming problems

In 1988 Kennedy and Chua introduced the dynamical canonical nonlinear programming circuit (NPC) to solve in real time nonlinear programming problems where the objective function and the constraints are smooth (twice continuously differentiable) functions. In this paper, a generalized circuit is introduced (G-NPC), which is aimed at solving in real time a much wider class of nonsmooth nonlinear programming problems where the objective function and the constraints are assumed to satisfy only the weak condition of being regular functions. G-NPC, which derives from a natural extension of NPC, has a neural-like architecture and also features the presence of constraint neurons modeled by ideal diodes with infinite slope in the conducting region. By using the Clarkes generalized gradient of the involved functions, G-NPC is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in the framework of nonsmooth analysis and the theory of differential inclusions. In the special important case of linear and quadratic programming problems, salient dynamical features of G-NPC, namely the presence of sliding modes , trajectory convergence in finite time, and the ability to compute the exact optimal solution of the problem being modeled, are uncovered and explained in the developed analytical framework.

Convergence of Neural Networks for Programming Problems via a Nonsmooth Lojasiewicz Inequality

This paper considers a class of neural networks (NNs) for solving linear programming (LP) problems, convex quadratic programming (QP) problems, and nonconvex QP problems where an indefinite quadratic objective function is subject to a set of affine constraints. The NNs are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic, which enable to implement an exact penalty method. A new method is exploited to address convergence of trajectories, which is based on a nonsmooth Lstrokojasiewicz inequality for the generalized gradient vector field describing the NN dynamics. The method permits to prove that each forward trajectory of the NN has finite length, and as a consequence it converges toward a singleton. Furthermore, by means of a quantitative evaluation of the Lstrokojasiewicz exponent at the equilibrium points, the following results on convergence rate of trajectories are established: 1) for nonconvex QP problems, each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points; 2) for convex QP problems, the same result as in 1) holds; moreover, the singleton belongs to the set of global minimizers; and 3) for LP problems, each trajectory converges in finite time to a singleton belonging to the set of global minimizers. These results, which improve previous results obtained via the Lyapunov approach, are true independently of the nature of the set of equilibrium points, and in particular they hold even when the NN possesses infinitely many nonisolated equilibrium points

Lyapunov Method and Convergence of the Full-Range Model of CNNs

This paper develops a Lyapunov approach for studying convergence and stability of a class of differential inclusions termed differential variational inequalities (DVIs). The DVIs describe the dynamics of a general system evolving in a compact convex subset of the state space. In particular, they include the dynamics of the full-range (FR) model of cellular neural networks (CNNs), which is characterized by hard-limiter nonlinearities with vertical segments in the i-v characteristic. The approach is based on the following two main tools: 1) a set-valued derivative, which enables to compute the evolution of a Lyapunov function along the solutions of the DVIs without involving integrations, and 2) an extended version of LaSalles invariance principle, which permits to study the limiting behavior of the solutions with respect to the invariant sets of the DVIs. Then, this paper establishes conditions for convergence (complete stability) of DVIs in the presence of multiple equilibrium points (EPs), global asymptotic stability (GAS), and global exponential stability (GES) of the unique EP. These conditions are applied to investigate convergence, GAS, and GES for FR-CNNs and some extended classes of FR-CNNs. It is shown that, by means of the techniques developed in this paper, the analysis of convergence and stability of FR-CNNs is no more difficult than that of the standard (S)-CNNs. In addition, there are significant cases, such as the symmetric FR-CNNs and the nonsymmetric FR-CNNs with a Lyapunov diagonally stable matrix, where the proof of convergence or global stability is much simpler than that of the S-CNNs.

Dynamical Analysis of Full-Range Cellular Neural Networks by Exploiting Differential Variational Inequalities

The paper considers the full-range (FR) model of cellular neural networks (CNNs) in the case where the neuron nonlinearities are ideal hard-comparator functions with two vertical straight segments. The dynamics of FR-CNNs, which is described by a differential inclusion, is rigorously analyzed by means of theoretical tools from set-valued analysis and differential inclusions. The fundamental property proved in the paper is that FR-CNNs are equivalent to a special class of differential inclusions termed differential variational inequalities. A sound foundation to the dynamics of FR-CNNs is then given by establishing the existence and uniqueness of the solution starting at a given point, and the existence of equilibrium points. Moreover, a fundamental result on trajectory convergence towards equilibrium points (complete stability) for reciprocal standard CNNs is extended to reciprocal FR-CNNs by using a generalized Lyapunov approach. As a consequence, it is shown that the study of the ideal case with vertical straight segments in the neuron nonlinearities is able to give a clear picture and analytic characterization of the salient features of motion, such as the sliding modes along the boundary of the hypercube defined by the hard-comparator nonlinearities. Finally, it is proved that the solutions of the ideal FR model are the uniform limit as the slope tends to infinity of the solutions of a model where the vertical segments in the nonlinearities are approximated by segments with finite slope.

Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.

A Sliding Manifold Approach to Satellite Attitude Control

Abstract A sliding manifold based control strategy is proposed for the attitude control of a satellite. By using the theory of singular perturbation a PD feedback control is designed. It exhibits robustness properties with respect to environmental disturbances and plant parametric uncertainties. The resulting control signal remains, after a fast transient, in a neighbourhood of the well-defined equivalent control. Then the control can take into account bounds on the available control signals and avoid the peaking phenomenon of high gain systems. Finally the proposed procedure is applied to the CARINA capsule, and simulations are performed by using the ESA-MIDAS dynamic simulator in presence of environmental disturbances and corrupted Earth-magnetic field measures

Robust control design with integral action and limited rate control

By means of singular perturbation methods, the authors propose two dynamical feedback controls in order to solve control tracking problems involving sliding manifolds. The first control strategy introduces an integral action to reject constant disturbances and follow constant references. The second approach, in addition to the integral action, allows the designer to take into account limits on the maximum allowable control rate. Simulations on a multi-input/multi-output (MIMO) system are considered to show the effectiveness of the proposed approach and to discuss the choice of the parameters involved.