Vladimir Răsvan

Functional differential equations of lossless propagation and almost linear behavior

Abstract Starting from various models in control and electrical engineering, there is obtained a rather general model of an IBVP(Initial Boundary Value Problem) for hyperbolic PDE(Partial Differential Equations) with the boundary conditions connected in feedback to some systems described by ODE(Ordinary Differential Equations - lumped parameter systems). To this model it is associated a special class of FDE(Functional Differential Equations) - delay equations coupled to continuous time difference equations. This survey examines stability and oscillation problems for such systems within the framework underlined by the dialectics Liapunov function/Popov frequency domain inequalities. Several open problems are equally mentioned.

Gradient Like Behavior and High Gain Design of KWTA Neural Networks

It is considered the static and dynamic analysis of an analog electrical circuit having the structure of the Hopfield neural network, the KWTA (K-Winners-Take-All) network. The mathematics of circuit design and operation is discussed via two basic tools: the Liapunov function ensuring the gradient like behavior and the rational choice of the weights that stands for network training to ensure order-preserving trajectories. Dynamics and behavior at equilibria are considered in their natural interaction, and some connections to the ideas in general dynamical systems of convolution type are suggested.

Delays and Propagation: Control Liapunov Functionals and Computational Issues

There are considered some controlled objects with distributed parameters described by partial differential equations of hyperbolic type inducing wave propagation, connected at its turn with propagation delays. The boundary conditions are non-standard being described by ordinary differential or integro-differential equations. Basic theory—existence, uniqueness, well posedness-, stability and stabilization and numerical computations are considered for the benchmark problem of the marine vessel crane: its model is very much alike not only to other cranes but also to the flexible manipulator or the oilwell drillstring. In the lossless case basic theory is associated to the basic theory for some functional differential equations of neutral type. Stabilization is achieved by synthesizing low order controllers via c.l.f. (Control Liapunov Functional) induced by the energy identity for the partial differential equations. The numerics are considered within the framework of the method of lines implemented by applying the paradigm of the Cellular Neural Networks, an applied issue of Neuromathematics. After an illustrating simulation for the closed loop model some general conclusions and open problems are enumerated.

Discrete Time Linear Periodic Hamiltonian Systems and Applications

Linear canonical (Hamiltonian) systems are familiar to the engineering community both from Rational Mechanics and Control. In Rational Mechanics an “evergreen” problem is that of the λ-zones of stability in connection with parametric resonance while in Control these systems belong to the field of Linear Quadratic Theory, being strongly connected to Matrix Riccati Equation. In both cases some robustness problems are met but they deal with different classes of systems: totally stable in the first and hyperbolic in the second case. The present survey gives an account of these topics especially of their discrete-time counterpart.

Delays. Propagation. Conservation Laws.

Since the very first paper of J. Bernoulli in 1728, a connection exists between initial boundary value problems for hyperbolic Partial Differential Equations (PDE) in the plane (with a single space coordinate accounting for wave propagation) and some associated Functional Equations (FE). The functional equations may be difference equations (in continuous time), delay-differential (mostly of neutral type) or even integral/integro-differential. It is possible to discuss dynamics and control either for PDE or FE since both may be viewed as self contained mathematical objects. A more recent topic is control of systems displaying conservation laws. Conservation laws are described by nonlinear hyperbolic PDE belonging to the class “lossless” (conservative). It is not without interest to discuss association of some FE. Lossless implies usually distortionless propagation hence one would expect here also lumped time delays. The paper contains some illustrating applications from various fields: nuclear reactors with circulating fuel, canal flows control, overhead crane, without forgetting the standard classical example of the nonhomogeneous transmission lines for distortionless and lossless propagation. Specific features of the control models are discussed in connection with the control approach wherever it applies.

Lossless propagation models describing the transients of combined heat/electricity generation

Abstract This paper addresses the stability analysis of some class of propagation models describing the transients of combined heat/electricity generation. Such models include two constant delays in the corresponding system representation. Function of the dependence between the delays, we are deriving several (closed-loop) (asymptotic) stability conditions, numerically tractable, in frequency-domain by using various techniques: frequency-sweeping tests, matrix pencil and singular perturbation approaches.

Control of Systems with Input Delay—An Elementary Approach

The stabilization by feedback control of systems with input delays may be considered in various frameworks; a very popular is the abstract one, based on the inclusion of such systems in the Pritchard-Salamon class. In this chapter we consider the elementary approach based on variants of the Smith predictor, make a system theoretic analysis of the compensator and suggest a computer control implementation. This implementation is based on piecewise constant control which associates a discrete-time finite dimensional control system; it is this system which is stabilized, thus avoiding unpleasant phenomena induced by the essential spectrum of other implementations

Time-delay systems with remarkable structural properties

The paper is concerned with a class of time-delay systems that are linearly equivalent to higher-order scalar equations. If the retarded case was considered some four decades ago in Halanay (1969), the nontrivial neutral case is considered here. The specific structure of these systems endows them with remarkable properties that are useful in various applications in dynamics and control. As an auxiliary necessary result, the variation of parameters formula is revisited what opens new perspectives for research progress in such directions as the Perron condition and forced linear oscillations.

On Structures with Emergent Computing Properties. A Connectionist versus Control Engineering Approach

This paper starts by revisiting some founding, classical ideas for Neural Networks as Artificial Intelligence devices. The basic functionality of these devices is given by stability related properties such as the gradient-like and other collective qualitative behaviors. These properties can be linked to the structural – connectionist – approach. A version of this approach is offered by the hyperstability theory which is presented in brief (its essentials) in the paper. The hyperstability of an isolated Hopfield neuron and the interconnection of these neurons in hyperstable structures are discussed. It is shown that the so-called “triplet” of neurons has good stability properties with a non-symmetric weight matrix. This suggests new approaches in developing of Artificial Intelligence devices based on the triplet interconnection of elementary systems (neurons) in order to obtain new useful emergent collective computational properties.

Stability and Control of Systems with Propagation

A natural way of introducing time delay equations is to consider boundary value problems for hyperbolic partial differential equations (PDEs) in two variables. Such problems account for the so-called propagation phenomena that may be found in several physical and engineering applications. Association of some functional (differential/integral) equations to the aforementioned boundary value problems represents a way of tackling basic theory (existence, uniqueness, data dependence, i.e. well posedness) but also some qualitative properties arising from ODEs (ordinary differential equations) such as stability, oscillations, dissipativeness, Perron condition, and others. On the other hand automatic feedback control for systems described by partial differential equations (PDEs), systems called also with distributed parameters, is often ensured by boundary control: the control signals appear as forcing signals at the boundaries. In control applications stability of the feedback structure is the very first requirement. In order to achieve stability, Lyapunov functionals are considered aiming to obtain simultaneously the control structure and stability of the controlled system.