Anthony N. Michel

Stability theory for hybrid dynamical systems

We first formulate a model for hybrid dynamical systems which covers a very large class of systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of an invariant set for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability). To demonstrate the applicability of the developed theory, we present specific examples of hybrid dynamical systems and we conduct a stability analysis of some of these examples.

Disturbance Attenuation Properties of Time-Controlled Switched Systems

In this paper, we investigate the disturbance attenuation properties of time-controlled switched systems consisting of several linear time-invariant subsystems by using an average dwell time approach incorporated with a piecewise Lyapunov function. First, we show that when all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than a positive scalar γ0, the switched system under an average dwell time scheme achieves a weighted disturbance attenuation level γ0, and the weighted disturbance attenuation approaches normal disturbance attenuation if the average dwell time is chosen sufficiently large. We extend this result to the case where not all subsystems are Hurwitz stable, by showing that in addition to the average dwell time scheme, if the total activation time of unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, then a reasonable weighted disturbance attenuation level is guaranteed. Finally, a discussion is made on the case for which nonlinear norm-bounded perturbations exist in the subsystems.

Qualitative analysis and synthesis of a class of neural networks

The dynamic properties of a class of neural networks (which includes the Hopfield model as a special case) are investigated by studying the qualitative behavior of equilibrium points. The results fall into one of two categories: results pertaining to analysis (e.g., stability properties of an equilibrium, asymptotic behavior of solutions, etc.) and results pertaining to synthesis (e.g. the design of a neural network with prespecified equilibrium points which are asymptotically stable). Most (but not all) of the results presented are global, and their applicability is demonstrated by an example. >

Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube

An investigation was conducted of the qualitative properties of a class of neural networks described by a system of first-order linear ordinary differential equations which are defined on a closed hypercube of the state space with solutions extended to the boundary of the hypercube. When solutions are located on the boundary of the hypercube, the system is said to be in a saturated mode. The class of systems considered retains the basic structure of the Hopfield model but is easier to analyze, synthesize, and implement. An efficient analysis method is developed which can be used to determine completely the set of asymptotically stable equilibrium points and the set of unstable equilibrium points. The latter set can be used to estimate the domains of attraction for the elements of the former set. The class of systems considered can easily be implemented in analog integrated circuits. The applicability of the results is demonstrated by means of several examples. >

Associative memories via artificial neural networks

Several design techniques that can be used for continuous-time and discrete-time neural networks to realize associative memories are presented. Associative memory is discussed, and neural network models are presented. Some stability concepts are outlined. The applicability of these techniques is demonstrated by means of specific examples that illustrate strengths and weaknesses.<<ETX>>

Towards a stability theory of general hybrid dynamical systems

In recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined on real time R^+=[0,~). The motions of this class of systems are in general discontinuous. This class of systems may be finite or infinite dimensional. For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems. In the present paper we continue the work described above. In doing so, we first develop a general comparison theory for the class of hybrid dynamical systems (resp., discontinuous dynamical systems) considered herein, making use of stability preserving mappings. We then show how these results can be applied to establish some of the Principal Lyaponov Stability Theorems. For the latter, we also state and prove a converse theorem not considered previously. Finally, to demonstrate the applicability of our results, we consider specific examples throughout the paper.

Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters

New results for an established for the global asymptotic stability of the equilibrium x=0 of nth order discrete-time systems with state saturations, x(k+1)=sat(Ax(k)), utilizing a class of positive definite and radially unbounded Lyapunov functions, v. When v is a quadratic form, necessary and sufficient conditions are obtained under which positive definite matrices H can be used to generate a Lyapunov function v(w)=w/sup T/Hw with the properties that v(Aw(k)) is negative semidefinite, and that v(sat(w)) >

Cellular neural networks for associative memories

A synthesis procedure for designing nonsymmetric cellular neural networks (CNN) with a predetermined local interconnection structure that will store a set of desired bipolar vectors as memory points is presented. A specific case of constructing Chinese characters is presented to demonstrate the applicability of the results. Simulation results show that all the vectors corresponding to 50 commonly used Chinese characters are reachable memory vectors of the synthesized CNN. >

A synthesis procedure for Hopfield's continuous-time associative memory

A new technique is presented for designing associative memories to be implemented by Hopfield neural networks. This technique guarantees that each desired memory is stored and is attractive. The procedure also guarantees that the resulting network can be implemented, a requirement often overlooked by other methods. This synthetic procedure does not require a symmetric interconnection matrix; instead, stability is guaranteed by use of the results presented by A.N. Michel et al. (1989). Two examples are presented that demonstrate the synthesis procedures storage ability and flexibility. >

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

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. >