The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective and easily computable control laws. These two aspects are complementary, since the latter approaches are, typically, based upon the inductive verification of the validity of the former system properties under compositions (in the terminology used in [62], the “activation” of theoretical concepts leads to “constructive” control). This expository presentation addresses the first of these aspects, and in particular the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability. We provide an intuitive and informal presentation of the main concepts. More precise statements, especially about older results, are given in the cited papers, as well as in several previous surveys such as [103] and [105] (of which the present paper represents an update), but we provide a little more detail about relatively recent work. Regarding applications and extensions of the basic framework, we give some pointers to the literature, but we do not focus on feedback design and specific engineering problems; for the latter we refer the reader to textbooks such as [43], [60], [58], [96], [66], [27], [44].

Input to State Stability: Basic Concepts and Results

For part I, see ibid., vol.42, p.1294-8, 1997. In this paper a new class of feedback control problems is introduced. Unlike classical models, the systems considered here have communication channel constraints. As a result, the issue of coding and communication protocol becomes an integral part of the analysis. Since these systems cannot be asymptotically stabilized if the underlying dynamics are unstable, a weaker stability concept called containability is introduced. A key result connects containability with an inequality equation involving the communication data rate and the rate of change of the state.

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Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback

This paper presents a converse Lyapunov function theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.

A Smooth Converse Lyapunov Theorem for Robust Stability

We consider a large-scale wireless network, but with a low density of nodes per unit area. Interferences are then less critical, contrary to connectivity. This paper studies the latter property for both a purely ad-hoc network and a hybrid network, where fixed base stations can be reached in multiple hops. We assume here that power constraints are modeled by a maximal distance above which two nodes are not (directly) connected. We find that the introduction of a sparse network of base stations does significantly help in increasing the connectivity, but only when the node density is much larger in one dimension than in the other. We explain the results by percolation theory. We obtain analytical expressions of the probability of connectivity in the 1D case. We also show that at a low spatial density of nodes, bottlenecks are unavoidable. Results obtained on actual population data confirm our findings.

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Connectivity in ad-hoc and hybrid networks

A state observer is proposed for nonlinear continuous time systems which extends the well known Luenberger observer. In particular, on the basis of simple assumptions on the regularity of the system equations (observability and the global Holder condition for suitable functions), which are generally satisfied for physically meaningful dynamic systems, the global asymptotic convergence of the estimated state towards the true state is shown. Finally, some examples of applications are also reported showing the effectiveness of the proposed observer.

A Luenberger-like observer for nonlinear systems

Investigates stability theory in terms of two different measures, treats the theory of a variety of inequalities, and demonstrates manifestations of the general Lyapunov method. Also covers the importance of utilizing different forms of nonlinear variation of parametric formulae, constructive methods generated by monotone iterative technique and th

Stability analysis of nonlinear systems

This is the first book that deals with practical stability and its development. It presents a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets and demonstrates the manifestations of general Lyapunov's method by showing how this effective technique can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.

Practical stability of nonlinear systems

Outline of the liapunov stability theory in general.- The stability theory of comparison systems.- Large-scale systems in general.- Singularly perturbed large-scale systems.- Large-scale power systems stability.- Postface.

Large Scale Systems Stability under Structural and Singular Perturbations

Stability Theory for Dynamic Equations on Time Scales

Introductory Remarks. Random Variables and Probability Distributions. Probability Processes and their Mathematical Description. Random Differential Equations. System with Random Structure. Stability Analysis Using Scalar Lyapunov Functions. Stability Concepts for Stochastic Systems. Random Scalar Lyapunov Functions. Conditions of Stability in Probability. Converse Theorems. Stability in Mean Square. Stability in Mean Square of Linear Systems. Stability Analysis Using Multi-component Lyapunov Functions. Vector Lyapunov Functions. Stochastic Matrix-Valued Lyapunov Functions. Stability Analysis in General. Stability Analysis of Systems in Ito's Form. Stochastic Singularly Perturbed Systems. Large-Scale Singularly Perturbed Systems. Stability Analysis by the First-Order Approximation. Stability Criterion by the First-Order Approximation. Stability with Respect to the First-Order Approximation. Stability by First-Order Approximation of Systems with Random Delay. Convergence of Stochastic Approximation Procedure. Stabilization of Controlled Systems with Random Structure. Problems of Stabilization. Optimal Stabilization. Linear-Quadratic Optimal Stabilization. Sufficient Stabilization Conditions for Linear Systems. Optimal Solution Existence. The Small Parameter Method Algorithm. Applications. A Stochastic Version of the Lefschetz Problem. Stability in Probability of Oscillating Systems. Stability in Probability of Regulation Systems. Price Stability in a Stochastic Market Model. References. Index. Lyapunov Functions. Stability Analysis Using Multicomponent Lyapunov Functions. Stability Analysis by the First-order Approximation. Stabilization of Controlled Systems with Random Structure. Applications References Index.

http://www.gbv.de/dms/goettingen/347560784.pdf

A. A. Martynyuk

Papers

Stability analysis of nonlinear systems

Practical stability of nonlinear systems

Large Scale Systems Stability under Structural and Singular Perturbations

Stability Theory for Dynamic Equations on Time Scales

Stability and Stabilization of Nonlinear Systems with Random Structure