Model reduction by moment matching for convergent Lur'e-type models

This paper proposes an approach to model order reduction of convergent Lur'e-type models, which consist of a linear time-invariant (LTI) block and a static nonlinear block that is placed in feedback with the LTI block. In the proposed approach, we match a finite number of moments of the LTI block and keep the static nonlinear block to approximate the moments of the Lur'e-type model. The benefits of this approach are that the Lur'e-type structure is preserved after reduction, that the reduction method has an interpretation in terms of the frequency response function of the LTI block and that global exponential stability properties of the full-order model are preserved. The effectiveness of the approach is illustrated in a numerical example

Optimal H∞ LMI-based model reduction by moment matching for linear time-invariant models

This paper proposes an approach to model order reduction of stable linear time-invariant (LTI) models. The proposed approach extends time-domain moment matching by the minimization of the H∞ norm of the error dynamics characterizing the difference between the full-order and reduced-order models given fixed interpolation points. The optimal H∞ moment matching problem is a constrained optimization problem with bilinear constraints. Introducing a novel numerical procedure, we minimize the approximation error, while respecting the constraints and, thereby, find a suboptimal H∞ reduced-order model. The effectiveness of the approach is illustrated in a numerical example

Observation of nonlinear systems via finite capacity channels:Part II: Restoration entropy and its estimates

\u3cp\u3eThe paper deals with the state estimation problem for nonlinear dynamical systems via communication channels with limited data rate. We introduce several minimum data-rate limits associated with various types of observability. A notion of the restoration entropy (RE) is also introduced and its relevance to the problem is outlined by a corresponding Data Rate Theorem. Theoretical lower and upper estimates for the RE are proposed in the spirit of the first and second Lyapunov methods, respectively. For three classic chaotic multi-dimensional systems, it is demonstrated that the lower and upper estimates for the RE coincide for one of them and are nearly the same for the others.\u3c/p\u3

Emergence of oscillations in networks of time-delay coupled inert systems

\u3cp\u3eWe discuss the emergence of oscillations in networks of single-input- single-output systems that interact via linear time-delay coupling functions.Although the systems itself are inert, that is, their solutions converge to a globally stable equilibrium, in the presence of coupling, the network of systems exhibits ongoing oscillatory activity.We address the problem of emergence of oscillations by deriving conditions for; 1. solutions of the time-delay coupled systems to be bounded, 2. the network equilibrium to be unique, and 3. the network equilibrium to be unstable. If these conditions are all satisfied, the time-delay coupled inert systems have a nontrivial oscillatory solution. In addition, we show that a necessary condition for the emergence of oscillations in such networks is that the considered systems are at least of second order.\u3c/p\u3

Data rate limitations for observability of nonlinear systems

The paper deals with observation of nonlinear and deterministic, though maybe chaotic, discrete-time systems via finite capacity communication channels. We consider various types of observability, and offer new tractable analytical techniques for both upper and lower estimation of the threshold that separates data rates for which reliable state observation is and is not possible, respectively. The main results are illustrated via their application to two celebrated samples of chaotic systems associated with the logistic and Lozi maps, respectively. In these cases, the thresholds attributed to some of the considered notions of observability are found in a closed form; they are shown to continuously depend on the parameters of the Lozi system

Stability analysis of PE systems via Steklov's averaging technique

\u3cp\u3eThe paper presents a novel approach to justification of asymptotic stability of linear time-varying systems with persistently excited right-hand side. This approach combines the direct Lyapunov method with the Steklov averaging technique; its distinguishing feature is closed-form construction of the Lyapunov functional, along with resultant explicit estimates of the rate of convergency.\u3c/p\u3

Observation of nonlinear systems via finite capacity channels:Constructive data rate limits

\u3cp\u3eThe paper deals with observation of nonlinear and deterministic, though maybe chaotic, discrete-time systems via finite capacity communication channels. We introduce several minimum data-rate limits associated with various types of observability, and offer new tractable analytical techniques for their both upper and lower estimation. Whereas the lower estimate is obtained by following the lines of the Lyapunov's linearization method, the proposed upper estimation technique is along the lines of the second Lyapunov approach. As an illustrative example, the potential of the presented results is demonstrated for the system which describes a ball vertically bouncing on a sinusoidally vibrating table. For this system, we provide an analytical computation of a closed-form expression for the threshold that separates the channel data rates for which reliable observation is and is not possible, respectively. Another illustration is concerned with the celebrated Hénon system. The offered sufficient data rate bound is accompanied with a constructive observer that works whenever the channel capacity fits this bound.\u3c/p\u3

Stability analysis via averaging functions

\u3cp\u3eA new class of Lyapunov functions is proposed for analysis of incremental stability for nonlinear systems. This class of Lyapunov functions allows to establish input-dependent incremental stability criteria. Two substantially different situations are considered: when incremental stability is guaranteed by the inputs of sufficiently small amplitude and when, similar to the excited van der Pol oscillator, the stability is induced by sufficiently large inputs.\u3c/p\u3