Output-based adaptive distributed observer for general linear leader systems over periodic switching digraphs

Some sufficient conditions for exponential stability of linear neutral functional differential equations

Adaptive distributed observer for general linear leader systems over periodic switching digraphs

This paper presents Lie-algebraic conditions for a type of exponential stability for linear impulsive systems that is uniform with respect to the set of impulse times. A Lie algebra is defined in terms the linear impulsive system data to which previously-derived Lie-algebraic stability criteria for switched linear systems are applied in order to yield sufficient conditions for uniform exponential stability of linear impulsive systems. An example is presented to illustrate the main ideas and some of the underlying computational aspects.

Sufficient conditions for stability of linear time-varying systems

This paper deals with the problem of stability for vehicles formations system with structured uncertainties and time delays. The vehicles are modeled as general linear continuous-time systems which exchange information according to an undirected communication graph. The feedback control is based only on relative information about vehicle states shared via the communication links. Delay-dependent robust stability of vehicles formations system with uncertainties is analyzed. The sufficient conditions for asymptotical stability are proposed. The analysis framework is based on tools from algebraic graph theory, matrix theory, control theory and linear inequality matrix theory. Finally, an illustrative example is used to show the validity of the theoretical results.

While the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated, the theory for nonlinear positive systems is considerably less well-developed. This paper presents a set of conditions for establishing delay-independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, when the time delays have a known upper bound, we derive necessary and sufficient conditions for exponential stability of (a) continuous-time positive systems whose vector fields are homogeneous and cooperative, and (b) discrete-time positive systems whose vector fields are homogeneous and order preserving. We then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization. Finally, we extend the results to general linear systems with time-varying delays.

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.

Finite-time stability and stabilization of linear discrete time-varying stochastic systems

International audience

In this paper we consider linear time-varying systems subject to norm bounded, one block, structured uncertainties. First, given an unforced system, we find necessary and sufficient conditions for quadratic stability; then, considered a controlled system, we provide necessary and sufficient conditions for quadratic stabilizability both for the state and the output feedback case.

In this paper the problem of linear control system stability is formulated in very general terms and then resolved by (i) establishing a necessary and sufficient condition for stability, (ii) exploring in some detail the nature of this condition, and (iii) developing a parametric characterization of all controllers which stabilize a given plant. The problem formulation developed herein is based on a ring theoretic setting for linear systems, and is distinguished by the use of a novel control system configuration. Any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems. Moreover, no other configuration has this property. Benefits and extensions accrued as a result of this property are thoroughly discussed, as are some promising directions for future research.

On the stability of certain systems under resonance conditions

Despite a rich and complete theory developed for robust control of linear systems, little work has been done for robust control of linear systems with constraints. In this paper, a synthesis method to design a model predictive controller which optimizes robust performance is proposed for a stable linear time-varying discrete-time system represented by a finite impulse response model. In the absence of constraints, we show that with this method robust bounded-input-bounded-output stability of the resulting closed-loop system is guaranteed. Both necessary and sufficient conditions for robust global asymptotic stability, i.e. offset free tracking for all plants in the set, are stated. Furthermore, robust global asymptotic stability is preserved for a class of asymptotically constant disturbances entering at the plant output. These results hold for any uncertainty description expressed in the time-domain. However, there is a trade-off between the generality of the uncertainty description and the computational complexity of the resulting optimization problem. For a broad class of uncertainty descriptions, we show that the optimization problem can be cast as a linear program of moderate size.

Necessary and Sufficient Conditions for Delay-Dependent Asymptotic Stability of Linear Discrete Time Delay Autonomous Systems

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.