Invariance-like theorems and “lim inf” convergence properties

International audienceSeveral theorems, inspired by the Krasovskii-LaSalle invariance principle, to establish “lim inf” convergence results are presented in a unified framework. These properties are useful to “describe” the oscillatory behavior of the solutions of dynamical systems. The theorems resemble “lim inf” Matrosov and Small-gain theorems and are based on a “lim inf” Barbalat's Lemma. Additional technical assumptions to have “lim” convergence are given: the “lim inf”/“lim” relation is discussed in-depth and the role of some of the assumptions is illustrated by means of examples

On reduction of differential inclusions and Lyapunov stability

In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is point-wise smaller (in the sense of set containment) than the original set-valued map. The corresponding reduced differential inclusion, defined by the reduced set-valued map, is utilized to develop a generalized notion of a derivative for locally Lipschitz candidate Lyapunov functions in the direction(s) of a set-valued map. The developed generalized derivative yields less conservative statements of Lyapunov stability theorems, invariance theorems, invariance-like results, and Matrosov theorems for differential inclusions. Included illustrative examples demonstrate the utility of the developed theory

Invariance Principles and Observability in Switched Systems with an Application in Consensus

Using any nonnegative function with a nonpositive derivative along trajectories to define a virtual output, the classic LaSalle invariance principle can be extended to switched nonlinear time-varying (NLTV) systems, by considering the weak observability (WO) associated with this output. WO is what the output informs about the limiting behavior of state trajectories (hidden in the zero locus of the output). In the context of switched NLTV systems, WO can be explored using the recently established framework of limiting zeroing-output solutions. Adding to this, an extension of the integral invariance principle for switched NLTV systems with a new method to guarantee uniform global attractivity of a closed set (without assuming uniform Lyapunov stability or dwell-time conditions) is proposed. By way of illustrating the proposed method, a leaderless consensus problem for nonholonomic mobile robots with a switching communication topology is addressed, yielding a new control strategy and a new convergence result

Robust stability theory for stochastic dynamical systems

In this work, we focus on developing analysis tools related to stability theory forcertain classes of stochastic dynamical systems that permit non-unique solutions. Thenon-unique nature of solutions arise primarily due to the system dynamics that aremodeled by set-valued mappings. There are two main motivations for studying suchclasses of systems. Firstly, understanding such systems is crucial to developing a robuststability theory. Secondly, such system models allow flexibility in control design problems.We begin by developing analysis tools for a simple class of discrete-time stochasticsystem modeled by set-valued maps and then extend the results to a larger class ofstochastic hybrid systems. Stochastic hybrid systems are a class of dynamical systemsthat combine continuous-time dynamics, discrete-time dynamics and randomness. Theanalysis tools are established for properties like global asymptotic stability in probabilityand global recurrence. We focus on establishing results related to sufficient conditions for stability, weak sufficient conditions for stability, robust stability conditions and converse Lyapunov theorems. In this work a primary assumption is that the stochastic system satisfies some mild regularity properties with respect to the state variable and random input. The regularity properties are needed to establish the existence of random solutions and results on sequential compactness for the solution set of the stochastic system.We now explain briefly the four main types of analysis tools studied in this work.Sufficient conditions for stability establish conditions involving Lyapunov-like functionssatisfying strict decrease properties along solutions that are needed to verify stability properties. Weak sufficient conditions relax the strict decrease nature of the Lyapunov like function along solutions and rely on either knowledge about the behavior of thesolutions on certain level sets of the Lyapunov-like function or use multiple nested non-strict Lyapunov-like functions to conclude stability properties. The invariance principleand Matrosov function theory fall in to this category. Robust stability conditions determinewhen stability properties are robust to sufficiently small perturbations of thenominal system data. Robustness of stability is an important concept in the presenceof measurement errors, disturbances and parametric uncertainty for the nominal system.We study two approaches to verify robustness. The first approach to establish robustnessrelies on the regularity properties of the system data and the second approach isthrough the use of Lyapunov functions. Robustness analysis is an area where the notionof set-valued dynamical systems arise naturally and it emphasizes the reason for ourstudy of such systems. Finally, we focus on developing converse Lyapunov theorems forstochastic systems. Converse Lyapunov theorems are used to illustrate the equivalencebetween asymptotic properties of a system and the existence of a function that satisfiesa decrease condition along the solutions. Strong forms of the converse theorem implythe existence of smooth Lyapunov functions. A fundamental way in which our resultsdiffer from the results in the literature on converse theorems for stochastic systems isthat we exploit robustness of the stability property to establish the existence of a smoothLyapunov function

Approximation, analysis and control of large-scale systems - Theory and Applications

This work presents some contributions to the fields of approximation, analysis and control of large-scale systems. Consequently the Thesis consists of three parts. The first part covers approximation topics and includes several contributions to the area of model reduction. Firstly, model reduction by moment matching for linear and nonlinear time-delay systems, including neutral differential time-delay systems with discrete-delays and distributed delays, is considered. Secondly, a theoretical framework and a collection of techniques to obtain reduced order models by moment matching from input/output data for linear (time-delay) systems and nonlinear (time-delay) systems is presented. The theory developed is then validated with the introduction and use of a low complexity algorithm for the fast estimation of the moments of the NETS-NYPS benchmark interconnected power system. Then, the model reduction problem is solved when the class of input signals generated by a linear exogenous system which does not have an implicit (differential) form is considered. The work regarding the topic of approximation is concluded with a chapter covering the problem of model reduction for linear singular systems. The second part of the Thesis, which concerns the area of analysis, consists of two very different contributions. The first proposes a new "discontinuous phasor transform" which allows to analyze in closed-form the steady-state behavior of discontinuous power electronic devices. The second presents in a unified framework a class of theorems inspired by the Krasovskii-LaSalle invariance principle for the study of "liminf" convergence properties of solutions of dynamical systems. Finally, in the last part of the Thesis the problem of finite-horizon optimal control with input constraints is studied and a methodology to compute approximate solutions of the resulting partial differential equation is proposed.Open Acces

A system-theoretic framework for privacy preservation in continuous-time multiagent dynamics

In multiagent dynamical systems, privacy protection corresponds to avoid disclosing the initial states of the agents while accomplishing a distributed task. The system-theoretic framework described in this paper for this scope, denoted dynamical privacy, relies on introducing output maps which act as masks, rendering the internal states of an agent indiscernible by the other agents as well as by external agents monitoring all communications. Our output masks are local (i.e., decided independently by each agent), time-varying functions asymptotically converging to the true states. The resulting masked system is also time-varying, and has the original unmasked system as its limit system. When the unmasked system has a globally exponentially stable equilibrium point, it is shown in the paper that the masked system has the same point as a global attractor. It is also shown that existence of equilibrium points in the masked system is not compatible with dynamical privacy. Application of dynamical privacy to popular examples of multiagent dynamics, such as models of social opinions, average consensus and synchronization, is investigated in detail.Comment: 38 pages, 4 figures, extended version of arXiv preprint arXiv:1808.0808