1,720,976 research outputs found
Converse Lyapunov results for switched systems with lower and upper bounds on switching intervals
Full text linkThe topic of this manuscript is the stability analysis of continuous-time switched nonlinear systems with constraints on the admissible switching signals. Our particular focus lies in considering signals characterized by upper and lower bounds on the length of the switching intervals. We adapt and extend the existing theory of multiple Lyapunov functions, providing converse results and thus a complete characterization of uniform stability for this class of systems. We specify our results in the context of switched linear systems, providing the equivalence of exponential stability and the existence of multiple Lyapunov norms. By restricting the class of candidate Lyapunov functions to the set of quadratic functions, we are able to provide semidefinite-optimization-based numerical schemes to check the proposed conditions. We provide numerical examples to illustrate our approach and highlight its advantages over existing methods
Converse Lyapunov results for stability of switched systems with average dwell-time
Full text linkThis article provides a characterization of stability for switched nonlinear systems under average dwell-time constraints, in terms of necessary and sufficient conditions involving multiple Lyapunov functions. Earlier converse results focus on switched systems with dwell-time constraints only, and the resulting inequalities depend on the flow of individual subsystems. With the help of a counterexample, we show that a lower bound that guarantees stability for dwell-time switching signals may not necessarily imply stability for switching signals with same lower bound on the average dwell-time. Based on these two observations, we provide a converse result for the average dwell-time constrained systems in terms of inequalities which do not depend on the flow of individual subsystems and are easier to check. The particular case of linear switched systems is studied as a corollary to our main result
Instability of dwell-time constrained switched nonlinear systems
Full text linkAnalyzing the stability of switched nonlinear systems under dwell-time constraints, this article investigates different scenarios where all the subsystems have a common globally asymptotically stable (GAS) equilibrium, but for the switched system, the equilibrium is not uniformly GAS for arbitrarily large values of dwell-time. We motivate our study with the help of examples showing that, if near the origin all the vector fields decay at a rate slower than the linear vector fields, then the trajectories are ultimately bounded for large enough dwell-time. On the other hand, if away from the origin, the vector fields do not grow as fast as the linear vector fields, then we can only guarantee local asymptotic stability for large enough dwell-times, with region of attraction depending on the dwell-time itself. We formalize our observations for homogeneous systems, and show that, even if the origin is not uniformly GAS with dwell-time switching for nonlinear systems, it still holds that the trajectories starting from a bounded set converge to a neighborhood of the origin if the dwell-time is large enough
MULTIPLE LYAPUNOV FUNCTIONS AND MEMORY: A SYMBOLIC DYNAMICS APPROACH TO SYSTEMS AND CONTROL
Full text linkWe propose a novel framework for the Lyapunov analysis of an important class of hybrid systems, inspired by the theory of symbolic dynamics and earlier results on the restricted class of switched systems. This new framework allows us to leverage language theory tools in order to provide a universal characterization of Lyapunov stability for this class of systems. We establish, in particular, a formal connection between multiple Lyapunov functions and techniques based on memorization and/or prediction of the discrete part of the state. This allows us to provide an equivalent (single) Lyapunov function, for any given multiple-Lyapunov criterion. By leveraging our language-theoretic formalism, a new class of stability conditions is then obtained when considering both memory and future values of the state in a joint fashion, providing new numerical schemes that outperform existing technique. Our techniques are then illustrated on numerical examples
Optimality of Vaccination for an SIR Epidemic with an ICU Constraint
Full text linkThis paper studies an optimal control problem for a class of SIR epidemic models, in scenarios in which the infected population is constrained to be lower than a critical threshold imposed by the intensive care unit (ICU) capacity. The vaccination effort possibly imposed by the health-care deciders is classically modeled by a control input affecting the epidemic dynamic. After a preliminary viability analysis, the existence of optimal controls is established, and their structure is characterized by using a state-constrained version of Pontryagin’s theorem. The resulting optimal controls necessarily have a bang-bang regime with at most one switch. More precisely, the optimal strategies impose the maximum-allowed vaccination effort in an initial period of time, which can cease only once the ICU constraint can be satisfied without further vaccination. The switching times are characterized in order to identify conditions under which vaccination should be implemented or halted. The uniqueness of the optimal control is also discussed. Numerical examples illustrate our theoretical results and the corresponding optimal strategies. The analysis is eventually extended to the infinite horizon by Γ-convergence arguments
Costruzione e validazione di un Questionario dei Comportamenti Interpersonali secondo il modello Circomplesso
No full textViability and control of a delayed SIR epidemic with an ICU state constraint
Full text linkThis paper studies viability and control synthesis for a delayed SIR epidemic. The model integrates a constant delay representing an incubation/latency time. The control inputs model nonpharmaceutical interventions, while an intensive care unit (ICU) state-constraint is introduced to reflect the healthcare systema's capacity. The arising delayed control system is analyzed via functional viability tools, providing insights into fulfilling the ICU constraint through feedback control maps. In particular, we consider two scenarios: first, we consider the case of general continuous initial conditions. Then, as a further refinement of our analysis, we assume that the initial conditions satisfy a Lipschitz continuity property, consistent with the considered model. The study compares the (in general, sub-optimal) obtained control policies with the optimal ones for the delay-free case, emphasizing the impact of the delay parameter. The obtained results are supported and illustrated, in a concluding section, by numerical examples
Max-Min Lyapunov Functions for Switching Differential Inclusions
Full text linkWe use a class of locally Lipschitz continuous Lyapunov functions to establish stability for a class of differential inclusions where the set-valued map on the right-hand-side comprises the convex hull of a finite number of vector fields. Starting with a finite family of continuously differentiable positive definite functions, we study conditions under which a function obtained by max-min combinations over this family of functions is a Lyapunov function for the system under consideration. For the case of linear systems, using the S-Procedure, our conditions result in bilinear matrix inequalities. The proposed construction also provides nonconvex Lyapunov functions, which are shown to be useful for systems with state-dependent switching that do not admit a convex Lyapunov function
Almost sure Stability of Stochastic Switched Systems: Graph lifts-based Approach
Full text linkIn this paper, we develop tools to establish almost sure stability of stochastic switched systems whose switching signal is constrained by an automaton. After having provided the necessary generalizations of existing results in the setting of stochastic graphs, we provide a characterization of almost sure stability in terms of multiple Lyapunov functions. We introduce the concept of lifts, providing formal expansions of stochastic graphs, together with the guarantee of conserving the underlying probability framework. We show how these techniques, firstly introduced in the deterministic setting, provide hierarchical methods in order to compute tight upper bounds for the almost sure decay rate. The theoretical developments are finally illustrated via a numerical example
Max–min Lyapunov functions for switched systems and related differential inclusions
Full text linkStarting from a finite family of continuously differentiable positive definite functions, we study conditions under which a function obtained by max-min combinations is a Lyapunov function, establishing stability for two kinds of nonlinear dynamical systems: (a) Differential inclusions where the set-valued right-hand-side comprises the convex hull of a finite number of vector fields, and (b) Autonomous switched systems with a state-dependent switching signal. We investigate generalized notions of directional derivatives for these max-min functions, and use them in deriving stability conditions with various degrees of conservatism, where more conservative conditions are numerically more tractable. The proposed constructions also provide nonconvex Lyapunov functions, which are shown to be useful for systems with state-dependent switching that do not admit a convex Lyapunov function. Several examples are included to illustrate the results
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