scholarly journals Safe Reachability Verification of Nonlinear Switched Systems via a Barrier Density

2019 ◽  
Author(s):  
Aysegul Kıvılcım ◽  
Ozkan Karabacak ◽  
Rafael Wisniewski
Keyword(s):  
Dynamical Systems ◽  
Density Function ◽  
Switched Systems ◽  
Autonomous Systems ◽  
Sufficient Condition ◽  
Temporal Properties ◽  
Nonlinear Switched Systems ◽  
Arbitrary Switching ◽  
Initial States ◽  
Set Of Initial States

One of the notable temporal properties of dynamical systems is that a set of initial states leads the solutions to reach desired states avoiding a predetermined unsafe set.This property, that we call safe reachability has been studied in literature for autonomous systems using Barrier functionand Barrier densities [1]. In this paper, we generalize a sufficient condition for safe reachability of autonomous systemto switched systems under arbitrary switching signals. The condition relies upon the existence of a common Barrier density function for each subsystem. We apply the condition using the sum of squares method together with Putinar Positivstellensatz.

scholarly journals Set Propagation Techniques for Reachability Analysis

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Matthias Althoff ◽  
Goran Frehse ◽  
Antoine Girard
Keyword(s):  
Hybrid Systems ◽  
Fundamental Problem ◽  
Autonomous Systems ◽  
Reachability Analysis ◽  
Controller Synthesis ◽  
Annual Review ◽  
Publication Date ◽  
Initial States ◽  
Set Of Initial States ◽  
Real World Problems

Reachability analysis consists in computing the set of states that are reachable by a dynamical system from all initial states and for all admissible inputs and parameters. It is a fundamental problem motivated by many applications in formal verification, controller synthesis, and estimation, to name only a few. This article focuses on a class of methods for computing a guaranteed overapproximation of the reachable set of continuous and hybrid systems, relying predominantly on set propagation; starting from the set of initial states, these techniques iteratively propagate a sequence of sets according to the system dynamics. After a review of set representation and computation, the article presents the state of the art of set propagation techniques for reachability analysis of linear, nonlinear, and hybrid systems. It ends with a discussion of successful applications of reachability analysis to real-world problems. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 3, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Observability and Distinguishability Properties and Stability of Arbitrarily Fast Switched Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Najah F. Jasim
Keyword(s):  
Asymptotic Stability ◽  
Dynamic Systems ◽  
Switched Systems ◽  
Sufficient Conditions ◽  
Stability Theorem ◽  
External Disturbances ◽  
Fast Switching ◽  
Nonlinear Switched Systems ◽  
Arbitrary Switching

This paper addresses sufficient conditions for asymptotic stability of classes of nonlinear switched systems with external disturbances and arbitrarily fast switching signals. It is shown that asymptotic stability of such systems can be guaranteed if each subsystem satisfies certain variants of observability or 0-distinguishability properties. In view of this result, further extensions of LaSalle stability theorem to nonlinear switched systems with arbitrary switching can be obtained based on these properties. Moreover, the main theorems of this paper provide useful tools for achieving asymptotic stability of dynamic systems undergoing Zeno switching.

scholarly journals A Sufficient Condition for the Almost Global Stability of Nonlinear Switched Systems with Average Dwell Time

2019 ◽  
Author(s):  
Ferruh İlhan ◽  
Ozkan Karabacak ◽  
Rafael Wisniewski
Keyword(s):  
Global Stability ◽  
Upper Bound ◽  
Switched Systems ◽  
Dwell Time ◽  
Average Dwell Time ◽  
Sufficient Condition ◽  
Nonlinear Switched Systems

A sufficient condition for the almost global sta-bility of nonlinear switched systems under average dwell timerestriction is obtained. This condition is derived leaning uponthe existence of multiple Lyapunov densities, which are associ-ated to subsystems and satisfy some compatibility conditions.An upper bound for the average dwell time that ensures almostglobal stability is obtained.

scholarly journals Almost Global Stability of Nonlinear Switched Systems with Time-Dependent Switching

2018 ◽  
Author(s):  
Ozkan Karabacak ◽  
Aysegul Kivilcim ◽  
Rafael Wisniewski
Keyword(s):  
Global Stability ◽  
Switched Systems ◽  
Lyapunov Functions ◽  
Dwell Time ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Multiple Lyapunov Functions ◽  
Koopman Operator ◽  
Nonlinear Switched Systems

For a dynamical system, it is known that the existence of a Lyapunov density implies almost global stability of an equilibrium. It is then natural to ask whether the existence of a common Lyapunov density for a nonlinear switched system implies almost global stability, in the same way as a common Lyapunov function implies global stability for nonlinear switched systems. In this work, the answer to this question is shown to be affirmative as long as switchings satisfy a dwell-time constraint with an arbitrarily small dwell time. As a straightforward extension of this result, we employ multiple Lyapunov densities in analogy with the role of multiple Lyapunov functions for the global stability of switched systems. This gives rise to a minimum dwell time estimate to ensure almost global stability of nonlinear switched systems, when a common Lyapunov density does not exist. The results obtained for continuous-time switched systems are based on some sufficient conditions for the almost global stability of discrete-time non-autonomous systems. These conditions are obtained using the duality between Frobenius-Perron operator and Koopman operator.

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