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Stability of Dynamical Systems with Discontinuous Motions:

Anthony N. MICHEL, Ling HOU
2008 SICE Journal of Control Measurement and System Integration  
We show that the stability results for DDS are in general less conservative than the corresponding well-known classical Lyapunov results for continuous dynamical systems and discrete-time dynamical systems  ...  In this paper we present a stability theory for discontinuous dynamical systems (DDS): continuous-time systems whose motions are not necessarily continuous with respect to time.  ...  Similarly as in the case of continuous dynamical systems (see Theorem 6), converse theorems for DDS involving continuous Lyapunov functions require that the motions x(·, x 0 , t 0 ) ∈ S of {R + , R n ,  ... 
doi:10.9746/jcmsi.1.411 fatcat:2nxl447rozafnlk6tr3ibwsvri

Classical converse theorems in Lyapunov's second method

Christopher M. Kellett
2015 Discrete and continuous dynamical systems. Series B  
Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems.  ...  This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions.  ...  Colleagues at the University of Bayreuth and the Australian National University were also kind enough to invite me to present seminars on the material contained herein and their feedback is also gratefully  ... 
doi:10.3934/dcdsb.2015.20.2333 fatcat:aquzwatsenaulm6b7gelwxbhqq

Impulsive and Hybrid Dynamcial Systems: Stability, Dissipativity and Control (W.M. Haddad et al.; 2008) [Bookshelf]

2008 IEEE Control Systems  
A comprehensive set of Lyapunov stability and boundedness results for DDSs, including converse theorems, are given in [3] .  ...  In doing so, the concepts of system storage function and system supply rate are extended to impulsive dynamical systems.  ...  A comprehensive set of Lyapunov stability and boundedness results for DDSs, including converse theorems, are given in [3] .  ... 
doi:10.1109/mcs.2008.4472383 fatcat:44hl23zdezgx5fmyll2yr34azi

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

M. Lazar, W.P.M.H. Heemels, A.R. Teel
2009 IEEE Transactions on Automatic Control  
In the presence of discontinuity of the dynamics one needs stronger properties, e.g., the one-step difference of the Lyapunov function should be upper bounded by a class K1 function with a minus sign in  ...  The first contribution of this note is to illustrate the precariousness of the second method of Lyapunov, as presented in [12], for discontinuous system dynamics.  ...  However, as shown above, the PWA system of Example 2 does admit a discontinuous USL function, which is conform with the converse stability result for discrete-time discontinuous systems presented in [  ... 
doi:10.1109/tac.2009.2029297 fatcat:y5ajvnbppremhe5bxav2cgafhe

Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks

Qing Hui, Wassim M. Haddad, Sanjay P. Bhat
2008 IEEE Transactions on Automatic Control  
Continuity of the settling-time function as well as Lyapunov and converse Lyapunov theorems for semistability are also developed.  ...  In particular, finite-time semistability for a continuum of equilibria of continuous autonomous systems is established.  ...  Unlike asymptotic stability, Lyapunov stability for autonomous dynamical systems does not imply the existence of a continuous Lyapunov function.  ... 
doi:10.1109/tac.2008.929392 fatcat:wzcqkmhg25asnjuajzedjnovpi

ISS-Lyapunov Functions for Discontinuous Discrete-Time Systems

Lars Grune, Christopher M. Kellett
2014 IEEE Transactions on Automatic Control  
Input-to-State Stability (ISS) and the ISS-Lyapunov function are useful tools for the analysis and design of nonlinear systems.  ...  We therefore propose a stronger implication-form ISS-Lyapunov function and provide a complete characterization of ISS-Lyapunov functions for discrete-time systems with discontinuous dynamics.  ...  ACKNOWLEDGEMENT The authors are grateful to Rob Gielen for helpful discussions on the result of Theorem 2.3.  ... 
doi:10.1109/tac.2014.2321667 fatcat:jaekve5kxvd6jgit6u6ath7ite

An alternative converse Lyapunov theorem for discrete-time systems

Roman Geiselhart, Rob H. Gielen, Mircea Lazar, Fabian R. Wirth
2014 Systems & control letters (Print)  
Furthermore, for specific classes of systems, the developed converse theorem can be used to establish non-conservatism of a particular type of Lyapunov functions.  ...  The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi-global practical Lyapunov functions for globally asymptotically  ...  In the monograph [12] converse Lyapunov theorems for continuous-time systems are obtained via piecewise linear Lyapunov functions and linear programming.  ... 
doi:10.1016/j.sysconle.2014.05.007 fatcat:iriu7ozbqzguvhbka7ggvapkgu

Lyapunov Functions and Feedback in Nonlinear Control [chapter]

Francis Clarke
2004 Lecture notes in control and information sciences  
The method of Lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems.  ...  For nonlinear systems, it turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth.  ...  The proof of the converse theorem (that a continuous Lyapunov function must exist when the system is globally asymptotically controllable) is more challenging.  ... 
doi:10.1007/978-3-540-39983-4_17 fatcat:a6dqa5bvhzghlcz5duq6lz4uey

Towards a stability theory of general hybrid dynamical systems

Anthony N. Michel, Bo Hu
1999 Automatica  
For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems.  ...  For such systems we introduced the usual concepts of Lyapunov and Lagrange stability.  ...  Continuing this work, we state and prove in the present paper a converse theorem for the case of exponential stability.  ... 
doi:10.1016/s0005-1098(98)00165-4 fatcat:anj3kselmfb6rlyhwjl6hmjjqu

Stability theory for hybrid dynamical systems

Hui Ye, A.N. Michel, Ling Hou
1998 IEEE Transactions on Automatic Control  
Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems).  ...  In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability).  ...  The proof of Theorem 4.5 follows along the same lines as the proof of an existing converse result for the uniform stability of continuous dynamical systems.  ... 
doi:10.1109/9.664149 fatcat:lco7cvfnl5csfkbkjguq6shokq

On the Equivalence Between Dissipativity and Optimality of Discontinuous Nonlinear Regulators for Filippov Dynamical Systems

Teymur Sadikhov, Wassim M. Haddad
2014 IEEE Transactions on Automatic Control  
In addition, we extend classical dissipativity theory to address the problem of dissipative discontinuous dynamical systems.  ...  for Filippov dynamical systems.  ...  ACKNOWLEDGMENT The authors would like to thank Dr. S. Bhat for several fruitful discussions and suggestions.  ... 
doi:10.1109/tac.2013.2283111 fatcat:yguzqghmrnfg3fccshkrlunw4q

A Robust Stabilization using State Feedback with Feedforward

Kumeresan A. Danapalasingam
2014 Jurnal Teknologi  
In a general nonlinear control system a stabilizing control strategy is often possible if complete information on external inputs affecting the system is available.  ...  Assuming that measurements of persistent disturbances are available it is shown that the existence of a smooth uniform control Lyapunov function implies the existence of a stabilizing state feedback with  ...  The main theorem of this paper, Theorem 2.1 as well as the converse Lyapunov theorem from Ref. 3, Theorem 2.2 are also stated here. Subsequently, a simulation example is given in Section 3.0.  ... 
doi:10.11113/jt.v67.2843 fatcat:x4xhyrlfjrfhjcilb6uk66gn4i

Perspectives and results on the stability and stabilizability of hybrid systems

R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson
2000 Proceedings of the IEEE  
In this endeavor, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable  ...  and finding conditions for the existence of switched controllers for stabilizing switched unstable systems.  ...  ACKNOWLEDGMENT The authors would like to thank the reviewers and the editor of this special issue for their highly constructive comments.  ... 
doi:10.1109/5.871309 fatcat:rto7r46chzba5anarvo34olfuu

The birth of the global stability theory and the theory of hidden oscillations [article]

Kuznetsov N.V., Lobachev M.Y., Yuldashev M.V., Yuldashev R.V., Kudryashova E.V., Kuznetsova O.A., Rosenwasser E.N., Abramovich S.M
2021 arXiv   pre-print
Boundaries of global stability in the space of parameters are limited by the birth of oscillations.  ...  The first mathematical problems of the global analysis of dynamical models can be traced back to the engineering problem of the Watt governor design.  ...  Krasovsky formulated a general theorem on global stability via Lyapunov functions for autonomous systems of ODEs [40] .  ... 
arXiv:2107.13889v1 fatcat:a3un7kjernar5ieb5ft7eevqy4

Stability analysis of discontinuous dynamical systems determined by semigroups

A.N. Michel, Ye Sun, A.P. Molchanov
2005 IEEE Transactions on Automatic Control  
We present Lyapunov stability results for discontinuous dynamical systems (DDS) determined by linear and nonlinear semigroups defined on Banach space.  ...  Index Terms-Asymptotic stability, 0 -semigroups, discontinuous dynamical systems (DDS), exponential stability, functional differential equations, heat equation, Lyapunov stability, nonlinear semigroups  ...  However, for cases where converse theorems are available for continuous dynamical systems, we propose a systematic procedure for constructing Lyapunov functions in the application of our results.  ... 
doi:10.1109/tac.2005.854582 fatcat:gc6mhlplfjc5vfr6olqfy7nw64
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