Weighted o-minimal hybrid systems

We consider weighted o-minimal hybrid systems, which extend classical o-minimal hybrid systems with cost functions. These cost functions are “observer variables” which increase while the system evolves but do not constrain the behaviour of the system. In this paper, we prove two main results: (i) optimal o-minimal hybrid games are decidable; (ii) the model-checking of WCTL, an extension of CTL which can constrain the cost variables, is decidable over that model. This has to be compared with the same problems in the framework of timed automata where both problems are undecidable in general, while they are decidable for the restricted class of one-clock timed automata.     

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system?s poles in the complex plane. We include several examples to show the utility of this theory.     

Hybrid decentralized maximum entropy control for large-scale dynamical systems

In the analysis of complex, large-scale dynamical systems it is often essential to decompose the overall dynamical system into a collection of interacting subsystems. Because of implementation constraints, cost, and reliability considerations, a decentralized controller architecture is often required for controlling large-scale interconnected dynamical systems. In this paper, a novel class of fixed-order, energy-based hybrid decentralized controllers is proposed as a means for achieving enhanced energy dissipation in large-scale lossless and dissipative dynamical systems. These dynamic decentralized controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that each subsystem–subcontroller pair of the hybrid closed-loop system is consistent with basic thermodynamic principles. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative combustion control example is given to demonstrate the efficacy of the proposed approach.     

Counteracting the dynamical degradation of digital chaos via hybrid control

Chaotic systems would degrade owing to finite computing precisions, and such degradation often seriously affects the performance of digital chaos-based applications. In this paper, a chaotification method is proposed to solve the dynamical degradation of digital chaotic systems based on a hybrid structure, where a continuous chaotic system is applied to control the digital chaotic system, and a unidirectional coupling controller that combines a linear external state control with a modular function is designed. Moreover, we proof rigorously that a class of digital chaotic systems can be driven to be chaotic in the sense that the system is sensitive to initial conditions. Different from the existing remedies, this method can recover the dynamical properties of system, and even make some properties better than those of the original chaotic system. Thus, this new approach can be applied to the fields of chaotic cryptography and secure communication.     

Floquet theory for a Volterra integro-dynamic system

In this paper, we study periodic linear Volterra systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We discuss the relationship between the solution of the Volterra integro-dynamic system and the limiting equation of the corresponding system. We also develop integrability conditions of the resolvent of Volterra integro-dynamic systems.     

A CSP Versus a Zonotope-Based Method for Solving Guard Set Intersection in Nonlinear Hybrid Reachability

Computing the reachable set of hybrid dynamical systems in a reliable and verified way is an important step when addressing verification or synthesis tasks. This issue is still challenging for uncertain nonlinear hybrid dynamical systems. We show in this paper how to combine a method for computing continuous transitions via interval Taylor methods and a method for computing the geometrical intersection of a flowpipe with guard sets, to build an interval method for reachability computation that can be used with truly nonlinear hybrid systems. Our method for flowpipe guard set intersection has two variants. The first one relies on interval constraint propagation for solving a constraint satisfaction problem and applies in the general case. The second one computes the intersection of a zonotope and a hyperplane and applies only when the guard sets are linear. The performance of our method is illustrated on examples involving typical hybrid systems.     

Limit cycles in switched single-server flow networks

The paper addresses the problem of qualitative analysis for a class of hybrid dynamical systems. This class consists of so-called switched flow networks which are used to model various communication, computer, and flexible manufacturing systems. We prove that any hybrid dynamical system from this class has a finite number of asymptotically stable limit cycles and any trajectory of the system converges to one of these cycles.     

Unifying theory for stability of continuous,discontinuous, and discrete-time dynamical systems

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuous-time dynamical systems whose motions are not necessarily continuous with respect to time (called discontinuous dynamical systems, or DDS). We show that the classical Lyapunov stability results for continuous dynamical systems are embedded in the authors’ stability results for DDS (given in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474]), in the following sense: if the hypotheses for a given Lyapunov stability result for continuous dynamical systems are satisfied, then the hypotheses of the corresponding stability result for DDS are also satisfied. This shows that the stability results for DDS in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than was previously known, and that the quality of the DDS results therein is consistent with that of the classical Lyapunov stability results for continuous dynamical systems.By embedding discrete-time dynamical systems into a class of DDS with equivalent stability properties, we also show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding DDS stability results are also satisfied. This shows that the results for DDS in [H. Ye, A.N. Michel, L. Hou Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than previously known, having connections even with discrete-time dynamical systems!Finally, we demonstrate by the means of a specific example that the stability results for DDS are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems.     

Modelling and Verification using Linear Hybrid Automata -- a Case Study

This paper discusses the use of hybrid automata to specify and verify embedded distributed systems, that consist of both discrete and continuous components. The basis of the evaluation is an automotive control system, which controls the height of an automobile by pneumatic suspension. It has been proposed by BMW AG as a case study taken from a current industrial development. Essential parts of the system have been modelled as hybrid automata and for appropiate ions several safety properties have been verified. The verification has been performed using HYTECH, a symbolic model checker for linear hybrid automata. The paper discusses the general appropiateness of hybrid automata to specify hybrid systems as well as advantages and drawbacks of the applied model-checking techniques.     

Sur la classification des systèmes dynamiques non commutatifs

The purpose of this paper is to extend certain results on commutative dynamical systems and on von Neumann algebras (provided with their inner automorphism groups) to general dynamical systems: “decompositions” into finite, semifinite, properly infinite, purely infinite, discrete and continuous systems; induced systems, system extensions, properties of invariant weights, etc.     

Approximability of nonlinear affine control systems

This paper focuses on a strong approximability property for nonlinear affine control systems. We consider control processes governed by ordinary differential equations (ODEs) and study an initial system and the associated generalized system. Our theoretical approach makes it possible to prove a strong approximability result for the above dynamical systems. The latter can be effectively applied to some classes of variable structure and hybrid control systems. In particular, this paper deals with applications of the strong approximability property obtained to the conventional sliding mode processes and to hybrid control systems with autonomous location transitions. We also take into consideration some optimal control problems for the above class of hybrid systems.     

Finite-time stabilization of nonlinear impulsive dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time, and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to reset the system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics. In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar and vector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems that are robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scale impulsive dynamical systems.     

Lyapunov-barrier characterization of robust reach–avoid–stay specifications for hybrid systems

Stability, reachability, and safety are crucial properties of dynamical systems. While verification and control synthesis of reach–avoid–stay objectives can be effectively handled by abstraction-based formal methods, such approaches can be computationally expensive due to the use of state–space discretization. In contrast, Lyapunov methods qualitatively characterize stability and safety properties without any state–space discretization. Recent work on converse Lyapunov-barrier theorems also demonstrates an approximate completeness for verifying reach–avoid–stay specifications of systems modeled by nonlinear differential equations. In this paper, based on the topology of hybrid arcs, we extend the Lyapunov-barrier characterization to more general hybrid systems described by differential and difference inclusions. We show that Lyapunov-barrier functions are not only sufficient to guarantee reach–avoid–stay specifications for well-posed hybrid systems, but also necessary for arbitrarily slightly perturbed systems under mild conditions. Numerical examples are provided to illustrate the main results.     

Invariance principles for hybrid systems with memory

Hybrid systems with memory are dynamical systems exhibiting both delayed and hybrid dynamics. Such systems can be described by hybrid functional inclusions. Classical invariance principles play an instrumental role in proving stability and convergence of dynamical systems. Invariance principles for general hybrid systems with delays, however, remain an open topic. In this paper, we prove invariance principles for hybrid systems with memory, using both Lyapunov–Razumikhin function and Lyapunov–Krasovskii functional methods. These invariance principles are then applied to derive two stability results as corollaries.     

The entropies of renewal systems

Renewal systems are symbolic dynamical systems originally introduced by Adler. IfW is a finite set of words over a finite alphabetA, then the renewal system generated byW is the subshiftX _W ⊂A ^Z formed by bi-infinite concatenations of words fromW. Motivated by Adler’s question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy. We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems. The authors were supported in part by NFS grants DMS-8706284, DMS-8814159 and DMS-8820716.     

Determination of the Chain-Like Non-Linear Multi-Degree-of-Freedom Systems Constant Parameters under Dynamical Complex Loads

A determination of mechanical properties of the real material systems under dynamical excitations is a very complex problem. The real system can be described by continuous model as well as a discrete model with a finite number of degree-of-freedom. Some mechanical systems like beams, rods, strings, tow-lines etc. have so-called chain-like configuration and may be treated as the dynamical one-dimensional cascade systems. In this paper, an identification method of internal interaction force, which can be arbitrary selected in discrete chain-like non-linear system, is presented. The method was tested on example of a two-mass system in which the interaction forces were modeled by spring-damper elements in a non-parallel arrangement. In experiment some non-sinusoidal complex loads were applied. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.     

Computing reachable sets for uncertain nonlinear monotone systems

We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work Ramdani (2008) [22], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of off-diagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields component-wise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, component-wise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.     

Persistence in systems of asymptotically autonomous difference equations

Asymptotically autonomous dynamical systems, both continuous and discrete, arise in the study of physical and biological systems that are modeled with explicit time-dependence.Convergence properties of such dynamical systems can be used to simplify analysis. In this paper, results are derived concerning the limiting behavior of a general asymptotically autonomous system of difference equations and its relationship to the dynamics of its limiting system. Examples from the biological literature are given.     

双符号等长代换系统的序列熵

系统称为null的,如果对任意序列,它的序列熵为零.双符号等长代换及其对应的代换极小系统可分成三类:有限的、离散的和连续的.容易看出离散的代换极小系统是null的,Goodman证明了连续的代换极小系统不是null的.本文将完全刻画所有的双符号等长代换极小系统的序列墒.