Scanning the Issue

REGULAR PAPERS

In this paper a model for hybrid dynamical systems is first formulated. The notion of invariant sets for hybrid dynamical systems is introduced and several types of (Lyapunov-like) stability concepts for an invariant set are defined. Sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems are established. In addition, sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability) are established. Specific examples of hybrid dynamical systems are presented, and a stability analysis is conducted for a class of sampleddata feedback control systems and a class of systems with impulse effects.

In this paper, the author introduces some analysis tools for switched and hybrid systems. The main concentration is on building the foundations of a Lyapunov-like stability theory that is applicable to hybrid systems: "multiple Lyapunov functions" are introduced as a tool for analyzing their Lyapunov stability, and iterated function systems are proposed for Lagrange stability. The author also discusses cases where the switched/hybrid systems are indexed by an arbitrary compact set. Finally, Bendixson's theorem is extended to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.

The uniqueness of solutions of hybrid systems is important as a check on model validity. Sufficient conditions for uniqueness of smooth solutions are obtained for so-called complementarity systems, using a combination of standard methods of input-output systems theory with the Linear Complementarity Problem of mathematical programming. The complementarity systems introduced form a class of hybrid systems carrying an additional structure that facilitates analysis. Several examples given in the paper indicate that the complementarity formalism applies to switching control schemes and to systems with piecewise linear elements, as well as to physical hybrid systems such as mechanical systems with unilateral constraints and electrical circuits with diodes.

The state space of a continuous system S is partitioned so as to define a set of continuous subsystems; the high-level transitions of the system between these subsystems are controlled by a discrete controller, while each is itself subject to state feedback control. This notion of hierarchical hybrid control is realized via the idea of dynamical consistency which is extended from its original formulation in the discrete case so as to define the set of hybrid partition machines associated with S. Following the formulation of between-block and in-block controllable hybrid partition machines, the lattice HIBC(S) of hybrid in-block controllable partition machines and hierarchical hybrid control systems are defined and investigated.

Proposals for a future Air Traffic Management system favor a decentralized control scheme called free flight, in which aircraft choose their own optimal route, altitude, and speed. In such an environment, the trajectories of different aircraft may come into conflict, in which case the aircraft may or may not cooperate in resolving the conflict. In this paper, the authors present methods for proving the safety of both cooperative and noncooperative collision avoidance maneuvers. Noncooperative methods are based on game theory. The solutions to the game partition the state space into safe and unsafe sets and the control strategies are abstracted into discrete protocols on hybrid automata. In cooperative methods, aircraft perform coordinated maneuvers in order to avoid conflict. Two examples of conflict resolution using both speed and heading changes are worked out in detail.

The objective of an Automated Highway System (AHS) is to increase the safety and throughput of highways by introducing traffic automation. An AHS is an example of a large scale, multiagent, complex dynamical system and is ideally suited for hierarchical hybrid control. In this paper the design of safe hybrid controllers for regulation of vehicles on an AHS is discussed. The design methodology makes use of game theoretic techniques to deal with the multi-agent and multi-objective nature of the problem. The result is a hybrid controller that guarantees safety by design, without the need for further verification. The results illustrate that, as expected, performance improves as the level of centralization increases at the cost of increased complexity of the communication protocols and the controllers.

This paper presents a methodology for algorithmically analyzing nonlinear hybrid systems by first translating to linear hybrid automata and then using automated model-checking tools. In a linear hybrid automaton, the analog environment is partitioned into a finite number of classes such that within each class, the analog variables are governed by a constant polyhedral differential inclusion. Two translation methods are presented. The first, called clock translation, replaces, when possible, constraints on nonlinear variables by constraints on variables with constant derivative equal to one. This method is efficient but has limited applicability. The second method, called linear phase-portrait approximation, overapproximates conservatively the phase-portrait of a nonlinear hybrid system using piecewiseconstant polyhedral differential inclusions.

TECHNICAL NOTES

In this paper the search for a piecewise quadratic Lyapunov function is stated as a convex optimization problem in terms of linear matrix inequalities. A compact parameterization of continuous piecewise quadratic functions is introduced. The method is derived for systems with piecewise affine dynamics, and the relation with frequency domain methods such as the circle and Popov criteria is explained. It is also shown how it is possible to search for discontinuous Lyapunov functions that are useful for analysis of hybrid systems with hysteresis effect. The use of piecewise quadratic Lyapunov functions is a powerful extension of quadratic stability that covers polytopic Lyapunov functions as a special case.

This paper presents a new task-level adaptive controller for the hybrid dynamic control of constrained motion systems. Using a hybrid dynamic model of the process, velocity constraints are derived from which satisfactory velocity commands are obtained. Due to modeling errors and parametric uncertainties, the velocity commands may be erroneous and may result in suboptimal performance. A tasklevel adaptive control scheme is used to change the model parameters from which the velocity commands are determined. Automated control of an assembly task is given as an example, and simulations and experiments for this task are presented. Properties for rapid convergence are also discussed.

In this paper a continuous plant is to be controlled via symbolic output feedback. The hybrid problem is first translated into a discrete problem by approximating the continuous plant model by a (nondeterministic) finite state machine. By taking into account past measurement and control symbols, approximation accuracy can be improved and adjusted to the specification requirements. Supervisory control theory for DES's is then applied to find the optimal controller which enforces the specifications. As the behavior of the approximating automaton is guaranteed to contain the behavior of the underlying continuous plant model, the controller also forces the latter to obey the specifications.

Differential Petri Nets are a new extension of Petri nets. Through the introduction of the differential place, the differential transition, and suitable evolution rules, it is possible to model concurrently discrete-event processes and continuous-time dynamic processes, represented by systems of linear ordinary differential equations. This model can contribute to the performance analysis and design of industrial supervisory control systems and of hybrid control systems in general.

This paper discusses fundamentals of hybrid system modeling. Emphasis is put on compositionality and the use of multiform time. Compositionality refers to the ability of freely composing hybrid systems. Since hybrid systems are considered, different time indexes occur, and modularity calls for considering "time" as local to each module; this is what is called multiform time. The proposed framework is behavioral, hence a way is provided to automatically synthesize proper scheduling constraints for the joint simulation of discrete-time, event-based, and continuous-time components.

SHIFT is a programming language for describing and simulating dynamic networks of hybrid automata. Such systems consist of components that can be created, interconnected, and destroyed as the system evolves. Components exhibit hybrid behavior, consisting of continuous-time phases separated by discrete-event transitions. SHIFT is currently used in two applications: automated highway systems and coordinated submarine systems. The SHIFT model offers the proper level of abstraction for describing these and other applications such as air trafffic control systems and robotic shop-floors whose dynamic reconfigurations cannot be captured easily by conventional models. A compiler and a run-time system for SHIFT have been implemented.