Towards Modelling of Hybrid Systems (original) (raw)
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Proceedings of 32nd IEEE Conference on Decision and Control
We discuss topological issues that arise when differential equations and finite automata interact (hybrid systems). In particular, we examine topologies for achieving continuity of maps from a set of measurements of continuous dynamics to a finite set of input symbols and from a finite set of output symbols into the control space for those continuous dynamics. Finding some anomalies in completing this loop, we discuss a new view of hybrid systems that may broach them and is more in line with traditional control systems. In fact, the most widely used fuzzy control system is related to this new view and does not possess these anomalies. Indeed, we show that fuzzy control leads to continuous maps (from measurements to controls) and that all such continuous maps may be implemented via fuzzy control.
Modal logics and topological semantics for hybrid systems
1997
Abstract: This paper introduces the logic of a control action S4F and the logic of a continuous control action S4C on the state space of a dynamical system. The state space is represented by a topological space (X, T) and the control action by a function f from X to X.
Abstractions of hybrid systems: formal languages to describe dynamical behaviour
We show how finite-state automata over finite and infinite words can capture key dynamical properties of hybrid systems. The purpose is to obtain a discrete abstraction of the main complex dynamical behaviour patterns that such systems exhibit. We will form a labelled transition system as an abstraction of the hybrid system, and, in order to differentiate between the key dynamical behaviours, we will associate a set of formal languages to each of the behaviours. We will use both finite regular and infinite ω-regular languages. These formal languages can be accepted by a generalised Muller automaton, which is a novel approach in the specification of dynamical properties of hybrid systems.
Discrete abstractions of hybrid systems
Proceedings of the IEEE, 2000
A hybrid system is a dynamical system with both discrete and continuous state changes. For analysis purposes, it is often useful to abstract a system in a way that preserves the properties being analyzed while hiding the details that are of no interest. We show that interesting classes of hybrid systems can be abstracted to purely discrete systems while preserving all properties that are definable in temporal logic. The classes that permit discrete abstractions fall into two categories. Either the continuous dynamics must be restricted, as is the case for timed and rectangular hybrid systems, or the discrete dynamics must be restricted, as is the case for o-minimal hybrid systems. In this paper, we survey and unify results from both areas.
Compositional Abstractions of Hybrid Control Systems
Discrete Event Dynamic Systems, 2004
Abstraction is a natural way to hierarchically decompose the analysis and design of hybrid systems. Given a hybrid control system and some desired properties, one extracts an abstracted system while preserving the properties of interest. Abstractions of purely discrete systems is a mature area, whereas abstractions of continuous systems is a recent activity. In this paper we present a framework for abstraction that applies to discrete, continuous, and hybrid systems. We introduce a composition operator that allows to build complex hybrid systems from simpler ones and show compatibility between abstractions and this compositional operator. Besides unifying the existing methodologies we also propose constructions to obtain abstractions of hybrid control systems.
Composing Abstractions of Hybrid Systems
Hybrid Systems, 2002
The analysis and design of hybrid systems must exploit their hierarchical and compositional nature of in order to tackle complexity. In previous work, we presented a hierarchical abstraction framework for hybrid control systems based on the notions of simulation and bisimulation. In this paper, we build upon our previous work and investigate the compositionality of our abstraction framework. We present a composition operator that allows synchronization on inputs and states of hybrid systems. We then show that the composition operator is compatible with our abstraction framework in the sense that abstracting subsystems will the result in an abstraction of the overall system.
On a Class of General Hybrid Dynamical Systems
IFAC Proceedings Volumes, 1996
Complex systems typically possess a hierarchical structure. characterized by continuous-variable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today perfonn computer-coded checks and issue logical as well as continuous-variable control commands. Such are "hybrid" systems. We introduce "general hybrid dynamical systems" as interacting collections of dynamical systems, evolving on continuous-variable state spaces, and subject to continuous controls and discrete phenomena. We discuss modeling issues, giving conditions for trajectories and providing a taxonomy for hybrid systems models. Wc review our hybrid systems analysis results, including topology, complexity/computation, stability tools, and examples. We summarize our hybrid control results. including optimal control theory, control algorithms, and examples.
Towards an integrated conception of hybrid dynamical systems
Proceedings of the 15th IFAC World Congress, 2002, 2002
Hybrid dynamical systems are composed of continuous-time dynamical parts, mixed with event-driven parts. Most of the time, both parts are designed separately using specific techniques of each domain, and integrated a posteriori in an application-specific manner. This approach is restrictive in that it does not exhibit a hybrid global model of the designed system, that would though be required for analysis and behavior-checking to take place. In this paper, we discuss and illustrate our approach of hybrid systems modeling, that is based on the obvious statement that both domains (dynamical and event-driven) must be clearly considered in an integrated manner from the very beginning of the design. In our example, we exhibit a draft formal framework for hybrid system modeling, that would allow for verification techniques. For that purpose, we take advantage of the recently developed techniques and tools, in both areas. The numerical computation laboratorymatlabthat we chose for the dynamical system part design, fits perfectly with our goals. But the reactive synchronous language chosen-Esterel-, if it actually fits with the event-driven part specification, exhibits some weaknesses when dealing with data and values, that are needed when interfacing both parts together.
2003
There is a growing recognition of the presence of dynamical systems in many fields. In areas ranging from economics to physics and computer science to biology, many open fundamental problems are known to relate to the central problems in dynamical systems theory. Physicists generally work more with continuous dynamical systems, rather than discrete dynamical systems which computer scientists may study. The difference is that a continuous dynamical system is almost always defined as a system with an evolution function that satisfies a certain set of differential equations, while a discrete dynamical system is described by a state space involving discrete states and events which trigger discrete transitions between them. However, as computer science has come into contact with problems such as traffic management, automotive control, and modeling biological cell networks, which involve both continuous and discrete dynamical systems, a new formalism has emerged. This new formalism, the concept of a hybrid system, combines both the continuous and discrete formalisms into one entity that has fueled much research in recent years. Here we will formally introduce the notion of a hybrid system as an extension (or special case) of the concept of a dynamical system that is appropriate for many problems of interest in a wide variety of fields. We will then present summaries of some papers written on hybrid systems in recent years and present ideas for future research in the field.