# Operational Semantics of Hybrid Systems

### Haiyang Zheng

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2007-68

May 18, 2007

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-68.pdf

Hybrid systems are heterogeneous systems that include continuous-time (CT) subsystems interacting with discrete-event (DE) subsystems. They are effective models for physical systems interacting with software or experiencing discrete mode changes. This dissertation discusses an interpretation of hybrid systems as executable programs written in a programming language with a hybrid system semantics. The semantic properties of such a programming language affect our ability to understand, execute, and analyze a hybrid system model. This dissertation focuses on a few semantic issues that come in defining such a programming language, such as the interpretation of discontinuities in CT signals and simultaneous discrete events in DE signals, liveness property, and the consequences of numerical ODE solver techniques. The interactions between CT and DE subsystems and between DE subsystems themselves are captured by discontinuities in continuous-time signals and simultaneous discrete events in discrete-event signals. In order to precisely represent them in compute execution results, a two-dimension domain, called ``super-dense time," is used as the domain for defining signals. This domain allows a signal to have multiple values at the same time point while keeping the values ordered. CT and DE subsystems are modeled as actors, which are functions that map a set of signals to another set of signals. In this way, a hybrid system model is just a network of actors interacting via signals. We can always transform a network of actors into a composite actor with feedback, where the function of the composition actor is the composition of functions of the component actors. The least fixed point solution to the function of the composite actor, which is a set of signals, gives the denotational semantics of the hybrid system model. The operational semantics takes the denotational semantics as a mathematical foundation and defines a set of rules for evaluating actors such that the least fixed point solution can be constructed. Rather than constructing the whole signals, the operational semantics only computes a discrete subset of the signals called a discrete representation of the signals. The constructive procedure is formalized with the Abstract State Machine semantics, where a hybrid system is treated as a state transition system and the rules specify how state transformations are performed. This operational semantics supports heterogeneous and hierarchical composition of different models of computation, such as CT, DE, finite state machines, and synchronous languages, and modular execution of the composition as a whole. This ability makes it easy to jointly model and design software controlled systems. The operational semantics proposed in this dissertation has been implemented in HyVisual, which is a software tool for modeling and simulating hybrid systems. HyVisual is part of the Ptolemy II software framework, which is available in open-source form at http://ptolemy.org.

**Advisor:** Edward A. Lee

BibTeX citation:

@phdthesis{Zheng:EECS-2007-68, Author = {Zheng, Haiyang}, Title = {Operational Semantics of Hybrid Systems}, School = {EECS Department, University of California, Berkeley}, Year = {2007}, Month = {May}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-68.html}, Number = {UCB/EECS-2007-68}, Abstract = { Hybrid systems are heterogeneous systems that include continuous-time (CT) subsystems interacting with discrete-event (DE) subsystems. They are effective models for physical systems interacting with software or experiencing discrete mode changes. This dissertation discusses an interpretation of hybrid systems as executable programs written in a programming language with a hybrid system semantics. The semantic properties of such a programming language affect our ability to understand, execute, and analyze a hybrid system model. This dissertation focuses on a few semantic issues that come in defining such a programming language, such as the interpretation of discontinuities in CT signals and simultaneous discrete events in DE signals, liveness property, and the consequences of numerical ODE solver techniques. The interactions between CT and DE subsystems and between DE subsystems themselves are captured by discontinuities in continuous-time signals and simultaneous discrete events in discrete-event signals. In order to precisely represent them in compute execution results, a two-dimension domain, called ``super-dense time," is used as the domain for defining signals. This domain allows a signal to have multiple values at the same time point while keeping the values ordered. CT and DE subsystems are modeled as actors, which are functions that map a set of signals to another set of signals. In this way, a hybrid system model is just a network of actors interacting via signals. We can always transform a network of actors into a composite actor with feedback, where the function of the composition actor is the composition of functions of the component actors. The least fixed point solution to the function of the composite actor, which is a set of signals, gives the denotational semantics of the hybrid system model. The operational semantics takes the denotational semantics as a mathematical foundation and defines a set of rules for evaluating actors such that the least fixed point solution can be constructed. Rather than constructing the whole signals, the operational semantics only computes a discrete subset of the signals called a discrete representation of the signals. The constructive procedure is formalized with the Abstract State Machine semantics, where a hybrid system is treated as a state transition system and the rules specify how state transformations are performed. This operational semantics supports heterogeneous and hierarchical composition of different models of computation, such as CT, DE, finite state machines, and synchronous languages, and modular execution of the composition as a whole. This ability makes it easy to jointly model and design software controlled systems. The operational semantics proposed in this dissertation has been implemented in HyVisual, which is a software tool for modeling and simulating hybrid systems. HyVisual is part of the Ptolemy II software framework, which is available in open-source form at http://ptolemy.org.} }

EndNote citation:

%0 Thesis %A Zheng, Haiyang %T Operational Semantics of Hybrid Systems %I EECS Department, University of California, Berkeley %D 2007 %8 May 18 %@ UCB/EECS-2007-68 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-68.html %F Zheng:EECS-2007-68