Introduction to Hybrid Systems - cse.chalmers.segersch/slides-talks/slides... · Introduction to Hybrid Systems – p.11/21. Discrete vs. Continuous Systems The state space of a discrete

Introduction to Hybrid Systems

GERARDO SCHNEIDER

[email protected]

IRISA/INRIA

EQUIPE LANDE

RENNES - FRANCE

Introduction to Hybrid Systems – p.1/21

Motivation

� Computers are everywhere

� Electronic commerce

� Education

� Thermostat

� Automated highway systems

� Air traffic management systems

� Automotive industry (robots)

� Chemical plants

�

� � �


Motivation


� Many of these systems have a “hybrid” nature.Systems exhibiting both:

� Continuous evolutions

� Discrete transitions


Motivation


� Many of these systems have a “hybrid” nature.Systems exhibiting both:

� Continuous evolutions

� Discrete transitions

� Some examples:

� Thermostat: Temperature + switch On/Off

� Robotic systems: Distance, speed, etc +switch direction

� Chemical plants: Chemical reactions +closing/opening valves

�

� � �


Hybrid Systems: Why a new theory?

� Two main reasons: Academic and practical



� Two main reasons: Academic and practicalAcademic reason: People competent in specific

domains of knowledge

� Control theoreticians

� Computer scientists

� Mathematicians







� MathematiciansPractical reason: Finding suitable abstract models

and analysis techniques for naturalphenomena







� MathematiciansPractical reason: Finding suitable abstract models

and analysis techniques for naturalphenomena

� Hybrid models offer clean modelling solutions forphenomena for which classical models areinadequate


Overview of the presentation

� Continuous models

� Discrete systems

� Hybrid automata

� Verification

� Discussion


Continuous Models

Traditional formalisms for describing system dynamicsare based on continuous dynamical systems


Continuous Models


� Initially: Conceived for predicting the behaviour ofuncontrolled systems (e.g. solar system)


Continuous Models



� Later: Adapted for systems with inputs - controlledsystems (e.g. robots)

� In the presence of disturbance or controlsignals: Need for input (or control) variables


Continuous Models



� Later: Adapted for systems with inputs - controlledsystems (e.g. robots)

� In the presence of disturbance or controlsignals: Need for input (or control) variables

� Such systems are specified by differential ordifference equations, describing the evolution ofthe state-variable


Continuous Models: Limitations

� The dynamics of many physical components ofplants cannot be modelled using purely-continuous models

� Behaviour of valves and switches are bestmodelled as discrete systems

� Continuous sensors and actuators aresaturated beyond certain values




� Some “intelligent” control might not be expressedin terms of continuous trajectories

� Movement in physical space may contain“objects” and “places”: Inherently discreteinvolving phenomena like collision





� Even in the presence of continuous models, thedynamics could be highly non-linear

� Many models based on a linear approximationare valid only in a certain region. Whenleaving such region a new linear model shouldbe used





� Even in the presence of continuous models, thedynamics could be highly non-linear

� Many control systems need interaction withentities other than continuous sensors: e.g. withcomputers or human operators

� Such entities may activate or suspend thecontroller execution or force it to switch toanother mode of operation Introduction to Hybrid Systems – p.6/21

Continuous Models: PracticalSolution

� Control Engineers know how to solve many of theabove problems:




� A continuous model is given for each “mode”of operation

� Control laws are synthesised for each of thesemodes and then “glue” together






� However, the transition between them is not apart of the “official” dynamics of the system






� However, the transition between them is not apart of the “official” dynamics of the system

� The formal notion of dynamical system isreserved only for the continuous modes; otherphenomena are treated as “extra-modelic”





� Hybrid automata

� Verification

� Discussion


Discrete Systems

� The design of reactive systems in ComputerScience has similar goals to Control Theory

� To design systems that interact with anexternal environment


Discrete Systems

� The design of reactive systems in ComputerScience has similar goals to Control Theory

� To design systems that interact with anexternal environment

� Example: A mechanism which controls theaccess of clients to some shared resources


Discrete Systems

� Main difference between Control Theory andComputer Science

� Control Theory:

� State variables are physical magnitudes(e.g. temperature)

� Interaction is done through measurementsof physical magnitudes

� Computer Science:

� State variables are non-numerical values(e.g. “ready”, “waiting”)

� Interaction via messages and events such as“request” or “release”


Discrete Systems: How to Model?

� State-transition dynamics: For each state andinput event, it defines what is the next state


Discrete Systems: How to Model?

� State-transition dynamics: For each state andinput event, it defines what is the next state

� Small state-space: It can be explicitly written in atable

� Larger systems are described using two methods:

� Composition, where interacting sub-systemsare described separately

� Implicit (symbolic) description (e.g. usingprogramming formalisms)


Discrete vs. Continuous Systems

The state space of a discrete system is much smallerthan that of a continuous one




� Are Discrete systems easier to analyse thanContinuous systems?




� Are Discrete systems easier to analyse thanContinuous systems?

Not Always!




� D.S. are defined on impoverish mathematicaldomains: Analysis and synthesis are more difficult

� Examples:

�

��

defined over the reals: Simple solutionusing division and subtraction

� Finding whether there is a Boolean vectorsatisfying a formula in propositional logic, is anNP-hard problem: We need to explore allpossible vectors!




� D.S. are defined on impoverish mathematicaldomains: Analysis and synthesis are more difficult

� Examples:

� In C.S.

�� : Knowledge about thetrajectories by inspecting

�

� In D.S.: No “holistic” way to capture the globalbehaviour of the systems. Sometimes, need toexplore all the possible trajectories.





� Hybrid automata

� Verification

� Discussion


Hybrid Automata

� Hybrid automata are a good formalism formodelling

� The continuous “modes” of operation, and

� The discrete switches between such modes

� They are a generalisation of a well-establishedformalism: Timed Automata


Hybrid Automata: Examples

� Frictionless movement of a particle in a boundedinterval

� � ��

subject to elastic collisions atthe endpoints of the interval

�

� ��



� Frictionless movement of a particle in a boundedinterval

� � ��

subject to elastic collisions atthe endpoints of the interval

�

� ��

� �

� � � � �

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� A heating system with external On/Off commands

� � � � �

��



� A heating system with a thermostat

guard reset

invariant

dynamics

label

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��

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��

�

��

��



� A heating system with a thermostat

label

invariant

dynamics

guard reset� � �

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� Hybrid automata

� Verification

� Discussion


Verification: Motivation

� How to build correct complex systems?




� Synthesis (from the specification)





� Build them and then

� Test

� Simulate





� Build them and then

� Test

� Simulate

� Alternative: Formal verification


What is Verification?

� Instance:

� �

: Program (e.g. Hw circuit, communicationprotocol, distributed system, C program,Real-time system, hybrid automata)

� �

: Specification

� Question:

� Does

�

satisfies�

?


What is Verification?

� Instance:

� �

: Program (e.g. Hw circuit, communicationprotocol, distributed system, C program,Real-time system, hybrid automata)

� �

: Specification

� Question:

� Does

�

satisfies�

?

� Example:

� �

: Thermostat

� �

: The temperature remains always between

� andIntroduction to Hybrid Systems – p.17/21

Verification of Discrete Systems:Methodology

� Modelling formalisms: Based on interactingautomata and other variants of transition systems




� Formalisms for specifying systems requirements:Automata, regular expressions or formulae intemporal logic




� Formalisms for specifying systems requirements:Automata, regular expressions or formulae intemporal logic

� Methods to verify that a controller, composed withits environment, generates only acceptablebehaviours

� Algorithmic: Explore the paths in the transitiongraph

� Deductive: Try to prove some claims about allsystem behaviours





� Hybrid automata

� Verification

� Discussion


Discussion

� Many natural phenomena and industrialapplications are hybrid by nature (continuous +discrete behaviours)

� Hybrid systems are studied by mathematicians,computer scientists and control theoreticians

� Hybrid automata are a good formalism formodelling hybrid systems

� A lot of work is still to be done for verifying andsynthesising hybrid systems!


MUITO OBRIGADO!


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Introduction to Hybrid Systems - cse.chalmers.segersch/slides-talks/slides... · Introduction to Hybrid Systems – p.11/21. Discrete vs. Continuous Systems The state space of a discrete