Post on 20-Aug-2020
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Introduction to Hybrid Systems
GERARDO SCHNEIDER
gerardo@irisa.fr
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
�
� � �
Introduction to Hybrid Systems – p.2/21
Motivation
� Computers are everywhere
� Many of these systems have a “hybrid” nature.Systems exhibiting both:
� Continuous evolutions
� Discrete transitions
Introduction to Hybrid Systems – p.2/21
Motivation
� Computers are everywhere
� 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
�
� � �
Introduction to Hybrid Systems – p.2/21
Hybrid Systems: Why a new theory?
� Two main reasons: Academic and practical
Introduction to Hybrid Systems – p.3/21
Hybrid Systems: Why a new theory?
� Two main reasons: Academic and practicalAcademic reason: People competent in specific
domains of knowledge
� Control theoreticians
� Computer scientists
� Mathematicians
Introduction to Hybrid Systems – p.3/21
Hybrid Systems: Why a new theory?
� Two main reasons: Academic and practicalAcademic reason: People competent in specific
domains of knowledge
� Control theoreticians
� Computer scientists
� MathematiciansPractical reason: Finding suitable abstract models
and analysis techniques for naturalphenomena
Introduction to Hybrid Systems – p.3/21
Hybrid Systems: Why a new theory?
� Two main reasons: Academic and practicalAcademic reason: People competent in specific
domains of knowledge
� Control theoreticians
� Computer scientists
� MathematiciansPractical reason: Finding suitable abstract models
and analysis techniques for naturalphenomena
� Hybrid models offer clean modelling solutions forphenomena for which classical models areinadequate
Introduction to Hybrid Systems – p.3/21
Overview of the presentation
� Continuous models
� Discrete systems
� Hybrid automata
� Verification
� Discussion
Introduction to Hybrid Systems – p.4/21
Continuous Models
Traditional formalisms for describing system dynamicsare based on continuous dynamical systems
Introduction to Hybrid Systems – p.5/21
Continuous Models
Traditional formalisms for describing system dynamicsare based on continuous dynamical systems
� Initially: Conceived for predicting the behaviour ofuncontrolled systems (e.g. solar system)
Introduction to Hybrid Systems – p.5/21
Continuous Models
Traditional formalisms for describing system dynamicsare based on continuous dynamical systems
� Initially: Conceived for predicting the behaviour ofuncontrolled systems (e.g. solar system)
� Later: Adapted for systems with inputs - controlledsystems (e.g. robots)
� In the presence of disturbance or controlsignals: Need for input (or control) variables
Introduction to Hybrid Systems – p.5/21
Continuous Models
Traditional formalisms for describing system dynamicsare based on continuous dynamical systems
� Initially: Conceived for predicting the behaviour ofuncontrolled systems (e.g. solar system)
� 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
Introduction to Hybrid Systems – p.5/21
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
Introduction to Hybrid Systems – p.6/21
Continuous Models: Limitations
� The dynamics of many physical components ofplants cannot be modelled using purely-continuous models
� 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
Introduction to Hybrid Systems – p.6/21
Continuous Models: Limitations
� The dynamics of many physical components ofplants cannot be modelled using purely-continuous models
� Some “intelligent” control might not be expressedin terms of continuous trajectories
� 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
Introduction to Hybrid Systems – p.6/21
Continuous Models: Limitations
� The dynamics of many physical components ofplants cannot be modelled using purely-continuous models
� Some “intelligent” control might not be expressedin terms of continuous trajectories
� 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:
Introduction to Hybrid Systems – p.7/21
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
Introduction to Hybrid Systems – p.7/21
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
Introduction to Hybrid Systems – p.7/21
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
� The formal notion of dynamical system isreserved only for the continuous modes; otherphenomena are treated as “extra-modelic”
Introduction to Hybrid Systems – p.7/21
Overview of the presentation
� Continuous models
� Discrete systems
� Hybrid automata
� Verification
� Discussion
Introduction to Hybrid Systems – p.8/21
Discrete Systems
� The design of reactive systems in ComputerScience has similar goals to Control Theory
� To design systems that interact with anexternal environment
Introduction to Hybrid Systems – p.9/21
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
Introduction to Hybrid Systems – p.9/21
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”
Introduction to Hybrid Systems – p.9/21
Discrete Systems: How to Model?
� State-transition dynamics: For each state andinput event, it defines what is the next state
Introduction to Hybrid Systems – p.10/21
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)
Introduction to Hybrid Systems – p.10/21
Discrete vs. Continuous Systems
The state space of a discrete system is much smallerthan that of a continuous one
Introduction to Hybrid Systems – p.11/21
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?
Introduction to Hybrid Systems – p.11/21
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?
Not Always!
Introduction to Hybrid Systems – p.11/21
Discrete vs. Continuous Systems
The state space of a discrete system is much smallerthan that of a continuous one
� D.S. are defined on impoverish mathematicaldomains: Analysis and synthesis are more difficult
� Examples:
�
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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!
Introduction to Hybrid Systems – p.11/21
Discrete vs. Continuous Systems
The state space of a discrete system is much smallerthan that of a continuous one
� 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.
Introduction to Hybrid Systems – p.11/21
Overview of the presentation
� Continuous models
� Discrete systems
� Hybrid automata
� Verification
� Discussion
Introduction to Hybrid Systems – p.12/21
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
Introduction to Hybrid Systems – p.13/21
Hybrid Automata: Examples
� Frictionless movement of a particle in a boundedinterval
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subject to elastic collisions atthe endpoints of the interval
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Introduction to Hybrid Systems – p.14/21
Hybrid Automata: Examples
� Frictionless movement of a particle in a boundedinterval
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subject to elastic collisions atthe endpoints of the interval
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Introduction to Hybrid Systems – p.14/21
Hybrid Automata: Examples
� A heating system with external On/Off commands
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Introduction to Hybrid Systems – p.14/21
Hybrid Automata: Examples
� A heating system with a thermostat
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Introduction to Hybrid Systems – p.14/21
Hybrid Automata: Examples
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Introduction to Hybrid Systems – p.14/21
Overview of the presentation
� Continuous models
� Discrete systems
� Hybrid automata
� Verification
� Discussion
Introduction to Hybrid Systems – p.15/21
Verification: Motivation
� How to build correct complex systems?
Introduction to Hybrid Systems – p.16/21
Verification: Motivation
� How to build correct complex systems?
� Synthesis (from the specification)
Introduction to Hybrid Systems – p.16/21
Verification: Motivation
� How to build correct complex systems?
� Synthesis (from the specification)
� Build them and then
� Test
� Simulate
Introduction to Hybrid Systems – p.16/21
Verification: Motivation
� How to build correct complex systems?
� Synthesis (from the specification)
� Build them and then
� Test
� Simulate
� Alternative: Formal verification
Introduction to Hybrid Systems – p.16/21
What is Verification?
� Instance:
� �
: Program (e.g. Hw circuit, communicationprotocol, distributed system, C program,Real-time system, hybrid automata)
� �
: Specification
� Question:
� Does
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satisfies�
?
Introduction to Hybrid Systems – p.17/21
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
Introduction to Hybrid Systems – p.18/21
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
Introduction to Hybrid Systems – p.18/21
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
� 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
Introduction to Hybrid Systems – p.18/21
Overview of the presentation
� Continuous models
� Discrete systems
� Hybrid automata
� Verification
� Discussion
Introduction to Hybrid Systems – p.19/21
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!
Introduction to Hybrid Systems – p.20/21
MUITO OBRIGADO!
Introduction to Hybrid Systems – p.21/21