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Theory of Hybrid Automata Sachin J Mujumdar Hybrid Automata A formal model for a dynamical system with discrete and continuous components Example – Temperature Control Formal Definition A Hybrid Automaton consists of following: Variables – Finite Set (real numbered) Continuous Change,

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theory of hybrid automata

Theory of Hybrid Automata

Sachin J Mujumdar

CS 367 - Theory of Hybrid Automata

hybrid automata
Hybrid Automata
  • A formal model for a dynamical system with discrete and continuous components
  • Example – Temperature Control

CS 367 - Theory of Hybrid Automata

formal definition
Formal Definition

A Hybrid Automaton consists of following:

  • Variables –
    • Finite Set (real numbered)
    • Continuous Change,
    • Values at conclusion at of discrete change,
  • Control Graph
    • Finite Directed Multigraph (V, E)
    • V – control modes (represent discrete state)
    • E – control switches (represent discrete dynamics)

CS 367 - Theory of Hybrid Automata

formal definition4
Formal Definition
  • Initial, Invariant & Flow conditions – vertex labeling functions
    • init(v) – initial condition whose free variable are from X
    • inv(v) – free variables from X
    • flow(v) – free variables from X U
  • Jump Conditions
    • Edge Labeling function, “jump” for every control switch, e Є E
    • Free Variables from X U X’
  • Events
    • Finite set of events, Σ
    • Edge labeling function, event: E  Σ, for assigning an event to each control switch
  • Continuous State – points in

CS 367 - Theory of Hybrid Automata

safe semantics
Safe Semantics
  • Execution of Hybrid Automaton – continuous change (flows) and discrete change (jumps)
  • Abstraction to fully discrete transition system
  • Using Labeled Transition Systems

CS 367 - Theory of Hybrid Automata

labeled transition systems
Labeled Transition Systems
  • Labeled Transition System, S
    • State Space, Q – (Q0 – initial states)
    • Transition Relations
      • Set of labels, A – possibly infinite
      • Binary Relations on Q,
    • Region, R Q
    • Transition – triplet of

CS 367 - Theory of Hybrid Automata

labeled transition systems7
Labeled Transition Systems
  • Two Labeled Transition Systems
    • Timed Transition System
      • Abstracts continuous flows by transitions
      • Retains info on source, target & duration of flow
    • Time-Abstract Transition System
      • Also abstracts the duration of flows
      • Called timed-abstraction of Timed Transition Systems

CS 367 - Theory of Hybrid Automata

live semantics
Live Semantics
  • Usually consider the infinite behavior of hybrid automaton. Thus, only infinite sequences of transitions considered
  • Transitions do not converge in time
  • Divergence of time – liveness
  • Nonzeno – Cant prevent time from diverging

CS 367 - Theory of Hybrid Automata

live transition systems
Live Transition Systems
  • Trajectory of S
    • (In)Finite Sequence of <ai, qi>i≥1
    • Condition –
    • q0 – rooted trajectory
    • If q0 is initial state, then intialized trajectory
  • Live Transition System
    • (S, L) pair
    • L infinite number of initialized trajectories of S
  • Trace
    • <ai, qi>i≥1 is finite initialized trajectory of S, or trajectory in L  corresponding sequence <ai>i≥1 of labels is a Trace of (S, L), i.e. the Live Transition System

CS 367 - Theory of Hybrid Automata

composition of hybrid automata
Composition of Hybrid Automata
  • Two Hybrid Automata, H1 & H2
  • Interact via joint events
  • a is an event of both  Both must synchronize on a-transitions
  • a is an event of only H1  each a-transition of H1 synchronizes with a 0-duration time transition of H2
  • Vice-Versa

CS 367 - Theory of Hybrid Automata

composition of hybrid automata11
Composition of Hybrid Automata
  • Product of Transition Systems
    • Labeled Transition Systems, S1 & S2
    • Consistency Check
      • Associative partial function
      • Denoted by
      • Defined on pairs consisting of a transition from S1 & a transition from S2
    • S1 x S2
      • w.r.t
      • State Space – Q1 x Q2
      • Initial States – Q01 x Q02
      • Label Set – range( )
      • Transition Condition
      • and 

CS 367 - Theory of Hybrid Automata

composition of hybrid automata12
Composition of Hybrid Automata
  • Parallel Composition
    • H1 and H2
    • of and of are consistent if one of the 3 is true
      • a1 = a2 consistency check yields a1
      • a1 belongs to Event space of H1 and a2 = 0  consistency check yields a1
      • a2 belongs to Event space of H2 and a1 = 0  consistency check yields a1
    • The Parallel Composition is defined to be the cross product w.r.t the consistency check

CS 367 - Theory of Hybrid Automata

railroad gate control example
Railroad Gate Control - Example
  • Circular track, with a gate – 2000 – 5000 m circumference
  • ‘x’ – distance of train from gate
  • speed – b/w 40 m/s & 50 m/s
  • x = 1000 m
    • “approach” event
    • may slow down to 30 m/s
  • x = -100 m (100m past the gate)
    • “exit event”
  • Problem
    • Train Automaton
    • Gate Automaton
    • Controller Automaton

CS 367 - Theory of Hybrid Automata

railroad gate control example14
Railroad Gate Control - Example

Train Automaton

CS 367 - Theory of Hybrid Automata

railroad gate control example15
Railroad Gate Control - Example

Gate Automaton

  • y – position of gate in degrees (max 90)
  • 9 degrees / sec

CS 367 - Theory of Hybrid Automata

railroad gate control example16
Railroad Gate Control - Example

Controller Automaton

  • u – reaction delay of controller
  • z – clock for measuring elapsed time

Question :

value of “u” so that,

y = 0, whenever -10 <= x <= 10

CS 367 - Theory of Hybrid Automata

verification
Verification

4 paradigmatic Qs about the traces of the H

  • Reachability
    • For any H, given a control mode, v, if there exists some initialized trajectory for its Labeled Transition System(LTS), can it visit the state of the form (v, x)?
  • Emptiness
    • Given H, if there exists a divergent initialized trajectory of the LTS?
  • (Finitary) Timed Trace Inclusion Problem
    • Given H1 & H2, if every (finitary) timed trace of H1 is also that of H2
  • (Finitary) Time-Abstract Trace Inclusion Problem
    • Same as above – consider time-abstract traces

CS 367 - Theory of Hybrid Automata

rectangular automata
Rectangular Automata
  • Flow Conditions are independent of Control Modes
  • First derivative, x dot, of each variable has fixed range of values, in every control mode
  • This is independent of the control switches
  • After a control switch – value of variable is either unchanged or from a fixed set of possibilities
  • Each variable becomes independent of other variables
  • Multirectangular Automata – allows for flow conditions that vary with control switches
  • Triangular Automata – allows for comparison of variables

CS 367 - Theory of Hybrid Automata

state space of hybrid automata
State Space of Hybrid Automata
  • State Space is infinite – cannot be ennumerated
  • Studied using finite symbolic representation
    • x – real numbered variable
    • 1 <= x <= 5  Finite symbolic representation of an infinite set of real numbers

CS 367 - Theory of Hybrid Automata

observational transition systems
Observational Transition Systems
  • Difficult to (dis)prove the assertion about behavior of H – sampling of only piecewise continuous trajectory of LTS’ at discrete time intervals
  • Reminder – Transition abstracts the information of all the intermediate states visited
  • Solution
    • Label each transition with a region
    • transition, t, is labeled with region, R, iff all intermediate & target states of t lie in R
    • i.e. Observational Transition System – from continuous observation of hybrid automaton

CS 367 - Theory of Hybrid Automata

summary
Summary
  • Introduction to Hybrid Systems
  • Formal Definition of Hybrid Systems
  • Change from hybrid to fully-discrete systems - Safe Semantics
  • Labeled transition Systems
  • Composition of Hybrid Automata
  • Properties of Hybrid Automata
  • Observational Transition Systems
  • Theorems & Theories presented in paper, for further reading – “The Theory of Hybrid Automata” – Thomas A. Henzinger

CS 367 - Theory of Hybrid Automata