Theory of Hybrid Automata

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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

Sachin J Mujumdar

CS 367 - Theory of 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

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 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
• 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 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 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
• 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
• 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
• 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 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 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

• 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

Train Automaton

CS 367 - Theory of Hybrid Automata

Gate Automaton

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

CS 367 - Theory of Hybrid Automata

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

• 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
• 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 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
• 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
• 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