- 566 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Theory of Hybrid Automata' - Olivia

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

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 - Example

Gate Automaton

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

CS 367 - Theory of Hybrid Automata

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

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

- 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

Download Presentation

Connecting to Server..