Hybrid automata and temporal logics

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Hybrid automata and temporal logics
Hybrid Systems PhD School Aalborg University
January 2007

• Anders P. Ravn
• Department of Computer Science,
• Aalborg University, Denmark

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Plan

• 930 -1000
• 1010 -1030
• 1040 -1120
• 1130 -1200

• Hybrid Automata
• Abstraction/Refinement
• Temporal Logics
• Model Checking

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Hybrid System
A dynamical system with a non-trivial interaction
of discrete and continuous dynamics

• autonomous
• switches
• jumps
• controlled
• switches
• jump
• between manifolds
• (Branicky 1995)

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Why are we here?
"Control Engineers will have to master computer
and software technologies to be able to build the
systems of the future, and software engineers
need to use control concepts to master
ever-increasing complexity of computing
systems. (IFAC Newsletter December 2005 No.6)
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Hybrid Systems in Control(take up of CS ideas
1990 - )

• Hybrid Automata is the Spec. Language
• Tools for simulation and model checking
  (Henzinger,Alur,Maler,Dang, )
• Bisimulation as abstraction technique
  (Pappas,Neruda,Koo, )
• Industrial Applications

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Hybrid Automaton - Syntax

• X x1, xn - variables
• (V, E) control graph
• init V ? pred(X)
• inv V ? pred(X)
• flow V ? pred(X ? X)
• jump E ? pred(X ? X)
• event E ? ?

.
?
x x-1
?
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Labelled Transition System

• Q states, e.g. (vOff,x 17.5)
• Q0 initial states, Q0 ? Q
• A labels
• ? transition relation, ?? Q ? A? Q

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Transition Semantics of HA
X x1, xn - variables (V, E) control
graph init V ? pred(X) inv V ? pred(X) flow V
? pred(X ? X) jump E ? pred(X ? X) event E ?
?
.
Q - states (v,x) v ? V and inv(v)X x
Q0 initial states - (v,x)? Q init(v)X x
A - labels - ? ? R?0
(v,x) ?? (v,x) e ? E(v,v) and
event(e) ? and jump(e) X x
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Trace Semantics

• Q - states, (v,x) v ? V and inv(v)X x
• Q0 initial states,
• A - labels ? ? R?0
• ? - transition relation, ?? Q ? A? Q

Trajectory ? lt(a0,q0)(ai,qi)gt where q0
? Q0 and qiai?qi1, i? 0

• Live Transition System (S, L ? ? infinite
  from S)
• Machine Closed ? finite from S, ? ? prefix(L)
• Duration of ? is sum of time labels.
• S is non-Zeno duration of ? ? L diverges,
  Machine closed

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Tree Semantics

• Q - states, (v,x) v ? V and inv(v)X x
• Q0 initial states,
• A - labels,
• ? - transition relation, ?? Q ? A? Q

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Classes of Hybrid Automata
.

• Rectangular init, inv, flow (x ? Iflow),
• jump (x x,y? I, x? I
  ,yy)
• Singular rectangular with Iflow a point
• Timed singular with Iflow 1,1n
• Multirectangular
• Triangular
• Stopwatch

Verification results pp. 11-12
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Composition of Transition Systems

• Q - states
• Q0 initial states,
• A - labels,
• ? - transition relation, ?? Q ? A? Q

S S1 S2 with ? A1 ? A2 ? A
Q Q1? Q2 Q0 Q10 ? Q20 (q1,q2) a? (q1,q2)
iff (qi ai? qi, i1,2 and a a1?a2 is
defined
Remark p 7
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Summary

• Hybrid Atomata Finite description through an
  intuitive syntax
• Clear semantics through Transition Systems
• Composition
• Specialization through restrictions on flow
  equations
• How to analyze them ?