Hybrid automata and temporal logics

Anders P. RavnDepartment of Computer Science,

Aalborg University, Denmark

Hybrid Systems – PhD SchoolAalborg University

January 2007

Plan

• 9:30 -10:00• 10:10 -10:30• 10:40 -11:20• 11:30 -12:00

Hybrid Automata

Abstraction/Refinement

Temporal Logics

Model Checking

Hybrid System

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

• autonomousswitchesjumps

•controlledswitchesjump

between manifolds(Branicky 1995)

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)

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

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

Hybrid Automaton - Syntax

.

x´ = x-1

Q – states, e.g. (v=”Off”,x = 17.5)

Q0 – initial states, Q0

Q

A – labels

– transition relation, Q A Q

Labelled Transition System

posta(R) = { q’ | q R and q q’}prea(R) = { q | q’ R and q q’}

a

a

Transition Semantics of HAX = {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]}

.

x’ = x-1

Q0 – initial states - {(v,x) Q | init(v)[X := x]}

A - labels - R0

{ (v,x) – (v’,x’) | e E(v,v’) and event(e) = and jump(e) [X :=

x]}{ (v,x) – (v,x’) | R0 and f: (0,) Rn s.t. f is diff. andf(0) = x and f() = x’ andflow(v)[X := f(t), X:= f(t)], t (0,) }

. .

Q - states, {(v,x) | v V and inv(v)[X := x]}

Q0 – initial states, …

A - labels R0

- transition relation, Q A Q

Trace Semantics

Trajectory: = <(a0,q0)…(ai,qi)…> where q0 Q0 and qi–aiqi+1, 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

Q - states, {(v,x) | v V and inv(v)[X := x]}

Q0 – initial states, …

A - labels, …

- transition relation, Q A Q

Tree Semantics

Computation tree: = q00

a

q10 q11 ... q1n1

…

q200 q201 q210 q211 q13

Classes of Hybrid AutomataX = {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

• Rectangular init, inv, flow (x Iflow), jump (x = x,y I, x’ I’ ,y’=y)

• Singular – rectangular with Iflow a point• Timed – singular with Iflow = [1,1]n

• Multirectangular …• Triangular …• Stopwatch …

.

Verification results pp. 11-12

Q - states

Q0 – initial states, …

A - labels, … - transition relation, Q A Q

Composition of Transition Systems

S = S1 || S2with

: A1 A2 A

Q = Q1 Q2Q0 = Q10 Q20

(q1,q2) –a (q1’,q2’) iff (qi –ai qi’, i=1,2 and a = a1a2 is defined

Remark p 7

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 ?