1. Hybrid automata and temporal logics
Hybrid Systems – PhD School
Aalborg University
January 2007
Anders P. Ravn
Department of Computer Science,
Aalborg University, Denmark

2. 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

3. Hybrid System A dynamical system with a non-trivial interaction of
discrete and continuous dynamics
autonomous
switches
jumps
controlled
jump
between manifolds
(Branicky 1995)

4. 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)

5. 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

6. 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   . 
Note: finite representation
x´ = x-1 

7. Labelled Transition System
Q – states, e.g. (v=”Off”,x = 17.5)
Q0 – initial states, Q0  Q
A – labels
 – transition relation,  Q  A Q
The intended models
posta(R) = { q’ | q  R and q  q’}
prea(R) = { q | q’  R and q  q’} a

8. 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  
x’ = x-1   .
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
Note intended model, generally infinite
{ (v,x) –  (v’,x’) | e  E(v,v’) and
event(e) =  and jump(e) [X := x]}
{ (v,x) –  (v,x’) |   R0 and f: (0,)  Rn s.t. f is diff. and
f(0) = x and f() = x’ and
flow(v)[X := f(t), X:= f(t)], t  (0,) }

9. 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:  = <(a0,q0)…(ai,qi)…>
where q0  Q0 and qi–aiqi+1, i 0
Check 2!
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

10. Tree Semantics Q - states, {(v,x) | v  V and inv(v)[X := x]}
Q0 – initial states, …
A - labels, …
 - transition relation,  Q  A Q
Computation tree:  =
q00 a
q q q1n1 …
q q q q q13
Can generate all traces -

11. Classes of Hybrid Automata
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   .
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

12. 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’, i=1,2 and
a = a1a2 is defined
Remark p 7

13. 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 ?