Transcript Hybrid automata and temporal logics

Hybrid automata
Hybrid Systems
October 9th 2009
Rafael Wisniewski
Automation and Control, Dept. of Electronic Systems
Aalborg University, Denmark
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 System
A dynamical system with a non-trivial interaction of
discrete and continuous dynamics
• autonomous
switches
jumps
•controlled
switches
jump
between manifolds
(Branicky 1995)
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
Hybrid Automaton
Syntax
.
X = {x1, … xn} - variables, X dotted variables, X’ primed variables
(V, E) – control graph
init: V  preds(X)
inv: V  preds(X)
.
flow: V  preds(X  X)

jump: E  preds(X  X´)
x´ = x-1
event: E  

Labelled Transition System
Q – states, e.g. (v=”Off”,x = 17.5)
Q0 – initial states, Q0  Q
A – labels
 – transition relation,  Q  A Q
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

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, 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,) }
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
q10
q11
...
q1n1
…
q200
q201
q210
q211 q13
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
• 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
(ompare with the two tank example)
Time Abstract Semantics
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]}
B - labels -   {} - finite !
{ (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,)}
Composition of Transition Systems
Q - states
Q0 – initial states, …
A - labels, …
 - transition relation,  Q  A Q
S = S1 || S2
with
 : A1  A2  A
Composition of hybrid automata :
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
Classes of Hybrid Automata
X = {x1, … xn} - variables
(V, E) – control graph
init: V  preds(X)
inv: V  preds(X)
.
flow: V  preds(X  X)
jump: E  preds(X  X’)
event: E  
.
(x  Iflow),
(x = (x,y) I, x’ = (x’,y’),
x’ I’ ,y’=y)
• Singular – rectangular with Iflow a point
• Timed – singular with
Iflow = [1,1]n
• Multirectangular …
• Rectangular init, inv, flow
jump
Timed Automaton
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’)
Init(v):
v = v0 and
X = 0, where v0  V
inv(v): X <= C , where C is rational
.
flow(v): X = 1
jump(e) : A boolean combination of X <= C, X < C
and Y = 0, where Y  X’’
Verification results
Trace Semantics
Q - states – {(v,x) | v  V and inv(v)[X := x]}
Q0 – initial states - {(v,x) Q | init(v)[X := x]}
B - labels -   {}
{ (v,x) –  (v’,x’) | e  E(v,v’), event(e) = , jump(e) [X := x]}
{ (v,x) –   (v,x’) | f(0) = x, f() = x’ , flow(v)[X := f(t), X:= f(t)], t  (0,)}
Trajectory:  = <(a0,q0)…(ai,qi)…>
where q0  Q0 and qi–aiqi+1, i 0
Symbolic Analysis
•
•
•
•
Q - states
Q0 – initial states, …
A - labels, …
a - transition relation, A 

Q Q
Theory: T = {p1, … pn … }, p is a predicate, e.g. pred(X  V)
Meaning of p: [p]  Q
q1  q2
iff
p(q1) = p(q2)
for all p  T
Symbolic Bisimilarity Computation
prea(R’)
R’
R