Hybrid Systems a lecture over: Tom Henzinger’s The Theory of Hybrid Automata

Hybrid Systemsa lecture over:Tom Henzinger’sThe Theory of Hybrid Automata Anders P. Ravn Aalborg University PhD-reading course November 2005

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

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 

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

 x’ = x-1 { (v,x) –  (v,x’) |  R0and 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,)} . .  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]}

 x’ = x-1 { (v,x) –  (v,x’) |  R0and 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,)} . .  Time Abstract 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]} B - labels -   {} - finite ! { (v,x) –  (v’,x’) | e E(v,v’)and event(e) =  and jump(e) [X := x]}

Trace Semantics Q - states, {(v,x) | v  V and inv(v)[X := x]} Q0– initial states, … A - labels, …  - transition relation, A  QQ a 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

Composition of Transition Systems Q - states Q0– initial states, … A - labels, …  - transition relation, A  QQ a 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 Remark p 7

Verification Tasks • Reachability of (v,x) – finitary, time-abstract trace inclusion • Emptiness – time-abstract trace inclusion • Trace (finitary) inclusion • Time-abstract (finitary) trace inclusion

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  Classes of Hybrid Automata . • 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

Symbolic Analysis Q - states Q0 – initial states, … A - labels, …  - transition relation, A  QQ a Theory: T = {p1, … pn … }, p is a predicate, e.g. pred(X  V) Meaning of p: [p]  Q q1 q2 iff p(q1) = r(q2) for all p, r  T

Symbolic Bisimilarity Computation prea R’ R

Mu-calculus // fixpoint computation

CTL

Further Work • Check the theorems and remarks • Experiment with tools • Investigate links with equivalences generated by Rafael’s homotopy (di-paths) • Compositionality, remarks on p. 7, 10, 17 – compositional model checking, abstraction-refinement • Build your own HA Application

Hybrid Systems a lecture over: Tom Henzinger’s The Theory of Hybrid Automata