1. Hybrid Systems a lecture over: Tom Henzinger’s The Theory of Hybrid Automata Anders P. Ravn Aalborg University PhD-reading course November 2005

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

3. X = {x 1, … x n } - 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  

4. Q – states, e.g. (v=”Off”,x = 17.5) Q 0 – initial states, Q 0  Q A – labels  – ransition relation, A  Q  Q Labelled Transition Systems a post a (R) = { q’ | q  R and q  q’} pre a (R) = { q | q’  R and q  q’} a a

5. Transition Semantics of HA X = {x 1, … x n } - 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   Q 0 – 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]} { (v,x) –  (v,x’) |   R  0 and f: (0,  )  R n s.t. f is diff. and f(0) = x and f(  ) = x’ and flow(v)[X := f(t), X:= f(t)], t  (0,  ) }.

6. Time Abstract Semantics of HA X = {x 1, … x n } - 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   Q 0 – 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,  )  R n s.t. f is diff. and f(0) = x and f(  ) = x’ and flow(v)[X := f(t), X:= f(t)], t  (0,  )}.

7. Q - states, {(v,x) | v  V and inv(v)[X := x]} Q 0 – initial states, … A - labels, …  - transition relation, A  Q  Q Trace Semantics a Trajectory:  = where q 0  Q 0 and q i –a i  q i+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

8. Q - states Q 0 – initial states, … A - labels, …  - transition relation, A  Q  Q Composition of Transition Systems a S = S1 || S2 with  : A1  A2  A Q = Q1  Q2 Q 0 = Q1 0  Q2 0 (q1,q2) –a  (q1’,q2’) iff (qi –ai  qi’, i=1,2 and a = a1  a2 Remark p 7

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

10. Classes of Hybrid Automata X = {x 1, … x n } - 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  I flow ), jump (x = x,y  I, x’  I’,y’=y) Singular – rectangular with I flow a point Timed – singular with I flow = [1,1] n Multirectangular … Triangular … Stopwatch …. Verification results pp. 11-12

11. Symbolic Analysis Q - states Q 0 – initial states, … A - labels, …  - transition relation, A  Q  Q a Theory: T = {p 1, … p n … }, p is a predicate, e.g. pred(X  V) Meaning of p: [p]  Q q 1  q 2 iff p(q 1 ) = r(q 2 ) for all p, r  T

12. Symbolic Bisimilarity Computation R R’ pre a

13. Mu-calculus // fixpoint computation

14. CTL

15. 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