Hybrid automata Rafael Wisniewski Automation and Control, Dept. of Electronic Systems Aalborg University, Denmark Hybrid Systems October 9th 2009

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 …) 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 = {x 1, … x n } - 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´) event: E   Hybrid Automaton - Syntax. x´ = x-1  .

Q – states, e.g. (v=”Off”,x = 17.5) Q 0 – initial states, Q 0  Q A – labels  – transition relation,  Q  A  Q Labelled Transition System

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]}. 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, 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,  ) }. x´ = x  

Q - states, {(v,x) | v  V and inv(v)[X := x]} Q 0 – initial states, … A - labels, …  - transition relation,  Q  A  Q Tree Semantics Computation tree:  = q 00 a q 10 q q 1n 1 … q 200 q 201 q 210 q 211 q 13

Q - states, {(v,x) | v  V and inv(v)[X := x]} Q 0 – initial states, … A - labels   R  0  - transition relation,  Q  A  Q Trace Semantics 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 (ompare with the two tank example)

Time Abstract Semantics 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]}. 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,  )}.

Q - states Q 0 – initial states, … A - labels, …  - transition relation,  Q  A  Q Composition of Transition Systems 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 is defined Remark p 7 Composition of hybrid automata  :

Classes of Hybrid Automata X = {x 1, … x n } - 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  . Rectangular init, inv, flow (x  I flow ), jump (x = (x,y)  I, x’ = (x’,y’), x’  I’,y’=y) Singular – rectangular with I flow a point Timed – singular with I flow = [1,1] n Multirectangular ….

Timed Automaton 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’). Init(v): v = v 0 and X = 0, where v 0  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 Trajectory:  = where q 0  Q 0 and q i –a i  q i+1, i  0 Q - states – {(v,x) | v  V and inv(v)[X := x]} Q 0 – 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,  )}

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 ) = p(q 2 ) for all p  T

Symbolic Bisimilarity Computation R R’ pre a (R’)