CIS 4930/6930: Principles of Cyber-Physical Systems Chapter 4: - PowerPoint PPT Presentation

CIS 4930/6930: Principles of Cyber-Physical Systems Chapter 4: Hybrid Systems - Hybrid Automata Hao Zheng Department of Computer Science and Engineering University of South Florida Ref.: An Introduction to Hybrid Automata http://link.springer.com/chapter/10.1007%2F0-8176-4404-0_21 Skip sec. 3.2, 4.2, skim sec. 5. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 1 / 31

Hybrid Automata: Syntax H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 2 / 31

A hybrid automata is defined with (ignoring discrete variables) • L : a finite set of locations. • l 0 ∈ L : the initial location. • X : a finite set of real-valued variables. • A : a finite set of actions. • E : a finite set of edges connecting locations. • Inv : location invariants. • Flow : definition of continuous evolution on ( X ∪ ˙ X ) in locations. • Init : initial values of X ∪ ˙ X . For each e ∈ E , e = ( l 1 , α, Jump , l 2 ) where • α ∈ A is an action, • Jump defines how X ∪ X ′ are updated when e happens. X ′ represents updates to X after e is taken. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 3 / 31

A Running Example H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 4 / 31

A Running Example • When the burner is Off , water temp. x decreses def’ed by x ( t ) = Ie − Kt when x ( t ) > 20. • I : initial water temp.. • K : heat transfer constant of tank. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 5 / 31

A Running Example • When the burner is Off , water temp. x decreses def’ed by x ( t ) = Ie − Kt when x ( t ) > 20. • I : initial water temp.. • K : heat transfer constant of tank. • When x ≤ 20, x stays constant. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 5 / 31

A Running Example • When the burner is Off , water temp. x decreses def’ed by x ( t ) = Ie − Kt when x ( t ) > 20. • I : initial water temp.. • K : heat transfer constant of tank. • When x ≤ 20, x stays constant. • When the burner is On , water temp. x decreses def’ed by x ( t ) = Ie − Kt + h (1 − e − Kt ) when x ( t ) < 100. • h : constant relative to the power of the burner. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 5 / 31

A Running Example • When the burner is Off , water temp. x decreses def’ed by x ( t ) = Ie − Kt when x ( t ) > 20. • I : initial water temp.. • K : heat transfer constant of tank. • When x ≤ 20, x stays constant. • When the burner is On , water temp. x decreses def’ed by x ( t ) = Ie − Kt + h (1 − e − Kt ) when x ( t ) < 100. • h : constant relative to the power of the burner. • When x = 100, x stays 100. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 5 / 31

A Possible Behavior of the Tank H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 6 / 31

Water Tank: Hybrid Automata t 1 t 2 B , x = 100 ∧ x ′ = x x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x Off , x = x ′ = 0 On , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x x = 20 20 ≤ x ≤ 100 H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 7 / 31

Hybrid Automata: Semantics H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 8 / 31

Transitions Let η : X − → R . • A state of a hybrid automata is ( l , η ). • The initial state is ( l 0 , η 0 ). e − → ( l 2 , η 2 ) Discrete transition: ( l 1 , η 1 ) • An edge e = ( l 1 , α, Jump , l 2 ) ∈ E is enabled/executable in a state ( l 1 , η 1 ) if • η 1 | = Jump ( X ), and • there is a matching synchronization action to α . • A new state ( l 2 , η 2 ) after executing e such that η 2 | = Jump ( X ′ ) . H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 9 / 31

Transitions (Cont’d) δ → ( l , η 2 ) , δ ∈ R + Continuous transition: ( l , η 1 ) − There is a differentiable function f : [0 , δ ] − → R m , with the first derivative ˙ → R m , such that f : [0 , δ ] − • f (0) = η 1 , • f ( δ ) = η 2 , = Inv ( l ) and ˙ • For all t ∈ [0 , δ ], f ( t ) | f ( t ) | = Flow ( l ). Intuitively, a hybrid automata can stay in a location by letting time pass by without violating the location invariant, and the valuation of X during that period of time is constrained by the flow condition labeled in that location. H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 10 / 31

Execution Traces e δ • Execution step: − → = − → ∪ − → • Execution trace: ( l 0 , u 0 ) − → ( l 1 , η 1 ) − → ( l 2 , η 2 ) . . . • Reachability: ( i , η ) is reachable if there exists a trace ( l 0 , η 0 ) − → ( l 1 , η 1 ) . . . − → ( l n , η n ) such that l = l n and u = η n . H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 11 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On ( t 4 , x = 20) − → ( t 1 , x = 20) H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On 10 ( t 4 , x = 20) − → ( t 1 , x = 20) − → ( t 1 , x = 88 . 59) H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On 10 2 . 74 ( t 4 , x = 20) − → ( t 1 , x = 20) − → ( t 1 , x = 88 . 59) − − → ( t 1 , x = 100) H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On 10 2 . 74 ( t 4 , x = 20) − → ( t 1 , x = 20) − → ( t 1 , x = 88 . 59) − − → ( t 1 , x = 100) B − → ( t 2 , x = 100) H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On 10 2 . 74 ( t 4 , x = 20) − → ( t 1 , x = 20) − → ( t 1 , x = 88 . 59) − − → ( t 1 , x = 100) B 5 − → ( t 2 , x = 100) − → ( t 2 , x = 100) H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On 10 2 . 74 ( t 4 , x = 20) − → ( t 1 , x = 20) − → ( t 1 , x = 88 . 59) − − → ( t 1 , x = 100) B 5 Off − → ( t 2 , x = 100) − → ( t 2 , x = 100) − − → ( t 3 , x = 100) H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

t 1 B , x = 100 ∧ x ′ = x t 2 x = K ( h − x ) ˙ x = 0 ˙ 20 ≤ x ≤ 100 x = 100 Off , x ′ = x On , x = x ′ Off , x = x ′ On , x ′ = x t 4 t 3 x = − Kx x = 0 ˙ ˙ C , x = 20 ∧ x ′ = x 20 ≤ x ≤ 100 x = 20 On 10 2 . 74 ( t 4 , x = 20) − → ( t 1 , x = 20) − → ( t 1 , x = 88 . 59) − − → ( t 1 , x = 100) B 5 Off − → ( t 2 , x = 100) → ( t 2 , x = 100) − − − → ( t 3 , x = 100) 8 − → ( t 3 , x = 54 . 88) , . . . H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 12 / 31

Composing Hybrid Automata H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 13 / 31

Parallel Composition of Hyrbid Automata Two HAs H 1 = ( L 1 , l 10 , X 1 , A 1 , E 1 , Inv 1 , Flow 1 , Init 1 ) and H 2 = ( L 2 , l 20 , X 2 , A 2 , E 2 , Inv 2 , Flow 2 , Init 2 ) such that L 1 ∩ L 2 = ∅ , their parallel composition, H 1 � H 2 is a HA ( L , l 0 , C , A , E , Inv ) where • L = L 1 × L 2 , • l 0 = ( l 10 , l 20 ); • X = X 1 ∪ X 2 , • A = A 1 ∪ A 2 , • E = { . . . } , defined in the next slide, • Inv ( l 1 , l 2 ) = Inv 1 ( l 1 ) ∧ Inv 2 ( l 2 ) for all ( l 1 , l 2 ) ∈ L , • Flow ( l 1 , l 2 ) = Flow 1 ( l 1 ) ∧ Flow 2 ( l 2 ) for all ( l 1 , l 2 ) ∈ L , • Init = Init 1 ∧ Init 2 . H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 14 / 31

Parallel Composition of Timed Automata E = { ( l 1 , l 2 ) , α, Jump , ( l ′ 2 ) } includes edges defined as follows. 1 , l ′ 1 ) ∈ E 1 2 ) ∈ E 2 ( l 1 , α, Jump 1 , l ′ ( l 2 , α, Jump 2 , l ′ Sync (( l 1 , l 2 ) , α, Jump 1 ∧ Jump 2 , ( l ′ 1 , l ′ 2 )) ∈ E 1 ) ∈ E 1 ∈ A 2 ( l 1 , α, Jump 1 , l ′ α / Async x ∈ X 2 − X 1 x ′ = x , ( l ′ (( l 1 , l 2 ) , α, Jump 1 ∧ � 1 , l 2 )) ∈ E ( l 2 , α, cc 2 , reset 2 , l ′ 2 ) ∈ E 2 α / ∈ A 1 Async x ∈ X 1 − X 2 x ′ = x , ( l 1 , l ′ (( l 1 , l 2 ) , α, Jump 2 ∧ � 2 )) ∈ E H. Zheng (CSE USF) CIS 4930/6930: Principles of CPS 15 / 31