A Uniform Theory of Hybrid Automata Renato Neves and Lu s S. - PowerPoint PPT Presentation

A Uniform Theory of Hybrid Automata Renato Neves and Lu´ ıs S. Barbosa November 6, 2018 INESC TEC (HASLab) & University of Minho 1

Introduction

Hybrid Systems Computational devices that interact with their physical environment 2

Preliminaries Hybrid Automata The standard formalism for designing hybrid systems 3

Preliminaries Hybrid Automata The standard formalism for designing hybrid systems They are classical automata enriched with machinery to specify continuous evolutions and discrete resets [Henzinger, 1996] 3

� � Preliminaries Hybrid Automata The standard formalism for designing hybrid systems They are classical automata enriched with machinery to specify continuous evolutions and discrete resets [Henzinger, 1996] Example Water level regulator; it raises the water level ( l ) periodically t ≥ c ˙ ˙ t :=0 l = 2 l = 0 ˙ ˙ t = 1 t = 1 t ≤ c t ≤ c t ≥ c t :=0 3

Motivation The notion of hybrid automata has several variants, e.g. • Deterministic [Henzinger, 1996] • Non-deterministic [Henzinger, 1996] • Probabilistic [Sproston, 2000] • Reactive [Liu et al., 1999] • Weighted [Bouyer, 2006] 4

Motivation The notion of hybrid automata has several variants, e.g. • Deterministic [Henzinger, 1996] • Non-deterministic [Henzinger, 1996] • Probabilistic [Sproston, 2000] • Reactive [Liu et al., 1999] • Weighted [Bouyer, 2006] No uniform framework for hybrid automata 4

Coalgebras Coalgebras can help us solving this issue 5

Coalgebras Coalgebras can help us solving this issue For this particular case we will see that they provide, 1. generic semantics, 5

Coalgebras Coalgebras can help us solving this issue For this particular case we will see that they provide, 1. generic semantics, 2. generic notions of bisimulation 5

Coalgebras Coalgebras can help us solving this issue For this particular case we will see that they provide, 1. generic semantics, 2. generic notions of bisimulation 3. and observational behaviour, 5

Coalgebras Coalgebras can help us solving this issue For this particular case we will see that they provide, 1. generic semantics, 2. generic notions of bisimulation 3. and observational behaviour, 4. and ‘regular expression’-like languages. 5

Hybrid Automata as Coalgebras

Coalgebras Definition Given a functor F : C → C, an F -coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type 6

Coalgebras Definition Given a functor F : C → C, an F -coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type Examples 1. (Kripke Frames) Powerset functor P : Set → Set 6

Coalgebras Definition Given a functor F : C → C, an F -coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type Examples 1. (Kripke Frames) Powerset functor P : Set → Set 2. (Deterministic Automata) ( − ) Σ × 2 : Set → Set 6

Coalgebras Definition Given a functor F : C → C, an F -coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type Examples 1. (Kripke Frames) Powerset functor P : Set → Set 2. (Deterministic Automata) ( − ) Σ × 2 : Set → Set 3. (Markov Chain) Distribution functor D : Set → Set 6

Hybrid Automata Definition ([Henzinger, 1996]) A hybrid automaton is a tuple ( M , E , X , dyn , inv , asg , grd ) where • M is a finite set of modes, E is a transition relation E ⊆ M × M , and X is a finite set of real-valued variables { x 1 , . . . , x n } . • dyn is a function that associates to each mode a predicate over the variables in X ∪ ˙ X , where ˙ X = { ˙ x 1 , . . . , ˙ x n } represents the first derivatives of the variables in X . • inv is a function that associates to each mode a predicate over the variables in X . • asg is a function that given an edge returns an assignment over X . The function grd associates each edge with a guard. 7

� Hybrid Automata Example The bouncing ball p = v ˙ p = 0 ∧ v ≤ 0, v = g ˙ v := v × − 0 . 5 p ≥ 0 8

� Hybrid Automata Example The bouncing ball p = v ˙ p = 0 ∧ v ≤ 0, v = g ˙ v := v × − 0 . 5 p ≥ 0 Its observable behaviour consists of continuous evolutions intercalated with resets 8

� � Hybrid Automata Example Water level regulator t ≥ c ˙ ˙ t :=0 l = 2 l = 0 ˙ ˙ t = 1 t = 1 t ≤ c t ≤ c t ≥ c t :=0 9

� � Hybrid Automata Example Water level regulator t ≥ c ˙ ˙ t :=0 l = 2 l = 0 ˙ ˙ t = 1 t = 1 t ≤ c t ≤ c t ≥ c t :=0 Its observable behaviour also consists of continuous evolutions intercalated with resets 9

A Surprisingly Useful Remark Hybrid automata are classical automata but with decorated states and edges. 10

A Surprisingly Useful Remark Hybrid automata are classical automata but with decorated states and edges. M → P ( M × Asg × Grd ) × DifEq × StInv 10

A Surprisingly Useful Remark Hybrid automata are classical automata but with decorated states and edges. M → P ( M × Asg × Grd ) × DifEq × StInv This immediately gives, • a uniform notion of hybrid automata, • bisimulation and languages 10

A Surprisingly Useful Remark Hybrid automata are classical automata but with decorated states and edges. M → P ( M × Asg × Grd ) × DifEq × StInv This immediately gives, • a uniform notion of hybrid automata, • bisimulation and languages We can now start studying hybrid automata in a uniform way 10

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv 11

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv Id ⇒ Deterministic hybrid automata 11

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata 11

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata 11

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata PD ⇒ Probabilistic hybrid automata 11

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata PD ⇒ Probabilistic hybrid automata W ⇒ Weighted hybrid automata 11

A Zoo of Hybrid Automata M → F ( M × Asg × Grd ) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata PD ⇒ Probabilistic hybrid automata W ⇒ Weighted hybrid automata We can additionally consider an input dimension 11

Semantics of Hybrid Automata

Semantics of Hybrid Automata We will show how to build a ‘semantics’ functor � − � : HybAt( F ) → Category of coalgebras 12

Semantics of Hybrid Automata We will show how to build a ‘semantics’ functor � − � : HybAt( F ) → Category of coalgebras Reminder The observable behaviour of hybrid automata consists of continuous evolutions intercalated with resets 12

Semantics of Hybrid Automata Notation Let X be a topological space. The set HX denotes � X [0 , r ] r ∈ [0 , ∞ ) the set of all continuous trajectories over intervals [0 , r ]. 13

Semantics of Hybrid Automata Notation Let X be a topological space. The set HX denotes � X [0 , r ] r ∈ [0 , ∞ ) the set of all continuous trajectories over intervals [0 , r ]. Assumption The function dyn only outputs differential equations with exactly one solution. This induces a function flow : M × R n × [0 , ∞ ) → R n 13

Semantics of Hybrid Automata Assumption (for simplicity) As soon as an edge is enabled the current state must switch Let us omit state invariants; they complicate the theory and can be added straightfowardly later on 14

Semantics of Hybrid Automata Assumption (for simplicity) As soon as an edge is enabled the current state must switch Let us omit state invariants; they complicate the theory and can be added straightfowardly later on Semantics M × R n → F ( M × Asg × Grd ) × DifEq ⇒ M × R n → F ( M × Asg × Grd ) × ( R n ) [0 , ∞ ) ⇒ M × R n → F � M × Asg × Grd × ( R n ) [0 , ∞ ) � ⇒ M × R n → F ( M × Asg × ( H ( R n ) + 1)) ⇒ M × R n → F ( M × Asg × H ( R n ) + M × Asg × 1) ⇒ M × R n → F ( M × R n × H ( R n ) + 1) 14

Semantics of Hybrid Automata Assumption (for simplicity) As soon as an edge is enabled the current state must switch Let us omit state invariants; they complicate the theory and can be added straightfowardly later on Semantics M × R n → F ( M × Asg × Grd ) × DifEq ⇒ M × R n → F ( M × Asg × Grd ) × ( R n ) [0 , ∞ ) ⇒ M × R n → F � M × Asg × Grd × ( R n ) [0 , ∞ ) � ⇒ M × R n → F ( M × Asg × ( H ( R n ) + 1)) ⇒ M × R n → F ( M × Asg × H ( R n ) + M × Asg × 1) ⇒ M × R n → F ( M × R n × H ( R n ) + 1) We obtain a coalgebra for F ( − × H ( R n ) + 1) 14

Semantics of Hybrid Automata The previous calculation determines a ‘semantics’ functor � − � : HybAt( F ) → CoAlg ( F ( − × H ( R n ) + 1)) 15