A coalgebraic approach to supervisory control of partially observed - PowerPoint PPT Presentation

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion A coalgebraic approach to supervisory control of partially observed Mealy automata Jun Kohjina 1 , Toshimitsu Ushio 1 , Yoshiki Kinoshita 2 1 Graduate School of Engineering Science, Osaka University, Japan 2 National Institute of Advanced Industrial Science and Technology, Japan CALCO 2011

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Outline 1 Introduction 2 Supervisory control (not using coalgebra) 3 Coalgebraic formulation 4 Solution to the problem 5 Conclusion

� � Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Introduction Control problem spec , design a Given a plant and a such that controller control spec . plant satisfies controller observe

� � Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Introduction Control problem spec , design a Given a plant and a such that controller control spec . plant satisfies controller observe Our interest When does a controller exist? How do we design the controller?

� � Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Supervisory control Control theory for discrete event systems [Ramadge and Wonham 1987] communication networks, manufacturing systems, traffic systems disabled event set spec supervisor plant generates observe the trace of plant plant deterministic partial automaton ( X, A, δ, x 0 ) spec non-empty prefix closed language over A supervisor function from a trace to a disabled event set S : A ∗ → P ( A )

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Settings Uncontrollable event [Ramadge and Wonham 1987] event set A = A c + A uc , supervisor S : A ∗ → P ( A c ) A c :controllable event set A uc :uncontrollable event set (not disabled by a supervisor) Partial observation [Ramadge and Wonham 1988, Cieslak et.al 1988] event set A = A o + A uo , supervisor S : ( A o ) ∗ → P ( A c ) A o :observable event set A uo :unobservable event set (not observed by a supervisor) Partially observed Mealy automata [Takai and Ushio 2009] plant modeled by a Mealy automaton supervisor S : ( B o ) ∗ → P ( A c ) input event: A = A c + A u output event: B = B o + B u

� � Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Our approach disabled event set spec supervisor plant generates observe the trace of plant M m → (1 + B × M ) A plant − partial Mealy automaton l spec → (1 + L ) A L − partial automaton ⟨ o,t ⟩ → P ( A c ) × S B o supervisor S − − Moore automaton

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Plant, Spec and Supervisor, coalgebraically Plant : M m → (1 + B × M ) A − � { } M is prefix- and length-preserving. M : A ∗ ⇀ B ∗ � M = � dom( M ) ̸ = ∅ . � m ( M )( a ) = if a ∈ dom( M ) then ⟨ M ( a ) , M a ⟩ else ⊥ . where M a ( w ) = tail ◦ M ( aw ) . l → (1 + L ) A Spec : L − L = { L ⊆ A ∗ | L is prefix-closed and nonempty. } l ( L )( a ) = if a ∈ L then L a else ⊥ . where L a := { w ∈ A ∗ | aw ∈ L } . ⟨ o,t ⟩ → P ( A c ) × S B o Supervisor : S − − S = { S : ( B o ) ∗ → P ( A c ) . } o ( S ) = S ( ε ) , t ( S )( b ) = S b , where S b ( w ) = S ( bw ) .

� � � Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Coinductive definition of supervisory composition ∃ ! / S × M L S = { S : ( B o ) ∗ → P ( A c ) } M = { M : A ∗ ⇀ B ∗ | · · · } spv l final L = { L ⊆ A ∗ | · · · } (id 1 + / ) A � (1 + L ) A (1 + S × M ) A spv ⟨ S, M ⟩ ( a ) =  a | b ⟨ S b , M a ⟩ if M − − → M a ∧ a / ∈ o ( S ) ∧ b ∈ B o ,    a | b ⟨ S, M a ⟩ if M − − → M a ∧ a / ∈ o ( S ) ∧ b ∈ B u ,   ⊥ otherwise .  / : S × M → L is the supervisory composition. S/M represents a language generated by the controlled plant.

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Formulation of supervisory control problem Supervisory control problem Given a plant M ∈ M and a specification K ∈ L , find a supervisor S ∈ S satisfying S/M = K. / : S × M → L S = { S : ( B o ) ∗ → P ( A c ) . } { � } M is prefix- and length-preserving M : A ∗ ⇀ B ∗ � M = � dom( M ) ̸ = ∅ . � L = { L ⊆ A ∗ | L is prefix-closed and non-empty . }

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Comparison Supervised product [Komenda & van Schuppen 2005] ( M/N ) a = a a  M a /N a if M − → ∧ N − → ,   (∪ ) if M ̸ a → ∧∃ M ′ ∈ DK : M ′ ≈ M s.t. M ′ a a  ⟨ M ′ ,M ⟩∈ Aux M ′ /N a − − → ∧ N − → ∧ a ∈ A c ∪ A o ,   a   if ( ∀ M ′ ∈ DK : M ′ ≈ M ) M ′ ̸ a a 0 /N a − → ∧ N − → ∧ a ∈ ( A uc ∩ A o ) ,  if M ̸ a a  M/N a − → ∧ N − → ∧ a ∈ A uc ∩ A uo ,      ∅ otherwise . Our work spv ⟨ S, M ⟩ ( a ) =  a | b ⟨ S b , M a ⟩ if M − − → M a ∧ a / ∈ o ( S ) ∧ b ∈ B o ,    a | b ⟨ S, M a ⟩ if M − − → M a ∧ a / ∈ o ( S ) ∧ b ∈ B u ,   ⊥ otherwise .  S ( w ) = A c \ { a ∈ A c | ∃ u ∈ A ∗ : ( K 0 ua − → ) ∧ ( P ◦ M 0 ( u ) = w ) }

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Partial bisimulation relation Definition Let ( X, ξ ) and ( Y, η ) be (1 + − ) A -coalgebras. A partial bisimulation relation is a binary relation R ⊆ X × Y satisfying (1), (2), and (3). (1)similarity ∀ a ∈ A , ∀ x, x ′ ∈ X , y ∈ Y , ∃ y ′ ∈ Y , → x ′ = → y ′ ∧ x ′ R y ′ . x R y ∧ x a ⇒ y a − − (2)controllability ∀ a ∈ A u , ∀ x ∈ X , ∀ y, y ′ ∈ Y , ∃ x ′ ∈ X , → y ′ = → x ′ ∧ x ′ R y ′ . x R y ∧ y a ⇒ x a − − (3)observability ∀ a ∈ A c , ∀ x ∈ X , ∀ y, y ′ ∈ Y , ∃ x ′ ∈ X , → y ′ ∧ ( ∃ q ∈ X, ( x ≈ q ) ∧ ( q a x R y ∧ y a − − → )) ⇒ x a → x ′ ∧ x ′ R y ′ . = − w ′ { � � ∃ w, w ′ ∈ A ∗ , x 0 } w ⟨ x, x ′ ⟩ → x ′ , P ◦ M ( w ) = P ◦ M ( w ′ ) . ≈ = − → x, x 0 − �

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion When does a supervisor exist? Theorem Given a plant M 0 ∈ M and a specification K 0 ∈ L , the following two conditions are equivalent. (1) ∃ S ∈ S , S/M 0 = K 0 (2) There exists a partial bisimuration relation R ⊆ L × L such that K 0 R dom( M 0 ) .

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion When does a supervisor exist? Theorem Given a plant M 0 ∈ M and a specification K 0 ∈ L , the following two conditions are equivalent. (1) ∃ S ∈ S , S/M 0 = K 0 (2) There exists a partial bisimuration relation R ⊆ L × L such that K 0 R dom( M 0 ) . (2) = ⇒ (1) S ( w ) = A c \ { a ∈ A c | ∃ u ∈ A ∗ : ( K 0 ua − → ) ∧ ( P ◦ M 0 ( u ) = w ) } is a desired supervisor.

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Modified normality Problem When no supervisor satisfies the specification, find the largest sublanguage of the specification. In general, there doesn’t exist the largest controllable and observable sublanguage. (not closed under the arbitrary union) Therefore, we introduce a notion of modified normality . (closed under the arbitrary union)

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Modified normality Problem When no supervisor satisfies the specification, find the largest sublanguage of the specification. In general, there doesn’t exist the largest controllable and observable sublanguage. (not closed under the arbitrary union) Therefore, we introduce a notion of modified normality . (closed under the arbitrary union) Compute the largest controllable and modified normal sublanguage of the specification.

Introduction Supervisory control Coalgebraic formulation Solution to the problem Conclusion Controllable and modified normal relation Definition Let ( X, ξ ) and ( Y, η ) be (1 + − ) A -coalgebras. A controllable and modified normal relation is a binary relation R ⊆ X × Y satisfying (1), (2), and (3). (1)similarity ∀ a ∈ A , ∀ x, x ′ ∈ X , ∀ y ∈ Y , ∃ y ′ ∈ Y , x R y ∧ x a → x ′ = ⇒ y a → y ′ ∧ x ′ R y ′ − − (2)controllability ∀ a ∈ A u , ∀ x ∈ X , ∀ y, y ′ ∈ Y , ∃ x ′ ∈ X , → x ′ ∧ x ′ R y ′ x R y ∧ y a → y ′ = ⇒ x a − − (3)modified normality ∀ a ∈ A c , ∀ x ∈ X , ∀ y, y ′ ∈ Y , ∃ x ′ ∈ X , → y ′ ∧ ( ∃ q ∈ X, ∃ a ′ ∈ A, ( x ≈ q ) ∧ ( q a ′ x R y ∧ y a − − → )) . ⇒ x a → x ′ ∧ x ′ R y ′ = −