Cooperative systems Centralized symbolic control Compositional approach Safety control with performance guarantees of cooperative systems using compositional abstractions Pierre-Jean Meyer Antoine Girard Emmanuel Witrant Universit´ e Grenoble-Alpes ADHS’15, October 16 th 2015 October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 1 / 25
Cooperative systems Centralized symbolic control Compositional approach Outline Cooperative control system 1 Centralized symbolic control 2 Compositional approach 3 October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 2 / 25
Cooperative systems Centralized symbolic control Compositional approach System description Nonlinear control system: x = f ( x , u , w ) ˙ x : state u : control input Trajectories: w : disturbance input x = Φ( · , x 0 , u , w ) x , u , w : time functions x x = Φ( · , x 0 , u , w ) x 0 x ( t ) f ( x ( t ) , u ( t ) , w ( t )) t October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 3 / 25
Cooperative systems Centralized symbolic control Compositional approach Cooperative system Definition (Cooperativeness) The system is cooperative if Φ preserves the componentwise inequality: u ≥ u ′ , w ≥ w ′ , x 0 ≥ x ′ 0 ⇒ ∀ t ≥ 0 , Φ( t , x , u , w ) ≥ Φ( t , x ′ , u ′ , w ′ ) u, w x Φ( · , x 0 , u , w ) w x 0 w ′ ⇒ x ′ Φ( · , x ′ 0 , u ′ , w ′ ) u 0 u ′ t t October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 4 / 25
Cooperative systems Centralized symbolic control Compositional approach Bounded inputs Control and disturbance inputs bounded in intervals: ∀ t ≥ 0 , u ( t ) ∈ [ u , u ] , w ( t ) ∈ [ w , w ] = ⇒ ∀ t ≥ 0 , Φ( t , x 0 , u , w ) ∈ [Φ( t , x 0 , u , w ) , Φ( t , x 0 , u , w )] x Φ( · , x 0 , u, w ) Φ( · , x 0 , u , w ) x 0 Φ( · , x 0 , u, w ) t October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 5 / 25
Cooperative systems Centralized symbolic control Compositional approach Outline Cooperative control system 1 Centralized symbolic control 2 Compositional approach 3 October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 6 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction-based synthesis Continuous state Uncontrolled x + = f ( x, u, w ) Synthesis w Controlled x + = f ( x, u, w ) x u Controller October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 7 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction-based synthesis Continuous state Discrete state u 1 Uncontrolled u 1 x + = f ( x, u, w ) u 2 u 2 Abstraction Controlled October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 7 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction-based synthesis Continuous state Discrete state u 1 Uncontrolled u 1 x + = f ( x, u, w ) u 2 u 2 Abstraction Synthesis u 1 Controlled u 1 October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 7 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction-based synthesis Continuous state Discrete state u 1 Uncontrolled u 1 x + = f ( x, u, w ) u 2 u 2 Abstraction Synthesis w u 1 Refining Controlled x + = f ( x, u, w ) u 1 x u Controller October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 7 / 25
Cooperative systems Centralized symbolic control Compositional approach Transition systems S = ( X , U , − → ) u Set of states X Set of inputs U x x ′ u ′ Transition relation − → u 1 u 2 u 3 Trajectories: x 1 − → x 2 − → x 3 − → . . . October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 8 / 25
Cooperative systems Centralized symbolic control Compositional approach Transition systems S = ( X , U , − → ) u Set of states X Set of inputs U x x ′ u ′ Transition relation − → u 1 u 2 u 3 Trajectories: x 1 − → x 2 − → x 3 − → . . . Sampled dynamics (sampling τ ) X = R n U = [ u , u ] u → x ′ ⇐ ⇒ ∃ w : [0 , τ ] → [ w , w ] | x ′ = Φ( τ, x , u , w ) x − Safety specification in [ x , x ] ⊆ R n October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 8 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction Discretization of the control space [ u , u ] Partition P of the interval [ x , x ] into symbols x s s x s October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 9 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction Discretization of the control space [ u , u ] Partition P of the interval [ x , x ] into symbols Over-approximation of the reachable set (cooperativeness) x Φ( τ, s, u, w ) s s Φ( τ, s, u, w ) x s October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 9 / 25
Cooperative systems Centralized symbolic control Compositional approach Abstraction Discretization of the control space [ u , u ] Partition P of the interval [ x , x ] into symbols Over-approximation of the reachable set (cooperativeness) Intersection with the partition u s u Obtain a finite abstraction S a = ( X a , U a , − → a ) October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 9 / 25
Cooperative systems Centralized symbolic control Compositional approach Alternating simulation Definition (Alternating simulation relation) H : X → X a is an alternating simulation relation from S a to S if: u → x ′ in S = u a H ( x ′ ) in S a ∀ u a ∈ U a , ∃ u ∈ U | x − ⇒ H ( x ) − → a October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 10 / 25
Cooperative systems Centralized symbolic control Compositional approach Alternating simulation Definition (Alternating simulation relation) H : X → X a is an alternating simulation relation from S a to S if: u → x ′ in S = u a H ( x ′ ) in S a ∀ u a ∈ U a , ∃ u ∈ U | x − ⇒ H ( x ) − → a Proposition The map H : X → X a defined by H ( x ) = s ⇐ ⇒ x ∈ s is an alternating simulation relation from S a to S: → x ′ in S = u a u a H ( x ′ ) in S a ∀ u a ∈ U a ⊆ U | x − ⇒ H ( x ) − → a October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 10 / 25
Cooperative systems Centralized symbolic control Compositional approach Safety synthesis Specification: safety of S a in P (the partition of the interval [ x , x ]) s ′ , s ′ ∈ Z } u F P ( Z ) = { s ∈ Z ∩ P | ∃ u , ∀ s − → a October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 11 / 25
Cooperative systems Centralized symbolic control Compositional approach Safety synthesis Specification: safety of S a in P (the partition of the interval [ x , x ]) s ′ , s ′ ∈ Z } u F P ( Z ) = { s ∈ Z ∩ P | ∃ u , ∀ s − → a Fixed-point Z a of F P reached in finite time Z a is the maximal safe set for S a , associated with the safety controller: s ′ , s ′ ∈ Z a } u C a ( s ) = { u | ∀ s − → a Theorem C a is a safety controller for S in Z a . October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 11 / 25
Cooperative systems Centralized symbolic control Compositional approach Performance criterion Minimize on a trajectory ( x 0 , u 0 , x 1 , u 1 , . . . ) of S : + ∞ � λ k g ( x k , u k ) k =0 with a cost function g and a discount factor λ ∈ (0 , 1) October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 12 / 25
Cooperative systems Centralized symbolic control Compositional approach Performance criterion Minimize on a trajectory ( x 0 , u 0 , x 1 , u 1 , . . . ) of S : + ∞ � λ k g ( x k , u k ) k =0 with a cost function g and a discount factor λ ∈ (0 , 1) Cost function on S a : g a ( s , u ) = max x ∈ s g ( x , u ) Focus the optimization on a finite horizon of N sampling periods Accurate approximation if λ N +1 ≪ 1 October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 12 / 25
Cooperative systems Centralized symbolic control Compositional approach Optimization Dynamic programming algorithm: J N a ( s ) = u ∈ C a ( s ) g a ( s , u ) min J k a s ′ J k +1 ( s ′ ) , ∀ k < N a ( s ) = min g a ( s , u ) + λ max a u u ∈ C a ( s ) − → s N � J 0 λ k g a ( s k , u k ) a ( s ) is the worst-case minimization of k =0 October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 13 / 25
Cooperative systems Centralized symbolic control Compositional approach Optimization Dynamic programming algorithm: J N a ( s ) = u ∈ C a ( s ) g a ( s , u ) min J k a s ′ J k +1 ( s ′ ) , ∀ k < N a ( s ) = min g a ( s , u ) + λ max a u u ∈ C a ( s ) − → s N � J 0 λ k g a ( s k , u k ) a ( s ) is the worst-case minimization of k =0 Receding horizon controller: C ∗ a s ′ J 1 a ( s ′ ) a ( s ) = arg min g a ( s , u ) + λ max u u ∈ C a ( s ) − → s October 16 th 2015 Pierre-Jean Meyer (Grenoble) Compositional abstractions 13 / 25
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