Monotonicity Invariance Symbolic Compositional Experiment Invariance and symbolic control of cooperative systems for temperature regulation in intelligent buildings Pierre-Jean Meyer Universit´ e Grenoble-Alpes PhD Defense, September 24 th 2015 September 24 th 2015 Pierre-Jean Meyer PhD Defense 1 / 45
Monotonicity Invariance Symbolic Compositional Experiment Motivations Develop new control methods for intelligent buildings Focus on temperature control in a small-scale experimental building September 24 th 2015 Pierre-Jean Meyer PhD Defense 2 / 45
Monotonicity Invariance Symbolic Compositional Experiment Outline Monotone control system 1 Robust controlled invariance 2 Symbolic control 3 Compositional approach 4 Control in intelligent buildings 5 September 24 th 2015 Pierre-Jean Meyer PhD Defense 3 / 45
Monotonicity Invariance Symbolic Compositional Experiment 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 September 24 th 2015 Pierre-Jean Meyer PhD Defense 4 / 45
Monotonicity Invariance Symbolic Compositional Experiment Monotone system Definition (Monotonicity) The system is monotone 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 September 24 th 2015 Pierre-Jean Meyer PhD Defense 5 / 45
Monotonicity Invariance Symbolic Compositional Experiment 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 September 24 th 2015 Pierre-Jean Meyer PhD Defense 6 / 45
Monotonicity Invariance Symbolic Compositional Experiment Characterization Proposition (Kamke-M¨ uller) The system ˙ x = f ( x , u , w ) is monotone if and only if the following implication holds for all i: u ≥ u ′ , w ≥ w ′ , x ≥ x ′ , x i = x ′ i ⇒ f i ( x , u , w ) ≥ f i ( x ′ , u ′ , w ′ ) Proposition (Partial derivatives) The system ˙ x = f ( x , u , w ) with continuously differentiable vector field f is monotone if and only if: ∀ i , j � = i , k , l , ∂ f i ∂ f i ≥ 0 , ∂ f i ≥ 0 , ≥ 0 ∂ x j ∂ u k ∂ w l September 24 th 2015 Pierre-Jean Meyer PhD Defense 7 / 45
Monotonicity Invariance Symbolic Compositional Experiment Outline Monotone control system 1 Robust controlled invariance 2 Symbolic control 3 Compositional approach 4 Control in intelligent buildings 5 September 24 th 2015 Pierre-Jean Meyer PhD Defense 8 / 45
Monotonicity Invariance Symbolic Compositional Experiment Definitions Definition (Robust Controlled Invariance) A set S is a robust controlled invariant if there exists a controller such that the closed-loop system stays in S for any initial state and disturbance: ∃ u : S → [ u , u ] | ∀ x 0 ∈ S , ∀ w ∈ [ w , w ] , ∀ t ≥ 0 , Φ u ( t , x 0 , w ) ∈ S September 24 th 2015 Pierre-Jean Meyer PhD Defense 9 / 45
Monotonicity Invariance Symbolic Compositional Experiment Definitions Definition (Robust Controlled Invariance) A set S is a robust controlled invariant if there exists a controller such that the closed-loop system stays in S for any initial state and disturbance: ∃ u : S → [ u , u ] | ∀ x 0 ∈ S , ∀ w ∈ [ w , w ] , ∀ t ≥ 0 , Φ u ( t , x 0 , w ) ∈ S Definition (Local control) Each control input affects a single state variable: ∀ k , ∃ ! i | ∂ f i � = 0 ∂ u k September 24 th 2015 Pierre-Jean Meyer PhD Defense 9 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust controlled invariance Theorem (Meyer, Girard, Witrant, CDC 2013) With the monotonicity and local control properties, the interval [ x , x ] ⊆ R n is robust controlled invariant if and only if � f ( x , u , w ) ≤ 0 f ( x , u , w ) ≥ 0 x f ( x, u, w ) f ( x, u, w ) x September 24 th 2015 Pierre-Jean Meyer PhD Defense 10 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust controlled invariance Theorem (Meyer, Girard, Witrant, CDC 2013) With the monotonicity and local control properties, the interval [ x , x ] ⊆ R n is robust controlled invariant if and only if � f ( x , u , w ) ≤ 0 f ( x , u , w ) ≥ 0 x Sides of [ x , x ] f ( x, u, w ) x ≤ x , x 1 = x 1 w ≤ w f ( x, u, w ) x Kamke-M¨ uller condition: f ( x, u, w ) f 1 ( x , u , w ) ≤ f 1 ( x , u , w ) ≤ 0 x September 24 th 2015 Pierre-Jean Meyer PhD Defense 10 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust controlled invariance Theorem (Meyer, Girard, Witrant, CDC 2013) With the monotonicity and local control properties, the interval [ x , x ] ⊆ R n is robust controlled invariant if and only if � f ( x , u , w ) ≤ 0 f ( x , u , w ) ≥ 0 x Other vertices of [ x , x ] f ( x, u, w ) f 1 ( x , u , w ) ≤ f 1 ( x , u , w ) ≤ 0 f ( x, u, w ) f 2 ( x , u , w ) ≥ f 2 ( x , u , w ) ≥ 0 f ( x, u, w ) Choice of u ? x Use local control property x September 24 th 2015 Pierre-Jean Meyer PhD Defense 10 / 45
Monotonicity Invariance Symbolic Compositional Experiment Choice of the interval x 2 x ∈ R 2 x 2 conditions on x f 1 ( x , u , w ) ≤ 0 f 2 ( x , u , w ) ≤ 0 2 conditions on x f 1 ( x , u , w ) ≥ 0 x f 2 ( x , u , w ) ≥ 0 x 1 September 24 th 2015 Pierre-Jean Meyer PhD Defense 11 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust set stabilization Definition (Stabilizing controller) A controller u : S 0 → [ u , u ] is a stabilizing controller from S 0 to S if ∀ x 0 ∈ S 0 , ∀ w ∈ [ w , w ] , ∃ T ≥ 0 | ∀ t ≥ T , Φ u ( t , x 0 , w ) ∈ S September 24 th 2015 Pierre-Jean Meyer PhD Defense 12 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust set stabilization Definition (Stabilizing controller) A controller u : S 0 → [ u , u ] is a stabilizing controller from S 0 to S if ∀ x 0 ∈ S 0 , ∀ w ∈ [ w , w ] , ∃ T ≥ 0 | ∀ t ≥ T , Φ u ( t , x 0 , w ) ∈ S Let S 0 = [ x 0 , x 0 ] and S = [ x , x ] ⊆ [ x 0 , x 0 ] Assumption ∃ X , X : [0 , 1] → R n , respectively strictly decreasing and increasing , such that [ X (1) , X (1)] = [ x 0 , x 0 ] , [ X (0) , X (0)] = [ x , x ] and satisfying f ( X ( λ ) , u , w ) > 0 , f ( X ( λ ) , u , w ) < 0 , ∀ λ ∈ [0 , 1] September 24 th 2015 Pierre-Jean Meyer PhD Defense 12 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust set stabilization f ( X ( λ ) , u , w ) > 0 , f ( X ( λ ) , u , w ) < 0 , ∀ λ ∈ [0 , 1] ∀ λ, λ ′ ∈ [0 , 1] , [ X ( λ ) , X ( λ ′ )] is a robust controlled invariant interval x 2 X (1) X X (0) X (0) X x 1 X (1) September 24 th 2015 Pierre-Jean Meyer PhD Defense 13 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust set stabilization Take the smallest interval of this family containing the current state x Apply any invariance controller in this interval x 2 x X (0) X (0) x 1 September 24 th 2015 Pierre-Jean Meyer PhD Defense 13 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust set stabilization � λ ( x ) = min { λ ∈ [0 , 1] | X ( λ ) ≥ x } λ ( x ) = min { λ ∈ [0 , 1] | X ( λ ) ≤ x } [ X ( λ ( x )) , X ( λ ( x ))] is the smallest interval containing the current state x September 24 th 2015 Pierre-Jean Meyer PhD Defense 14 / 45
Monotonicity Invariance Symbolic Compositional Experiment Robust set stabilization � λ ( x ) = min { λ ∈ [0 , 1] | X ( λ ) ≥ x } λ ( x ) = min { λ ∈ [0 , 1] | X ( λ ) ≤ x } [ X ( λ ( x )) , X ( λ ( x ))] is the smallest interval containing the current state x Invariance controller Candidate stabilizing controller X i ( λ ( x )) − x i u i ( x ) = u i + ( u i − u i ) x i − x i u i ( x ) = u i + ( u i − u i ) (1) x i − x i X i ( λ ( x )) − X i ( λ ( x )) Theorem (Meyer, Girard, Witrant, prov. accepted in Automatica) (1) is a stabilizing controller from [ X (1) , X (1)] to [ X (0) , X (0)] . September 24 th 2015 Pierre-Jean Meyer PhD Defense 14 / 45
Monotonicity Invariance Symbolic Compositional Experiment Outline Monotone control system 1 Robust controlled invariance 2 Symbolic control 3 Compositional approach 4 Control in intelligent buildings 5 September 24 th 2015 Pierre-Jean Meyer PhD Defense 15 / 45
Monotonicity Invariance Symbolic Compositional Experiment Abstraction-based synthesis Continuous state Uncontrolled x + = f ( x, u, w ) Synthesis w Controlled x + = f ( x, u, w ) x u Controller September 24 th 2015 Pierre-Jean Meyer PhD Defense 16 / 45
Monotonicity Invariance Symbolic Compositional Experiment Abstraction-based synthesis Continuous state Discrete state u 1 Uncontrolled u 1 x + = f ( x, u, w ) u 2 u 2 Abstraction Controlled September 24 th 2015 Pierre-Jean Meyer PhD Defense 16 / 45
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