Experimental implementation of UFAD regulation based on robust - PowerPoint PPT Presentation

Experimental implementation of UFAD regulation based on robust controlled invariance Pierre-Jean Meyer Hosein Nazarpour Antoine Girard Emmanuel Witrant Universit´ e de Grenoble ECC 2014, June 26 th 2014 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 1 / 18

Outline Temperature model and monotonicity 1 Invariance (CDC13) 2 Stabilization 3 Experimental implementation 4 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 2 / 18

UnderFloor Air Distribution June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 3 / 18

Model Temperature variations in room i : energy conservation; mass conservation. June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 4 / 18

Model Temperature variations in room i : dT i � dt = j a i , j ( T j − T i ) Conduction through walls + b i u i ( T u − T i ) Controlled fan air flow u i + � j δ d ij c i , j ∗ h ( T j − T i ) Open doors (flow hot → cold) + δ s i d i ( T 4 s i − T 4 i ) Radiation from heat sources June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 4 / 18

Model Temperature variations in room i : dT i � dt = j a i , j ( T j − T i ) Conduction through walls + b i u i ( T u − T i ) Controlled fan air flow u i + � j δ d ij c i , j ∗ h ( T j − T i ) Open doors (flow hot → cold) + δ s i d i ( T 4 s i − T 4 i ) Radiation from heat sources a , b , c , d > 0; δ s , δ d : discrete state of the disturbances (heat sources and doors); � h ( x ≤ 0) = 0 : door heat transfer only in the colder room. h ( x > 0) = x 3 / 2 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 4 / 18

Monotonicity Generic system ˙ x = f ( x , v ) with trajectories Φ( t , x , v ). Definition (Monotonicity) The system Φ is monotone if its trajectories preserve some partial orders: v � v v ′ , x � x x ′ ⇒ ∀ t ≥ 0 , Φ( t , x , v ) � x Φ( t , x ′ , v ′ ) June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 5 / 18

Monotonicity Generic system ˙ x = f ( x , v ) with trajectories Φ( t , x , v ). Definition (Partial order) x � x x ′ ⇔ ∀ i , ( − 1) ε i ( x i − x ′ i ) ≥ 0 , with ε i ∈ { 0 , 1 } June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 6 / 18

Monotonicity Generic system ˙ x = f ( x , v ) with trajectories Φ( t , x , v ). Definition (Partial order) x � x x ′ ⇔ ∀ i , ( − 1) ε i ( x i − x ′ i ) ≥ 0 , with ε i ∈ { 0 , 1 } Proposition (Angeli and Sontag, 2003) The system defined by ˙ x = f ( x , v ) is monotone if and only if, ( − 1) ε i + ε j ∂ f i  ( x , v ) ≥ 0 , ∀ i , ∀ j � = i ,   ∂ x j ∀ x ∈ R n , ∀ v ∈ R m , ( − 1) ε i + γ k ∂ f i ( x , v ) ≥ 0 , ∀ i , ∀ k .   ∂ v k Where ε ∈ { 0 , 1 } n and γ ∈ { 0 , 1 } m define the partial orders for x and v . June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 6 / 18

Monotonicity Our model: ˙ T = f ( T , u , w , δ ) T : state (temperature); u : controlled input (fan air flow); w : exogenous input (other temperatures); δ : discrete disturbance embedded in a continuous space. June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 7 / 18

Monotonicity Our model: ˙ T = f ( T , u , w , δ ) T : state (temperature); u : controlled input (fan air flow); w : exogenous input (other temperatures); δ : discrete disturbance embedded in a continuous space. T � T T ′ ⇔ ∀ i , T i ≥ T ′ i u � u u ′ ⇔ ∀ t ≥ 0 , ∀ k , u k ( t ) ≤ u ′ k ( t ) w � w w ′ ⇔ ∀ t ≥ 0 , ∀ k , w k ( t ) ≥ w ′ k ( t ) δ � δ δ ′ ⇔ ∀ t ≥ 0 , ∀ k , δ k ( t ) ≥ δ ′ k ( t ) Φ( t , T , u , w , δ ) � T Φ( t , T ′ , u ′ , w ′ , δ ′ ) June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 7 / 18

Outline Temperature model and monotonicity 1 Invariance (CDC13) 2 Stabilization 3 Experimental implementation 4 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 8 / 18

Robust Controlled Invariance Definition (Robust Controlled Invariance) The system is Robust Controlled Invariant in [ T , T ] if, ∀ T 0 ∈ [ T , T ] , ∀ w ∈ [ w , w ] , ∀ δ ∈ [ δ, δ ] , ∃ u ∈ [ u , u ] | ∀ t ≥ 0 , Φ( t , T 0 , u , w , δ ) ∈ [ T , T ] . June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 9 / 18

Robust Controlled Invariance Definition (Robust Controlled Invariance) The system is Robust Controlled Invariant in [ T , T ] if, ∀ T 0 ∈ [ T , T ] , ∀ w ∈ [ w , w ] , ∀ δ ∈ [ δ, δ ] , ∃ u ∈ [ u , u ] | ∀ t ≥ 0 , Φ( t , T 0 , u , w , δ ) ∈ [ T , T ] . June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 9 / 18

Robust Controlled Invariance Definition (Robust Controlled Invariance) The system is Robust Controlled Invariant in [ T , T ] if, ∀ T 0 ∈ [ T , T ] , ∀ w ∈ [ w , w ] , ∀ δ ∈ [ δ, δ ] , ∃ u ∈ [ u , u ] | ∀ t ≥ 0 , Φ( t , T 0 , u , w , δ ) ∈ [ T , T ] . June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 9 / 18

Robust Controlled Invariance Definition (Robust Controlled Invariance) The system is Robust Controlled Invariant in [ T , T ] if, ∀ T 0 ∈ [ T , T ] , ∀ w ∈ [ w , w ] , ∀ δ ∈ [ δ, δ ] , ∃ u ∈ [ u , u ] | ∀ t ≥ 0 , Φ( t , T 0 , u , w , δ ) ∈ [ T , T ] . June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 9 / 18

Robust Controlled Invariance Proposition The system is Robust Controlled Invariant in [ T , T ] if and only if � f i ( T , u i , w , δ ) ≤ 0 ∀ i , f i ( T , u i , w , δ ) ≥ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 10 / 18

Robust Controlled Invariance Proposition The system is Robust Controlled Invariant in [ T , T ] if and only if � f i ( T , u i , w , δ ) ≤ 0 ∀ i , f i ( T , u i , w , δ ) ≥ 0 Definition (Decentralized Linear Saturated Controller)  T i ≥ T i ⇒ u i = u i    T i ≤ T i ⇒ u i = u i = 0 ∀ i , T i − T i  T i ∈ [ T i , T i ] ⇒ u i = u i ∗   T i − T i June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 10 / 18

Controllable Spaces (2-room example) f 1 ( T , u 1 , w , δ ) ≥ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 11 / 18

Controllable Spaces (2-room example) f 1 ( T , u 1 , w , δ ) ≥ 0 f 2 ( T , u 2 , w , δ ) ≥ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 11 / 18

Controllable Spaces (2-room example) f 1 ( T , u 1 , w , δ ) ≤ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 11 / 18

Controllable Spaces (2-room example) f 1 ( T , u 1 , w , δ ) ≤ 0 f 2 ( T , u 2 , w , δ ) ≤ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 11 / 18

Controllable Spaces (2-room example) f 1 ( T , u 1 , w , δ ) ≤ 0 f 2 ( T , u 2 , w , δ ) ≤ 0 f 1 ( T , u 1 , w , δ ) ≥ 0 f 2 ( T , u 2 , w , δ ) ≥ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 11 / 18

Controllable Spaces (2-room example) f 1 ( T , u 1 , w , δ ) ≤ 0 f 2 ( T , u 2 , w , δ ) ≤ 0 f 1 ( T , u 1 , w , δ ) ≥ 0 f 2 ( T , u 2 , w , δ ) ≥ 0 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 11 / 18

Outline Temperature model and monotonicity 1 Invariance (CDC13) 2 Stabilization 3 Experimental implementation 4 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 12 / 18

Stabilization June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 13 / 18

Stabilization June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 13 / 18

Stabilization June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 13 / 18

Stabilization June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 13 / 18

Stabilization June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 13 / 18

Stabilization June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 13 / 18

Outline Temperature model and monotonicity 1 Invariance (CDC13) 2 Stabilization 3 Experimental implementation 4 June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 14 / 18

Identification Identification (least-squares) over 57079 data points ( ≈ 16 h ) Evaluation on another scenario: June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 15 / 18

Control Robust controlled invariant interval: T = 21 , T = [24; 26; 26; 27] Initial state above the interval: T 0 ≈ [26; 27; 28; 28] Decentralized linear saturated controller June 26 th 2014 Pierre-Jean Meyer (Univ. de Grenoble) Robust controlled invariance 16 / 18