Optimal Control of Hybrid Systems with Regional Dynamics - PowerPoint PPT Presentation

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Optimal Control of Hybrid Systems with Regional Dynamics – Presentation at the Measurement and Control Laboratory, ETH Z¨ urich – Angela Sch¨ ollig January 30th, 2008 Σ 2 ξ T ξ 0 Σ 1 Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Hybrid? Something of mixed origin or composition. Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Hybrid? Something of mixed origin or composition. Combining different elements. Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Hybrid? Something of mixed origin or composition. Combining different elements. Definition: Hybrid System A dynamic system which exhibits both continuous and discrete dynamic behavior. Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Hybrid Systems with Regional Dynamics Definition: Hybrid System with Regional Dynamics x = f i ( x , u ) vary A dynamic system, where the governing (continuous) dynamics ˙ depending on the region D i , the continuous state x is evolving in. Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Hybrid Systems with Regional Dynamics Definition: Hybrid System with Regional Dynamics x = f i ( x , u ) vary A dynamic system, where the governing (continuous) dynamics ˙ depending on the region D i , the continuous state x is evolving in. Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Hybrid Systems with Regional Dynamics Definition: Hybrid System with Regional Dynamics x = f i ( x , u ) vary A dynamic system, where the governing (continuous) dynamics ˙ depending on the region D i , the continuous state x is evolving in. (1) (3) (2) Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Outline The Multiregional Point-To-Point Problem 1 The Basic Idea 2 A Dynamic Programming Approach 3 Σ 2 Conclusions 4 ξ T ξ 0 Σ 1 Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions Outline The Multiregional Point-To-Point Problem 1 The Basic Idea 2 A Dynamic Programming Approach 3 Σ 2 Conclusions 4 ξ T ξ 0 Σ 1 Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

� The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions Regions and Geometric Framework D 1 D 4 D 3 D 3 X D 2 state space X ⊂ R n : q X = ( D i ∪ ∂ D i ) , D i ∩ D j = / 0 , i � = j i = 1 Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

� The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions Regions and Geometric Framework D 1 D 4 m (1 , 4) = m (4 , 1) D 3 m (1 , 3) = m (3 , 1) m (1 , 2) = m (2 , 1) m (2 , 3) = m (3 , 2) m (3 , 4) = m (4 , 3) D 3 X D 2 state space X ⊂ R n : q X = ( D i ∪ ∂ D i ) , D i ∩ D j = / 0 , i � = j i = 1 switching manifolds m ( i , j ) : m ( i , j ) = ∂ D i ∩ ∂ D j Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

� The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions Regions and Geometric Framework D 1 D 4 m (1 , 4) = m (4 , 1) Σ 1 : ˙ x = f 1 ( x, u ) D 3 m (1 , 3) = m (3 , 1) m (1 , 2) = m (2 , 1) Σ 4 : ˙ x = f 4 ( x, u ) m (2 , 3) = m (3 , 2) Σ 2 : ˙ x = f 2 ( x, u ) Σ 3 : ˙ x = f 3 ( x, u ) m (3 , 4) = m (4 , 3) D 3 X D 2 state space X ⊂ R n : q X = ( D i ∪ ∂ D i ) , D i ∩ D j = / 0 , i � = j i = 1 switching manifolds m ( i , j ) : m ( i , j ) = ∂ D i ∩ ∂ D j continuous dynamics Σ i : x ( t ) = f i ( x ( t ) , u ( t )) ˙ if x ( t ) ∈ D i Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions Transition Behavior ξ s = lim t → t s x ( t ) ∈ m ( i , j ) , If there are Σ 2 : ˙ x = f 2 ( x, u ) two possibilities of further execution: (i) ( t s , ξ s ) (i) “passing through” the switching manifold (ii) x (0) = ξ 0 (ii) “bouncing back” into the original X Σ 1 : ˙ x = f 1 ( x, u ) region Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions Transition Behavior ξ s = lim t → t s x ( t ) ∈ m ( i , j ) , If there are Σ 2 : ˙ x = f 2 ( x, u ) two possibilities of further execution: (i) ( t s , ξ s ) (i) “passing through” the switching manifold (ii) x (0) = ξ 0 (ii) “bouncing back” into the original X Σ 1 : ˙ x = f 1 ( x, u ) region discrete control input e ∈ E is needed specifying the behavior at a switching point mode-transitions triggered by events in the continuous state space Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions The Hybrid Execution Σ 2 : ˙ x = f 2 ( x, u ) D 2 ( t 1 s , ξ 1 s ) x ( T ) = ξ T ( t 2 s , ξ 2 s ) x (0) = ξ 0 ( t 3 s , ξ 3 s ) D 1 X Σ 1 : ˙ x = f 1 ( x, u ) continuous-valued state x ( t ) ∈ X continuous dynamics within D i x ( t ) = f i ( x ( t ) , u ( t )) ˙ continuous-valued control signal u ( · ) ∈ U ( U , L ∞ ([ 0 , T ])) Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions The Hybrid Execution Σ 2 : ˙ x = f 2 ( x, u ) D 2 e 22 e 22 q ( t ) ( t 1 s , ξ 1 s ) q 2 e 12 x ( T ) = ξ T ( t 2 s , ξ 2 s ) q 1 x (0) = ξ 0 ( t 3 s , ξ 3 s ) D 1 t 1 t 2 t 3 X 0 T t s s s Σ 1 : ˙ x = f 1 ( x, u ) continuous-valued state x ( t ) ∈ X discrete-valued state q ( t ) ∈ Q continuous dynamics within D i discrete dynamics on boundaries � � x ( t ) = f i ( x ( t ) , u ( t )) q j = Γ ˙ q i , e ij continuous-valued control signal discrete input sequence u ( · ) ∈ U ( U , L ∞ ([ 0 , T ])) w = e i 1 j 1 e i 2 j 2 e i 3 j 3 ... Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem Regions and Geometric Framework The Basic Idea Dynamics and Executions A Dynamic Programming Approach The Optimization Problem Conclusions The Optimization Problem D 1 D 4 Σ 1 : ˙ x = f 1 ( x, u ) x (0) = ξ 0 Σ 4 : ˙ x = f 4 ( x, u ) D 3 x ∗ ( · ) ( t 2 s , ξ 2 s ) ∗ x ( T ) = ξ T ( t 1 s , ξ 1 s ) ∗ ( t 3 s , ξ 3 s ) ∗ Σ 2 : ˙ x = f 2 ( x, u ) Σ 3 : ˙ x = f 3 ( x, u ) D 3 X D 2 ✁ T “Given a specific cost function J = 0 ℓ ( x ( t ) , u ( t )) dt , determine the optimal path x ∗ ( · ) of going from a given initial state x ( 0 ) = ξ 0 to a fixed final state x ( T ) = ξ T during an a priori specified time horizon T assuming an upper bound N on the number of transitions along the hybrid trajectory.” Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig

The Multiregional Point-To-Point Problem The Basic Idea A Dynamic Programming Approach Conclusions Outline The Multiregional Point-To-Point Problem 1 The Basic Idea 2 A Dynamic Programming Approach 3 Σ 2 Conclusions 4 ξ T ξ 0 Σ 1 Optimal Control of Hybrid Systems with Regional Dynamics Angela Sch ¨ ollig