Abstractions of Dynamical Systems Colas Le Guernic October 28, 2010 - PowerPoint PPT Presentation

Abstractions of Dynamical Systems Colas Le Guernic October 28, 2010 Colas Le Guernic CMACS meeting – 1 / 25

Motivations A typical example: Introduction Motivations x = f ( x ) , f : R d → R d Hybrid Systems a differential equation ˙ ■ Outline an initial point x 0 ■ State of the Art a set of “bad” states F ■ Abstraction Conclusion Colas Le Guernic CMACS meeting – 2 / 25

Motivations A typical example: Introduction Motivations x = f ( x ) , f : R d → R d Hybrid Systems a differential equation ˙ ■ Outline an initial point x 0 ■ State of the Art a set of “bad” states F ■ Abstraction Conclusion Colas Le Guernic CMACS meeting – 2 / 25

Motivations A typical example: Introduction Motivations x = f ( x ) , f : R d → R d Hybrid Systems a differential equation ˙ ■ Outline an initial set X 0 ■ State of the Art a set of “bad” states F ■ Abstraction Conclusion Colas Le Guernic CMACS meeting – 2 / 25

Motivations A typical example: Introduction Motivations x ∈ f ( x ) , f : R d → P Hybrid Systems � R d � a differential inclusion ˙ ■ Outline an initial set X 0 ■ State of the Art a set of “bad” states F ■ Abstraction Conclusion Colas Le Guernic CMACS meeting – 2 / 25

Motivations A typical example: Introduction Motivations x ∈ f ( x ) , f : R d → P Hybrid Systems � R d � a differential inclusion ˙ ■ Outline an initial set X 0 ■ State of the Art a set of “bad” states F ■ Abstraction Conclusion Colas Le Guernic CMACS meeting – 2 / 25

Hybrid Systems Introduction Motivations x ∈ f 4 ( x ) ˙ Hybrid Systems Outline State of the Art Abstraction x ∈ G 2 , 4 x ← R 2 , 4 ( x ) Conclusion x ∈ G 1 , 2 x ← R 1 , 2 ( x ) x ∈ f 1 ( x ) ˙ x ∈ f 2 ( x ) ˙ x ∈ G 3 , 2 x ← R 3 , 2 ( x ) x ∈ G 3 , 1 x ∈ G 2 , 3 x ← R 3 , 1 ( x ) x ← R 2 , 3 ( x ) x ∈ f 3 ( x ) ˙ Colas Le Guernic CMACS meeting – 3 / 25

Outline A few reflexions on: Introduction Motivations Hybrid Systems Reachability for some specific classes of functions f . ■ Outline Abstractions of arbitrary systems using these specific ■ State of the Art functions. Abstraction Conclusion Including some ongoing work: On Linear Parameter Varying systems with Matthias Althoff ■ and Bruce Krogh. On multi-affine systems with Radu Grosu, Flavio Fenton, ■ James Glimm, Scott Smolka and Ezio Bartocci. Colas Le Guernic CMACS meeting – 4 / 25

Introduction State of the Art f : R 0 → P “ R 0 ” f ( x ) = { 1 } f ( x ) = P f ( x ) = A { x } ⊕ U f ( x ) = A{ x } f : R d → P “ R d ” State of the Art Abstraction Conclusion Colas Le Guernic CMACS meeting – 5 / 25

f : R 0 → P ( R 0 ) Introduction State of the Art f : R 0 → P “ R 0 ” S4 f ( x ) = { 1 } f ( x ) = P f ( x ) = A { x } ⊕ U f ( x ) = A{ x } f : R d → P “ R d ” Abstraction S1 S2 Conclusion S3 Colas Le Guernic CMACS meeting – 6 / 25

f ( x ) = { 1 } Introduction State of the Art x = 1 ˙ f : R 0 → P “ R 0 ” f ( x ) = { 1 } f ( x ) = P x ∈ G 2 , 4 f ( x ) = A { x } ⊕ U ∀ i ∈ R 2 , 4 f ( x ) = A{ x } x ∈ G 1 , 2 f : R d → P “ R d ” x i ← 0 ∀ i ∈ R 1 , 2 Abstraction x i ← 0 x = 1 ˙ x = 1 ˙ Conclusion x ∈ G 3 , 2 ∀ i ∈ R 3 , 2 x i ← 0 x ∈ G 3 , 1 x ∈ G 2 , 3 ∀ i ∈ R 3 , 1 ∀ i ∈ R 2 , 3 x i ← 0 x i ← 0 x = 1 ˙ Colas Le Guernic CMACS meeting – 7 / 25

f ( x ) = P Linear Hybrid Automata Introduction State of the Art f : R 0 → P “ R 0 ” simple continuous dynamics: conjunctions of linear ■ f ( x ) = { 1 } a ∈ Z n , b ∈ Z constraints a · ˙ x ≤ b, f ( x ) = P All sets defined by Boolean combinations of linear constraints f ( x ) = A { x } ⊕ U ■ f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Colas Le Guernic CMACS meeting – 8 / 25

f ( x ) = P Linear Hybrid Automata Introduction State of the Art f : R 0 → P “ R 0 ” simple continuous dynamics: conjunctions of linear ■ f ( x ) = { 1 } a ∈ Z n , b ∈ Z constraints a · ˙ x ≤ b, f ( x ) = P All sets defined by Boolean combinations of linear constraints f ( x ) = A { x } ⊕ U ■ f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Post c : letting time ellapse Colas Le Guernic CMACS meeting – 8 / 25

f ( x ) = P Linear Hybrid Automata Introduction State of the Art f : R 0 → P “ R 0 ” simple continuous dynamics: conjunctions of linear ■ f ( x ) = { 1 } a ∈ Z n , b ∈ Z constraints a · ˙ x ≤ b, f ( x ) = P All sets defined by Boolean combinations of linear constraints f ( x ) = A { x } ⊕ U ■ f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Post c : letting time ellapse Post d : discrete transition Colas Le Guernic CMACS meeting – 8 / 25

f ( x ) = P Linear Hybrid Automata Introduction State of the Art f : R 0 → P “ R 0 ” simple continuous dynamics: conjunctions of linear ■ f ( x ) = { 1 } a ∈ Z n , b ∈ Z constraints a · ˙ x ≤ b, f ( x ) = P All sets defined by Boolean combinations of linear constraints f ( x ) = A { x } ⊕ U ■ f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Post c : letting time ellapse Post d : discrete transition Colas Le Guernic CMACS meeting – 8 / 25

f ( x ) = P Linear Hybrid Automata Introduction State of the Art f : R 0 → P “ R 0 ” simple continuous dynamics: conjunctions of linear ■ f ( x ) = { 1 } a ∈ Z n , b ∈ Z constraints a · ˙ x ≤ b, f ( x ) = P All sets defined by Boolean combinations of linear constraints f ( x ) = A { x } ⊕ U ■ f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Post c : letting time ellapse Post d : discrete transition Colas Le Guernic CMACS meeting – 8 / 25

f ( x ) = P Linear Hybrid Automata Introduction State of the Art f : R 0 → P “ R 0 ” simple continuous dynamics: conjunctions of linear ■ f ( x ) = { 1 } a ∈ Z n , b ∈ Z constraints a · ˙ x ≤ b, f ( x ) = P All sets defined by Boolean combinations of linear constraints f ( x ) = A { x } ⊕ U ■ f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Post c : letting time ellapse Post d : discrete transition Colas Le Guernic CMACS meeting – 8 / 25

f ( x ) = A { x } ⊕ U Introduction More expressive than LHA: f ( x ) = 0 { x } ⊕ P State of the Art f : R 0 → P “ R 0 ” Continuous dynamics: ˙ x ∈ A q { x } ⊕ U q ■ f ( x ) = { 1 } Switching hyperplanes or Polyhedral guards. ■ f ( x ) = P f ( x ) = A { x } ⊕ U f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Colas Le Guernic CMACS meeting – 9 / 25

f ( x ) = A { x } ⊕ U Reachability for LTI: Introduction State of the Art f : R 0 → P Time discretization: ˙ x ∈ A { x } ⊕ U − → x k +1 ∈ Φ { x k } ⊕ V “ R 0 ” ■ f ( x ) = { 1 } Computation of the N first terms of: ■ f ( x ) = P f ( x ) = A { x } ⊕ U Ω n +1 = ΦΩ n +1 ⊕ V f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Colas Le Guernic CMACS meeting – 10 / 25

f ( x ) = A { x } ⊕ U Reachability for LTI: Introduction State of the Art f : R 0 → P Time discretization: ˙ x ∈ A { x } ⊕ U − → x k +1 ∈ Φ { x k } ⊕ V “ R 0 ” ■ f ( x ) = { 1 } Computation of the N first terms of: ■ f ( x ) = P f ( x ) = A { x } ⊕ U Ω n +1 = ΦΩ n +1 ⊕ V f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Colas Le Guernic CMACS meeting – 10 / 25

f ( x ) = A { x } ⊕ U Reachability for LTI: Introduction State of the Art f : R 0 → P Time discretization: ˙ x ∈ A { x } ⊕ U − → x k +1 ∈ Φ { x k } ⊕ V “ R 0 ” ■ f ( x ) = { 1 } Computation of the N first terms of: ■ f ( x ) = P f ( x ) = A { x } ⊕ U Ω n +1 = ΦΩ n +1 ⊕ V f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Φ Colas Le Guernic CMACS meeting – 10 / 25

f ( x ) = A { x } ⊕ U Reachability for LTI: Introduction State of the Art f : R 0 → P Time discretization: ˙ x ∈ A { x } ⊕ U − → x k +1 ∈ Φ { x k } ⊕ V “ R 0 ” ■ f ( x ) = { 1 } Computation of the N first terms of: ■ f ( x ) = P f ( x ) = A { x } ⊕ U Ω n +1 = ΦΩ n +1 ⊕ V f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Colas Le Guernic CMACS meeting – 10 / 25

f ( x ) = A { x } ⊕ U Reachability for LTI: Introduction State of the Art f : R 0 → P Time discretization: ˙ x ∈ A { x } ⊕ U − → x k +1 ∈ Φ { x k } ⊕ V “ R 0 ” ■ f ( x ) = { 1 } Computation of the N first terms of: ■ f ( x ) = P f ( x ) = A { x } ⊕ U Ω n +1 = ΦΩ n +1 ⊕ V f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion ⊕ Colas Le Guernic CMACS meeting – 10 / 25

f ( x ) = A { x } ⊕ U Reachability for LTI: Introduction State of the Art f : R 0 → P Time discretization: ˙ x ∈ A { x } ⊕ U − → x k +1 ∈ Φ { x k } ⊕ V “ R 0 ” ■ f ( x ) = { 1 } Computation of the N first terms of: ■ f ( x ) = P f ( x ) = A { x } ⊕ U Ω n +1 = ΦΩ n +1 ⊕ V f ( x ) = A{ x } f : R d → P “ R d ” Abstraction Conclusion Colas Le Guernic CMACS meeting – 10 / 25