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

Colas Le Guernic CMACS meeting – 1 / 25

Abstractions of Dynamical Systems

Colas Le Guernic

October 28, 2010

Motivations

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 2 / 25

A typical example:

■

a differential equation ˙ x = f(x), f : Rd → Rd

■

an initial point x0

■

a set of “bad” states F

Motivations

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 2 / 25

A typical example:

■

a differential equation ˙ x = f(x), f : Rd → Rd

■

an initial point x0

■

a set of “bad” states F

Motivations

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 2 / 25

A typical example:

■

a differential equation ˙ x = f(x), f : Rd → Rd

■

an initial set X0

■

a set of “bad” states F

Motivations

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 2 / 25

A typical example:

■

a differential inclusion ˙ x ∈ f(x), f : Rd → P

• Rd

■

an initial set X0

■

a set of “bad” states F

Motivations

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 2 / 25

A typical example:

■

a differential inclusion ˙ x ∈ f(x), f : Rd → P

• Rd

■

an initial set X0

■

a set of “bad” states F

Hybrid Systems

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 3 / 25

˙ x ∈ f1(x) ˙ x ∈ f2(x) ˙ x ∈ f3(x) ˙ x ∈ f4(x)

x ∈ G1,2 x ← R1,2(x) x ∈ G2,4 x ← R2,4(x) x ∈ G2,3 x ← R2,3(x) x ∈ G3,2 x ← R3,2(x) x ∈ G3,1 x ← R3,1(x)

Outline

Introduction Motivations Hybrid Systems Outline State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 4 / 25

A few reflexions on:

■

Reachability for some specific classes of functions f.

■

Abstractions of arbitrary systems using these specific functions. 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.

State of the Art

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 5 / 25

f : R0 → P (R0)

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 6 / 25

S1 S2 S3 S4

f(x) = {1}

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 7 / 25

˙ x = 1 ˙ x = 1 ˙ x = 1 ˙ x = 1

x ∈ G1,2 ∀i ∈ R1,2 xi ← 0 x ∈ G2,4 ∀i ∈ R2,4 xi ← 0 x ∈ G2,3 ∀i ∈ R2,3 xi ← 0 x ∈ G3,2 ∀i ∈ R3,2 xi ← 0 x ∈ G3,1 ∀i ∈ R3,1 xi ← 0

f(x) = P

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 8 / 25

Linear Hybrid Automata

■

simple continuous dynamics: conjunctions of linear constraints a · ˙ x ≤ b, a ∈ Zn, b ∈ Z

■

All sets defined by Boolean combinations of linear constraints

f(x) = P

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 8 / 25

Linear Hybrid Automata

■

simple continuous dynamics: conjunctions of linear constraints a · ˙ x ≤ b, a ∈ Zn, b ∈ Z

■

All sets defined by Boolean combinations of linear constraints Postc: letting time ellapse

f(x) = P

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 8 / 25

Linear Hybrid Automata

■

simple continuous dynamics: conjunctions of linear constraints a · ˙ x ≤ b, a ∈ Zn, b ∈ Z

■

All sets defined by Boolean combinations of linear constraints Postc: letting time ellapse Postd: discrete transition

f(x) = P

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 8 / 25

Linear Hybrid Automata

■

simple continuous dynamics: conjunctions of linear constraints a · ˙ x ≤ b, a ∈ Zn, b ∈ Z

■

All sets defined by Boolean combinations of linear constraints Postc: letting time ellapse Postd: discrete transition

f(x) = P

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 8 / 25

Linear Hybrid Automata

■

simple continuous dynamics: conjunctions of linear constraints a · ˙ x ≤ b, a ∈ Zn, b ∈ Z

■

All sets defined by Boolean combinations of linear constraints Postc: letting time ellapse Postd: discrete transition

f(x) = P

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 8 / 25

Linear Hybrid Automata

■

simple continuous dynamics: conjunctions of linear constraints a · ˙ x ≤ b, a ∈ Zn, b ∈ Z

■

All sets defined by Boolean combinations of linear constraints Postc: letting time ellapse Postd: discrete transition

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 9 / 25

More expressive than LHA: f(x) = 0{x} ⊕ P

■

Continuous dynamics: ˙ x ∈ Aq{x} ⊕ Uq

■

Switching hyperplanes or Polyhedral guards.

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

Φ

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

Φ ⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

Φ ⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

Φ ⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

Φ ⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

Φ ⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

. . .

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V Ωn−1 may have more than (2n)d−1

√ d

vertices.

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = ΦΩn+1 ⊕ V

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

Φ ⊕

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

α( )

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

α(Φγ( )⊕ )

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

α(Φγ( )⊕ )

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V)

. . .

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V) The approximation error can be exponential in the number of steps! − → wrapping effect

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 10 / 25

Reachability for LTI:

■

Time discretization: ˙ x ∈ A{x} ⊕ U − → xk+1 ∈ Φ{xk} ⊕ V

■

Computation of the N first terms of: Ωn+1 = α(Φγ(Ωn+1) ⊕ V) The approximation error can be exponential in the number of steps! − → wrapping effect T : X → ΦX ⊕ V (α ◦ T ◦ γ)n = α ◦ T n ◦ γ (α ◦ T ◦ γ) ◦ α = α ◦ T

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 11 / 25

Ωn = ΦnΩ0 ⊕

n−1

• i=0

ΦiV

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 11 / 25

Ωn = ΦnΩ0 ⊕

n−1

• i=0

ΦiV A0 = Ω0 An+1 = ΦAn V0 = V Vn+1 = ΦVn S0 = {0} Sn+1 = Sn ⊕ Vn Then: Ωn = An ⊕ Sn

■

Ai and Vi have a constant representation size.

■

We can exploit redundancies of Si (zonotopes, support functions).

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 11 / 25

Ωn = ΦnΩ0 ⊕

n−1

• i=0

ΦiV A0 = Ω0 An+1 = ΦAn V0 = V Vn+1 = ΦVn S0 = {0} Sn+1 = Sn ⊕ Vn Approximations can still be interesting:

■

We are only interested in one individual Ωi.

■

We want to use a tool that can not exploit the redundancies.

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 11 / 25

Ωn = ΦnΩ0 ⊕

n−1

• i=0

ΦiV A0 = Ω0 An+1 = ΦAn V0 = V Vn+1 = ΦVn S0 = {0} Sn+1 = α(γ(Sn) ⊕ Vn) Approximations can still be interesting:

■

We are only interested in one individual Ωi.

■

We want to use a tool that can not exploit the redundancies.

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 11 / 25

Ωn = ΦnΩ0 ⊕

n−1

• i=0

ΦiV A0 = Ω0 An+1 = ΦAn V0 = V Vn+1 = ΦVn S0 = {0} Sn+1 = α(γ(Sn) ⊕ Vn) T : (X , Y, Z) → (ΦX , ΦY, Z ⊕ Y) (α ◦ T ◦ γ)n = α ◦ T n ◦ γ α(γ(α(Z)) ⊕ Y) = α(Z ⊕ Y)

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 12 / 25

f(x) = A{x} ⊕ U

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 13 / 25

f(x) = {Ax | A ∈ A}

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 14 / 25

More expressive than f(x) = A{x} ⊕ U (in smaller dimension): f(x) = A u x 1

• | u ∈ U
• ■

Time discretization: ˙ x ∈ Ax − → xn+1 ∈ Mxn

■

Use of set representations in the space of Matrices.

f(x) = {Ax | A ∈ A}

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 15 / 25

Phi Phi_t zAxis yAxis yAxis xAxis dotPsi beta vel delta

■

8 variables

■

3 discrete locations

f(x) = {Ax | A ∈ A}

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 15 / 25

−0.1 0.1 −0.1 −0.05 0.05 x5 x6

■

8 variables

■

3 discrete locations

f : Rd → P

• Rd

Introduction State of the Art f : R0 → P “ R0” f(x) = {1} f(x) = P f(x) = A{x} ⊕ U f(x) = A{x} f : Rd → P “ Rd” Abstraction Conclusion

Colas Le Guernic CMACS meeting – 16 / 25

If we want to use similar techniques:

■

Adapt integration schemes: X → X ⊕ δf(X ) ⊕ E

■

Abstract

Abstraction

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 17 / 25

¯ f : R0 → P (R0)

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 18 / 25

Rectangular partition.

?

We need to know if fi(G) ∩ R+ is empty.

¯ f : R0 → P (R0)

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 19 / 25

Smooth partition.

■

Sign conditions on a set of functions and their derivatives.

■

No transition from (x > 0, ˙ x > 0) to (x < 0, ˙ x > 0) We need to check emptiness of the cells.

¯ f(x) = {1}

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 20 / 25

Timed automata.

■

Partition of the state space in slices

■

Clocks measure time to get from one slice to the other

■

We need to know upper and lower bounds for fi(S).

■

Easier when Lyapunov functions are availables

¯ f(x) = P

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 21 / 25

LHA

■

Polyhedral partition

■

For each cell C of the partition, we need to know f(C)

¯ f(x) = P

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 21 / 25

LHA

■

Polyhedral partition

■

For each cell C of the partition, we need to know f(C)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x1 x2

initial states final states reachable final states reachable states R1

¯ f(x) = P

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 21 / 25

LHA

■

Polyhedral partition

■

For each cell C of the partition, we need to know f(C)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x1 x2

restricted states of H2 reachable final states initial states R2 final states

¯ f(x) = P

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 21 / 25

LHA

■

Polyhedral partition

■

For each cell C of the partition, we need to know f(C)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x1 x2

initial states reachable states vanish at last iteration final states R3

¯ f(x) = A{x} ⊕ U

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 22 / 25

For each cell C of the partition:

■

Choose linearization A

■

Compute U = {y − Ax | x ∈ C, y ∈ f(x)} We want U to be as small as possible, how do we choose A?

¯ f(x) = A{x} ⊕ U

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 22 / 25

For each cell C of the partition:

■

Choose linearization A

■

Compute U = {y − Ax | x ∈ C, y ∈ f(x)} We want U to be as small as possible, how do we choose A? We do not really know...

¯ f(x) = A{x} ⊕ U

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 22 / 25

For each cell C of the partition:

■

Choose linearization A

■

Compute U = {y − Ax | x ∈ C, y ∈ f(x)} We want U to be as small as possible, how do we choose A? We do not really know... One guess is to take the Jacobian at the center of the cell.

¯ f(x) = {Ax | A ∈ A}

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 23 / 25

¯ f(x) = {Ax | A ∈ A}

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 23 / 25

One guess is to take the Jacobians at every point of the cell.

¯ f(x) = {Ax | A ∈ A}

Introduction State of the Art Abstraction ¯ f : R0 → P “ R0” ¯ f(x) = {1} ¯ f(x) = P ¯ f(x) = A{x} ⊕ U ¯ f(x) = A{x} Conclusion

Colas Le Guernic CMACS meeting – 23 / 25

One guess is to take the Jacobians at every point of the cell. If we find a subset of variables such that:

■

f is linear in these variables

■

no product of two of these variables appear in f We do not need to partition along these variables.

Conclusion

Introduction State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 24 / 25

Choosing the right abstraction is rarely easy.

■

choice of the partition

■

choice of the class of abstraction

■

choice of the abstraction in this class

Conclusion

Introduction State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 24 / 25

Choosing the right abstraction is rarely easy.

■

choice of the partition

■

choice of the class of abstraction

■

choice of the abstraction in this class

■

modifying the number of continuous variables

■

combining different classes of abstractions

Thank you

Introduction State of the Art Abstraction Conclusion

Colas Le Guernic CMACS meeting – 25 / 25