Computing Invariance Kernels of Polygonal Hybrid Systems G ERARDO S - - PowerPoint PPT Presentation

▶
computing invariance kernels of polygonal hybrid systems
SMART_READER_LITE
LIVE PREVIEW

Computing Invariance Kernels of Polygonal Hybrid Systems G ERARDO S - - PowerPoint PPT Presentation

Jun 03, 2023 447 likes •1.02k views

Computing Invariance Kernels of Polygonal Hybrid Systems G ERARDO S CHNEIDER gerardos@it.uu.se U PPSALA U NIVERSITY D EPARTMENT OF I NFORMATION T ECHNOLOGY U PPSALA , S WEDEN Computing Invariance Kernels of Polygonal Hybrid Systems p.1/26

slide-1
SLIDE 1

Computing Invariance Kernels of Polygonal Hybrid Systems

GERARDO SCHNEIDER

gerardos@it.uu.se

UPPSALA UNIVERSITY DEPARTMENT OF INFORMATION TECHNOLOGY UPPSALA, SWEDEN

Computing Invariance Kernels of Polygonal Hybrid Systems – p.1/26

slide-2
SLIDE 2

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.2/26

slide-3
SLIDE 3

Motivation and Related Work

  • For Hybrid Systems
  • Verification (reachability, ...):
  • Qualitative behavior (Phase Portrait, ...)

Computing Invariance Kernels of Polygonal Hybrid Systems – p.3/26

slide-4
SLIDE 4

Motivation and Related Work

  • For Hybrid Systems
  • Verification (reachability, ...):
  • Qualitative behavior (Phase Portrait, ...)
  • For a class of non-deterministic systems (SPDI)
  • Verification (HSCC’01)
  • Undecidability of some extensions

(CONCUR’02)

  • Phase Portrait (HSCC’02):
  • Viability Kernel
  • Controllability Kernel
  • Invariance Kernels

Computing Invariance Kernels of Polygonal Hybrid Systems – p.3/26

slide-5
SLIDE 5

Why Invariance Kernels?

  • Important objects for giving SPDIs
  • Crucial for proving termination of a BFS

reachability algorithm for SPDI

Computing Invariance Kernels of Polygonal Hybrid Systems – p.4/26

slide-6
SLIDE 6

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.5/26

slide-7
SLIDE 7

Hybrid Systems

  • Hybrid Systems: interaction between discrete and

continuous behaviors

  • Examples: thermostat, automated highway

systems, air traffic management systems, robotic systems, chemical plants, etc.

Computing Invariance Kernels of Polygonal Hybrid Systems – p.6/26

slide-8
SLIDE 8

Hybrid Systems

Model: Hybrid Automata

label invariant dynamics guard reset

x = M x ≤ M ˙ x = 3 − x x ≥ m ˙ x = −x

Off On

x = m /γ

Computing Invariance Kernels of Polygonal Hybrid Systems – p.6/26

slide-9
SLIDE 9

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

slide-10
SLIDE 10

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

slide-11
SLIDE 11

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

slide-12
SLIDE 12

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

slide-13
SLIDE 13

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

slide-14
SLIDE 14

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

slide-15
SLIDE 15

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.8/26

slide-16
SLIDE 16

Polygonal Differential Inclusion Systems (SPDIs)

  • A partition of the plane into convex

polygonal regions

  • A constant differential inclusion for each region

˙ x ∈ ∠b

a if x ∈ Ri

e3 e2 e4 e5 e9 e12 e1 e8 e11 e7 e6 e10

Computing Invariance Kernels of Polygonal Hybrid Systems – p.9/26

slide-17
SLIDE 17

Polygonal Differential Inclusion Systems (SPDIs)

  • A partition of the plane into convex

polygonal regions

  • A constant differential inclusion for each region

˙ x ∈ ∠b

a if x ∈ Ri

x′ Ri x b a

b ∠b

a

a

Computing Invariance Kernels of Polygonal Hybrid Systems – p.9/26

slide-18
SLIDE 18

Polygonal Differential Inclusion Systems (SPDIs)

  • The “swimmer” is a hybrid system
  • Hybrid Automata?

Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

slide-19
SLIDE 19

Polygonal Differential Inclusion Systems (SPDIs)

  • The “swimmer” is a hybrid system
  • Hybrid Automata?

e3 e2 e4 e5 e9 e12 e1 e8 e11 e7 e6 e10

Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

slide-20
SLIDE 20

Polygonal Differential Inclusion Systems (SPDIs)

  • The “swimmer” is a hybrid system
  • Hybrid Automata?

e2 e3 e9 e12 e4 e3 e1 e2 e12 e11 e1 e8 e7 e8 e11 e7 e6 e10 e6 e5 e4 e5 e9 e10 Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

slide-21
SLIDE 21

Polygonal Differential Inclusion Systems (SPDIs)

  • The “swimmer” is a hybrid system
  • Hybrid Automata?

˙ x = a7 ˙ x = a8 ˙ x = a4 Invℓ2 ˙ x = a2 x = e3

R2

˙ x = a1 x = e2 ˙ x ∈ ∠b a x = e7 x = e6 x = e8 x = e1 x = e10 x = e11 x = e12 x = e9 x = e4 x = e5 Invℓ4 Invℓ3 Invℓ1 Invℓ8 Invℓ7 Invℓ6 ˙ x = a6 Invℓ5 ˙ x = a5

R1 R5 R8 R7 R6 R3 R4

Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

slide-22
SLIDE 22

Polygonal Differential Inclusion Systems (SPDIs)

  • The “swimmer” is a hybrid system
  • Hybrid Automata?

We will use the “geometric” representation instead of the hybrid automata

Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

slide-23
SLIDE 23

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.11/26

slide-24
SLIDE 24

Successor Operators

For a signature σ = e1 . . . e8e1:

R6 R8 R3 R7 R4 R2 R1 R5

e2 e3 e1 e7 e6 e4 e8 e5

l u

Succσ([l, u]) Succσ([l, u])

Computing Invariance Kernels of Polygonal Hybrid Systems – p.12/26

slide-25
SLIDE 25

Successor Operators

  • Successors have the form

Succσ(l, u) = [a1l + b1, a2u + b2] ∩ J if [l, u] ⊆ S

  • Fixpoint equations

[a1l∗ + b1, a2u∗ + b2] = [l∗, u∗]

can be explicitely solved (without iterating).

Computing Invariance Kernels of Polygonal Hybrid Systems – p.13/26

slide-26
SLIDE 26

Predecessor Operators

For σ = e1 . . . e8e1:

R6 R8 R3 R7 R4 R2 R1 R5

e2 e3 e1 e7 e6 e4 e8 e5

u l

f Preσ([l, u])) f Preσ([l, u]))

Computing Invariance Kernels of Polygonal Hybrid Systems – p.14/26

slide-27
SLIDE 27

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.15/26

slide-28
SLIDE 28

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-29
SLIDE 29

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

STAY:

L ≤ l∗ ≤ u∗ ≤ U

DIE:

u∗ < L ∨ l∗ > U

EXIT-BOTH:

l∗ < L ∧ u∗ > U

EXIT-LEFT:

l∗ < L ≤ u∗ ≤ U

EXIT-RIGHT:

L ≤ l∗ ≤ U < u∗

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-30
SLIDE 30

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

STAY:

e1 L U l∗ u∗

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-31
SLIDE 31

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

DIE:

e1 L U u∗ l∗

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-32
SLIDE 32

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

EXIT-BOTH:

e1 L l∗ u∗ U

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-33
SLIDE 33

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

EXIT-LEFT:

e1 L U l∗ u∗

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-34
SLIDE 34

Classification of Simple Cycles

Given a cyclic signature σ = e1 . . . e8e1. Let e1 = [L, U].

Succ∗

σ = [l∗, u∗]

Succσ([l, u]) ⊆ [l∗, u∗]

EXIT-RIGHT:

e1 L U l∗ u∗

Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

slide-35
SLIDE 35

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.17/26

slide-36
SLIDE 36

Phase Portrait

Phase Portrait: a picture of important objects of a dynamical system

Computing Invariance Kernels of Polygonal Hybrid Systems – p.18/26

slide-37
SLIDE 37

Phase Portrait

Phase Portrait: a picture of important objects of a dynamical system

e3 e9 e12 e2 e6 e7 e4 e5 e8 e1 e10 e11

Computing Invariance Kernels of Polygonal Hybrid Systems – p.18/26

slide-38
SLIDE 38

Phase Portrait

Phase Portrait: a picture of important objects of a dynamical system

e3 e9 e12 e2 e6 e7 e4 e5 e8 e1 e10 e11

Computing Invariance Kernels of Polygonal Hybrid Systems – p.18/26

slide-39
SLIDE 39

Phase Portrait

Phase Portrait: a picture of important objects of a dynamical system

e3 e9 e12 e2 e6 e7 e4 e5 e8 e1 e10 e11

Computing Invariance Kernels of Polygonal Hybrid Systems – p.18/26

slide-40
SLIDE 40

Phase Portrait

Phase Portrait: a picture of important objects of a dynamical system

e3 e9 e12 e2 e6 e7 e4 e5 e8 e1 e10 e11

Computing Invariance Kernels of Polygonal Hybrid Systems – p.18/26

slide-41
SLIDE 41

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.19/26

slide-42
SLIDE 42

Invariance Kernel

Inv(σ): Is the greatest set of points such that every

trajectory starting in such points remains in the set forever.

Computing Invariance Kernels of Polygonal Hybrid Systems – p.20/26

slide-43
SLIDE 43

Invariance Kernel

Inv(σ): Is the greatest set of points such that every

trajectory starting in such points remains in the set forever. Example: σ = e1e2 . . . e8e1

R6 R8 e2 R4 R3 R7 R1 R5 e6 e7 R2 e5 e1 e3 e8 e4 Computing Invariance Kernels of Polygonal Hybrid Systems – p.20/26

slide-44
SLIDE 44

Invariance Kernel

Inv(σ): Is the greatest set of points such that every

trajectory starting in such points remains in the set forever. Theorem: If σ is STAY: Inv(σ) =

Preσ( Preσ(J))

Computing Invariance Kernels of Polygonal Hybrid Systems – p.20/26

slide-45
SLIDE 45

Overview of the presentation

  • Motivation
  • Introduction: Hybrid System
  • Polygonal Differential Inclusion System (SPDI)
  • Successors and Predecessors
  • Classification of Simple Cycles
  • Phase Portrait of SPDIs
  • Invariance Kernels
  • Conclusions

Computing Invariance Kernels of Polygonal Hybrid Systems – p.21/26

slide-46
SLIDE 46

Conclusions

ACHIEVEMENTS:

  • Algorithm for obtaining a new object of SPDI’s

Phase Portrait: Invariance Kernel APPLICATIONS:

  • Find “sinks” of non-linear differential equations
  • IK are important for proving termination of a BFS

reachability algorithm for SPDIs FUTURE WORK:

  • Extend the tool SPeeDI to compute Invariance

Kernels

Computing Invariance Kernels of Polygonal Hybrid Systems – p.22/26

slide-47
SLIDE 47

Auxiliary Slides

Computing Invariance Kernels of Polygonal Hybrid Systems – p.23/26

slide-48
SLIDE 48

Viability Kernel

Viab(σ): Is the greatest set of initial points of

trajectories which can cycle forever in σ

Computing Invariance Kernels of Polygonal Hybrid Systems – p.24/26

slide-49
SLIDE 49

Viability Kernel

Viab(σ): Is the greatest set of initial points of

trajectories which can cycle forever in σ Example: σ = e1e2 . . . e8e1

R6 R8 R4 R3 R7 R1 R5 R2

e5 e4 e3 e2 e1 e8 e7 e6

Computing Invariance Kernels of Polygonal Hybrid Systems – p.24/26

slide-50
SLIDE 50

Viability Kernel

Viab(σ): Is the greatest set of initial points of

trajectories which can cycle forever in σ Example: σ = e1e2 . . . e8e1

R6 R8 R4 R3 R7 R1 R5 R2

e5 e4 e3 e2 e1 e8 e7 e6

Theorem: Viab(σ) = Preσ(Dom(Succσ))

Computing Invariance Kernels of Polygonal Hybrid Systems – p.24/26

slide-51
SLIDE 51

Controllability Kernel

Cntr(σ): Is the greatest set of mutually reachable

points via trajectories that remain in the cycle

Computing Invariance Kernels of Polygonal Hybrid Systems – p.25/26

slide-52
SLIDE 52

Controllability Kernel

Cntr(σ): Is the greatest set of mutually reachable

points via trajectories that remain in the cycle Example: σ = e1e2 . . . e8e1

R6 R8 R3 R7 R4 R2 R1 R5

l u e2 e3 e1 e7 e6 e4 e8 e5

Computing Invariance Kernels of Polygonal Hybrid Systems – p.25/26

slide-53
SLIDE 53

Controllability Kernel

Cntr(σ): Is the greatest set of mutually reachable

points via trajectories that remain in the cycle Example: σ = e1e2 . . . e8e1

R6 R8 R3 R7 R4 R2 R1 R5

l u e2 e3 e1 e7 e6 e4 e8 e5

Theorem: Cntr(σ) = (Succσ ∩ Preσ)(CD(σ))

Computing Invariance Kernels of Polygonal Hybrid Systems – p.25/26

slide-54
SLIDE 54

Phase Portrait of SPDIs

Algorithm: phase portrait for SPDIs for each simple cycle σ do Compute Viab(σ) (viability kernel) Compute Cntr(σ) (controllability kernel)

Computing Invariance Kernels of Polygonal Hybrid Systems – p.26/26

slide-55
SLIDE 55

Phase Portrait of SPDIs

Algorithm: phase portrait for SPDIs for each simple cycle σ do Compute Viab(σ) (viability kernel) Compute Cntr(σ) (controllability kernel) Both kernels are exactly computed by non-iterative algorithms!

Computing Invariance Kernels of Polygonal Hybrid Systems – p.26/26