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Algorithmic Analysis of Polygonal Hybrid Systems G ERARDO S CHNEIDER V ERIMAG G RENOBLE Algorithmic Analysis of Polygonal Hybrid Systems p.1/66 Hybrid Systems Hybrid Systems: interaction between discrete and continuous behaviors

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SLIDE 1

Algorithmic Analysis of Polygonal Hybrid Systems

GERARDO SCHNEIDER

VERIMAG GRENOBLE

Algorithmic Analysis of Polygonal Hybrid Systems – p.1/66

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SLIDE 2

Hybrid Systems

  • Hybrid Systems: interaction between discrete and

continuous behaviors

  • Examples: thermostat, automated highway

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.2/66

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SLIDE 3

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.2/66

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SLIDE 4

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

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SLIDE 5

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

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SLIDE 6

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

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SLIDE 7

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

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SLIDE 8

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

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SLIDE 9

Hybrid Systems

Example: Swimmer in a whirlpool

e10 e9 e12 e11 e2 e4 e5 e8 e1

x0

e6 e7 e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

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SLIDE 10

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.4/66

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SLIDE 11

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.4/66

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SLIDE 12

Polygonal Differential Inclusion Systems (SPDIs)

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

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SLIDE 13

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

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SLIDE 14

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

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SLIDE 15

Polygonal Differential Inclusion Systems (SPDIs)

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

˙ x = a7 ˙ x = a8 ˙ x = a4 Inv(ℓ2) ˙ 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 Inv(ℓ4) Inv(ℓ3) Inv(ℓ1) Inv(ℓ8) Inv(ℓ7) Inv(ℓ6) ˙ x = a6 Inv(ℓ5) ˙ x = a5

R1 R5 R8 R7 R6 R3 R4

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

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SLIDE 16

Polygonal Differential Inclusion Systems (SPDIs)

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

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

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SLIDE 17

Overview of the presentation

  • Motivation and Contributions
  • Algorithm for Reachability Problem (SPDIs)
  • Implementation – SPeeDI
  • Algorithm for Phase Portrait construction

(SPDIs)

  • Other 2 dim Hybrid Systems
  • Between Decidability and Undecidability
  • Undecidability results
  • Summary of Results and Perspectives

Algorithmic Analysis of Polygonal Hybrid Systems – p.6/66

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SLIDE 18

Motivation and Contributions

Algorithmic Analysis of Polygonal Hybrid Systems – p.7/66

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SLIDE 19

Motivation and Contributions

Scientific Context (Reachability)

dim reachability Planar 2−dim Decidable Undecidable 3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 20

Motivation and Contributions

Scientific Context (Reachability)

dim reachability

PCDs

Planar 2−dim Decidable Undecidable 3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 21

Motivation and Contributions

Scientific Context (Reachability)

dim reachability

PCDs

Planar 2−dim Decidable Undecidable 3−dim

Piecewise Constant Derivative System

e3 e2 e4 e5 e9 e12 e1 e8 e11 e7 e6 e10 Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 22

Motivation and Contributions

Scientific Context (Reachability)

dim reachability

PCDs

Planar 2−dim Decidable Undecidable 3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 23

Motivation and Contributions

Scientific Context (Reachability)

dim reachability

PCDs PCDs

Planar 2−dim Decidable Undecidable 3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 24

Motivation and Contributions

Scientific Context (Reachability)

dim reachability

PCDs PCDs

Planar 2−dim Decidable Undecidable 3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 25

Motivation and Contributions

Challenge

dim reachability

Non deterministic

PCDs

Extensions

  • f

PCDs PCDs PCDs

Planar 2−dim Decidable Undecidable 3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 26

Motivation and Contributions

Contributions (Reachability)

dim

SPDIs PCDs

Extensions

  • f

Extensions

  • f

PCDs PCDs PCDs

Planar 2−dim Decidable Undecidable 3−dim Problem Open reachability

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

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SLIDE 27

Motivation and Contributions

Scientific Context (Phase Portrait)

  • Phase Portrait for PCDs
  • Numerical algorithms for computing Viability

Kernels

Algorithmic Analysis of Polygonal Hybrid Systems – p.9/66

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SLIDE 28

Motivation and Contributions

Contributions (Phase Portrait)

  • Phase Portrait for SPDIs
  • Viability Kernel
  • Controllability Kernel

Algorithmic Analysis of Polygonal Hybrid Systems – p.9/66

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SLIDE 29

Reachability Analysis for SPDIs

Algorithmic Analysis of Polygonal Hybrid Systems – p.10/66

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SLIDE 30

The Reachability Problem for SPDIs

e3 e10 e9 e12 e11 e2 e4 e5 e8 e1 xf x0 e7 e6

Algorithmic Analysis of Polygonal Hybrid Systems – p.11/66

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SLIDE 31

The Reachability Problem for SPDIs

e3 e10 e9 e12 e11 e2 e4 e5 e8 e1 xf x0 e7 e6

Reachability problem: Is there a trajectory from x0 to xf?

Algorithmic Analysis of Polygonal Hybrid Systems – p.11/66

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SLIDE 32

Solving the Reachability Prob- lem

  • 1. From trajectories to simplified trajectories

Algorithmic Analysis of Polygonal Hybrid Systems – p.12/66

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SLIDE 33
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 34
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 35
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11 R8 x′

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 36
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11 R8

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 37
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 38
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 39
  • 1. Simplification of trajectories

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

Theorem: If there is an arbitrary trajectory between two points then it always exists a straightened non–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

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SLIDE 40

Solving the Reachability Prob- lem

  • 1. From trajectories to simplified trajectories
  • 2. From simplified trajectories to signatures

Algorithmic Analysis of Polygonal Hybrid Systems – p.14/66

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SLIDE 41
  • 2. Abstraction into signatures

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

σ = e1e2e3 . . . e5e6e7 . . . e13e6e7e8e15

Algorithmic Analysis of Polygonal Hybrid Systems – p.15/66

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SLIDE 42

Solving the Reachability Prob- lem

  • 1. From trajectories to simplified trajectories
  • 2. From simplified trajectories to signatures
  • 3. From signatures to factorized signatures

Algorithmic Analysis of Polygonal Hybrid Systems – p.16/66

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SLIDE 43
  • 3. Factorization of Signatures

For σ = e1e2e3 . . . e5e6e7 . . . e13e6e7e8e15

e11 e10 e9 e8 e7 e6 e5 e3 e4 e13 e14 e15 e2 e1 e12 x′ x R3 R1 R2 R4 R5 R6 R7 R8 R9 R10 R12 R11

We obtain the representation: σ = e1e2e3 (e4e1e2e3)2e5e6e7e8 (e9 · · · e13e6e7e8)2 e15

Algorithmic Analysis of Polygonal Hybrid Systems – p.17/66

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SLIDE 44

3. Canonical Factorization of Signatures

Representation Theorem: Any edge signature σ = e1, e2, . . . , en can be represented as σ = r1(s1)k1r2(s2)k2 . . . rn(sn)knrn+1

Algorithmic Analysis of Polygonal Hybrid Systems – p.18/66

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SLIDE 45

3. Canonical Factorization of Signatures

Representation Theorem: Any edge signature σ = e1, e2, . . . , en can be represented as σ = r1(s1)k1r2(s2)k2 . . . rn(sn)knrn+1

  • Properties:
  • ri is a seq. of pairwise different edges;
  • si is a simple cycle;
  • ri and rj are disjoint
  • si and sj are different

Proof based on topological properties of the plane

Algorithmic Analysis of Polygonal Hybrid Systems – p.18/66

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SLIDE 46

Solving the Reachability Prob- lem

  • 1. From trajectories to simplified trajectories
  • 2. From simplified trajectories to signatures
  • 3. From signatures to factorized signatures
  • 4. From factorized signatures to types of signatures

Algorithmic Analysis of Polygonal Hybrid Systems – p.19/66

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SLIDE 47
  • 4. Types of Signatures

Abstraction: Any edge signature σ = r1(s1)k1r2(s2)k2 . . . rn(sn)knrn+1 belongs to a type type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

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SLIDE 48
  • 4. Types of Signatures

Abstraction: Any edge signature σ = r1(s1)k1r2(s2)k2 . . . rn(sn)knrn+1 belongs to a type type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

s1 s2 sn rn rn+1 r3 r2 r1

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

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SLIDE 49
  • 4. Types of Signatures

Abstraction: Any edge signature σ = r1(s1)k1r2(s2)k2 . . . rn(sn)knrn+1 belongs to a type type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1 In the previous example: type(σ) = e1e2e3, e4e1e2e3, e5e6e7e8, e9 · · · e13e6e7e8, e15

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

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SLIDE 50
  • 4. Types of Signatures

Abstraction: Any edge signature σ = r1(s1)k1r2(s2)k2 . . . rn(sn)knrn+1 belongs to a type type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

  • Prop. The set of types of signatures is finite

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

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SLIDE 51

Solving the Reachability Prob- lem

  • 1. From trajectories to simplified trajectories
  • 2. From simplified trajectories to signatures
  • 3. From signatures to factorized signatures
  • 4. From factorized signatures to types of signatures
  • 5. Analysis of each type of signature (computing

successors)

Algorithmic Analysis of Polygonal Hybrid Systems – p.21/66

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SLIDE 52

Computing Successors (for σ)

One step (σ = e1e2)

e4 e5 e11 e12 e9 e10 e2 e3 e1 e13 e8 e7 e6 x I2

[a1x + b1, a1x + b1]

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

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SLIDE 53

Computing Successors (for σ)

Several steps (σ = e1e2e3)

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

e4 e5 e11 e12 e9 e10 e2 e3 e1 e13 e8 e7 e6 I3 x

I3 = Succσ(x) = [a2x + b2, a′

2x + b′ 2] ∩ e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

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SLIDE 54

Computing Successors (for σ)

Several steps (σ = e1e2e3e4e5)

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

x e5 e4 e1 e10 e9 e12 e11 e6 e7 e8 e13 e2 e3

I5 = Succσ(x) = [a4x + b4, a′

4x + b′ 4] ∩ e5

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

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SLIDE 55

Computing Successors (for σ)

One cycle (σ = s = e1e2 · · · e8e1)

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

x e5 e4 e8 e11 e10 e9 e12 e13 e7 e6 e3 e2 e1 I9

I9 = Succσ(x) = [a8x + b8, a′

8x + b′ 8] ∩ e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

slide-56
SLIDE 56

Computing Successors (for σ)

One cycle (σ = s = e1e2 · · · e8e1)

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

x e5 e4 e8 e11 e10 e9 e12 e13 e7 e6 e3 e2 e1 I∗ u∗ l∗

l∗ = a1l∗ + b1 u∗ = a2u∗ + b2 I∗ = Succ∗

σ(x) = [l∗, u∗] ∩ e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

slide-57
SLIDE 57

Computing Successors (for σ)

σ = (s)∗e13 (s = e1e2 · · · e8e1)

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄

e8 e13 e1 e2 e4 e6 e7 e11 e10 e9 e12 e5

x I′

e3

One cycle iterated: solution of fixpoint equation (acceleration): I′ = Succe8e13 ◦ Succe1···e8 ◦ Succ∗

s(x)

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

slide-58
SLIDE 58

Computing Successors

Lemma: Successors have the form Succσ(l, u) = [a1l + b1, a2u + b2] ∩ J if [l, u] ⊆ S Lemma: Fixpoint equations [a1l∗ + b1, a2u∗ + b2] = [l∗, u∗] can be explicitely solved (without iterating). We have that (I = [l, u]): Succ∗

σ(I) = [l∗, u∗] ∩ J

Algorithmic Analysis of Polygonal Hybrid Systems – p.23/66

slide-59
SLIDE 59

Reachability Algorithm

for each type of signature τ do check whether Reachτ(x0, xf) To test whether Reachτ(x0, xf) for τ = r1(s1)∗ · · · (sn)∗rn+1 Compute Succr Accelerate (Succs)∗

Algorithmic Analysis of Polygonal Hybrid Systems – p.24/66

slide-60
SLIDE 60

Reachability: Main Result

  • The capability of computing fixpoints for simple

cycles (acceleration)

  • The set of types of signatures is finite

Reachability is decidable for SPDI

Algorithmic Analysis of Polygonal Hybrid Systems – p.25/66

slide-61
SLIDE 61

SPeeDI: a Tool for SPDIs

Algorithmic Analysis of Polygonal Hybrid Systems – p.26/66

slide-62
SLIDE 62

Implementation: SPeeDI

  • We have implemented the reachability algorithm

for SPDIs: SPeeDI (joint work with Gordon Pace)

  • Language: Haskell

Algorithmic Analysis of Polygonal Hybrid Systems – p.27/66

slide-63
SLIDE 63

Implementation: SPeeDI

  • We have implemented the reachability algorithm

for SPDIs: SPeeDI (joint work with Gordon Pace)

  • Language: Haskell

<trace> <type_of_signature> simsig2fig simsig reachable <file.fig>

NO YES

<file.spdi> <input interval> <exit interval>

Algorithmic Analysis of Polygonal Hybrid Systems – p.27/66

slide-64
SLIDE 64

Implementation: SPeeDI

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

slide-65
SLIDE 65

Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

slide-66
SLIDE 66

Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

slide-67
SLIDE 67

Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

slide-68
SLIDE 68

Phase Portrait of SPDIs

Algorithmic Analysis of Polygonal Hybrid Systems – p.29/66

slide-69
SLIDE 69

Phase Portrait

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

slide-70
SLIDE 70

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

slide-71
SLIDE 71

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

slide-72
SLIDE 72

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

slide-73
SLIDE 73

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

slide-74
SLIDE 74

Viability Kernel

Viab(σ): Is the greatest set of initial points of trajectories which can cycle forever in σ

Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

slide-75
SLIDE 75

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

slide-76
SLIDE 76

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))

Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

slide-77
SLIDE 77

Controllability Kernel

Cntr(σ): Is the greatest set of mutually reachable points via trajectories that remain in the cycle

Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

slide-78
SLIDE 78

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

slide-79
SLIDE 79

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))

Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

slide-80
SLIDE 80

Viability Kernel

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

Algorithmic Analysis of Polygonal Hybrid Systems – p.33/66

slide-81
SLIDE 81

Viability Kernel

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!

Algorithmic Analysis of Polygonal Hybrid Systems – p.33/66

slide-82
SLIDE 82

Properties of the Kernels

Theorem: Any viable trajectory in σ converges to Cntr(Kσ)

R6 R8 R4 R2 R3 R7 R5 R1

e6 e7 e8 e1 e2 e3 e4 e5

  • Controllability Kernel: “

Weak”analog of limit cycle

  • Viability Kernel: Its “

local”attraction basin

Algorithmic Analysis of Polygonal Hybrid Systems – p.34/66

slide-83
SLIDE 83

Convergence Properties

Every trajectory with infi nite signature without self-crossings converges to the controllability kernel of some simple edge-cycle

R5 R4 R3 R2 R11 R12 e12 R13 R14 R15 R1 R8 R7 R6

e5 e4 e3 e2 e1 e8 e7 e6 e15 e14 e13 e11 e10

Algorithmic Analysis of Polygonal Hybrid Systems – p.35/66

slide-84
SLIDE 84

Between Decidable and Undecidable

Algorithmic Analysis of Polygonal Hybrid Systems – p.36/66

slide-85
SLIDE 85

More complex 2-dim systems

What happens if ...

  • ...we allow jumps?
  • ...the PCD is on a 2-dim surface/manifold?
  • ...?

Algorithmic Analysis of Polygonal Hybrid Systems – p.37/66

slide-86
SLIDE 86

More complex 2-dim systems

What happens if ...

  • ...we allow jumps?
  • ...the PCD is on a 2-dim surface/manifold?
  • ...?

Answer: Reachability is equivalent to a well known

  • pen problem

Algorithmic Analysis of Polygonal Hybrid Systems – p.37/66

slide-87
SLIDE 87

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-88
SLIDE 88

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

I3 R I5 I2 a1x + b1 I4 I1

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-89
SLIDE 89

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

I3 R I5 I2 a1x + b1 a5x + b5 I4 I1

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-90
SLIDE 90

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

I3 R I5 I2 a4x + b4 a1x + b1 a5x + b5 I4 I1

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-91
SLIDE 91

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

I3 R a2x + b2 I5 I2 a4x + b4 a1x + b1 a5x + b5 I4 I1

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-92
SLIDE 92

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

I3 R a2x + b2 I5 I2 a4x + b4 a1x + b1 a5x + b5 I4 I1

Reachability?

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-93
SLIDE 93

Our Reference Model

1-dim Piecewise Affine Maps (PAMs): f : R → R, f(x) = aix + bi for x ∈ Ii

I3 R a2x + b2 I5 I2 a4x + b4 a1x + b1 a5x + b5 I4 I1

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

slide-94
SLIDE 94

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-95
SLIDE 95

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-96
SLIDE 96

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Reachability?

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-97
SLIDE 97

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Reachability?

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-98
SLIDE 98

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Reachability?

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-99
SLIDE 99

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Reachability?

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-100
SLIDE 100

PCD on 2-dim manifolds (PCD2m)

Example: Torus

R2 R1 R3 R4

Reachability? Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

slide-101
SLIDE 101

Hierarchical PCDs (HPCD)

x := ax + b y := 0

R1 R4 R3

(a1, b1) (a4, b4) (a3, b3) (a2, b2)

R2 PCD2 ℓ3 ℓ2 ℓ1 ℓ2 PCD3 PCD1 g g

x := ax + b

g

y := 0 x := ax + b y := 0

Algorithmic Analysis of Polygonal Hybrid Systems – p.40/66

slide-102
SLIDE 102

Hierarchical PCDs (HPCD)

x := ax + b y := 0

R1 R4 R3

(a1, b1) (a4, b4) (a3, b3) (a2, b2)

R2 PCD2 ℓ3 ℓ2 ℓ1 ℓ2 PCD3 PCD1 g g

x := ax + b

g

y := 0 x := ax + b y := 0

Reachability? Theorem: HPCD ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.40/66

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SLIDE 103

Undecidable 2-dim Systems

Algorithmic Analysis of Polygonal Hybrid Systems – p.41/66

slide-104
SLIDE 104

Undecidability Results

  • HPCDs with One Counter (HPCD1c)
  • HPCDs with Infinite Partition (HPCD∞)
  • Origin-dependent rate HPCDs (HPCDx)

Algorithmic Analysis of Polygonal Hybrid Systems – p.42/66

slide-105
SLIDE 105

Undecidability Results

  • HPCDs with One Counter (HPCD1c)
  • HPCDs with Infinite Partition (HPCD∞)
  • Origin-dependent rate HPCDs (HPCDx)

Reachability?

Algorithmic Analysis of Polygonal Hybrid Systems – p.42/66

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SLIDE 106

Undecidability Results

  • HPCDs with One Counter (HPCD1c)
  • HPCDs with Infinite Partition (HPCD∞)
  • Origin-dependent rate HPCDs (HPCDx)

Reachability? UNDECIDABLE! Theorem: HPCD1c, HPCD∞ and HPCDx simulate Turing machines

Algorithmic Analysis of Polygonal Hybrid Systems – p.42/66

slide-107
SLIDE 107

Summary of Results

Algorithmic Analysis of Polygonal Hybrid Systems – p.43/66

slide-108
SLIDE 108

Summary of Results

PCD A B "A is a particular case of B" SPDI

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-109
SLIDE 109

Summary of Results

PCD A B "A is a particular case of B" SPDI

decidable Controllability kernel Convergence properties Viability kernel Exact computation Abstraction Acceleration Poincare map SPeeDI

Reachability analysis Phase Portrait

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-110
SLIDE 110

Summary of Results

HPCD PCD A B "A is a particular case of B" SPDI

decidable

Reachability analysis Phase Portrait

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-111
SLIDE 111

Summary of Results

HPCD PCD A B "A is a particular case of B" SPDI

decidable

HPCDx HPCD∞ HPCD1c

Reachability analysis Phase Portrait

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-112
SLIDE 112

Summary of Results

TM HPCD PCD A A B B "A is a particular case of B" "A is simulated by B" SPDI

decidable undecidable

HPCDx HPCD∞ HPCD1c

Reachability analysis Phase Portrait

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-113
SLIDE 113

Summary of Results

TM HPCD PCD A A B B "A is a particular case of B" "A is simulated by B" SPDI

decidable undecidable

HPCDx HPCD∞ HPCD1c RA2cl PCD2m HPCDiso RA1sk1sl LAst RA1cl1mc

Reachability analysis Phase Portrait

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-114
SLIDE 114

Summary of Results

TM HPCD PCD A A PAM B B "A is a particular case of B" "A is simulated by B" SPDI

decidable undecidable do not know

HPCDx HPCD∞ HPCD1c PAMinj RA2cl PCD2m HPCDiso RA1sk1sl LAst RA1cl1mc

Reachability analysis Phase Portrait

Algorithmic Analysis of Polygonal Hybrid Systems – p.44/66

slide-115
SLIDE 115

Perspectives

  • SPDI to approximate non-linear differential

equations

  • Conditions for decidability of PCDs on 2-dim

manifolds

  • Application of the geometric method to higher

dimensions

  • Extension of SPeeDI: algorithm for viability and

controllability kernels

  • SPeeDI: “Topological” optimizations

Algorithmic Analysis of Polygonal Hybrid Systems – p.45/66

slide-116
SLIDE 116

Merci! Gracias! Obrigado!

(Brasil Penta-Campeão!)

Thank you!

Algorithmic Analysis of Polygonal Hybrid Systems – p.46/66

slide-117
SLIDE 117

Theorem de Poincaré-Bendixson

A non-empty compact limit set of C1 planar dynamical system that contains no equilibrium points is a close orbit.

Algorithmic Analysis of Polygonal Hybrid Systems – p.47/66

slide-118
SLIDE 118

Comparison with HyTech

Example:

(1, −2) (−1, −2) (−1,

9 10 )

(1, 1) (−1,

1 10 )

Fixpoint: I∗ = (200

9 ; 200)

Algorithmic Analysis of Polygonal Hybrid Systems – p.48/66

slide-119
SLIDE 119

Comparison with HyTech

Final Point HyTech SPeeDI Reachable 199

  • verflow

0.05 sec Yes 200

  • verflow

0.05 sec No 201

  • verflow

0.01 sec No 210

  • verflow

0.05 sec No 5 0.04 sec 0.05 sec No 20 0.07 sec 0.05 sec No

200 9

0.10 sec 0.05 sec Yes

201 9

  • verflow

0.03 sec Yes

199 9

0.07 sec 0.04 sec Yes

Algorithmic Analysis of Polygonal Hybrid Systems – p.48/66

slide-120
SLIDE 120

Comparison with HyTech

Simulation of reachability for xf = 201

9

l1 = 203 10 l∗ = 200 9 If = 201 9 u1 = 118 5 3 4 Algorithmic Analysis of Polygonal Hybrid Systems – p.48/66

slide-121
SLIDE 121

Kσ

e6 e7 e2 e3 e4 e5

R6 R7 R8 R1 R2 R3 R4 R5

e8 e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.49/66

slide-122
SLIDE 122

Composition of TAMFs

TAMFs are closed under composition: For F1(x) = F1({x} ∩ S1) ∩ J1 and F2(x) = F2({x} ∩ S2) ∩ J2 we have that F2 ◦ F1(x) = FF ′,S′,J′(x) with F ′ = F2 ◦ F1, J′ = J2 ∩ F2(J1 ∩ S2) and S′ = S1 ∩ F −1

1 (J1 ∩ S2)

Algorithmic Analysis of Polygonal Hybrid Systems – p.50/66

slide-123
SLIDE 123

Reachability Algorithm (Exam- ple)

e5 e4 e3 e2 e1 e8 e7 e6

R1 R5 R4 R8 R6

x0 = 1

2

R2 R3 R7

xf = 4

5

  • Type of signature: σ = (e1 · · · e8)∗
  • Successor for the loop s = e1 . . . e8:

Succs(l, u) = [ l

2 − 1 20, u 2 + 23 60] ∩ (1 5, 1)

if [l, u] ⊆ (0, 1)

Algorithmic Analysis of Polygonal Hybrid Systems – p.51/66

slide-124
SLIDE 124

Reachability Algorithm (Exam- ple)

  • Fixpoint equation: Succe1...e8(I∗) = I∗
  • Solution: I∗ = [l∗, u∗] = [1

5, 23 30]

  • Hence: Succe1...e8(x0) ⊆ [1

5, 23 30]

...

e4 e3 e2 e6 e1 e8 e7 e5 u∗ = 23

30

xf = 4

5 1 5

x0 = 1

2

Conclusion: xf ∈ [1

5, 23 30]).

Hence,

Algorithmic Analysis of Polygonal Hybrid Systems – p.52/66

slide-125
SLIDE 125

Viability Kernel

K

A

Algorithmic Analysis of Polygonal Hybrid Systems – p.53/66

slide-126
SLIDE 126

Viability Kernel

K

A

Algorithmic Analysis of Polygonal Hybrid Systems – p.53/66

slide-127
SLIDE 127

Viability Kernel

K

A

Algorithmic Analysis of Polygonal Hybrid Systems – p.53/66

slide-128
SLIDE 128

Viability Kernel

K

B A

w

x

z

Viab(K) = A ∪ B

  • M is a viability domain if ∀x ∈ M, ∃ at least one

trajectory ξ, starting in x and remaining in M

  • Viab(K): Viability kernel of K is the largest

viability domain M contained in K

Algorithmic Analysis of Polygonal Hybrid Systems – p.53/66

slide-129
SLIDE 129

Viability Kernel for SPDIs

  • We can easily compute the Viability Kernel for
  • ne cycle, which is a polygon

Algorithmic Analysis of Polygonal Hybrid Systems – p.54/66

slide-130
SLIDE 130

Viability Kernel for SPDIs

  • We can easily compute the Viability Kernel for
  • ne cycle, which is a polygon

e6 e7 e2 e3 e4 e5

R6 R7 R8 R1 R2 R3 R4 R5

e8 e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.54/66

slide-131
SLIDE 131

Viability Kernel for SPDIs

  • We can easily compute the Viability Kernel for
  • ne cycle, which is a polygon

R6 R8 R4 R3 R7 R1 R5 R2

e5 e4 e3 e2 e1 e8 e7 e6

  • Theorem: Viab(Kσ) = Preσ(Dom(Succσ))

Algorithmic Analysis of Polygonal Hybrid Systems – p.54/66

slide-132
SLIDE 132

Controllability Kernel

A

K

Algorithmic Analysis of Polygonal Hybrid Systems – p.55/66

slide-133
SLIDE 133

Controllability Kernel

A

K

Algorithmic Analysis of Polygonal Hybrid Systems – p.55/66

slide-134
SLIDE 134

Controllability Kernel

A

K

Algorithmic Analysis of Polygonal Hybrid Systems – p.55/66

slide-135
SLIDE 135

Controllability Kernel

A

K

x w z

Cntr(K) = A

  • M is controllable if ∀x, y ∈ M, ∃ a trajectory

segment ξ starting in x that reaches an arbitrarily small neighborhood of y without leaving M

  • Controllability kernel of K, denoted Cntr(K), is

the largest controllable subset of K

Algorithmic Analysis of Polygonal Hybrid Systems – p.55/66

slide-136
SLIDE 136

Controllability Kernel for SPDIs

e6 e7 e2 e3 e4 e5

R6 R7 R8 R1 R2 R3 R4 R5

e8 e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.56/66

slide-137
SLIDE 137

Controllability Kernel for SPDIs

R6 R8 R3 R7 R4 R2 R1 R5

l u e2 e3 e1 e7 e6 e4 e8 e5

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

(We know how to compute the special interval CD(σ) = [l, u])

Algorithmic Analysis of Polygonal Hybrid Systems – p.56/66

slide-138
SLIDE 138

PAM simulate HPCD

e1 e5 e2 e0 e4 e3 I1 I2 I3 x′ = a3x + b3 x e1 e2 e3 I1 I2 I3 A2x + B2 A3x + B3 A4x + B4 e0 R

Algorithmic Analysis of Polygonal Hybrid Systems – p.57/66

slide-139
SLIDE 139

HPCD simulate PAM

e′ e γ(e′, x, y) = (e, aix + bi, 0) Ii

Algorithmic Analysis of Polygonal Hybrid Systems – p.58/66

slide-140
SLIDE 140

RA1cl1mc equivalent to PAM

˙ x = 0 ˙ y = 1

0 ≤ y ≤ 1 y = 1 ∧ x ∈ Ii x := aix + bi; y := 0

Algorithmic Analysis of Polygonal Hybrid Systems – p.59/66

slide-141
SLIDE 141

RA2cl equivalent to PAM

˙ x = 1 ˙ y = 1

0 ≤ y ≤ 1

y = 1 ∧ x − 1 ∈ Ii x := ai(x − 1) + bi; y := 0

e′ e γ(e′, x, y) = (e, ai(x − 1) + bi, 0) Ii + 1

Algorithmic Analysis of Polygonal Hybrid Systems – p.60/66

slide-142
SLIDE 142

RA1sk1sl equivalent to PAM

(a) (b) aj(uj − lj) ai(ui − li)

e′

i

ei (−1, ai) ej (1, aj) e′

j

PCDj PCDi uj lj li ui

x = li y := 0, x := y + fi(li) ˙ x = 1 ˙ y = aj 0 ≤ y ≤ aj(uj − lj) lj ≤ x ≤ uj 0 ≤ y ≤ ai(ui − li) ˙ x = −1 ˙ y = ai li ≤ x ≤ ui γ(e′

i, x, y) = (ej, y + fi(li), 0)

ℓj ℓi

Algorithmic Analysis of Polygonal Hybrid Systems – p.61/66

slide-143
SLIDE 143

From RA1sk1sl to LAst

Ci ≤ y ≤ Di Cj ≤ y ≤ Dj ˙ x = 1 ˙ y = ai ˙ x = 1 ˙ y = aj Aj ≤ x ≤ Bj Ai ≤ x ≤ Bi

˙ x = 0 ˙ y = 1 where bi = Ci + Biai and bj = Cj + Bjaj ˙ x = 1 ˙ y = 0 ˙ x = 0 ˙ y = 1 ˙ x = 1 ˙ y = 0 y = −aix + bi y = −ajx + bj y ≥ −aix + bi Ci ≤ y ≤ Di x ≤ Bi y ≤ −aix + bi y ≥ Ci Ai ≤ x ≤ Bi y ≤ −ajx + bj Aj ≤ x ≤ Bj y ≥ Cj y ≥ −ajx + bj Cj ≤ y ≤ Dj x ≤ Bj

y := Cj, x := y + di x = Bi

y := Cj, x := y + di x = Bi

Algorithmic Analysis of Polygonal Hybrid Systems – p.62/66

slide-144
SLIDE 144

PCD2m simulate PAMinj

(a) (b) (c)

aix − aili

Ij

i

Il

i

Ik

i

Ii

h

Ii

m aix + bi v e′ e li ui v′ f(li) −f(li)

Ii Ij

i

Il

i

Ik

i

Ij Ik Il

Algorithmic Analysis of Polygonal Hybrid Systems – p.63/66

slide-145
SLIDE 145

HPCD1c simulate TM

R2 e1 e2 R4 R3 R1 R2 R1 R4 R3 e4 e3 e1 e5 e6 e7 R6 R5 e2 v1 v2 e3 e4 e8

PCDi PCD′

i; PCD′′ i

Algorithmic Analysis of Polygonal Hybrid Systems – p.64/66

slide-146
SLIDE 146

HPCD1c simulate TM

TM-state qi:

PCDi PCD′

i

PCD′′

i

γ(e1, λ, c) = (e2, λ + 1, c − 1) g ≡ e1 ∧ c > 0 γ(e1, λ, c) = (e2, λ + 1, c − 1) g ≡ e1 ∧ c > 0 γ(e2, λ, c) = (e2, λ, c) g ≡ e2 PCDj PCDk γ(e1, λ, c) = (e4, f′′(λ), c) g ≡ e1 ∧ c = 0 g ≡ e1 ∧ c = 0 g ≡ e1 γ(e1, λ, c) = (e4, λ − 1, c + 1) γ(e3, λ, c) = (e2, λ, c) g ≡ e3

qi ℓi ℓ′′

i

ℓ′

i

ℓj ℓk

γ(e1, λ, c) = (e4, f′(λ), c)

Algorithmic Analysis of Polygonal Hybrid Systems – p.65/66

slide-147
SLIDE 147

HPCDx simulate TM

e2 e3 e1 (0, f(x0) = (−1)⌊2x0⌋) (0, f(x0) = (−1)⌊2x0⌋)

f(x) =

  • 1 if fracx < 1

2

−1 otherwise

Algorithmic Analysis of Polygonal Hybrid Systems – p.66/66