Formal Verification of Cyber-Physical Systems Matthew Chan , Daniel - - PowerPoint PPT Presentation

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Formal Verification of Cyber-Physical Systems Matthew Chan , Daniel Ricketts, Sorin Lerner, Gregory Malecha University of California, San Diego veridrone.ucsd.edu Cyber-Physical Systems Cyber-Physical Systems Cyber-Physical Systems

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SLIDE 1

Formal Verification of Cyber-Physical Systems

Matthew Chan, Daniel Ricketts, Sorin Lerner, Gregory Malecha University of California, San Diego veridrone.ucsd.edu

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SLIDE 2

Cyber-Physical Systems

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SLIDE 3

Cyber-Physical Systems

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SLIDE 4

Cyber-Physical Systems

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SLIDE 5

Cyber-Physical Systems

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SLIDE 6

Cyber-Physical Systems

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SLIDE 7

Cyber-Physical Systems

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SLIDE 8

Cyber-Physical Systems

Program

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SLIDE 9

Cyber-Physical Systems

World Program

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SLIDE 10

Cyber-Physical Systems

World Program

Sensor

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SLIDE 11

Cyber-Physical Systems

World Program

Actuator Sensor

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SLIDE 12

Cyber-Physical Systems

Program World

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SLIDE 13

Program

World

v t

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SLIDE 14

Program

World

v t

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SLIDE 15

Program

World

v t

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SLIDE 16

Program

World

v t

{

∆

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SLIDE 17

Program

World

x t

{

∆

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SLIDE 18

Program

World

v t

{

∆

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SLIDE 19

Program World

v t

{

∆

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SLIDE 20

Outline

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SLIDE 21

Outline

  • How we formalize CPSs in Coq
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SLIDE 22

Outline

  • How we formalize CPSs in Coq
  • Stability (graphically)
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SLIDE 23

Outline

  • How we formalize CPSs in Coq
  • Stability (graphically)
  • Lyapunov Stability
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SLIDE 24

Outline

  • How we formalize CPSs in Coq
  • Stability (graphically)
  • Lyapunov Stability
  • Exponential Stability
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SLIDE 25

Outline

  • How we formalize CPSs in Coq
  • Stability (graphically)
  • Lyapunov Stability
  • Exponential Stability
  • Proving Stability using Lyapunov Functions
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SLIDE 26

Program

Actuator Sensor

World

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SLIDE 27

Program

Actuator Sensor

World

a! = − v ∆

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SLIDE 28

Program

Actuator Sensor

World

a! = − v ∆

v

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SLIDE 29

Program

Actuator Sensor

World

a! = − v ∆ a!

v

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SLIDE 30

Program

Actuator Sensor

World

a! = − v ∆

∂v ∂t = a a!

v

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SLIDE 31

Program World

a! = − v ∆

∂v ∂t = a

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SLIDE 32

Program World

∨

a! = − v ∆

∂v ∂t = a

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SLIDE 33

Program World

∨

⇤( )

a! = − v ∆

∂v ∂t = a

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SLIDE 34

`

P

Program World

∨

⇤( )

a! = − v ∆ ˙ x = ∂v ∂t = a

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SLIDE 35

`

Safe

Program World

∨

⇤( ) ⇤(

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SLIDE 36

`

Safe

Program World

∨

⇤( )

[Ricketts et al. MEMOCODE 15]

Velocity Height Boundary box …etc

⇤(

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SLIDE 37

`

Program World

∨

⇤( )

Stable

⇤(

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SLIDE 38

Stability

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SLIDE 39
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SLIDE 40
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SLIDE 41
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SLIDE 42
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SLIDE 43

Lyapunov stability

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SLIDE 44

Lyapunov stability

v t

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SLIDE 45

Lyapunov stability

∀α > 0 → ∃

{

α

v t

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SLIDE 46

Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β)

{

α

{

β

v t

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SLIDE 47

Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β) → ⇤(|x| < α) v t

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SLIDE 48

Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β) → ⇤(|x| < α) v t

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SLIDE 49

Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β) → ⇤(|x| < α) v t

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SLIDE 50

Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β) → ⇤(|x| < α) v t

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SLIDE 51

Proving Lyapunov stability

v t

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SLIDE 52

Proving Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β) → ⇤(|x| < α) v t

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SLIDE 53

Proving Lyapunov stability

∀α > 0 → ∃β > 0 ∧ (|x| < β) → ⇤(|x| < α) v t

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SLIDE 54

Proof Sketch

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SLIDE 55

Proof Sketch

Spec = Ctrl ∨ World

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SLIDE 56

Proof Sketch

Spec = Ctrl ∨ World

a! = − v ∆

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SLIDE 57

Proof Sketch

Spec = Ctrl ∨ World

a! = − v ∆ ∂v ∂t = a

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SLIDE 58

Proof Sketch

Spec = Ctrl ∨ World

a! = − v ∆ ∂v ∂t = a

Inv = v < 0 → −vt ≤ x ∧ v ≥ 0 → vt ≤ −x

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SLIDE 59

Proof Sketch

Spec = Ctrl ∨ World

a! = − v ∆ ∂v ∂t = a

Spec ` (Inv ! next(Inv))

Inv = v < 0 → −vt ≤ x ∧ v ≥ 0 → vt ≤ −x

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SLIDE 60

Improving Lyapunov stability

v t

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SLIDE 61

Exponential stability

Displacement (x)

`

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 62

Exponential stability

Displacement (x)

`

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 63

Exponential stability

Displacement (x)

`

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 64

Displacement (x)

`

Exponential stability

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 65

Exponential stability

Displacement (x)

`

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 66

Exponential stability

Displacement (x)

`

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 67

Proving Exponential stability

v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

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SLIDE 68

Proving Exponential stability

where v t

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

α = c|v0| γ = 1 ∆

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SLIDE 69

Proving Exponential stability

v t where

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

α = c|v0| γ = 1 ∆

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SLIDE 70

Proving Exponential stability

v t where

∃α > 0, ∃γ > 0 ∧ ⇤(|v| ≤ αe−γ(t−t0))

α = c|v0| γ = 1 ∆

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SLIDE 71

Proving Stability with Lyapunov Functions

The difficulty of proving stability manually is

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SLIDE 72

Proving Stability with Lyapunov Functions

The difficulty of proving stability manually is

  • explicit reasoning about time
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SLIDE 73

Proving Stability with Lyapunov Functions

The difficulty of proving stability manually is

  • explicit reasoning about time
  • complicated inductive invariants
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SLIDE 74

Proving Stability with Lyapunov Functions

The difficulty of proving stability manually is

  • explicit reasoning about time
  • complicated inductive invariants
  • manual proof of Lyapunov stability is 190 lines;

46 lines using Lyapunov function

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SLIDE 75

Lyapunov functions

E(v) v

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SLIDE 76

Lyapunov functions

E(v) v

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SLIDE 77

Lyapunov functions

E(0) = 0

E(v) 0, v 6= 0

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SLIDE 78

Lyapunov functions

E(0) = 0

E(v) 0, v 6= 0

Lyapunov stable: ˙ E(v) ≤ 0

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SLIDE 79

Lyapunov functions

E(0) = 0

E(v) 0, v 6= 0

Lyapunov stable: ˙ E(v) ≤ 0

˙ E(v) ≤ αE(v)

Exponentially stable:

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SLIDE 80

Recap

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SLIDE 81

Recap

  • How we formalize CPSs in Coq
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SLIDE 82

Recap

  • How we formalize CPSs in Coq
  • Stability (graphically)
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SLIDE 83

Recap

  • How we formalize CPSs in Coq
  • Stability (graphically)
  • Lyapunov Stability
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SLIDE 84

Recap

  • How we formalize CPSs in Coq
  • Stability (graphically)
  • Lyapunov Stability
  • Exponential Stability
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SLIDE 85

Recap

  • How we formalize CPSs in Coq
  • Stability (graphically)
  • Lyapunov Stability
  • Exponential Stability
  • Proving Stability with Lyapunov Functions
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SLIDE 86

Lessons learned

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SLIDE 87

Lessons learned

  • Coq can be applied to continuous domains
  • I learned some Coq
  • I learned some control theory
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SLIDE 88

Lessons learned

  • Coq can be applied to continuous domains
  • I learned some Coq
  • I learned some control theory
  • Better solvers for real numbers
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SLIDE 89

Lessons learned

  • Coq can be applied to continuous domains
  • I learned some Coq
  • I learned some control theory
  • Better solvers for real numbers
  • Real analysis library could be modernized and better
  • rganized
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SLIDE 90

Thanks!

veridrone.ucsd.edu