Formal Verification of Cyber-Physical Systems
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1 Formal Verification of Cyber-Physical Systems Matthew Chan, Daniel Ricketts, Sorin Lerner, Gregory Malecha University of California, San Diego veridrone.ucsd.edu
2 Cyber-Physical Systems
3 Cyber-Physical Systems
4 Cyber-Physical Systems
5 Cyber-Physical Systems
6 Cyber-Physical Systems
7 Cyber-Physical Systems
8 Cyber-Physical Systems Program
9 Cyber-Physical Systems Program World
10 Cyber-Physical Systems Program World Sensor
11 Cyber-Physical Systems Actuator Program World Sensor
12 Cyber-Physical Systems Program World
13 Program World v t
14 Program World v t
15 Program World v t
16 Program World v { t
17 Program World x { t
18 Program World v { t
19 Program World v { t
20 Outline
21 Outline How we formalize CPSs in Coq
22 Outline How we formalize CPSs in Coq Stability (graphically)
23 Outline How we formalize CPSs in Coq Stability (graphically) Lyapunov Stability
24 Outline How we formalize CPSs in Coq Stability (graphically) Lyapunov Stability Exponential Stability
25 Outline How we formalize CPSs in Coq Stability (graphically) Lyapunov Stability Exponential Stability Proving Stability using Lyapunov Functions
26 Actuator Program World Sensor
27 Actuator Program World a! = v Sensor
28 Actuator Program World Sensor a! = v v
29 a! Actuator Program World a! = v Sensor v
30 a! Actuator Program World Sensor a! = a v
31 Program World a! = a
32 Program _ World a! = a
33 ( Program _ World ) a! = a
34 ( Program _ World ) ` P a! = v = a
35 ( Program _ World ) ` ( Safe
36 ( Program _ World ) ` ( Safe Velocity Height Boundary box [Ricketts et al. MEMOCODE 15] etc
37 ( Program _ World ) Stable ` (
38 Stability
39
40
41
42
43 Lyapunov stability
44 v Lyapunov stability t
45 { v Lyapunov stability 8 > 0! t
46 { { v Lyapunov stability 8 > 0!9 > 0 ^ ( x < ) t
47 v Lyapunov stability 8 > 0!9 > 0 ^ ( x < )! ( x < ) t
48 v Lyapunov stability 8 > 0!9 > 0 ^ ( x < )! ( x < ) t
49 v Lyapunov stability 8 > 0!9 > 0 ^ ( x < )! ( x < ) t
50 v Lyapunov stability 8 > 0!9 > 0 ^ ( x < )! ( x < ) t
51 v Proving Lyapunov stability t
52 Proving Lyapunov stability v 8 > 0!9 > 0 ^ ( x < )! ( x < ) t
53 Proving Lyapunov stability v 8 > 0!9 > 0 ^ ( x < )! ( x < ) t
54 Proof Sketch
55 Proof Sketch Spec = Ctrl _ World
56 Proof Sketch a! = v Spec = Ctrl _ World
57 Proof Sketch a! = a Spec = Ctrl _ World
58 Proof Sketch a! = a Spec = Ctrl _ World Inv = v<0! vt apple x ^ v 0! vt apple x
59 Proof Sketch a! = a Spec = Ctrl _ World Inv = v<0! vt apple x ^ v 0! vt apple x Spec ` (Inv! next(inv))
60 v Improving Lyapunov stability t
61 v Exponential stability 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) Displacement (x) ` t
62 v Exponential stability 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) ` t
63 v Exponential stability 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) Displacement (x) ` t
64 v Exponential stability 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) Displacement (x) ` t
65 v Exponential stability 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) Displacement (x) ` t
66 v Exponential stability 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) ` t
67 Proving Exponential stability v 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) t
68 Proving Exponential stability v 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) where = c v 0 = 1 t
69 Proving Exponential stability v 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) where = c v 0 = 1 t
70 Proving Exponential stability v 9 > 0, 9 > 0 ^ ( v apple e (t t 0 ) ) where = c v 0 = 1 t
71 Proving Stability with Lyapunov Functions The difficulty of proving stability manually is
72 Proving Stability with Lyapunov Functions The difficulty of proving stability manually is explicit reasoning about time
73 Proving Stability with Lyapunov Functions The difficulty of proving stability manually is explicit reasoning about time complicated inductive invariants
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
75 Lyapunov functions E(v) v
76 Lyapunov functions E(v) v
77 Lyapunov functions E(0) = 0 E(v) 0,v 6= 0
78 Lyapunov functions E(0) = 0 E(v) 0,v 6= 0 Lyapunov stable: Ė(v) apple 0
79 Lyapunov functions E(0) = 0 E(v) 0,v 6= 0 Lyapunov stable: Ė(v) apple 0 Exponentially stable: Ė(v) apple E(v)
80 Recap
81 Recap How we formalize CPSs in Coq
82 Recap How we formalize CPSs in Coq Stability (graphically)
83 Recap How we formalize CPSs in Coq Stability (graphically) Lyapunov Stability
84 Recap How we formalize CPSs in Coq Stability (graphically) Lyapunov Stability Exponential Stability
85 Recap How we formalize CPSs in Coq Stability (graphically) Lyapunov Stability Exponential Stability Proving Stability with Lyapunov Functions
86 Lessons learned
87 Lessons learned Coq can be applied to continuous domains I learned some Coq I learned some control theory
88 Lessons learned Coq can be applied to continuous domains I learned some Coq I learned some control theory Better solvers for real numbers
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 organized
90 Thanks! veridrone.ucsd.edu
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