An Algorithmic Approach to Global Asymptotic Stability Verification - - PowerPoint PPT Presentation

October, 2016

An Algorithmic Approach to Global Asymptotic Stability Verification of Hybrid Systems

Miriam García Soto & Pavithra Prabhakar

IMDEA Software Institute & Kansas State University

1

EMSOFT’16 Pittsburgh, PA, USA

Hybrid Systems

Cyber-Physical Systems

Systems controlled by computer-based algorithms integrated in the physical world.

Automotive Robotics Process control

System exhibiting a mixed continuous and discrete behaviour. Hybrid System

Combine control, communication and computation. Design methodology for building high-confidence systems. Discrete and continuous behaviour.

Medical Devices

Cruise control and automatic gearbox

Automatic gearbox

Drive the vehicle velocity to a desired velocity.

Kq Kq τ Z (vd − v)dv

q

T

+ +

+

–

vd v ωhigh ωlow

CRUISE CONTROLLER

GEARBOX

Continuous controller Discrete controller

Integral

Kq(vd − v)

Proportional

˙ v = pr

qT

M

Automatic gearbox: a hybrid system

1 2 3 4

E = 1 p4 ωlow E = 1 p3 ωlow E = 1 p2 ωlow E = 1 p1 ωhigh E = 1 p2 ωhigh E = 1 p3 ωhigh x = ✓ E TI ◆ ˙ x = A1x ˙ x = A2x ˙ x = A3x ˙ x = A4x E = vd − v Difference between desired and current velocity TI Integral part of the torque ˙ E = − pq MrKqE − pq MrTI ˙ TI = −Kq τ E Dynamical equations

Automatic gearbox: a hybrid system

1 2 3 4

E = 1 p4 ωlow E = 1 p3 ωlow E = 1 p2 ωlow E = 1 p1 ωhigh E = 1 p2 ωhigh E = 1 p3 ωhigh ˙ x = A1x ˙ x = A2x ˙ x = A3x ˙ x = A4x

TI E

Dynamics Executions

TI

4 to 3

• 3

to 2

• 2

to 1

• E

1 to 2

• 2

to 3

• 3

to 4

• x3

x0 x1 x2

Stability Notions

Lyapunov Stability (LS)

8

A system is Lyapunov stable with respect to 0 if for every ε > 0 there exists δ > 0 such that every execution σ starting from Bδ(0) implies σ ∈ Bε(0).

✏

9

δ ✏

A system is Lyapunov stable with respect to 0 if for every ε > 0 there exists δ > 0 such that every execution σ starting from Bδ(0) implies σ ∈ Bε(0).

Lyapunov Stability (LS)

10

δ σ(0)

σ

✏

A system is Lyapunov stable with respect to 0 if for every ε > 0 there exists δ > 0 such that every execution σ starting from Bδ(0) implies σ ∈ Bε(0).

Lyapunov Stability (LS)

11

δ σ(0)

σ

✏

A system is Lyapunov stable with respect to 0 if for every ε > 0 there exists δ > 0 such that every execution σ starting from Bδ(0) implies σ ∈ Bε(0).

Lyapunov Stability (LS)

Asymptotic Stability (AS)

12

A system is AS with respect to 0 if it is Lyapunov stable and there exists a value δ > 0 such that every execution σ starting from Bδ(0) converges to 0.

δ σ(0)

σ

Global Asymptotic Stability (GAS)

13

A system is GAS with respect to 0 if it is Lyapunov stable and every execution σ converges to 0. Global asymptotic stability Asymptotic stability

Region Stability (RS)

14

A system is RS with respect to R if for every execution σ there exists a value T ≥ 0 such that σ at time T belongs to R.

R

Global Asymptotic Stability Verification

GAS verification

16

Step 1 : Asymptotic Stability (AS) verification Step 2 : Stability zone construction Step 3 : Region Stability (RS) verification

LHA

PSS PSS G

Z

True

False

True/False

H

,

GAS verification AS verification

RS verification

Hybridization

Stability zone construction

Polyhedral Switched System (PSS)

17

q1 q2 q3 q4 q5 q6 q7 q8 q9 q10

Dynamics are modelled by polyhedral inclusions. Invariants and guards are polyhedral sets.

Step 1: AS verification

18

q1 q2 q3 q4 q5 q6 q7 q8 q9 q10

Local analysis is reduced to the switching predicates passing through the equilibrium point.

Concrete system H

19

q7 q8 q9 q10

Local analysis is reduced to the switching predicates passing through the equilibrium point.

Step 1: AS verification

Concrete system H0

Predicate Abstraction

20

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0

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Predicate Abstraction

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0

22

Predicate Abstraction

= ⇒

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0 Abstract system A(H0, F)

f1 f2 f3 f4

23

Predicate Abstraction

= ⇒

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0 An edge between facets indicates the existence of an execution. Abstract system A(H0, F)

f1 f2 f3 f4

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= ⇒

f1 f2 f3 f4 f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0 An edge between facets indicates the existence of an execution.

Predicate Abstraction

Abstract system A(H0, F)

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= ⇒

f1 f2 f3 f4 f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0 An edge between facets indicates the existence of an execution.

Predicate Abstraction

Abstract system A(H0, F)

26

= ⇒

f1 f2 f3 f4 f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0 An edge between facets indicates the existence of an execution.

Predicate Abstraction

Abstract system A(H0, F)

27

Predicate Abstraction

= ⇒

f1 f2 f3 f4 f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4} Concrete system H0 An edge between facets indicates the existence of an execution. Abstract system A(H0, F)

28

Quantitative Predicate Abstraction

= ⇒

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4}

1

2

Concrete system H0 An edge between facets indicates the existence of an execution. The weight refers to the variation of distance from equilibrium.

f1 f2 f3 f4

Abstract system A(H0, F)

29

Quantitative Predicate Abstraction

= ⇒

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4}

1

2

Concrete system H0 An edge between facets indicates the existence of an execution. The weight refers to the variation of distance from equilibrium.

f1 f2 f3 f4

2

Abstract system A(H0, F)

30

Quantitative Predicate Abstraction

= ⇒

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4}

3 −1

Concrete system H0 An edge between facets indicates the existence of an execution. The weight refers to the variation of distance from equilibrium.

f1 f2 f3 f4

2

Abstract system A(H0, F)

31

Quantitative Predicate Abstraction

= ⇒

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4}

3 −1

Concrete system H0 An edge between facets indicates the existence of an execution. The weight refers to the variation of distance from equilibrium.

f1 f2 f3 f4

2 1 3

Abstract system A(H0, F)

32

Quantitative Predicate Abstraction

= ⇒

f1 f2 f3 f4

2 1 3 1 3 1

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4}

3 −1

Concrete system H0 An edge between facets indicates the existence of an execution. The weight refers to the variation of distance from equilibrium. Abstract system A(H0, F)

33

Quantitative Predicate Abstraction

= ⇒

f1 f2 f3 f4

2 1 3 1 3 1

f1 f2 f3 f4 u1 u2 u3 u4

Facets F = {f1, f2, f3, f4}

3 −1

Concrete system H0 An edge between facets indicates the existence of an execution. The weight refers to the variation of distance from equilibrium. W(π) = 2 · 1 3 · 1 3 · 1 = 2 9 < 1

π

Abstract system A(H0, F)

Model-checking

34

Theorem (Soundness)

Let be a quantitative abstraction. The hybrid system is asymptotically stable if: All executions which eventually remain in a region converge to the origin. Every simple cycle has product of weights on the edges less than 1.

A(H, F)

H

AS verification for the gearbox

35

TI

4 to 3

• 3

to 2

• 2

to 1

• E

1 to 2

• 2

to 3

• 3

to 4

AS verification for the gearbox

36

W(π) = 0.0746 · 2.678 · 1 · 0.0746 · 2.678 · 1 = 0.03991 < 1 ⇒ AS TI l E

l+

l− T +

I

T −

I

0.0746 0.0746 2.678 2.678 1

1

E−

E+

Step 2: Stability zone computation

37

is a stability zone with respect to if every execution starting at will remain forever inside .

q7 q8 q9 q10

R

Z

q7 q8 q9 q10

R

Z

Z Z ⊆ R R R Stability zone Not stability zone

Stability zone computation

38

q7

q8

q9

q10

R

Center region of H

R

Stability zone computation

39

q7 q8 q9 q10

R

d Md

f1

f2

f3

f4

2

1 3 1 3 1

Center region of M = max {1, W(%): % path in }

A(H, F)

H

R

M = 2

Stability zone computation

40

Extract the center region of M = max {1, W(%): % path in }

A(H, F)

H

R

M = 2 Shrink the center region by a factor of M: Z

q7 q8 q9 q10

R

Md r Z r/2M

f1

f2

f3

f4

2

1 3 1 3 1

Stability zone computation for the gearbox

41

TI

Center region Stability zone

E

1 to 2

• 2

to 3

• 3

to 4

• 4

to 3

• 3

to 2

• 2

to 1

Step 3: RS verification

42

Quantitative predicate abstraction. Graph transformation. Termination analysis.

43

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

Quantitative Predicate Abstraction

RS verification for the gearbox

44

RS verification for the gearbox

Graph Transformation

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

45

Delete nodes in the interior of stability zone.

RS verification for the gearbox

Graph Transformation

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

46

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

Delete nodes in the interior of stability zone. Delete non-reachable nodes from initial nodes.

E

RS verification for the gearbox

Graph Transformation

47

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

RS verification for the gearbox

Termination Analysis

48

Existence of an edge with weight ∞ ⇒ RS False.

RS verification for the gearbox

Termination Analysis

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

49

Existence of an edge with weight ∞ ⇒ RS False. Existence of a cycle ⇒ RS inconclusive.

RS verification for the gearbox

Termination Analysis

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

50

Existence of an edge with weight ∞ ⇒ RS False. Existence of a cycle ⇒ RS inconclusive. Existence of nodes with no outgoing edges different to the nodes

• n the boundary of the stability zone ⇒ RS inconclusive.

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

RS verification for the gearbox

Termination Analysis

51

Existence of an edge with weight ∞ ⇒ RS False. Existence of a cycle ⇒ RS inconclusive. Existence of nodes with no outgoing edges different to the nodes

• n the boundary of the stability zone ⇒ RS inconclusive.

TI q4 q4, q3 q3 q3, q2 q2

q2, q1 q1

E

RS verification for the gearbox

Termination Analysis

Region stability established

52

Summary

LHA

PSS PSS

G G0

Z True

False

True/False

H

,

GAS verification

AS verification

RS verification

Hybridization Abstraction & Model-checking Termination analysis

Abstraction

Stability zone construction

G00

Graph transformation

53

Future research

Extension of the algorithmic stability verification to non-linear systems. Compositional analysis for input-output stability verification. Synthesis of state based switching control for a family of dynamical systems.

Pavithra Prabhakar and Miriam García Soto, Counterexample Guided Abstraction Refinement for Stability Analysis, CAV 2016 ———, Hybridization for Stability Analysis of Switched Linear Systems, HSCC 2016 ———, Foundations of Quantitative Predicate Abstraction for Stability Analysis of Hybrid Systems, VMCAI 2015 ———, An algorithmic approach to stability verification of polyhedral switched systems, ACC 2014 ———, Abstraction Based Model-Checking of Stability of Hybrid Systems, CAV 2013

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Thank you