[PPT] - Invariant-Based Verification and Synthesis for Hybrid Systems PowerPoint Presentation

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Invariant-Based Verification and Synthesis for Hybrid Systems

Naijun Zhan

Institute of Software, Chinese Academy of Sciences (Joint work with Hengjun Zhao, Jiang Liu, Deepak Kapur, Kim G. Larsen, Liang Zou, etc.)

IFIP WG 2.2 Scientific Meeting, IMS, Singapore

Sept. 12-16, 2016

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Outline

Background
Invariant and Verification
Invariant-Based Synthesis
Case Studies
Conclusion

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Outline

Background
Invariant and Verification
Invariant-Based Synthesis
Case Studies
Conclusion

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Classification of Dynamical Systems

Discrete
Continuous

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ON

) ( d d x f t x  x ) (t x

OFF

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Hybrid System

Continuous + Discrete

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by Heer Rami http://www.benettonplay.com/toys/flipbookdeluxe/player.php?id=294504

Universal Law of Gravitation

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Hybrid Automata

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) (

1 x

f x  ) (

2 x

f x 

domain guard transition Initial

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HSs in Engineering

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Electrical Circuits Chemical Process

http://people.ee.ethz.ch/~mpt/2/docs/demos/twotanks.php

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Embedded Control Systems

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logic/computation sensor actuator physical discipline

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Safety Critical Systems

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Motivation

Develop formal methods for enhancing the

trustworthiness of safety critical embedded systems

Problems: Verification and Design System Requirements: mainly safety Techniques: symbolic/rigorous computation

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Outline

Background
Invariant and Verification
Invariant-Based Synthesis
Case Studies
Conclusion

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Deductive Verification

Program

x:=1; while (x<=1000000000) { x:=x+1; } x≦0

Inductive Invariant

 x=1  x≧1  x≧1  x+1≧1  x≧1  ﹁(x≦0)

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Continuous system

) ( d d x f t x 

Inductive Invariant

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Inductiveness

Discrete
Inductiveness
Transition relation

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Continuous
Inductiveness
Transition relation

I x I x

k k

  

1

) (

1 k k

x x  



Δt

I t t x I t x    

 )

( ) (

t t x t x t t x

 

    ) ( ) ( ) (

'

I I

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Lie Derivatives and Invariant

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) ( d d x f t x 

p(x) > 0 p(x) = 0

d )) ( ( d  t t x p d )) ( ( d  t t x p d )) ( ( d  t t x p

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Higher-Order Lie Derivatives

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p(x) > 0 p(x) = 0

d )) ( ( d  t t x p

d d

1 1

 t p d d d d

2 2 1 1

    t p t p d d d d d d

3 3 2 2 1 1

      t p t p t p         d d d d d d

3 3 2 2 1 1

t p t p t p

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Criterion for Invariant

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      0 d d

1 1

t p     d d d d

2 2 1 1

t p t p           d d d d d d

2 2 1 1 N N

t p t p t p 

) ( d d x f t x 

f(x) and p(x) are polynomials
Compute an upper bound N s.t.
p(x) ≥ 0 is an inductive invariant of

iff

 0 p   

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Main Result

Semi-algebraic set

,

First-order theory of real numbers is decidable

Quantifier Elimination

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Checking whether a semi-algebraic set is an inductive invariant of a polynomial continuous dynamical systems is decidable

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Parametric Case

Parametric polynomials p(u,x)
p(u,x) ≥ 0 is an inductive invariant of

iff u satisfies

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      0 d d

1 1

t p     d d d d

2 2 1 1

t p t p           d d d d d d

2 2 1 1 N N

t p t p t p   0 ) , ( x u p   

Use parametric polynomials and quantifier elimination (or other compuation techniques) to automatically discovering inductive invariants

) ( d d x f t x 

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Inductive Invariant of HSs

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2 1, Inv

Inv

Init Inv1 Inv2

1

Inv Init 

2 12 1

Inv G Inv  

1 21 2

Inv G Inv  

) (

1 x

f x 

) (

2 x

f x  

12

G

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G

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Safety Verification

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S Inv

                      y x x y y x 3

3

 

Try to generate an

invariant that implies the safety property

Example

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Outline

Background
Invariant and Verification
Invariant-Based Synthesis
Case Studies
Conclusion

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Problem Description

Given an initial specification of a hybrid

system and a safety requirement, construct a refined hybrid system such that the safety requirement is satisfied

domains guards

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Nuclear Reactor

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http://commons.wikimedia.org/wiki/File:Control_rods_schematic.svg

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Hybrid Automata Model

x: temperature of the reactor
p: fraction of the rod immersed into the reactor

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Violation of Safety

510 x 550

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x p

550 510 1

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Invariant for Refinement

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Domain1 Domain2 Guard12

S

Inv

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Result

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92 . 547 12 6575   x

Inv

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Optimization

Further refine the hybrid system according to

certain optimization criteria

polynomial objective function +

semi-algebraic feasible region

Symbolic optimization

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Outline

Background
Invariant and Verification
Invariant-Based Synthesis
Case Studies

Oil pump Lunar lander

Conclusion

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Oil Pump Switching

First studied in

[Cassez et al. HSCC09, 45% improvement]

Provided by the German

company HYDAC

Determine the time points

to switch the pump on/off s.t.

Safety: Optimality:

minimize

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Synthesized Switching Controller

v0 is the initial volume of oil

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n off
n
ff

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Safety
Improve the optimal value of [HSCC09] by 7.5%
The synthesized controller is correct, also optimal 32

Performance

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Soft Landing

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15km 3km 2.4km 100m 30m 0m Braking Adjustment Approach Obstacle avoidance Slow descent

Lunar surface

Hovering

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Slow Descent Phase

Trajectory control
Sampling period：∆T = 0.128s
Control objective: v = -2m/s

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Hybrid Automata Model

Dynamics
Replace the non-polynomial term by a new

variable: a = Fc/m

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Verification

Safety requirement: |v – (-2)|

0.05

Generated Invariant:

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Kong, H., He, F., Song, X., Hung,W., Gu, M.: Exponential-condition-based barrier certificate generation for safety verification of hybrid systems. In: CAV’13. pp. 242–257 (2013)

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Conclusion

Hybrid systems attracts more and more interests with

the development of safety critical embedded systems

Invariant plays an important role in the study (formal

verification, controller synthesis) of hybrid systems

Semi-algebraic inductive invariant checking for

polynomial continuous/hybrid systems is decidable

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Conclusion

Use parametric polynomials and symbolic computation

to automatically discover invariants, and to perform

ptimization

 rigorous  high complexity (may be combined with numeric computation)  Non-polynomial systems transformed to polynomials ones

Case studies show good prospect of proposed methods

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Related references

Hengjun Zhao, Mengfei Yang, Naijun Zhan, Bin Gu, Liang Zou and Yao Chen (2014):

Formal verification of a descent guidance control program of a lunar lander, in

Proc. of FM 2014, Lecture Notes in Computer Science 8442, pp.733-748.
Hengjun Zhao, Naijun Zhan and Deepak Kapur (2013): Synthesizing switching

controllers for hybrid systems by generating invariants, in Proc. of the Jifeng Festschrift, Lecture Notes in Computer Science 8051, pp.354-373.

Hengjun Zhao, Naijun Zhan, Deepak Kapur, and Kim G. Larsen (2012): A “hybrid”

approach for synthesizing optimal controllers of hybrid systems: A Case study of the oil pump industrial example, in Proc. of FM 2012, Lecture Notes in Computer Science 7436, pp.471-485, 2012.

Jiang Liu, Naijun Zhan and Hengjun Zhao (2011):Computing semi-algebraic

invariants for polynomial dynamical systems, in Proc. of EMSOFT 2011, pp.97-106, ACM Press.

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Thanks! Questions?

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