Statistical Model Checking as Feedback Control
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1 Statistical Model Checking as Feedback Control, MSc Vienna University of Technology Supervisor: Radu Grosu Co-supervisor: Ezio Bartocci
2 Analysis of CPS: Challenges State-space explosion: Open, physical part, uncertain and distributed Model is generally not known: Basic laws of physical part (or controller) only partially available Current-state is generally not known: Output is a function of only a subset of the state variables How to steer towards rare events (RE) is a challenge: Relation between RE and the CPS behavior is not known
3 Outline o o o o Learning State Estimation Control Future
4 Model Checking as Feedback Control???? 2 3 T T 2 T a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / a: /5 Controller (imp. splitting) state Estimator (imp. sampling) action CP System (hidden MM) output / 2 / 2 / 2 / a: 4/5 b: /5 a: /5 a: /5
5 Learning a DTMC: Input Trace(s):,, 2, 3,, 2, 2, 2, 3, 3,, 2, 3, 3, 3,... Unknown model: 2 assume DTMC x 3 T H x x ( x4 x3) O H x P 0 0 P 0 x (4 )
6 Learning a DTMC: Output Learned model x x x3 x T H x x O H x
7 Learning a DTMC: Input Trace(s):,, 2, 3,, 2, 2, 2, 3, 3,, 2, 3, 3, 3,... Unknown model: 2,3 assume HMM x,2 3,4,4 T H x O H x x
8 Learning a DTMC: Output Learned model 0.84 x P()=0.64 P(2)= P(3)= P(4)= Initial O H big impact! P(4)= T H x x O H x
9 Discrete-time Markov Chain / 2 / 2 / 2 / 2 / Learning curve with Matlab HMM Toolbox
10 State Estimation (IS) Given P(X X ) T, P( X P( y t trace t t H X t) OH y y,, y,t t ) T H T H T H X t- X t X t+ O H O H O H yt yt y t T H ( X y ) Compute P t+ t+ :
11 State Estimation (IS) 0.6 Initial distribution of the particles P(a) = 0.2 P(b) = 0.8 x P(c) =.0 P(a) = 0.6 P(b) = P(d) =.0.0 P(a) = 0-6 P(b) = x5
12 State Estimation (IS) 0.6 Simulate the CPS P(a) = 0.2 P(b) = 0.8 x P(c) =.0 P(a) = 0.6 P(b) = P(d) =.0.0 P(a) = 0-6 P(b) = x5
13 State Estimation (IS) 0.6 New configuration of the particles P(a) = 0.2 P(b) = 0.8 x P(c) =.0 P(a) = 0.6 P(b) = P(d) =.0.0 P(a) = 0-6 P(b) = x5
14 State Estimation (IS) 0.6 Observe a and resample the particles P(a) = 0.2 P(b) = 0.8 x 0.4 P( x ) P(c) =.0 P( x ) 0 2 P(a) = 0.6 P(b) = 0.4 P(d) = P( x ) P( x ) P(a) = 0-6 P(b) = x5 P( x ) 0.5 5
15 Property Decomposition A nested sequence of temporal logic properties: A set of increasing levels: n n T Reaching a level implies having reached all the lower levels: P( i ) P( i i) P( i), P 0 ( ), P( ) n The shorter trace satisfying more intermediate properties is given a higher score The probability of the rare event: n P( i i) i0 Levels are chosen such that to minimize the relative variance of the final estimate
16 Adaptive Levels for Control (ISp) / 2 / 2 / 2 / 2 / 2 / Check the property of reaching state N within N- transitions T H p p p 0 p p p O H
17 Adaptive Levels for Control (ISp) Simulation with 0 particles for checking the property of reaching state 25
18 Model Checking as Feedback Control???? 2 3 T T Controller (imp. splitting) a: 4/5 b: /5 / 2 / 2 / a: /5 / 2 Estimator (imp. sampling) a: /5 action: s CP System output: a
19 Model Checking as Feedback Control???? 2 3 T T Controller (imp. splitting) a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / Estimator (imp. sampling) a: /5 action: s CP System output: a
20 Model Checking as Feedback Control???? 2 3 T T Controller (imp. splitting) a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / Estimator (imp. sampling) a: /5 action: s CP System output: a
21 Model Checking as Feedback Control???? 2 3 4/5 a: 4/5 b: /5 / 2 / 2 T T 2 /5 a: /5 / 2 / a: /5 Controller (imp. splitting) state: Estimator (imp. sampling) action: s CP System output: a
22 Model Checking as Feedback Control???? 2 3 T T a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / a: /5 Controller (imp. splitting) state: Estimator (imp. sampling) action: s CP System output: ab
23 Model Checking as Feedback Control???? 2 3 T T a: 4/5 b: /5 / 2 / 2 / a: /5 / 2 a: /5 Controller (imp. splitting) state: Estimator (imp. sampling) action: s CP System output: ab
24 Model Checking as Feedback Control???? 2 3 T T /5 a: 4/5 b: /5 / 2 / 2 2 4/5 a: /5 / 2 / a: /5 Controller (imp. splitting) state: 2 Estimator (imp. sampling) action: s CP System output: ab
25 Model Checking as Feedback Control???? 2 3 T T a: 4/5 b: /5 / 2 / 2 / a: /5 / 2 a: /5 Controller (imp. splitting) state: 2 Estimator (imp. sampling) action: sp CP System output: ab
26 Model Checking as Feedback Control?? 2 3 T T a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / a: /5 Controller (imp. splitting) state Estimator (imp. sampling) action CP System output
27 Model Checking as Feedback Control 2 / 0?? 2 3 T T a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / a: /5 Controller (imp. splitting) state Estimator (imp. sampling) action CP System output
28 Model Checking as Feedback Control 2 / 0?? 2 3 T T 2 T a: 4/5 b: /5 / 2 / 2 a: /5 / 2 / a: /5 Controller (imp. splitting) state Estimator (imp. sampling) action CP System output
29 Future Research Testing on real case studies of CPS Efficient scoring for importance splitting Optimal derivation of the levels for importance splitting Importance sampling gives the beliefs and not actual states Optimal control from the belief-states
30 Q&A Thank you!
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