Georgios E. Fainekos and George J. Pappas

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1 FORMATS 2007 Georgios E. Fainekos and George J. Pappas Department of Computer and Information Science University of Pennsylvania seas.upenn.edu

2 Motivation a study of transient dynamics Black-box controller tuning Mathematical model

3 Motivation a study of transient dynamics Desired Performance Characteristics 1. Overshoot 2. Rise time 3. Delay time 4. Settling time 5. Constraints on input/states 6. Response sensitivity Can be captured with Metric Interval Temporal Logic

4 Example s p 1 MITL φ = [0,9π/2] (p 1 Øí [π,2π] p 2 ) Time p A/D σ Time

5 Boolean Monitoring / Testing 2 σ MITL φ = [0,9π/2] (p 1 Øí [π,2π] p 2 ) Time Monitoring Algorithm Truth Value {0,1} [Maler and Nickovic 04] [Thati and Rosu 04] [Rosu and Havelund 05] [Geilen 01] others

6 Example Bad sampling s p 1 MITL φ = [0,9π/2] (p 1 Øí [π,2π] p 2 ) Time p 2

7 Problem formulation s MITL φ = [0,9π/2] (p 1 Øí [π,2π] p 2 ) Monitor/Tester (signal (signal dynamics dynamics + + sampler) sampler) s = C φ iff (sëτ,τ) = D φ

8 Signals & Timed State Sequences (TSS) A signal is a function s : RöX, RŒ + A sampling function is a function τ : Nö +, NŒ A discrete time signal is a function σ :NöXwith σ = sëτ A timed state sequence μ is the pair (σ,τ) (σ(0),τ(0)) (σ(2),τ(2)) s X (σ(1),τ(1)) O(p)

9 Metric Interval Temporal Logic (MITL) Syntax: Φ + ::= p Ÿ p φ 1 φ 2 φ 1 φ 2 φ 1 U I φ 2 φ 1 R I φ 2 I can be of any bounded or unbounded interval of +, but I «i.e. I = [0,+ ), I = [2.5,9.8]

10 Boolean Continuous-time MITL Semantics C

11 MITL Discrete-time Semantics Timed state sequence μ = (σ,τ), where σ = sëτ i [ ]

12 Observation Specification : íp = T U p X O(p)

13 Metric Spaces A metric space (X, d) is a set X with a metric d A metric on a set X is a positive function d: X x XY +, such that the three following properties hold for all x 1,x 2,x 3 X it is d(x 1,x 3 ) d(x 1,x 2 )+d(x 2,x 3 ) for all x 1,x 2 X it is d(x 1,x 2 )=0 iff x 1 =x 2 for all x 1,x 2 X it is d(x 1,x 2 )=d(x 2,x 1 ) Given a metric d, a radius ε + and a point x X, then the open ε- ball centered at x is defined as B d (x,ε) = { y X d(x,y)<ε } 2ε x X C

14 (Signed) Distance Let x X be a point, C X be a set and d be a metric. Then we define dist d (x, C) := inf{d(x, y) y cl(c)} depth d (x, C):=dist d (x, X\C) Dist d (x, C):= è à dist d (x, C) depth d (x, C) if x 6 C if x C dist d (x,c) 2ε x 2ε x depth d (x,c) B d (x,ε) X C

15 Discrete-time Robust Semantics for MITL D

16 Intuition Example Specification : íp = T U p X O(p)

17 Observation Specification : íp = T U p j X O(p) If "t œ [τ(j)-δτ, τ(j)+δτ], where Δτ = sup i τ(i+1)-τ(i), the distance d(s(t),σ(j)) is bounded and smaller than depth d (σ(j),o(p)), then both s(t) and σ(j) satisfy p.

18 Observation 1D Specification : íp = T U p X j d(s(t),σ(j)) O(p) Δτ Δτ τ(j) time

19 Assumption on signal dynamics

20 In order to use induction Specification : í I p = T U I p X j O(p) i Δτ Δτ τ(i) τ(i)+i [ ] Δτ Δτ time

21 Strengthening MITL formulas t+i t [ [ ] ] t+c(i,δτ) t [ t+i [ ] t+e(i,δτ) ]

22 The importance of the sampling function Specification: I p = ^R I p τ(i)+i i [] τ -1 (τ(i)+i) = «τ(i)+i i R [] τ -1 (τ(i)+ R I) = «D μ D μ D μ Pnueli, Development of Hybrid Systems, FTRTFT 1994

23 Sampling Assumptions Assumptions 2&3 2&3 imply that that ττ -1-1 (τ(i)+i) «

24 Main Result Theorem: Let Φ be an MITL formula, sœf(r,x) be a continuous time signal, τœf si (N,R) be a sampling function and let Assumptions 1-3 hold. Let μ = (sëτ,τ), then implies

25 Relationship of discrete and continuous time semantics Proposition: Let Φ be an MITL formula and μ be a TSS, then implies Proposition: Let Φ be an MITL formula and μ be a TSS, then Corollary: Let Φ be an MITL formula, sœf(r,x) be a continuous time signal, τœf si (N,R) be a sampling function and let Assumptions 1-3 hold. Let μ = (sëτ,τ), then implies

26 Example 1 2 p 11 s Time Also, p 12 MITL Φ 1 = [0,9π/2] (p 11 Øí [π,2π] p 12 ) 2 σ thus and We compute* Time thus * Fainekos, Pappas, Robustness of temporal logic specifications, FATES/RV 2006

27 Example MITL σ 2 0 p 22 Φ 2 = [0,4π] p 21 í [3π,4π] p Time In this case, thus We compute* thus * Fainekos, Pappas, Robustness of temporal logic specifications, FATES/RV 2006

28 Example 3 linear system with nonlinear feedback p 31 s MITL Φ 3 = í [6,8] [0,10] p sec and

29 Example 3 Determination of E x x 1

30 Related Research 1. [de Alfaro & Manna] Verification in Continuous Time by Discrete Reasoning 2. [Furia & Rossi] Integrating Discrete and Continuous Time Metric Temporal Logics Through Sampling 3. [Henzinger; Manna & Pnueli] What Good Are Digital Clocks?

31 Conclusions / Future Work Continuous time satisfiability using discrete time reasoning Derive conditions on the dynamics of the signal Derive conditions on the sampling function Derive bounds on the continuous time robustness from the discrete time robustness of the signal Future work Use methods from optimization theory to determine E Design on-line monitoring algorithm improve bounds apply to hybrid systems use approximate metrics to compute bounds

32 Thank You! Questions?

Robustness of Temporal Logic Specifications for Continuous-Time Signals

Robustness of Temporal Logic Specifications for Continuous-Time Signals Georgios E. Fainekos a, George J. Pappas a,b a Department of Computer and Information Science, University of Pennsylvania, 3330 Walnut