Sy Symb mboli lic c and St Statis istica tical l Model el Checkin Ch king g in in UPP PPAAL AL Alexandre Al xandre Dav avid id Kim Ki m G. Lar . Larsen n nis, Peter Bulychev, Marius us Mikuc ucio ioni Axel Legay, Dehui Du, Guangyuan Li, Danny B. Poulsen, Amélie Stainer, Zheng Wang CAV11, FORMA RMATS11, TS11, PDMC11, C11, QAP APL12, 12, LPAR1 AR12, 2, NFM12, 12, iWIG WIGP12, 12, RV12, FORMA RMATS12, TS12, HBS12, BS12, ISOLA OLA12, 12, SCIE IENCE CE China na
Ov Overview view Stochastic Hybrid id Automata Biological Oscillator Continuous vs. Stochastic Models Parameter Optimization – ANOVA Energy Aware Building Controller Synthesis for Hybrid Systems Greno noble e Summ mmer er Schoo hool Alexand andre David [2]
Sto Stochas chastic tic Hyb Hybri rid Au Auto tomata mata
Sto toch chastic astic Semantics antics of of TA TA Exponential Distribution Uniform Distribution 1 0.5 Let’s make this hybrid. 2 4 1 3 5 What happens to the semantics if you add differential equations? Compositi tion on = Input enabled Repeated races between components for output utti ting ng Greno noble e Summ mmer er Schoo hool Alexand andre David [4]
Sto toch chastic astic Hy Hybrid rid Systems tems A Bouncing Ball Player r 1 Ball Player r 2 simulate 1 [<=20]{Ball1.p, Ball2.p} Pr[<=20](<>(time>=12 && Ball.p>4)) Greno noble e Summ mmer er Schoo hool Alexand andre David [5]
UPPAAL AAL SMC MC Un Unif iform m dis istri tribu bution tions s (bound unded ed dela lay) y) Exponential distributions (unbounded delay) Discrete probabilistic choices Distribution on successor state – random Hybrid flow by use of ODEs + usual UPPAAL features Logic: MITL support. Greno noble e Summ mmer er Schoo hool Alexand andre e David [6]
UPPAAL AAL SMC MC Uniform distributions (bounded delay) Exponen nentia ial l dis istrib ibuti utions ons (unb nbounde unded d dela lay) y) Discrete probabilistic choices Distribution on successor state – random Hybrid flow by use of ODEs + usual UPPAAL features Logic: MITL support. Greno noble e Summ mmer er Schoo hool Alexand andre e David [7]
UPPAAL AAL SMC MC Uniform distributions (bounded delay) Exponential distributions (unbounded delay) Dis iscret ete e probab abil ilistic istic choic ices Distribution on successor state – random Hybrid flow by use of ODEs + usual UPPAAL features Logic: MITL support. Greno noble e Summ mmer er Schoo hool Alexand andre e David [8]
UPPAAL AAL SMC MC Uniform distributions (bounded delay) Exponential distributions (unbounded delay) Discrete probabilistic choices Dis istribut ibution ion on successo essor state e – random Hybrid flow by use of ODEs + usual UPPAAL features Logic: MITL support. Greno noble e Summ mmer er Schoo hool Alexand andre e David [9]
UPPAAL AAL SMC MC Uniform distributions (bounded delay) Exponential distributions (unbounded delay) Discrete probabilistic choices Distribution on successor state – random Hybrid id flo low by use of OD ODEs + usual UPPAAL features Logic: MITL support. Greno noble e Summ mmer er Schoo hool Alexand andre e David [10 10]
UPPAAL AAL SMC MC Uniform distributions (bounded delay) Exponential distributions (unbounded delay) Discrete probabilistic choices Distribution on successor state – random Hybrid flow by use of ODEs + usual UPPAAL features Logi gic: MITL su supp pport. t. Greno noble e Summ mmer er Schoo hool Alexand andre e David [11 11]
Hy Hybrid rid Aut utom omata ata H=(L, L, l 0 , § , , X,E,F,I ,Inv nv) I/O – broadcast sync Ball where input-enabled L set of locations l 0 initial location § = § i [ § o o set of actions X set of continuous variables valuation º : X ! R (=R X ) E set of edges (l,g,a, Á ,l ’) with g µ R X and X and a 2 § Á µ R X £ R Player r 2 For each l a Player r 1 delay function F(l): R >0 £ R X ! R X For each l an invariant Inv(l) µ R X Greno noble e Summ mmer er Schoo hool Alexand andre David [12 12]
Hy Hybrid rid Aut utom omata ata H=(L, L, l 0 , § , , X,E,F,I ,Inv nv) Ball where L set of locations l 0 initial location § = § i [ § o o set of actions X set of continuous variables valuation º : X ! R (=R X ) E set of edges (l,g,a, Á ,l ’) with g µ R X and X and a 2 § Á µ R X £ R Player r 2 For each l a Player r 1 delay function General “delay”. F(l): R >0 £ R X ! R X Handles clock rates. For each l an invariant Inv(l) µ R X Greno noble e Summ mmer er Schoo hool Alexand andre David [13 13]
Hybrid Hy rid Aut utom omata ata Semantics ntics Ball States tes (l, (l, º ) ) where º 2 R X Transit ansition ions (l, (l, º ) ) ! d (l, (l, º ’) where º ’=F(l)(d, º ) provided º ’ 2 Inv(l) d ! ( p = 10 ¡ 9 : 81 = 2 d 2 ; v = ¡ 9 : 81 d ) ( p = 10 ; v = 0) bounce! (l, (l, º ) ) ! a a ( l’, º ’) if ! ( p = 0 ; v = 14 : 02 ¢ 0 : 83) at d = 1 : 43 d there exists (l,g,a, Á ,l ’) 2 E ! ( p = 6 : 92 ; v = 0) at d = 1 : 18 d with º 2 g and ! ( p = 0 ; v = 11 : 51) at d = 1 : 18 bounce! ( º , º ’) 2 Á and ! : : : º ’ 2 Inv (l’) Greno noble e Summ mmer er Schoo hool Alexand andre David [14 14]
Sto toch chastic astic Hy Hybrid rid Aut utoma omata ta Ball Stoc ochasti tic Seman emanti tics cs For each state s=(l, º ) ) Delay density function * ¹ s : R >0 ! R Output Probability Function d ! ( p = 10 ¡ 9 : 81 = 2 d 2 ; v = ¡ 9 : 81 d ) ( p = 10 ; v = 0) ° s : § o ! [0,1] bounce! ! ( p = 0 ; v = 14 : 02 ¢ 0 : 83) at d = 1 : 43 Next-state density function * Player r 1 Player r 2 ´ a s : St ! R R where a 2 § . 𝑢=1.43 𝑢=1.43 1 3 2.5 𝑓 −2.5𝑢 𝑒𝑢 𝑄𝑠 1 ℎ𝑗𝑢! 𝑐𝑝𝑣𝑜𝑑𝑓! = 𝑄𝑠 2 ℎ𝑗𝑢! 𝑐𝑝𝑣𝑜𝑑𝑓! = 𝑒𝑢 𝑢=0 𝑢=0 * Dirac’s delta functions for 1.43 = 0.97 1.43 = 0.48 = −𝑓 −2.5𝑢 1 3 = 𝑢 0 0 deterministic delays / next state Greno noble e Summ mmer er Schoo hool Alexand andre David [15 15]
Sol olving ving OD ODEs/S s/Stoc tocha hastic stic Semantic antics Processes Fixed delay dt clock updates. <Integrator> Delay given by distribution hit! Player Fixed delay to reach p==0 bounce. Ball Time Race between processes. Choice of dt and clock updates can be changed (solver). Greno noble e Summ mmer er 16 Scho hool
Bi Biol olog ogical ical Os Oscillato cillator
A Bi Biolo ological gical Os Oscillator illator Circadian oscillator. N. Barkai and S. Leibler. Biological rhythms: Circadian clocks limited by noise. Nature, 403:267 – 268, 2000 Two ways to model: 1. Stochastic model that follow the reactions. 2. Continuous model solving the ODEs. Analysis: Evaluate time between peaks. The continuous model is the limit behavior of the stochastic model. Use frequency analysis for comparison. Grenoble Summer School 18
Sto toch chastic astic Mo Model el 19 Grenoble Summer School
Con ontinuou tinuous s Mo Model el 20 Grenoble Summer School
Results ults of of Simulation mulations 21 Grenoble Summer School
Fre requenc quency y Doma main in An Anal alysis ysis (Fou ourrier rrier Tran ansform) sform) 22 Grenoble Summer School
Time me Betw tween n Peaks ks Use the MITL formula 1100 true U[<=1000] (A>1100 & true U[<=5] A<=1000) . 1000 5 Generate monitors (one shown). Run SMC. Grenoble Summer School 23
En Energ ergy y Awa Aware re Bu Buil ildings dings
Wha hat t Thi his s Wor ork k is Abou out Find optimal parameters for, e.g., a controller. Applied to stochastic hybrid systems. Suitable for different domains: biology, avionics… Technique: statistical model-checking. This work: Apply ANOVA to reduce the number of needed simulations. Greno noble e Summ mmer er Scho hool 25 25
Ov Overview view Energy aware buildings The case-study in a nutshell Choosing the parameters Naïve approach Efficiently choosing the (best) parameters ANOVA Greno noble e Summ mmer er Scho hool 26 26
Ene nergy gy Awa ware re Bui uildings ldings The case: Building with rooms ms separated by doors or walls. Contact with the envir iron onment ment by windows or walls. Few transportable hea eat source rces between the rooms. Objective: mainta ntain in the temperat perature ure within range. Greno noble e Summ mmer er Scho hool 27 27
Ene nergy gy Awa ware re Bui uildings ldings Model: Matrix of coefficients for heat transfer between rooms . Environment temperature weather model. Different controllers user profiles. Goal: Op Optimiz imize e the co control troller er. Grenoble Summer School 28
Mo Model el Ov Overview view Room Controller Global Heater controller. Room User Profiles (per room) Heater Room Monitor Local bang-bang controllers. Weather model Grenoble Summer School 29
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