Optimal Scheduling and Controller Synthesis. Kim G. Larsen

Transcription

1 Optimal Scheduling and Controller Synthesis Kim G. Larsen

2 Collaborators Wang Yi Paul Pettersson John Håkansson Anders Hessel Pavel Krcal Leonid Mokrushin Kim G Larsen Gerd Behrman Arne Skou Brian Nielsen Alexandre David Jacob Illum Rasmussen Marius Mikucionis Emmanuel Fleury, Didier Lime, Johan Bengtsson, Fredrik Larsson, Kåre J Kristoffersen, Tobias Amnell, Thomas Hune, Oliver Möller, Elena Fersman, Carsten Weise, David Griffioen, Ansgar Fehnker, Frits Vandraager, Theo Ruys, Pedro D Argenio, J-P Katoen, Jan Tretmans, Judi Romijn, Ed Brinksma, Martijn Hendriks, Klaus Havelund, Franck Cassez, Magnus Lindahl, Francois Laroussinie, Patricia Bouyer, Augusto Burgueno, H. Bowmann, D. Latella, M. Massink, G. Faconti, Kristina Lundqvist, Lars Asplund, Justin Pearson...

3 Real Time Systems Informationsteknologi Plant Continuous Eg.: Realtime Protocols Pump Control Air Bags Robots Cruise Control ABS CD Players Production Lines sensors actuators Controller Program Discrete Real Time System A system where correctness not not only only depends on on the the logical order of of events but but also also on on their their timing!!

4 Real Time Model Checking Informationsteknologi Plant Continuous Model of environment (user-supplied / non-determinism) b a c b a c sensors actuators UPPAAL Model 4 Controller Program Discrete Model of tasks (automatic?) 1 2 SAT φ a?? b c 3 4

5 Real Time Scheduling & Control Synthesis Informationsteknologi Plant Continuous Model of environment (user-supplied) b a c b a c sensors actuators Controller Program Discrete?? Partial UPPAAL Model Synthesis of tasks/scheduler (automatic) SAT φ!! SAT a φ!! b c

6 Overview Informationsteknologi Real Time Scheduling Timed Automata Verification Engine Optimal Scheduling Priced Timed Automata Applications: Task Graph Scheduling Optimal FPGA layout Dynamic Voltage Scaling Real Time Controller Synthesis Timed Game Automata Controller Synthesis Applications: Brick Sorting Climate Control CLASSIC CORA TIGA

7 Timed Automata [Alur & Dill 89] Informationsteknologi Resource Reset Invariant Synchronization Guard Semantics: ( Idle, x=0 ) ( Idle, x=2.5) ( InUse, x=0 ) ( InUse, x=5) ( Idle, x=5) ( Idle, x=8) ( InUse, x=0 ) d(2.5) use? d(5) done! d(3) use?

8 Composition Informationsteknologi Resource Synchronization Shared variable Semantics: ( Idle, Init, B=0, x=0) ( Idle, Init, B=0, x= ) ( InUse, Using, B=6, x=0 ) ( InUse, Using, B=6, x=6 ) ( Idle, Done, B=6, x=6 ) Task d(3) use d(6) done

9 Jobshop Scheduling Informationsteknologi J O B s Kim Maria Nicola Sport 2. 5 min min 4. 1 min R E S O U R C E S Economy 4. 1 min min min Local News 3. 3 min 3. 1 min min Problem: compute the minimal MAKESPAN Comic Stip min 4. 1 min min

10 Jobshop Scheduling in UPPAAL Informationsteknologi Kim Maria Sport 2. 5 min min Economy 4. 1 min min Local News 3. 3 min 3. 1 min Comic Stip min 4. 1 min Nicola 4. 1 min min min min

11 Experiments [TACAS 2001] B-&-B algorithm running for 60 sec. Informationsteknologi m=5 m=10 j=10 j=15 j=20 j=10 j=15 Lawrence Lawrence Job Job Shop Shop Problems Problems

12 Symbolic Transitions using Zones Informationsteknologi x>3 a y:=0 n m y y x x 1<=x<=4 1<=y<=3 delays to conjuncts to projects to y y x x 1<=x, 1<=y -2<=x-y<=3 3<x, 1<=y -2<=x-y<=3 3<x, y=0 Thus Thus (n,1<=x<=4,1<=y<=3) =a =a => => (m,3<x, y=0) y=0)

13 Forward Rechability Init -> Final? Informationsteknologi Waiting n,z Init n,z m,u Passed Final INITIAL Passed := Ø; Waiting := {(n0,z0)} REPEAT - pick (n,z) in Waiting - if for some Z Z (n,z ) in Passed then STOP -else/explore/ add { (m,u) : (n,z) => (m,u) } to Waiting; Add (n,z) to Passed UNTIL Waiting = Ø or Final is in Waiting

14 Canonical Datastructures for Zones Informationsteknologi x1-x2<=4 x1-x2<=4 x2-x1<=10 x2-x1<=10 x3-x1<=2 x3-x1<=2 x2-x3<=2 x2-x3<=2 x0-x1<=3 x0-x1<=3 x3-x0<=5 x3-x0<=5 3 x1 x x2 x3 Shortest Path Reduction O(n^3) x1 2 Shortest Path Closure O(n^3) -4 x2 2 3 x1 DBMs Bellmann x x0 1 x3 5 2 Space worst O(n^2) practice O(n) x0 x3 Minimal Constraint Form RTSS 97

15 Datastructures for Zones Informationsteknologi Difference Bounded Matrices (DBMs) Minimal Constraint Form [RTSS97] Clock Difference Diagrams [CAV99] PW List [SPIN03] x1 x2 x x3 2

16 Datastructures for Zones Informationsteknologi Difference Bounded Matrices (DBMs) Minimal Constraint Form [RTSS97] Clock Difference Diagrams Alexandre David [CAV99] PW List [SPIN03] -4 x1 x Elegant x0 RUBY 1bindings x3 for easy implementations 5

17 Informationsteknologi Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration Δ : (P M) N < : p.o. on P (pred.) A task can be executed only if all predecessors have completed Each machine can process at most one task at a time Task cannot be preempted. 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

18 Informationsteknologi Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration Δ : (P M) N < : p.o. on P (pred.) 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

19 Experimental Results Informationsteknologi Abdeddaïm, Kerbaa, Maler

20 Optimal Task Graph Scheduling Power-Optimality Informationsteknologi Energy-rates: C : M NxN P 2 P 1 16,10 2,3 2,3 cost ==1 6,6 10,16 P 6 P 3 P 4 cost ==4 P 7 P 5 2,2 8,2 1W 4W 2W 3W

21 Priced Timed Automata Timed Automata + COST variable Behrmann, Fehnker, et all (HSCC 01) Alur, Torre, Pappas (HSCC 01) Informationsteknologi cost rate TRACES l l 1 2 x 2 l 3 3 y 0 y 4 c =4 c =2 x:=0 c+=4 c+=1 ε(3) x:=0 y 4 (l 1,x=y=0) (l 1,x=y=3) (l 2,x=0,y=3) (l 3,_,_) ε(2.5) cost update (l 1,x=y=0) (l 1,x=y=2.5) (l 2,x=0,y=2.5) (l 2,x=0.5,y=3) (l 3,_,_) ε(3) (l 1,x=y=0) (l 2,x=0,y=0) (l 2,x=3,y=3) (l 2,x=0,y=3) (l 3,_,_) ε(.5) c=17 c=16 c=11

22 Priced Timed Automata Timed Automata + COST variable Behrmann, Fehnker, et all (HSCC 01) Alur, Torre, Pappas (HSCC 01) Informationsteknologi cost rate TRACES l l 1 2 x 2 l 3 3 y 0 y 4 c =4 c =2 x:=0 c+=4 c+=1 ε(3) x:=0 y 4 (l 1,x=y=0) (l 1,x=y=3) (l 2,x=0,y=3) (l 3,_,_) ε(2.5) Problem : Find the minimum cost of of reaching location ll 3 cost update (l 1,x=y=0) (l 1,x=y=2.5) (l 2,x=0,y=2.5) (l 2,x=0.5,y=3) (l 3,_,_) ε(3) (l 1,x=y=0) (l 2,x=0,y=0) (l 2,x=3,y=3) (l 2,x=0,y=3) (l 3,_,_) ε(.5) Efficient Implementation: CAV 01 and and TACAS c=17 c=16 c=11

23 Aircraft Landing Problem Informationsteknologi cost e*(t-t) E d+l*(t-t) Planes have to keep separation distance to avoid turbulences caused by preceding planes T L t E earliest landing time T target time L latest time e cost rate for being early l cost rate for being late d fixed cost for being late Runway

24 Modeling ALP with PTA Informationsteknologi Planes have to keep separation distance to avoid turbulences caused by preceding planes 129: Earliest landing time 153: Target landing time 559: Latest landing time 10: Cost rate for early 20: Cost rate for late Runway handles 2 types of planes Runway

25 Symbolic A*

26 Zones Operations Informationsteknologi y Z x

27 Priced Zone CAV 01 y Informationsteknologi Δ 4 2 Cost ( x, y) -1 Z x = 2 y x + 2

28 Branch & Bound Algorithm Informationsteknologi

29 Branch & Bound Algorithm Informationsteknologi Z' Z Z Z is isbigger & cheaper cheaperthan thanzz is is a well-quasi well-quasi ordering orderingwhich guarantees guarantees termination! termination!

30 Experimental Results

31 Aircraft Landing CAV 01 Source of examples: Baesley et al 2000 Informationsteknologi

32 Branch & Bound Algorithm Informationsteknologi Zone based Linear Programming Problems

33 Zone LP Min Cost Flow Exploiting duality Informationsteknologi minimize 3x 1-2x 2 +7 when x 1 -x x 2 3 x 2 1 minimize 3y 2,0 -y 0,2 +y 1,2 y 0,1 when y 2,0 -y 0,1 -y 0,2 =1 y 0,2 +y 1,2 =2 y 0,1 -y 1,2 =-3

34 Aircraft Landing Using MCF/Netsimplex Rasmussen, Larsen, Subramani TACAS04 Informationsteknologi

35 Dynamic Voltage Scaling UC b

36 Performance vs Ressource-Efficiency? Consumer constantly demand better functionality, flexibility, availability,.. increase in resources needed: Time Energy Memory Bandwidth.. Application of CUPPAAL to modeling, analysis and synthesis of resourceefficient schedules for realtime systems.

37 Power Management Dynamic Voltage Scaling

38 Energy in Processor Power consumption mainly by dynamic power energy A non-experts understanding of CMOS P E dynamic dynamic = C L pr. cycle V = 2 dd C L f clk V 2 dd Supply voltage reduction => decreased frequency delay V dd f clk ~ V ( α -1) dd ; α > 1 V dd We may miss deadlines

39 Task Scheduling utilization of CPU P(i), [E(i), L(i)],.. : period or earliest/latest arrival or.. for T i C(i): execution time for T i D(i): deadline for T i T 1 T 2 ready done Scheduler T n stop run T 2 is running { T 4, T 1, T 3 } ready ordered according to some given priority: (e.g. Fixed Priority, Earliest Deadline,..)

40 Modeling Task T 1 1 T 2 2 T n n ready done stop run Scheduler

41 Modeling Scheduler T 1 1 T 2 2 T n n ready done stop run Scheduler

42 Modeling Queue T 1 1 T 2 2 T n n ready done stop run Scheduler

43 Schedulability = Safety Property May be extended with preemption (Task0.Error or Task1.Error or ) A (Task0.Error or Task1.Error or )

44 Energy Optimal Scheduling Using Priced Timed Automata T 1 1 T 2 2 ready done Scheduler T n n stop run F:=?? ; V:=?? Choose Scaling/Cost (Freq/Voltage) C

45 Cost Optimal Scheduling = Optimal Infinite Path Informationsteknologi σ c 3 c n c 1 c 2 t 3 t n t 1 t 2 (Car0.Err or Car1.Err or ) Accumulated cost Accumulated time Value of path σ: val(σ) = lim n c n /t n Optimal Schedule σ * : val(σ * ) = inf σ val(σ)

46 Cost Optimal Scheduling = Optimal Infinite Path σ c 3 c n t 1 t 2 t 3 t n (Car0.Err or Car1.Err or ) Accumulated cost Accumulated time THEOREM: THEOREM: σ σ * * is is computable computable Bouyer, Brinksma, Larsen HSCC 04 Value of path σ: val(σ) = lim n c n /t n Informationsteknologi Optimal Schedule σ * : val(σ * ) = inf σ val(σ)

47 Approximate Optimal Schedule T<N T<N X X T<N X Optimal infinite schedule modulo cost-horizon T>=N T>=N C=M C=M C=M C=M E[] (not (Task0.Error or Task1.Error or Task2.Error) and (cost>=m imply time >= N)) = E[] φ(m,n) σ ² [] φ(m,n) imply val(σ) M/N

48 Preliminary Results P 1 =D 1 =32 C 1 =6 P 2 =D 2 =48 C 2 =18 P 3 =D 3 =64 C 3 =12 EDF w preemption no DVS: avr.: 48

49 Preliminary Results Cost horizon: 2,000 P 1 =D 1 =32 C 1 =6 P 2 =D 2 =48 C 2 =18 P 3 =D 3 =64 C 3 =12 EDF w preemption w DVS: avr.: 43.37

50 Optimal Reconfiguration of FPGA Introducing new features of UPPAAL 4.0 User-defined functions Types Select UC b

51 Field Programmable Gate Array

52 The Problem

53 The Problem

54 Example

55 Example

56 Example

57 Example 1

58 Example 1

59 UPPAAL Model Informationsteknologi

60 UPPAAL Model Informationsteknologi

61 UPPAAL Model Informationsteknologi

62 UPPAAL Model Informationsteknologi

63 TIGA Real Time Controller Synthesis with Franck Cassez, Agnes Counard, Alexandre David Emmanuel Fleury, Didier Lime

64 Controller Synthesis and Timed Games Production Cell GIVEN System moves S, Controller moves C, and property φ FIND strategy s C such that s C S ² φ A Two-Player Game

65 Untimed and Timed Games Reachability / Safety Games x>1 1 x<1 x 1 2 x 2 x<1 3 x:=0 x 1 4 Uncontrollable Controllable

66 Untimed Games Reachability / Safety Games Strategy: F : Run(A) E c Memoryless strategy: F : Q E c Winning Run: States(ρ) G Ø States(ρ) G = Ø Winning Strategy: Runs(F) WinRuns Uncontrollable Controllable

67 Untimed Games Reachability / Safety Games Strategy: F : Run(A) E c Memoryless strategy: F : Q E c Winning Run: States(ρ) G Ø States(ρ) G = Ø Winning Strategy: Runs(F) WinRuns Uncontrollable Controllable

68 Untimed Games Reachability / Safety Games Strategy: F : Run(A) E c Memoryless strategy: F : Q E c Winning Run: States(ρ) G Ø States(ρ) B = Ø Winning Strategy: Runs(F) WinRuns Loosing (memoryless) strategy Uncontrollable Controllable

69 Untimed Games Reachability / Safety Games Strategy: F : Run(A) E c Memoryless strategy: F : Q E c Winning Run: States(ρ) G Ø States(ρ) B = Ø Winning Strategy: Runs(F) WinRuns Winning (memoryless) strategy) Uncontrollable Controllable

70 Untimed Games Backwards Fixed-Point Computation cpred(x) = { q Q q X. q c q } upred(x) = { q Q q X. q u q } π(x) = cpred(x) \ upred(x C ) ] Theorem: The set of winning states is obtained as the least fixpoint of the function: X a π(x) Goal Uncontrollable Controllable

71 Untimed Games Backwards Fixed-Point Computation cpred(x) = { q Q q X. q c q } upred(x) = { q Q q X. q u q } π(x) = cpred(x) \ upred(x C ) ] Theorem: The set of winning states is obtained as the least fixpoint of the function: X a π(x) Goal Uncontrollable Controllable

72 Timed Games Reachability / Safety Games Strategy: F : Run(A) E c λ Memoryless strategy: F : Q E c λ x>1 1 x 1 x<1 Winning Run: States(ρ) G Ø States(ρ) G = Ø 2 3 x<1 x 2 x:=0 Winning Strategy: Runs(F) WinRuns x 1 4 Uncontrollable Controllable

73 Timed Games Reachability / Safety Games Strategy: F : Run(A) E c λ Memoryless strategy: F : Q E c λ x>1 1 x 1 x!= 1 : λ x=1 : c x<1 Winning Run: States(ρ) G Ø States(ρ) G = Ø x<2 : λ x 2 : c 2 3 x<1 x 2 x:=0 Winning Strategy: Runs(F) WinRuns x 1 4 x<1 : λ x 1 : c Winning (memoryless) strategy) Uncontrollable Controllable x!= 1 : λ x=1 : c

74 Timed Games State-of-the-Art Backwards Fixed-Point Computation Definitions cpred(x) = { q Q q X. q c q } upred(x) = { q Q q X. q u q } Pred t (X,Y) = { q Q t. q t X and s t. q s Y C } π(x) = Pred t [ X cpred(x), upred(x C ) ] X Pred t (X,Y) Y Theorem: The set of winning states is obtained as the least fixpoint of the function: X a π(x) Goal

75 Timed Games State-of-the-Art Backwards Fixed-Point Computation x>1 1 x<1 x 1 2 x 2 x<1 3 x:=0 x We want Forward and On-The-Fly Algorithm in order to avoid constructing all (backwards) reachable state-space and to allow for discrete variables (e.g. in UPPAAL)

76 On-the-fly Algorithms for Timed Games UPPAAL Tiga = on-the-fly algorithm for timed games [CONCUR 05] S Win(S)

77 Running Example

78 Production Cell

79 Experimental Results

80 New Experimental Results Using UPPAAL 4.0 architecture Tricks (Alexandre): - UPPAAL pipeline architecture, which implies * active clock reduction * PW-list * UPPAAL optimizations (successor computation, postponed evaluation, reduced copies..) * improved DBM library * improved copy-on-write implementations * improved subtraction (vital) * enormously improved merge (between DBMs) (vital)

81 A Buggy Brick Sorting Program First UPPAAL model Sorting of Lego Boxes eject Ken Tindell Boxes Piston eject remove 99 Conveyer Belt Conveyer Belt Blck Yel Black Controller MAIN PUSH Yellow Exercise: Design Controller so that only yellew boxes are being pushed out MCD 2001, Twente Kim G. Larsen 16

82 Brick Sorting Generic Plate Piston Controller

83 Brick Sorting Generic Plate Strategy for EJECT Piston Controller

84 Balancing Plates / Timed Automata A Plate The Joggler E (Plate1.Bang or Plate2.Bang or )

85 Balancing Plates / Time Uncertainty Strategy BDD/CDD

86 Timed Games and Test Generation Tidle=20 Tsw=4 + Time Uncertainty + Uncontrollable output + Input Enabledness + Determinism Off-line test-case generation = Compute winning strategy for reaching Bright

87 A trick light control Tidle=20 Tsw=4 How to test for Bright? E<> (control: A<> Bright) or <<c,u>> (<<c>> Bright)

88 By Jan J. Jessen Jacob I. Rasmussen Climate Control

89 Climate Control

90 Climate Control / Neighbor Neighboring zone Neighbor wants to receive flow? Temperature in neighbor zone (lower/higher)

91 Zone Controller Climate Control / Controller

92 Zone Controller Climate Control / Controller

93 Obtaining executable code Stragegy get0 give0 give0 1 temp0 temp0 temp0 get1 get1 get1 get1 give1 give1 give1 give1 temp1 temp1 temp1 objective hottest objective hottest humid0 humid0 humid1 humid1 humid1 humid1 have0 want0 have1 want1 inlet outlet heater dec_humid have0 have0 want0 want0 have1 want1 inlet outlet heater have1 want1 inlet outlet humid0 objective humid1 humid1 dec_humid have0 have0 dec_humid want0 want0 have1 want1 have1 want1 humid0 humid1 humid1 dec_humid have0 want0 have1 want1 inlet objective hottest humid0 humid1 humid1 dec_humid have0 want0 have1 temp1 temp1 temp1 hottest hottest hottest objective humid0 humid1 humid1 dec_humid hottest give1 give1 give1 give1 temp1 temp1 temp1 objective hottest objective hottest humid0 humid1 humid1 dec_humid have0 have0 want0 want0 dec_humid humid0 objective humid1 humid1 dec_humid have0 have0 dec_humid want0 want0 dec_humid humid0 humid1 humid1 dec_humid have0 want0 dec_humid objective hottest humid0 humid1 humid1 dec_humid have0 want0 dec_humid temp1 temp1 temp1 hottest hottest hottest hottest objective objective humid0 humid1 humid1 humid0 humid1 humid1 dec_humid objective objective objective humid0 humid1 humid1 dec_humid have0 dec_humid want0 have1 want1 humid0 humid0 humid1 humid1 dec_humid dec_humid humid1 dec_humid dec_humid hottest hottest objective objective humid0 humid1 humid1 dec_humid have0 want0 have1 humid0 humid1 humid1 dec_humid get1 get1 give1 give1 give1 give1 temp1 temp1 temp1 objective hottest objective hottest humid0 humid1 humid1 dec_humid have0 humid0 objective humid1 humid1 dec_humid have0 dec_humid humid0 humid1 humid1 dec_humid have0 objective hottest humid0 humid1 humid1 dec_humid have0 temp1 temp1 temp1 hottest hottest hottest objective humid0 humid1 humid1 dec_humid hottest 1296 cases BDD 289 nodes

94 u u u Obtaining executable code [T2] [H1] [H3] [H2] SafeT1 SafeH1 SafeT2 SafeH2 T1 T2 [Q_12]^+_2 [Q_12]^ _2 T objective H1 H2 H objective T1 T2 T3 [Q_12]^+_1 [Q_12]^ _1 [Q_23]^+_3 [Q_23]^ _3 T objective H1 H2 H3 H objective state Qout Qin [Q_12]^+_1 [Q_12]^ _1 Left zone controller state Qout Qin [Q_12]^+_2 [Q_12]^ _2 [Q_23]^+_2 [Q_23]^ _2 Middle Zone Controller [u2] [T2] T2 [T3] T3 state [Q_23]^+_2 Qout [Q_23]^ _2 Qin T objective [H2] SafeT3 H2 [H3] H3 [Q_23]^+_3 [u3] SafeH3 H objective [Q_23]^ _3 Right Zone Controller 16.5 [H1] [H2] [T2] [T1] [T3] [u1] 15 Simulink T 1 T 2 T 3 [T1] qout1 Qout1 qin1 q12p1 Qin1 q12m1 qout2 Q12 qin2 q12p2 Qout2 q12m2 Qin2 q23p2 q23m2 Q23 qout3 qin3 Qin3 q23p3 q23m3 Qout3 Flow Calc [T1] [T2] [T3] Ttotal [H1] [H2] [H3] Htotal q12 q23 [T2] Amb T&H amb [T1] [H1] [T3] [H3] [T2] [H2] [u1] [u2] [H2] [u3] qout1 qin1 T1 q12 T2 H2 Amb H1 u1 First Zone Dynamics q12 qout2 qin2 T2 u2 q23 T1 H1 T3 H3 H2 Amb Middle Zone Dynamics q23 T2 T3 H2 qin3 qout3 Amb H3 u3 Third Zone Dynamics [T1] [H1] [T2] [H2] [T3] [H3]

95 Reading material UPPAAL Cora (Priced Timed Automata): Buhrmann, Larsen, Rasmussen: Optimal Scheduling using Priced Timed Automata, ACM SIGMETRICS Performance Evaluation Review, vol. 32, nb. 4, 2005, pp , ACM Press. Priced Timed Automata: Algorithms, and Applications G. Behrmann, K. G. Larsen, J. I. Rasmussen. In proc. of FMCO'04, LNCS vol. 3657, pp , Springer Verlag, Patricia Bouyer. Weighted Timed Automata: Model-Checking and Games. In Proceedings of the 22nd Conference on Mathematical Foundations of Programming Semantics (MFPS'06), Genova, Italy, May 2006, ENTCS 158, pages Elsevier Science Publishers. UPPAAL Tiga (Timed Games & Controller Synthesis): Franck Cassez, Alexandre David, Emmanuel Fleury, Kim G. Larsen, and Didier Lime. Efficient on-the-fly algorithms for the analysis of timed games. CONCUR05, volume 3653 of Lecture Notes in Computer Science. Patricia Bouyer and Fabrice Chevalier. On the Control of Timed and Hybrid Systems. EATCS Bulletin 89, pages 79-96, Guided Controller Synthesis for Climate Controller using UPPAAL TIGA. Under Submission.

96