EECE Hybrid and Embedded Systems: Computation T. John Koo, Ph.D. Institute for Software Integrated Systems Department of Electrical Engineering and Computer Science Vanderbilt University 300 Featheringill Hall April 20,

2 Summary

3 Hybrid System A system built from atomic discrete components and continuous components by parallel and serial composition, arbitrarily nested. The behaviors and interactions of components are governed by models of computation (MOCs). Discrete Components Finite State Machine (FSM) Discrete Event (DE) Synchronous Data Flow (SDF) Continuous Components Ordinary Differential Equation (ODE) Partial Differential Equation (PDE)

4 Why Hybrid Systems? Modeling abstraction of Continuous systems with phased operation (e.g. walking robots, mechanical systems with collisions, circuits with diodes) Continuous systems controlled by discrete inputs (e.g. switches, valves, digital computers) Coordinating processes (multi-agent systems) Important in applications Hardware verification/CAD, real time software Manufacturing, communication networks, multimedia Large scale, multi-agent systems Automated Highway Systems (AHS) Air Traffic Management Systems (ATM) Uninhabited Aerial Vehicles (UAV) Power Networks

5 Topics Modeling Finite State Machines Time Automata Ordinary Differential Equations Hybrid Automata Analysis Reachability - Discrete Reachability - Continuous Reachability - Hybrid Tool Ptolemy II HyTech Requiem d/dt Checkmate Verification Temporal Logic Model Checking Time Automata

6 Hybrid Automaton

7 Hybrid Automaton (Lygeros, 2003)

8 Hybrid Automaton Q X Execution

9 Examples: Thermostat

10 Examples: Bouncing Ball

11 Motivating Examples:Two Tanks

12 Hybrid Automaton t i

13 Hybrid Automaton i t

14 Hybrid Automaton i t

15 Hybrid Automaton

16 Examples: Bouncing Ball

17 Hybrid Automaton i t finite i t infinite

18 Hybrid Automaton i t finite i t Zeno

19 Hybrid Automaton Zeno of Elea, 490BC Ancient Greek philosopher The race of Achilles and the turtle Achilles, a renowned runner, was challenged by the turtle to a race. Being a fair sportsman, Achilles decided to give the turtle a 10 meter head-start. To overtake the turtle, Achilles will have to first cover half the distance separating them. To cover the remaining distance, he will have to cover half that distance, and so on. No matter how fast Achilles is, he can never overtake the turtle. Why??? Ans: Covering each one of the segments in this series requires a non zero amount of time. Since there is an infinite number of segments, Achilles will never overtake the turtle.

20 Hybrid Automaton Non-Determinism Multiple Executions for the same initial condition Sources of non-determinism Non-Lipschitz continuous vectorfields, f Multiple discrete transition destinations, E & G Choice between discrete transition and continuous evolution, D & G Non-unique continuous state assignment, R Definition: A hybrid automaton H is deterministic if for all initial conditions there exists a unique maximal sequence

21 Hybrid Automaton Blocking No Infinite executions for some initial states Source of blocking Cannot continue in domain due to reaching the boundary of the domain where no guard is defined Have no place to make discrete transition to Definition: A hybrid automaton H is non-blocking if for every initial condition there exists at least one infinite execution ?

22 Hybrid Automaton Zeno Executions Infinite execution defined over finite time Infinite number of transitions in finite time Transition times converge Definition: A hybrid automaton H is zeno if there exists an initial condition for which all infinite executions are Zeno

23 Exercise

24 Examples: Bouncing Ball Is this model: Deterministics? Non-Blocking? Zeno?

25 Examples: Bouncing Ball Is this model: Deterministics? Yes, the Guard and Domain contains only one element. Reset maps from one point to exactly another point. Also, the vector field is Lipschitz continuous. Non-Blocking? Zeno?

26 Examples: Bouncing Ball Is this model: Deterministics? Non-Blocking? Yes, the guard is always reachable from any initial condition within the domain and also the reset makes the state start within the domain. Zeno?

27 Examples: Bouncing Ball

28 Examples: Bouncing Ball

29 Examples: Bouncing Ball

30 Examples: Bouncing Ball Is this model: Deterministics? Non-Blocking? Zeno? Yes, it is Zeno since the time sequence converges.

31 Thermostat Is this model: Deterministics? Non-Blocking? Zeno?

32 Thermostat Is this model: Deterministics? No. Non-Blocking? Yes. Zeno? No.

33 Two Tanks Is this model: Deterministics? Yes. Non-Blocking? Yes. Zeno? Yes.

34 Zeno—infinitely many jumps in finite time If Water Tank Automaton

35 Timed Automata Is this model: Deterministics? Non-Blocking? Zeno?

36 Timed Automata Is this model: Deterministics? No. Non-Blocking? Yes. Zeno? No.

37 In Summary DeterministicNon-BlockingZeno Thermostat NOYESNO Bouncing Ball YES Two Tanks YES Time Automaton NOYESNO

38 In Summary DeterministicNon-BlockingZeno Thermostat NOYESNO Bouncing Ball YES Two Tanks YES Time Automaton NOYESNO Mapping Verification Special Attention in Simulation Verification

39 Computational Tools

40 Computational Tools Simulation Ptolemy II: ptolemy.eecs.berkeley.eduptolemy.eecs.berkeley.edu Modelica: SHIFT: Dymola: OmSim: ABACUSS: yoric.mit.edu/abacuss/abacuss.htmlyoric.mit.edu/abacuss/abacuss.html Stateflow: CHARON: Masaccio:

41 Computational Tools Simulation Models of Computation System Complexity Ptolemy II Dymola Modelica ABACUSS SHIFT OmSim Masaccio CHARON StateFlow/Simulink

42 Computational Tools Verification Finite Automata Timed Automata Linear Automata Linear Hybrid Systems Nonlinear Hybrid Systems d/dt CheckMate Timed COSPAN KRONOS Timed HSIS VERITI UPPAAL HYTECHCOSPAN SMV VIS … Requiem

43 End