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Introduction to Hybrid Automata

Introduction to Hybrid Automata. Arijit Mondal Kapil Modi Arnab Sinha. Introduction. A hybrid automaton is a formal model for a mixed discrete continuous system. Systems with ‘ discrete jumps ’ & ‘ continuous flow ’ can be modeled into Hybrid Automata. Bouncing Ball Example:.

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Introduction to Hybrid Automata

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  1. Introduction to Hybrid Automata Arijit Mondal Kapil Modi Arnab Sinha

  2. Introduction • A hybrid automaton is a formal model for a mixed discrete continuous system. • Systems with ‘discrete jumps’ & ‘continuous flow’ can be modeled into Hybrid Automata. • Bouncing Ball Example: Here, the following properties hold:

  3. Bouncing Ball: Properties • States: In air (Assumption: Rebound time is negligible) • Continuous Variable: height (h), velocity (v) • Guard Condition : height=0, velocity=negative. • Effect (Reset Map): velocity changes due to restitution coefficient (e) We are ready for the Model !!!

  4. Bouncing Ball Model: Guard Condition State Continuous variables Domain (Fly) Reset condition

  5. An Illustration: Water Tank Problem

  6. Water Tank: Properties • The supplier can supply water at a rate of w to only one reservoir at a time. [Discrete Behavior] • The current levels are x1 and x2 respectively. [Continuous Variables] • The minimum threshold to be maintained are r1 and r2 respectively. [Guard Conditions] • It is assumed that while transition between reservoirs none of the level changes. [Reset Property] Hence we can model it with Hybrid Automata!!!

  7. Water Tank Problem state Continuous variables Domain(q2) Domain(q1) Reset Property Guard Condition

  8. The Automaton Where, Q = set of discrete states. X = set of continuous variables, Where, E is the set of edges. G is the guard condition, and, R is the Reset Map

  9. An Illustration: Water Tank Problem

  10. Water Tank Problem: Formal Model

  11. Water Tank Problem: Formal Model (Contd.)

  12. Water Tank Problem

  13. Hybrid time set It is a sequence of finite or infinite intervals such that

  14. Bouncing Ball: Hybrid time-set The bouncing ball: The first half is upward movement and the next half is downwards. The first run is interval and the next run is in and so on.

  15. Hybrid Trajectory (t, q, x) • A hybrid trajectory is a triple (t, q, x) consisting of a hybrid time set, t and two sequences of functions q and x such that

  16. Hybrid Execution An execution ofa hybrid automation H is hybrid trajectory, (t, q, x), which satisfies the following conditions. • Initial condition: • Discrete evolution:

  17. Hybrid Execution (contd.) • Continuous evolution: such that is the solution to the diff. equation starting at over

  18. Water Tank Problem: Hybrid Execution

  19. Water Tank Problem: Hybrid Execution (Contd.) Initial Condition Discrete Evolution

  20. Water Tank Problem: Hybrid Execution (Contd.) Continuous Evolution

  21. Classification of Executions • Finite, iftis a finitesequence and the last interval in t is closed. • Infinite, iftis a infinitesequence, or if, • Zeno, ifit is infinite but the sum of intervals is finite. Real life designs are mostly non-zeno i.e. time-diverging e.g. bouncing ball system. • Maximal, if it is not a strict prefix of any other execution of H.

  22. 0-Transition • We know, • Hence we define an event which triggers transition iff there exists an edge e= (q, q’) such that for some , • Hence we can say for all states q, of a hybrid automaton i.e. we can always construct an edge such that 0

  23. Composition of Automata • For two hybrid automata,and then we can define the semantics of parallel composition as • But for composition, the transitions have to be consistent. • The transitions, and are consistent if any of the following three conditions are true, • and . • and .

  24. Composition: Water Tank Model • We develop two independent models of the 2 reservoirs. 0 0 holds when water is supplied to tank 1.

  25. Composition: Water Tank Model • The complete model.

  26. Example: Buck Converter Buck converter driving variable load • Switch S1 remain on for 6 secs and off for 4 secs • Switch S2 alternate between R1 and R2 in every 4 secs

  27. Discrete states and State variables • Four discrete states • S1 on and S2—R1 (A) • S1 on and S2—R2 (B) • S1 off and S2—R2 (C) • S1 off and S2—R1 (D) • For circuit dynamics: • Current through inductor (i) • Voltage across capacitor (v) • Clock variables: • S1: denotes the duration of on/off state of switch S1 • S2: denotes the duration of connection of switch S2 with R1 or R2

  28. Dynamic activities For states (A) and (B) For states (C) and (D) For clock variable S1 and S2 for all locations

  29. Hybrid model of Buck converter

  30. Example (Buck converter) [Santosh]

  31. Descriptions • Zero pulse – Generates –ve square pulse when input crosses zero volt from any +ve voltage • Monoshot – Generates +ve square pulse with Ton and it is triggered by a –ve edge at the input. • Startup pulse – Generates –ve pulse to trigger the monoshot. • Zero crossing detector – It toggles output when the input crosses zero volt. Initial output logic zero. • Drivers – To drive power MOS switches.

  32. Hysteresis comparator • Outputs logic high if input is below threshold • Outputs logic low if input is above threshold Vout Vin

  33. Determination of discrete states • This systems can be modeled as hybrid system and dynamics behavior of each state depends on the following • State of PMOS • State of NMOS • Control signal to PMOS • Control signal to NMOS • Dynamic behavior of each state will depend on the following: • Vcx : PMOS drain voltage • Vout : Output voltage

  34. Hybrid automata

  35. Linear hybrid systems (LHS) • For all locations activity (vector field) can be defined as follows: • For all locations invariant (domain) is defined by a linear formula over continuous states (X). • For all transitions, guarded set of nondeterministic assgn.

  36. Example (x+y>4)→{x:=[3x+y,2y], y:=[7,5x]} v:(x=3,y=12) x=23 y=9 v(αx)=21 v(βx)=24 v(αy)=7 v(βy)=15 x=3 y=12

  37. Special cases Discrete variable Discrete system – All variable are discrete variable Proposition – x is discrete variable and Clock

  38. Special cases (contd.) • Timed automaton – Linear hybrid system all of whose variables are propositions or clocks and linear expression are Boolean combination of inequalities. (x#c or x-y#c) • Skewed clock: • Multirate timed system – LHS whose variables are propositions and skewed clocks • n-rate timed system – Multirate timed system whose skewed clocks proceed at n different rates

  39. Special cases (contd.) • Integrator • Parameter - x discrete variable • Simple LHS – Domains (invariants) and Guards are of the form x≤k or x≥k

  40. Reachability results • The reachability problem is decidable for simple multirate timed system. • The reachability problem is undecidable for 2-rate timed system. • The reachability problem is undecidable for simple integrator systems

  41. Verification of Hybrid Automata • A hybrid automata specification can be encoded as a set of desirable hybrid trajectories, H. • The given model is said to meet the given specification if the set of execution of the model is a subset of H. • Safety Property:- • where F is the set of safe states in which we wish to remain always. • Liveness Property:- • where T is the set of states in which we visit eventually.

  42. Example • Say we model a traffic system with a hybrid automata, then the set of safe states F, are those, in which no two cars collide. • Set of live states T, are those, in which the cars eventually reach their destination.

  43. Transition System from a hybrid automaton • H = (Q, X, Init, f, Dom, E, G, R) be a hybrid automaton with a distinguished set of final states, F, • S: set of states (finite or infinite) • A transition relation • A set of initial states • A set of final states Hybrid Automata transformed into a transition system.

  44. Transition System from a hybrid automaton (contd.) The transition relation can be divided into a discrete transition relation and a continuous transition relation. For each edge, For the continuous transition relation, Where, x(.) is the solution of the differential equation. Hence,

  45. Backward Reachability Algorithm: Initialization: repeat if return ” reachable “ endif until return “ not reachable“

  46. Backward Reachability: Example q0 q2 q1 q6 q4 q5 q3

  47. Backward Reachability: Example q0 q2 q1 q6 q4 q5 q3

  48. Backward Reachability: Example q0 q2 q1 q6 q4 q5 q3

  49. Bisimulation: Example • We can check, is a bisimulation of the given system, but is not. q0 q2 q1 q6 q4 q5 q3

  50. Bisimulation: Example q0 q2 q1 q6 q4 q5 q3

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