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Theory of Hybrid Automata

Theory of Hybrid Automata Sachin J Mujumdar Hybrid Automata A formal model for a dynamical system with discrete and continuous components Example – Temperature Control Formal Definition A Hybrid Automaton consists of following: Variables – Finite Set (real numbered) Continuous Change,

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Theory of Hybrid Automata

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  1. Theory of Hybrid Automata Sachin J Mujumdar CS 367 - Theory of Hybrid Automata

  2. Hybrid Automata • A formal model for a dynamical system with discrete and continuous components • Example – Temperature Control CS 367 - Theory of Hybrid Automata

  3. Formal Definition A Hybrid Automaton consists of following: • Variables – • Finite Set (real numbered) • Continuous Change, • Values at conclusion at of discrete change, • Control Graph • Finite Directed Multigraph (V, E) • V – control modes (represent discrete state) • E – control switches (represent discrete dynamics) CS 367 - Theory of Hybrid Automata

  4. Formal Definition • Initial, Invariant & Flow conditions – vertex labeling functions • init(v) – initial condition whose free variable are from X • inv(v) – free variables from X • flow(v) – free variables from X U • Jump Conditions • Edge Labeling function, “jump” for every control switch, e Є E • Free Variables from X U X’ • Events • Finite set of events, Σ • Edge labeling function, event: E  Σ, for assigning an event to each control switch • Continuous State – points in CS 367 - Theory of Hybrid Automata

  5. Safe Semantics • Execution of Hybrid Automaton – continuous change (flows) and discrete change (jumps) • Abstraction to fully discrete transition system • Using Labeled Transition Systems CS 367 - Theory of Hybrid Automata

  6. Labeled Transition Systems • Labeled Transition System, S • State Space, Q – (Q0 – initial states) • Transition Relations • Set of labels, A – possibly infinite • Binary Relations on Q, • Region, R Q • Transition – triplet of CS 367 - Theory of Hybrid Automata

  7. Labeled Transition Systems • Two Labeled Transition Systems • Timed Transition System • Abstracts continuous flows by transitions • Retains info on source, target & duration of flow • Time-Abstract Transition System • Also abstracts the duration of flows • Called timed-abstraction of Timed Transition Systems CS 367 - Theory of Hybrid Automata

  8. Live Semantics • Usually consider the infinite behavior of hybrid automaton. Thus, only infinite sequences of transitions considered • Transitions do not converge in time • Divergence of time – liveness • Nonzeno – Cant prevent time from diverging CS 367 - Theory of Hybrid Automata

  9. Live Transition Systems • Trajectory of S • (In)Finite Sequence of <ai, qi>i≥1 • Condition – • q0 – rooted trajectory • If q0 is initial state, then intialized trajectory • Live Transition System • (S, L) pair • L infinite number of initialized trajectories of S • Trace • <ai, qi>i≥1 is finite initialized trajectory of S, or trajectory in L  corresponding sequence <ai>i≥1 of labels is a Trace of (S, L), i.e. the Live Transition System CS 367 - Theory of Hybrid Automata

  10. Composition of Hybrid Automata • Two Hybrid Automata, H1 & H2 • Interact via joint events • a is an event of both  Both must synchronize on a-transitions • a is an event of only H1  each a-transition of H1 synchronizes with a 0-duration time transition of H2 • Vice-Versa CS 367 - Theory of Hybrid Automata

  11. Composition of Hybrid Automata • Product of Transition Systems • Labeled Transition Systems, S1 & S2 • Consistency Check • Associative partial function • Denoted by • Defined on pairs consisting of a transition from S1 & a transition from S2 • S1 x S2 • w.r.t • State Space – Q1 x Q2 • Initial States – Q01 x Q02 • Label Set – range( ) • Transition Condition • and  CS 367 - Theory of Hybrid Automata

  12. Composition of Hybrid Automata • Parallel Composition • H1 and H2 • of and of are consistent if one of the 3 is true • a1 = a2 consistency check yields a1 • a1 belongs to Event space of H1 and a2 = 0  consistency check yields a1 • a2 belongs to Event space of H2 and a1 = 0  consistency check yields a1 • The Parallel Composition is defined to be the cross product w.r.t the consistency check CS 367 - Theory of Hybrid Automata

  13. Railroad Gate Control - Example • Circular track, with a gate – 2000 – 5000 m circumference • ‘x’ – distance of train from gate • speed – b/w 40 m/s & 50 m/s • x = 1000 m • “approach” event • may slow down to 30 m/s • x = -100 m (100m past the gate) • “exit event” • Problem • Train Automaton • Gate Automaton • Controller Automaton CS 367 - Theory of Hybrid Automata

  14. Railroad Gate Control - Example Train Automaton CS 367 - Theory of Hybrid Automata

  15. Railroad Gate Control - Example Gate Automaton • y – position of gate in degrees (max 90) • 9 degrees / sec CS 367 - Theory of Hybrid Automata

  16. Railroad Gate Control - Example Controller Automaton • u – reaction delay of controller • z – clock for measuring elapsed time Question : value of “u” so that, y = 0, whenever -10 <= x <= 10 CS 367 - Theory of Hybrid Automata

  17. Verification 4 paradigmatic Qs about the traces of the H • Reachability • For any H, given a control mode, v, if there exists some initialized trajectory for its Labeled Transition System(LTS), can it visit the state of the form (v, x)? • Emptiness • Given H, if there exists a divergent initialized trajectory of the LTS? • (Finitary) Timed Trace Inclusion Problem • Given H1 & H2, if every (finitary) timed trace of H1 is also that of H2 • (Finitary) Time-Abstract Trace Inclusion Problem • Same as above – consider time-abstract traces CS 367 - Theory of Hybrid Automata

  18. Rectangular Automata • Flow Conditions are independent of Control Modes • First derivative, x dot, of each variable has fixed range of values, in every control mode • This is independent of the control switches • After a control switch – value of variable is either unchanged or from a fixed set of possibilities • Each variable becomes independent of other variables • Multirectangular Automata – allows for flow conditions that vary with control switches • Triangular Automata – allows for comparison of variables CS 367 - Theory of Hybrid Automata

  19. State Space of Hybrid Automata • State Space is infinite – cannot be ennumerated • Studied using finite symbolic representation • x – real numbered variable • 1 <= x <= 5  Finite symbolic representation of an infinite set of real numbers CS 367 - Theory of Hybrid Automata

  20. Observational Transition Systems • Difficult to (dis)prove the assertion about behavior of H – sampling of only piecewise continuous trajectory of LTS’ at discrete time intervals • Reminder – Transition abstracts the information of all the intermediate states visited • Solution • Label each transition with a region • transition, t, is labeled with region, R, iff all intermediate & target states of t lie in R • i.e. Observational Transition System – from continuous observation of hybrid automaton CS 367 - Theory of Hybrid Automata

  21. Summary • Introduction to Hybrid Systems • Formal Definition of Hybrid Systems • Change from hybrid to fully-discrete systems - Safe Semantics • Labeled transition Systems • Composition of Hybrid Automata • Properties of Hybrid Automata • Observational Transition Systems • Theorems & Theories presented in paper, for further reading – “The Theory of Hybrid Automata” – Thomas A. Henzinger CS 367 - Theory of Hybrid Automata

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