Hybrid Systems Presented by: Arnab De Anand S

An Intuitive Introduction to Hybrid Systems Discrete program with an analog environment. What does it mean? Sequence of discrete steps – in each step the system evolves continuously according to some dynamical law until a transition occurs. Transitions are instantaneous.

A Motivating Example: Thermostat The heater can be on or off. When the heater is on, the temperature increases continuously according to some formula. When the heater is off, the temperature decreases. Thermostat keeps the temperature within some limit by putting the heater on or off.

Formal Model of Hybrid Systems Model Hybrid Systems as graphs:  Vertices represent continuous activities.  Edges represent transition.

Formal Model cont’d… H = (Loc, Var, Lab, Edg, Act, Inv)  Loc: finite set of vertices (locations)  Var: finite set of real-valued variables.  A valuation v(x) assignes a real value to each variable. V is the set of valuations.  A state is a pair (l, v), l є Loc, v є V.

Formal Model cont’d…  Lab: finite set of synchronization labels, containing the stutter label τ  Edg: finite set of edges (transitions). e = (l, a, µ, l’) Stutter transition (l, µ, Id Con, l).  Act: set of activities, maps non-negative reals to valuations.  Inv: set of invariants at a location.

Time-deterministic hybrid system There is at most one activity for each location and each valuation such that f(0) = v Denoted by φ l [v].

Runs of a Hybrid System A state can change in two ways:  Discrete and Instantaneous transition that changes both l and v.  Time delay that changes only v according to activities of the location.  Some transition must be taken before the invariant becomes false. Run:

Thermostat example revisited

Hybrid Systems as Transition Systems

Composition of Hybrid Systems

Linear Hybrid System A time-deterministic hybrid system is linear if: 1. The activity functions are of the form 2. The invariant for each location is defined by a linear formula over Var.

Linear Hybrid System cont’d… 3. For all transitions, the transition relation µ is defined by a guarded set of non-deterministic assignments If α x = β x, we write

Special Cases of Linear Hybrid Systems If Act(l,x) = 0 for all locations, then x is a discrete variable. A discrete variable x is a proposition if for all transitions. A finite-state system is a linear hybrid system all of whose variables are propositions.

Special cases cont’d… If Act(l,x) = 1 for each location and for each transition, then x is a clock. A timed automaton is a LHS all of whose variables are either propositions or clocks and the linear expressions are boolean combination of inequalities of the form x#c or x-y#c (c non-negative integer).

Special cases cont’d… If for each location and for each edge, then x is an integrator. An integrator system is a LHS all of whose variables are propositions or integrators.

Example of LHS: Leaking Gas Burner

Reachability problem Given two states, does there exist any run that starts at first state and ends at another. Verification of some invariant property is equivalent to the reachability question. Reachability is undecidable in general… but decidable for some special cases.

Verification of Linear Hybrid Systems H=(Loc,Var,lab,Edg,Act,Inv) Do a reachability analysis Iteratively find out the reachable states Forward analysis – computes step successors of a given set of states Backward analysis

Forward analysis Forward time closure Set of valuations reachable from some v єP by letting time progress. (l,v)  t (l’,v’) Post condition of P w.r.t an edge e, The set of valuations reachable from v є P by executing transition e. (l,v)  a (l’,v’)

Forward Analysis (contd…) Region: A set of states Define (l,P) = {(l,v) | v є P } Extension to regions: for R=U lєLoc (l,R l )

Forward Analysis (contd…) A symbolic run on H is (in)infinite sequence ρ: (l 0,P 0 )(l 1,P 1 ),……(l i,P i ). The region (l i,P i ) is the set of states reachable from (l 0,v 0 ) after executing e 0,….e i-1 Every run of H can be represented by some symbolic run of H Given I (subset of Σ), the reachable region (I  *) is the set of states reachable from I.

Forward Analysis (contd…) Reachable region is least fixed point of. Or R l of valuations for l є Loc if lfp of. [ψ] = set of valuations that satisfy ψ Ψ is a linear formula P  v is linear if P=[ψ] for some ψ

Forward Analysis (contd…) For linear H, if P is linear, then so is l  and post e [P] pc  Var is a control var with range Loc A region R is linear of all R l ([ψ l ]) are linear Region R is defined by Do successive approx. Terminate for simple mutirated timed systems

Example : leaking gas burner.

Backward Analysis Backward time closure. Precondition. Extension

Backward Analysis (contd…) Initial region. Equations Initial region if lfp. l  and pre e [P] are linear In example, we find set of states from which ψ R =y ≥60  20z ≤y is reachable. We get null set

Model Checking (Timed CTL) Check if H satisfies a requirement expressed in real-time temporal logic Define C (disjoint with Var) State predicate is a linear formula over Var U C The grammer. Ψ is state predicate and zєC Formulas of TCTL are interpreted over state space of H

Timed CTL (contd…) Clocks can be used to express timing constraints. A run ρ=σ 0  t0 σ 1  t1 For a state ρ i =(li,vi), position  =(i,t) (0≤t ≤t i ) Positions are lexicographically ordered.

TCTL (contd…) For all positions  =(i,t) Clock valuation ξ: C  R ≥0 ξ+t and ξ[z=0] Extended state (σ, ξ)

Model Checking (contd…) (σ, ξ) ╞ Φ, if

Model Checking algorithm σ ╞ Φ, of (σ,ξ) Φ for all ξ evaluations Computes Characteristic set [Φ] (l,v) є (R ► R’) iff Single step until operator If R and R’ are linear so is R ► R’ Thus the modalities can be computed iteratively using ► Will terminate in simple multirate timed system

Examples Φ  U Φ’ computed as U i R i with  ◊ ≤c Φ computed as ¬U i R i [z=0] with

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