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, IdCon, 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 φl[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 αx = βx, 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=UlєLoc(l,Rl)

Forward Analysis (contd…) A symbolic run on H is (in)infinite sequence

ρ: (l0,P0)(l1,P1),……(li,Pi) .

The region (li,Pi) is the set of states reachable from (l0,v0) after executing e0,….ei-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 Rl of valuations for l є Loc if lfp of .

[ψ] = set of valuations that satisfy ψ Ψ is a linear formula

Pv is linear if P=[ψ] for some ψ

Forward Analysis (contd…) For linear H, if P is linear, then so is

<P>l and poste[P]

pc Var is a control var with range Loc A region R is linear of all Rl([ψl]) 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

. .

<P>l and pree[P] are linear

In example, we find set of states from which ψR=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=σ0 t0 σ1 t1 For a state ρi=(li,vi), position =(i,t)

(0≤t ≤ti) Positions are lexicographically ordered

.

TCTL (contd…)

For all positions =(i,t)

Clock valuation ξ: CR≥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

ΦU Φ’ computed as UiRi with

◊≤c Φ computed as ¬UiRi[z=0] with

Thank you