Model Checking LTL over (discrete time) Controllable Linear System is Decidable

P. Tabuada and G. J. Pappas

Michael, RoozbehPh.D. Course November 2005

Overview

• Transition system with observations

• Linear Temporal Logic (LTL)

• Simulation/bisimulation relations

• Construction of finite abstraction– Transform system into Brunovsky normal form– Bisimulation with denumerable state space Zn

• LTL control of linear control systems

Transition Systems - Revisited

Notation: X : set of all infinte strings formed by elements of X

Transition Systems as LTL Models

Formally represents temporal properties of dynamical and control systems.

Specification formulas are built from atomic propositions belonging to a finiteSet

Use of LTL formulas to specify the sequency of observations (desired behavior)

Means ”next”: The formula 1 will be true in the next time step

Means ”until”: The formula 1 must hold until 2 holds

Transition Systems as LTL Models

PS: O can be infinte while is finite.

The sequence satisfies formula iff (0) ²

LTL Example

Relationship between Transition Systems

Relationship between Transitiom Systems - II

Important: Language equivalence preserves properties expressible in LTL

Important: Bisimilarity also preserves properties expressible in LTL

Linear Control Systems as Transition Systems

Requirement: The (discrete time) linear systems that are controllable are considered

Note: The set of observations O and the observation map h are defined later.

Brunovsky Normal Form

0

r = rank(B)

Brunovski Normal Form

This is refered to as shift register form

Example

Consider the controllable linear system with n=3 and m=2

Shift register formBrunovsky normal form

Bisimulation I between T and T’

T bisimilar to T’ (’ and are isomorphic) Observation map

New Transition System - I

The new transition system T, (with state-space Zn) which is bisimilar to T´, is

constructed

where

Quantization map:

where

New Transition Map - IIControlled evolution on the space of blocks – under appropiate inputs blocks will move into other blocks of the grid

Example:

Bisimulation II between T’ and T

T’ bisimilar to TObservation map

Pre Operator

Given a state q 2 Q, we denote by Pre(q) the set of states in Q that can reach q in one step, that is

Example – Pre Operator

Language Equivalent Finite Abstraction

Assumption: Set of observations O is finite.

Language Equivalent Finite Abstraction - II

This finite abstraction requires the following subset of the state space, defined for any a 2 S

Covers the state-space

Language Equivalent Finite Abstraction - III

The finite transition system

Where the transition relation is constructed as follows

Language Equivalent Finite Abstraction - IV

Decidability of Model Checking

Canonical Projection

Example - Construction of T

Finite set of atomic propositions S = a = {(0,0)} 2 Z2

Finite observation space O = S [ {}

Since k1 = 2 we need to compute the following sets:

Construction of T

Summary

Relationship between transition systems

Relationship between observation space

Atomic proposition

(Brunovsky Set) (Quantization Block) (Point)

LTL Control of Linear Control Systems

Implementation

Brunovskynormal form

Original linearcontrol system

Supervisor (FSM)

Symbols

Continuousinput