model checking ltl over (discrete time) controllable linear system is decidable p. tabuada and g. j....
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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 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
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.
Example
Consider the controllable linear system with n=3 and m=2
Shift register formBrunovsky normal form
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:
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
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
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:
Summary
Relationship between transition systems
Relationship between observation space
Atomic proposition
(Brunovsky Set) (Quantization Block) (Point)