Model Checking LTL over (discrete time) Controllable Linear System is Decidable P. Tabuada and G. J. Pappas Michael, Roozbeh Ph.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 Z n 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 finite Set 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 form Brunovsky 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 Z n ) which is bisimilar to T  ´, is constructed where Quantization map: where

New Transition Map - II Controlled 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 T  Observation 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 relationis 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 Z 2 Finite observation space O = S [ {  } Since k 1 = 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 Brunovsky normal form Original linear control system Supervisor (FSM) Symbols Continuous input