application: discrete event systems (pre lecture) · discoveries & inventions of the nineteenth...

Application: Discrete Event Systems (Pre Lecture)

Dr. Neil T. Dantam

CSCI-561, Colorado School of Mines

Fall 2019

Dantam (Mines CSCI-561) Application: Discrete Event Systems (Pre Lecture) Fall 2019 1 / 63

On Specifications

Introduction

Discrete Event Systems

I Discrete Event System (DES): asystem that exhibits discrete(non-continuous change)

I The control engineer’s view oflanguage and automata theory

I Applications:I Embedded SystemsI RoboticsI Verification

OutcomesI Broad introduction to DES

I Construct DES models of physicalprocesses / systems

I Construct specifications as formallanguages

I Control DES to achieve a desiredspecification

Outline

Introduction to Control

Discrete Event SystemsDefinitionsDES Languages & Properties

Supervisory Control

Outline

Supervisory Control

Physical vs. Computational Laws

A Physical Law

dt= g(x,u)

I x: current state

I u: current input

I dxdt : change in state

I g: process function

A Computational Law

qk+1 = δ(qk , σk)

I qk : current state

I σk : current input

I qk+1: successor state

I δ: transition function

Continuous Dynamics. Discrete Dynamics.

Ye Olde Controller

Flyball Governer

Discoveries & Inventions of the Nineteenth Century.R. Routledge, 13th edition, published 1900.

Control: modify system’s input to achieve desired property

The World According to Controls Engineers

x = F (x , u)u = G (r)

Controller Plant

Symbol Description

x ∈ X stateu ∈ U input

r referenceX state spaceF PlantG Control Law

Feedforward and Feedback Control

Feedforward / Open-Loop Control

x = F (x , u)u = G (r)

Controller Plant

Feedback / Closed-Loop Control

x = F (u)u = G (r , x)

Controller Plant

Example: The Classic Inverted Pendulum

(mc + m`) x −m``θ cos θ + m`θ2 sin θ = f`θ − g sin θ = x cos θ

Gf x , x

θ, θx = 0, x = 0θ = 0, θ = 0

Outline

Supervisory Control

Supervisory Control Theory

Peter Ramadge

I Princeton

Walter Wonham

I Toronto (emeritus)

P. J. Ramadge and W. M. Wonham. Supervisory control of a class ofdiscrete event processes. SIAM J. Control and Optimization, Vol. 25,No. 1, pp. 206-230, 1987.

Discrete Event Systems Definitions

Events

Switch On/Off

Clock edge

Contact

|x | ≥ k

Threshold

A discrete item of a finite set.Dantam (Mines CSCI-561) Application: Discrete Event Systems (Pre Lecture) Fall 2019 13 / 63

Physical Interpretation of Events

Quantum Mechanics

Maxwell’s Equations ∇× B = µ0

(J + ε0

δEδt

)Linear Circuits V = IR

Switch

Event {on,off}

Events are a convenient abstractionDantam (Mines CSCI-561) Application: Discrete Event Systems (Pre Lecture) Fall 2019 14 / 63

The Discrete Event System

Definition (Discrete Event System)

A discrete-state, event-drive system. That is, its state evolutiondepends entirely on the occurrence of asynchronous discrete eventsover time.D = (X ,E , f , Γ, x0,Xm):

I X is the set of states

I E is the finite set of events

I f : X × E 7→ X is the transition function

I Γ : X 7→ P (E ) is the active event function.Γ(x) = {e ∈ E | f (x , e) is defined}

I x0 is the initial state

I Xm is the set of marked states

DES vs. DFA

D = (X ,E , f , Γ, x0,Xm):

I X is the set of states

I E is the finite set of events

I f : X × E 7→ X is the transition function

I Γ : X 7→ P (E ) is the active eventfunction.Γ(x) = {e ∈ E | f (x , e) is defined}

I x0 is the initial state

I Xm is the set of marked states

M = (Q,Σ, δ, q0,F ), where

I Q is a finite set call the states

I Σ is a finite set call the alphabet

I δ : Q ×Σ 7→ Q is the transition function

I q0 ∈ Q is the start state

I F ⊆ Q is the set of accept states

Example: Going for a walk

I Scenario:I You may go for a walk.I It could rain.I If it rains, you’ll get wet.

I Events:I go-outsideI sunI rainI get-wetI go-home

go-outside

get-wet

go-home

Example: Carry an Umbrella

I Scenario:I You may go for a walk.I It could rain.I If it rains, you’ll get wet.I You can pack an umbrella.

I Events:I go-outsideI sunI rainI get-wetI go-homeI pack-umbrella

startgo-outside

rain get-wet

go-home

go-outside sun

packumbrella

go-home

Exercise: Robot Vacuum

I Drive till bump sensor

I Then turnI Events:

I startI stopI driveI bumpI turn

startstart drive bump

Exercise: Robot Vacuum, Redux

I Add IR proximitysensor

I Events:

I startI stopI driveI bumpI turnI Proximity

startstart drive proximity

driveturn

Discrete Event Systems DES Languages & Properties

Extended Transition Function

Transition Function f : X × E 7→ X

Extended Transition Function f : X × E ∗ 7→ X

base: f (x , ε) = xinductive: f (x , αe) = f (f (x , α), e)

where α ∈ E ∗ and e ∈ E

Define transitions recursively over strings

Language Marked

Definition (Language Marked)

The language marked by D = (X ,E , f , Γ, x0,Xm) is the set of stringsthat take D to a final marked state:

Lm (D) ={s ∈ E ∗ | f (x0, s) ∈ Xm

Different word for acceptance

Language Generated for DES

Definition (Language Generated)

The language generated by D = (X ,E , f , Γ, x0,Xm) is the set ofstrings that have defined transitions in D:

Lg (D) ={s ∈ E ∗ | f (x0, s) is defined

Behavior that is possible, but not necessarily “acceptable.”

Example: Marking and Generation 0

0 1 2 3

a b c Lm (D) = {ab}

Lg (D) = {ε, a, ab, abc}

Example: Marking and Generation 1

startb a

D I Lm (D) :I Strings ending in aI (a|b)∗a

I Lg (D) :I All strings in EI (a|b)∗

Exercise: Marking and Generation

starta

I Lm (D) :

I Sequences of a’s with isolated b’s,starting and ending with a

I a (a| (ba))∗

I Lg (D) :

I Sequences a’s and b’s with at most oneconsecutive b

I (a|ab)∗

Prefix-Closure

Definition

The prefix closure of language L is the set of all prefixes of strings in L:

L = {α ∈ E ∗ | ∃β ∈ E ∗, αβ ∈ L},

where E is the event set (alphabet) for L.

Example: Prefix Closure 0

0 1 2 3

Lm (D) = {ab}

Lm (D) = {ε, a, ab}

Example: Prefix Closure 1

Prefix Closure Algorithm

Algorithm 1: prefix-closure

Input: M = (Q,Σ, δ, q0,F ) ; // states,alphabet,transition,start,accept

Output: M ′ = (Q ′,Σ, δ′, q′0,F′) ; // states,alphabet,transition,start,accept

1 (Q ′, δ′, q′0,F′)← (Q, δ, q0,F );

2 function visit(q,p) is // new-state× visited-states

3 if q 6∈ p then

4 p′ ← p ∪ {q};5 if q ∈ F then F ′ ← F ′ ∪ p′;6 forall σ ∈ Σ do // Visit all neighbors of q

7 visit(δ(q, σ), p′);

8 visit(q0, ∅);

Exercise: Prefix Closure

Deadlock

Generally: Lm (D) ⊆ Lm (D) ⊆ Lg (D)

Definition

Automaton reaches a state from which no further execution is possible:

∃x ∈ X , σ ∈ Lg (D),

(x = f (x0, σ))

︸︷︷︸Reach x on σ

∧ (Γ(x) 6= ∅)︸︷︷︸Some active event at x

Example: Deadlock

Livelock

Definition

Automaton reaches a cycle from which a marked (accept) state is notreachable.

∃(x ∈ X ,σ ∈ Lg (D), α ∈ E+),

∀β ∈ E ∗,(x = f (x0, σ)

)︸︷︷︸

Reach x on σ

∧(x = f (x , α)

)︸︷︷︸

∧(f (x , β) 6∈ Xm

)︸︷︷︸No path to accept

Example: Livelock

Blocking

Generally: Lm (D) ⊆ Lm (D) ⊆ Lg (D)

Definition

An automaton D is blocking if it can deadlock or livelock. Anautomaton D is nonblocking if neither deadlock nor livelock arepossible.

Blocking: Lm (D) ⊂ Lg (D)

I We can generate a string that is not a prefix to amarked state.

Nonblocking: Lm (D) = Lg (D)

I Every string we can generate is a prefix to a markedstate.

Supervisory Control

Outline

Supervisory Control

Supervisor Function

I Dynamically enable/disable events in E

I S : Lg (D) 7→ P (E )

DSS(σ)

SupervisorUncontrolled

System

AllowedEvents Generated

String

Restricts the DES to desirable behavior.

Supervisory Control

Supervised Generation: Lg (S/D)

Formal Description

Base: Contains the empty string:ε ∈ Lg (S/D)

Inductive: Next event e in recursively-allowed string σ is allowed by the supervisor:

(σe ∈ Lg (S/D)) ⇐⇒

(σ ∈ Lg (S/D))︸︷︷︸recurse

∧ (σe ∈ Lg (D))︸︷︷︸uncontrolled

∧ (e ∈ S(σ))︸︷︷︸allowed

where e ∈ E and σ ∈ E ∗

Supervisory Control

Supervised Marking: Lm (S/D)

Lm (S/D)︸︷︷︸supervised marked

= Lm (D)︸︷︷︸system marked

∩ Lg (S/D)︸︷︷︸supervised generated

Supervisory Control

Supervised Blocking

∅ ⊆ Lm (S/D) ⊆ ˜Lm (S/D) ⊆ Lg (S/D) ⊆ Lg (D)

Blocking Lg (S/D) 6= ˜Lm (S/D)

I Generated strings not prefixes of marked stringsI Can generate unmarked strings

Nonblocking Lg (S/D) = ˜Lm (S/D)

I All generated strings are prefixes of marked strings

Supervisory Control

Example: Supervision

0 1 2 3

a b cD

Lm (D) = {ab}

Lm (D) = {ε, a, ab}

I S(ε) = a

I S(a) = b

I S(ab) = ∅

0 1 2 3

a bS/D

Lm (D) = {ab}

Lm (D) = {ε, a, ab}

Lg (D) = {ε, a, ab}

Supervisory Control

Supervisor Functions are Languages

Language of Supervisor

Supervisor functions define languages:

L (S) =

σe | (σ ∈ L (S))︸︷︷︸recurse

∧ (e ∈ S(σ))︸︷︷︸allowed

Supervisor of Language

Languages define supervisor functions:

SL(σ) = {e | σe ∈ L}

Lg (S/D) ⊆ L (S)

Supervisory Control

Example: Supervisor

Function

I S(ε) = a

I S(a) = b

I S(ab) = ∅

Language

L (S) = {ε, a, ab}

Automaton

s0 s1 s2

Supervisory Control

Exercise: Supervision

start a

I S(ε) = {a}I S(a) = ∅

L (S) = {ε, a}

start a

Supervisory Control

Specifications

I Express desired behavior:I “Don’t do x”I “Don’t do x after you do y”I “Do x , then do y”

I Represent (specify) language of acceptable/admissible behavior:La

I System should only generate admissible behaviors:Lg (La/D) ⊆ La

Supervisory Control

Example: Specifications

For E = {x , y , z}, write the following as regular expressions:I “Don’t do x”

I (y |z)∗

I “Don’t do x after you do y”I (x |z)∗

(y (y |z)∗

I (x |z)∗(ε|y (y |z)∗

)I “Do x , then do y”

I xy(x |y |z)∗

Supervisory Control

Exercise: Specifications

For E = {a, b, c}, write the following as regular expressions:I “Repeat a until b”

I a∗b(a|b|c)∗

I “Avoid a until b”

I c∗b(a|b|c)∗

I “Do a at most once”

I (b|c)∗a?(b|c)∗

I (b|c)∗(a|ε)(b|c)∗

Supervisory Control

Controllable and Uncontrollable Events

controllable︷︸︸︷Ec ∪

uncontrollable︷︸︸︷Euc

Controllable Events: events we can prevent from happening.Conversely, an action we may choose to take.

I Honk hornI Turn left

Uncontrollable Events: events we cannot prevent from happening

Other agent: Stoplight turns redSensed condition: Icy roads

Fault: Brakes fail

Supervisory Control

Fully Controllable Case

I All events are controllable:I Ec = EI Euc = ∅

I Supervision is a direct intersection operate:I Lg (S/D) = Lg (D) ∩ L (S)

Supervisory Control

Example: Weather

S = (E \ wet)∗

= (home|out|sun|rain)∗

home,out,sun,rain

start dwet

home,out,sun,rain,wet

Supervisory Control

Example: Weather (continued)

0,s 1,s

3,s 4,s

start 0,d 1,d

3,d 4,d

homewet

What’s wrong?

Modeling: can’t control the rain

Supervisory Control

Partially Controllable Case

I Some events are uncontrollable:I Ec ⊂ EI Euc 6= ∅

I A Simple Controller Synthesis Algorithm:1. Find “bad” states:

I base: Disallowed and blocking states in SI recursive: states with uncontrollable transitions to “bad” states

2. Avoid “bad” states

Supervisory Control

Example: Weather (continued)

Euc = {sun, rain,wet}

0,s 1,s

3,s 4,s

start 0,d 1,d

3,d 4,d

homewet

0,d 1,d

3,d 4,d3,s

1,s 0,d 1,d

3,d 4,d3,s

Supervisory Control

Example: Umbrella

0start 1

3 4out

rainget-wet

5 6 7out sun

packumbrella

0start 1

get-wet

5 6out

packumbrella

sun,rain

simplify

Supervisory Control

Example: Umbrella(continued)

0,s 1,s

5,s 6,s

start 0,d 1,d

5,d 6,d

packumbrella

sun,rain

packumbrella

sun,rain

0,d 1,d

5,d 6,d

3,s1,s

Supervisory Control

Fixed PointReview

Definition: Fixed Point

The fixed point of function is avalue where the function’s inputand output are equal.

For f : X 7→ X , the fixpoint issome value x ∈ X where f (x) = x .

Examples

I f : Z 7→ ZI f (x) = x2

I 0 is a fixpoint: f (0) = 0I 1 is a fixpoint: f (1) = 1I 2 is NOT a fixpoint: f (2) = 3

I g : R 7→ RI g(σ) = σR

I aa is a fixpoint: g(aa) = aaI ab is NOT fixpoint: g(ab) = ba

Supervisory Control

Partition Bad States

Algorithm 2: partition-bad-states(D,S)

Input: D = (XD ,E , fD , ΓD , x0,D ,Xm,D); // DES

Input: S = (XS ,E , fS , ΓS , x0,S ,Xm,S); // Specification

Input: Xbad; // Bad states

1 B ← blocking states in S ;2 X ′ ← XD × XS ;

3 Xbad ←

(xD , xS) ∈ X ′ |uncontrollable︷︸︸︷∃e ∈ Euc , e ∈ ΓD(xD)︸︷︷︸

allowed in D

fS(xS , e) = ∅︸︷︷︸not in S

∨ fS(xS , e) ∈ B︸︷︷︸blocking in S

4 XOK ← X ′ \ Xbad;

5 f ′((xD , xS), e) , (fD(xD , e), fS(xS , e));6 Xbad ← partition-fixpoint(XOK,Xbad, f

Initial Xbad: defined transitions to blocking states

Supervisory Control

Partition Fixpoint

Function partition-fixpoint(XOK,Xbad,E , f )

/* Uncontrollable transitions to bad states */

1 X ′ = {x ∈ XOK | ∃e ∈ Euc , f (x , e) ∈ Xbad};2 if X ′ = ∅ then // base: fixpoint

3 return Xbad;4 else // Recurse

5 return partition-fixpoint(XOK \ X ′,Xbad ∪ X ′, f );

Supervisory Control

Exercise: Robot Vacuum Supervision

stop drive proximitydrive

bumpturn

I Don’t bump

E \ bump

Supervisory Control

Exercise: Robot Vacuum Supervisioncontinued

1,s 2,s 3,s

4,s5,s

0,d 1,d 2,d 3,d

4,d5,dstart

stop drive proximity

bumpturn

0,d 1,d 2,d 3,d

4,d5,d

Supervisory Control

Why don’t I just write the code?

I Can you prove the code does what you want?

I How do you say (specify) what you want?

“Program testing can be usedto show the presence of bugs,but never to show their absence!”–Edsger W. Dijkstra (EWD249)

“A design without specificationscannot be right or wrong,it can only be surprising!”–William D. Young (paraphrased)Young, Boebert, and Kain.Proving a computer system secure.

Supervisory Control

References and Further Reading

Discrete Event Systems: Christos Cassandras and Stephane Lafortune.Introduction to Discrete Event Systems.

Verification: Christel Baier and Joost-Pieter Katoen. Principles of Model Checking.

Switching and Hybrid Systems: Daniel Liberzon. Switching in Systems and Control.

Principles of Model CheckingChristel Baier and Joost-Pieter Katoen

application: discrete event systems (pre lecture) · discoveries & inventions of the nineteenth...

Documents

inventions. minor inventions and industrial designs...

(routledge philosophy companions) berys gaut, dominic...

theodolites - routledge handbooks

entrepreneurship.aui.ac.ir · 2018-09-12 · twenty-fint...

heidegger in the routledge

action research - routledge

depression - routledge

hosted by miss routledge

the routledge handbook of applied...

myths & legends - routledge

phosphorus - routledge handbooks

principles of mathematics (routledge classics) - bertrand...

burke behne_everyday inventions powerpoint_everyday...

ecology - routledge handbooks

tesol 11 routledge pptho

feminist publishing at routledge

[] routledge urban_regeneration_management(book_fi.org)

routledge major works politics ir political...

routledge intensive german course

workbook - depression - routledge