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UNIVERSITY OF MICHIGANDEPARTMENT OF ELECTRICAL ENGINEERING AND

COMPUTER SCIENCELECTURE NOTES FOR EECS 661

CHAPTER 1: INTRODUCTION TO DISCRETE EVENTSYSTEMS

Stephane Lafortune

August 2006

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' $EECS 661 - Chapter 1

References for Chapter 1: Textbook, Chapter 1: Section 1.3

Discrete Event Systems

A Multidisciplinary Area:

Systems & Control

DES

OperationsResearch

ComputerScience

What:

• Discrete State Space (logical, symbolic variables)

• Event-driven Dynamics

Why:

• Technological Systems, Computer Control

−→ Large, Complex Systems: they need to be analyzed, diagnosed, controlled, and optimized

S. Lafortune - Last revision: August 2006 1& %


Where:

• Inherently Discrete Systems:

computer systems, communication networks, automated manufacturing systems (cell and

factory levels), software systems.

• Systems with Continuous and Discrete Variables (hybrid systems), modeled as DES at a

certain level of abstraction, e.g., for the higher level control logic:

automated manufacturing systems (machine and cell levels), process control,

transportation systems.

• Embedded systems ; networked systems.

How:

• Mathematical Modeling, Analysis, Verification, Diagnosis, Controller Design,

Optimization, Simulation



Conceptual Control

System Architecture:

COORDINATION

REAL-TIME

CONTROLDIAGNOSTICS

FAILURERECOVERY

SUPERVISORY CONTROLLER

INTERFACE

EQUIPMENT

CONTROLLERSCONTROLLER

SYSTEM

Commands Observable events



Some Examples

The Heating System of a Heating, Ventilation, and Air Conditioning (HVAC)

Unit

FAN HTG. COIL

PUMP

BOILERCONTROLLER

VALVE

• The operation of the unit is monitored by a set of sensors.

• The issue of interest: Fault Diagnosis.

• Specifically: diagnose occurrence of “sharp” faults during the on-line operation of the unit.

• Examples of faults: stuck failures of valves, on-off failures of pumps, controllers, sensors,

etc.

• Implementation: diagnostics module in the control logic.



Models

of the Components of

the HVAC System:

L1

BOFF

B2

BON

FON S P I OV PON BON

BOFF POFF

S P D

CV

S P IS P D

FOFF

C1 C2 C3 C4 C5 C6

C7C8C9C10

CONTROLLER

FOFF

F1 F2

FON

PON

POFF SO1

CV, OV

OV

SC1 SC2

SO2

CV

L2

S P I

PUMP VALVE

FAN BOILER

LOAD

P1 P2POFF PONV3

V1 V2

V4

CV, OV

CV OVL0

FOFF FON

B1

BOFF BON

S P I

FOFF FOFF

S P D

S P D

C21

CFOFF

FONSPD

FOFF

FOFF

SPISPI SPDC22

C23

C24

SPIOV PON BON

OV PON BONSPD

SPISPD

CFON

C16

FONC13

C12C11

C20 C19 C17

C15

C18

C14



Part of the Diagnoser

for the Heating System

(HVAC Unit)

F 1: SO

F 2: SC

F 3: CFON

F 4: CFOFF

7 N 8 F1 9 F237 F3 38 F1 F3 39 F2 F3

85 F4 86 F1 F4 87 F2 F4

10 N 11 F1 12 F2 40 F3 41 F1 F3 42 F2 F3

13 N 14 F1 43 F3 44 F1 F3

16 N 17 F1 46 F3 47 F1 F3

19 N 20 F1 58 F3 59 F1 F3

22 N

25 N 27 F2

28 N 29 F1 30 F2

1 N 2 F1 3 F2 7 N 8 F1 9 F2

10 N 11 F1 12 F2

13 N 14 F1

16 N 17 F1

19 N 20 F1

55 F3 56 F1 F3

67 F3 68 F1 F3

70 F3 71 F1 F3

73 F3 74 F1 F3

76 F3 77 F1 F3

46 F3 47 F1 F3

64 F3 65 F1 F3

61 F3 62 F1 F3

58 F3 59 F1 F3

4 N 5 F1 6 F234 F3 35 F1 F3 36 F2 F3

82 F4 83 F1 F4 84 F2 F4

28 N 29 F1 30 F249 F3 50 F1 F3 51 F2 F3

88 F4 89 F1 F4 90 F2 F4

52 F3 53 F1 F3 54 F2 F3

1 N

1 N 2 F1 3 F2

79 F4 80 F1 F4 81 F2 F4

57 F2 F3

< FON, NF >

< SPI, NF >

< SPD, NF >< OV, NF >

< PON, F >

< BON, F >

< POFF, NF >

< CV, N F >

< SPD, F >

< BOFF, NF >

< FOFF, NF >< SPI, NF >

< OV, NF >

< PON, F >

< BON, F >

< SPD, F >

< CV, N F >

< OV, NF >

< PON, F >

< BON, F >

< PON, F >

< BON, F >

< SPD, F >

< OV, F >

< PON, F >

< BON, F >

< BON, NF >

< SPI, NF >

< OV, NF >

< PON, NF >

< BON, NF >

< SPD, NF >

< OV, NF >

< PON, NF >

< BON, NF >

< FON, NF >

< FOFF, NF >

< FON, NF >

< SPI, F >

< OV, F >

< PON, NF >

69 F2 F3

72 F2 F3

48 F2 F3

75 F2 F3

78 F2 F3

66 F2 F3

63 F2 F3

60 F2 F3

A

B

C

D

E



A “Small” Telephone System

1 2

0

1 2

0

• The network has screening, forwarding, and multi-way calling capabilities.

• The issue of interest: Feature Interactions.

• Specifically: detection and resolution of logical conflicts (interactions) between options

(features).

• Implementation: correct design of the (modular) software programs that run at the

switches.



Model of User 0 in a Telephone System:

req01

offh0onh0

fh0

req02

nocon02con02

nocon01con01

req00

nocon00

fwd012

req01req00 req02

fwd010

fwd001

REQ_0 REQ_1 REQ_2

FWD_TO_2FWD_TO_1FWD_TO_0

CON

INIT

fwd002 fwd020 fwd021

dfh0

nocon0

nocon0 nocon0

Model of User 1 at Switch 0 in Telephone

System:

INIT

NOT_REQ

REQ

fh1

req10

con10

nocon10

fwd101

fwd102

onh1 offh1

dfh1



A Control Architecture for

Approaching this

Problem:

G0C 0

.

.

.

TCSS

OCSS

SPOTS-4

.

.

.

.

.

.

...

G



Other examples:

Railway Connections and Time Tables1

• The network of railway connections is closed and each line has a fixed number of trains.

The inter-station travel times are known and deterministic.

• The objective is to design “satisfactory” time tables for the trains.

• Specifications include: certain trains have to wait for one another to allow change overs.

• Constraints: want system to operate fast, but also want perturbations to completely

disappear in finite time.

• Issues of interest: how do perturbations to the time table propagate, what limits the

minimum operation time, where would it be helpful to add trains, etc.

• Approach: write equations for the departure times of the trains, using “maximum” and

“addition.”

1Example due to G. J. Olsder



Dispatching Control in an Elevator System2

• Events: hall call, car call, car arrives at floor i, etc.

• States: position of car k, number of passengers waiting at floor i, etc. (very large state

space!)

• Control problem: which car to send where so as to achieve “satisfactory”

performance?

• Performance measures: average waiting time (until car comes), average service time (until

car delivers to desired floor), fraction of passengers waiting more (on average) than one

minute, etc.

• Probabilistic formulation: passenger arrival rates at floors, probability distribution for

destination floors, load times and travel times, etc.

• Common solution: threshold-based control, i.e., hold a car until a threshold is reached.

→ The issue is then to determine this threshold and “automatically” adjust it in

real-time, based on observed passenger arrival rates.

2Example due to C. Cassandras



The Three Levels of Abstraction in Modeling DES

Sample Paths of Discrete Event Systems

x(t)

t

e

t t t t t t t

e e e e e e e

x

x

x

x

x

x

7654321

7654321

1

2

3

4

5

6

Describe this sample path by the timed sequence of events that it contains:

ste = (e1, t1)(e2, t2)(e3, t3)(e4, t4)(e5, t5)(e6, t6)(e7, t7)



The behavior of a given DES is described as follows:

• Timed Language: set of all timed sequences of events that the DES can

generate/execute

• Stochastic Timed Language: a timed language with a probability distribution

function defined over it

• Language: a timed language where the timing information has been deleted, i.e., it is a

set of sequences, or traces, of events.

se = e1e2e3e4e5e6e7

Formal language theory:

– Finite set of events E : {e1, e2, . . . , en}

– Set of all finite strings of event in E: E∗ - Kleene-closure

– A language L is a subset of E∗: L ⊆ E∗



This leads to the three complemetary levels of abstraction at which DES are studied.

• Logical level: the language model is used to study properties that concern event

ordering only; e.g., consider the telephone system example, as well as the HVAC unit

example (diagnosis).

Priorities, mutual exclusion, deadlock, livelock, occurrence of unobservable events, etc.

• Temporal level: the timed language model is used to study properties that concern the

timing of the events; e.g., consider the railway network example.

Deadlines, cycle times, effect of perturbations, etc.

• Stochastic level: the stochastic timed language model is used to study properties that

concern the expected behavior of the system under the given statistical information; e.g.,

consider the elevator example.

Average delay, throughput, and other relevant performance measures.

N.B.: Discrete Event Simulation usually refers to the stochastic level.

Question: How to represent [(stochastic) timed] languages?



Discrete Event Modeling Formalisms

• Formal classes of models that represent [(stochastic) timed] languages

• “State-based” formalisms: define a state space and specify the state transition structure

(i.e., (out state, event, in state) triples) that represents the language.

Automata (or State Machines) and Petri Nets are widely used.

• “Trace-based” formalisms: use (recursive) algebraic equations on the events to represent

the traces in the language (i.e., no explicit “state”). Often referred to as Process Algebras.

Communicating Sequential Processes (CSP) is a well-know formalism in this category.

• We will study:

– (untimed and timed) automata [modeling, analysis, diagnosis, supervisory control]

– (untimed and timed) Petri nets [modeling, analysis, some control]

– timed event graphs, a special case of timed Petri nets [analysis using max-plus

algebra]

→ We illustrate the above modeling formalisms for the (familiar) example of the dining

philosophers.



Automata models of two philosophers (P 1, P 2) and two forks (F 1, F 2)

P1 P2

F1 F2

1E

1f2

1f2 1f1

1f

2I12f1 2f2

2f2 2f1

1f1,2f1

1f,2f

1f2,2f2

1f,2f

1f1

1I2

1I1

1U 2U

2I2

2E2f

1A

1T 2T

2A



Composition of the four automata: P 1||P 2||F 1||F 2

2f22f

1f

1f2

1f1

1f2

1f2

2f2

1f1

2f1

1f1

(1I2,2I1,1U,2U)

(1I1,2I2,1U,2U)

(1T,2E,1U,2U)

2f1

2f2

2f1(1E,2T,1U,2U)

(1T,2T,1A,2A)



Petri net model of one philosopher and two forks

holding fork 1

fork 2 available

fork 1 available

eating

if1 if2

if

if2 if1

holding fork 2

thinking



Petri net model of two philosophers and two forks

fork 1 availablephilosopher 1 philosopher 2

1f1 1f2

1f

1f2 1f1

2f12f2

2f

2f22f1

fork 2 available



Recursive equation model of two philosophers and two forks

P 1 = (1f1 → 1f2 → E1 | 1f2 → 1f1 → E1)

E1 = (1f → P 1)

P 2 = (2f1 → 2f2 → E2 | 2f2 → 2f1 → E2)

E2 = (2f → P 2)

F 1 = (1f1 → 1f → F 1 | 2f1 → 2f → F 1)

F 2 = (1f2 → 1f → F 2 | 2f2 → 2f → F 2)

SY STEM = P 1||P 2||F 1||F 2

In general, we get a set of equations of the form:

X = f(X)

Y = g(X)

where X is a vector of processes and f must contain →.



How to Compare Modeling Formalisms?

Descriptive Power: Language complexity or class of languages that a (finite) model can

represent.

• Finite-state automata: Regular Languages R

• Labeled Petri Nets: PNL ⊃ R.

Algebraic Structure: Formal operations that permit to build complex systems by

interconnecting simple systems and that allow to “manipulate” a model for analysis and

synthesis purposes.

• R has nice properties: closed under union, concatenation, intersection, parallel

composition, complementation w.r.t. E∗.

These operations can be “implemented” using finite-state automata.

• PNL does not enjoy such nice properties.

However, Petri nets have intrinsically modular structure: e.g., system decomposition

by means of place-bordered Petri nets.


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