01-system_models_slides.pdf

Introduction to Discrete Event Systems
Soulimane MAMMAR
ENPO-MA
Soulimane MAMMAR (ENPO-MA) Introduction to Discrete Event Systems 1 / 65

Introduction
Evolution of:
computing
communication
sensor technologies
Proliferation of new dynamic sys-
tems
mostly technological
often highly complex
Examples :
Computer and communication networks
Automated manufacturing systems
Air traffic control systems
Monitoring and control systems in automobiles or large buildings
Activity governed by operational rules designed by humans
Characterized by asynchronous occurrences of discrete events
Controlled : hitting a keyboard key, turning a piece of equipment “on”
Not controlled : spontaneous equipment failure, packet loss, etc.

Introduction
The mathematics used to model and study the time-driven processes
governed by the laws of nature is inadequate for discrete event systems.
The challenge:
Develop new:
I Modeling frameworks
I Analysis techniques
I Design tools
I Testing methods

Systems and Models
Systems and Models

Systems and Models
Introduction
Historically, focus was on studying natural phenomena which are
well-modeled by:
Laws of gravity
Classical and non-classical mechanics
Physical chemistry
Typically quantities are "continuous variables"
Displacement, velocity, acceleration,. . . etc.
Ordinary and partial differential equations provides the main infrastructure
for system analysis and control

Systems and Models
Introduction
But in the day-to-day life . . .
Increasingly computer-dependent
Deal with "discrete" quantities
I Parts in an inventory
I Planes on a runway
I Active telephone calls
Processes depends on and are driven by instantaneous "events"
I Pushing a button
I Hitting a keyboard key
I Traffic light turning green

System and Control Basics

The Concept of System
An aggregation or assemblage of things so combined by nature or man
as to form an integral or complex whole (Encyclopedia Americana).
A regularly interacting or interdependent group of items forming a
unified whole (Webster’s Dictionary)
A combination of components that act together to perform a function
not possible with any of the individual parts (IEEE Standard
Dictionary of Electrical and Electronic Terms).

The Input–Output Modeling Process
Scientists and engineers concerns:
Quantitative analysis
Development of techniques for :
I Design
I Control
I Performance measurement
We need a model of an actual system
A device that duplicates the behavior of the system

Defining a set of measurable variables associated with a given system.
positions, velocities, voltages, current
We select a subset of these variables that we have the ability to vary them
over time
This defines a set of time functions which we shall call the input
variables
{u1(t), · · · up(t)} t0 ≤ t ≤ tf
We select another set of variables which we assume we can directly
measure while varying the input and define them as output variables
{y1(t), · · · yp(t)} t0 ≤ t ≤ tf

The system model in the mathematical form
y = g(u) = [g1(u1(t), · · · up(t)), · · · , gm(u1(t), · · · up(t))]T

Example 1 (A voltage-divider system)

Example 2 (A spring-mass system)
mÿ = −ky
u(t) =
(
u0 t = 0
0 otherwise

Static and Dynamic Systems
We define a static system to be one where the output y(t) is independent
of past values of the input u(τ), τ < t for all t
A dynamic system is one where the output generally depends on past
values of the input.

Time-Varying and Time-Invariant Dynamic Systems
Question: Is the output always the same when the same input is
applied ?
I Not always "YES"
Example: Suppose in example 2 that the mass attached to the spring
is actually a container full of some liquid, and that a leak develops
immediately after time t = 0
A more general representation would be
y = g(u, t)
A time-invariant system(stationary): If an input u(t) results in an
output y(t), then the input u(t − τ) results in the output y(t − τ),
for any τ

Time-Varying and Time-Invariant Dynamic Systems

The Concept of State
The state of a system at a time instant t
Describe its behavior at that instant
In measurable way
Suppose:
u(t) is completely specified for all t ≥ t0
y(t) is observed at some time t = t1 ≥ t0
Is this information adequate to uniquely predict all future output
y(t), t > t1 ?

For the example 2:
Can we uniquely determine y(t1 + τ) for some time increment τ > 0?
I Answer: No
I What is missing here is information regarding the velocity of the mass
I Mathematically, we cannot solve for y(t1 + τ) with only one initial
condition, i.e., y(t1)
need information about velocity, ẏ(t1)

Definition
The state of a system at time t0 is the information required at t0 such
that the output y(t), for all t ≥ t0, is uniquely determined from this
information and from u(t), t ≥ t0
The state is also generally a vector, which we shall denote by x(t). The
components of this vector, x1(t), · · · , xn(t), are called state variables.

The State Space Modeling Process
The modeling process:
Determine suitable mathematical relationships (dynamics)
I involving the input u(t), the output y(t), and the state x(t)
Definition
The set of equations required to specify the state x(t) for all t ≥ t0 given
x(t0) and the function u(t), t ≥ t0 , are called state equations
Definition
The state space of a system, usually denoted by X, is the set of all
possible values that the state may take.
The state equations could take several different forms, mostly the form:
ẋ(t) = f(x(t), u(t), t)

We have a state space model of a system when we can specify the
following set of equations:
ẋ(t) = f(x(t), u(t), t), x(t0) = x0
y(t) = g(x(t), u(t), t)
Remark
For a static system, the state equation reduces to ẋ(t) = 0 for all t

Example 3
Consider the spring-mass system
I Let y(t) be the output measuring the mass displacement at time t
I u(t) the input function (as previously defined)
Suppose we define the mass displacement as a state variable
I y(t) = x1(t)
To obtain a state equation, we have at our disposal the differential equation mÿ = −ky
which, since y(t) = x1(t), we rewrite as
mẍ1 = −kx1
To obtain first-order differential equations, we introduce an additional state variable,
x2(t), which we define as
ẋ1 = x2
We now have a complete model with two state variables and a single output variable.
h
ẋ1
ẋ2
i
=

0 1
− k
m
0

,
h
x1(0)
x2(0)
i
=
h
u0
0
i
y =

0 1
h
x1
x2
i

Example 4
v̇C = −
1
RC
vC +
1
RC
V
v = V − vC

Linear and Nonlinear Systems
Definition
The function g is said to be linear if and only if
g(a1u1 + a2u2) = a1g(u1) + a2g(u2)
Definition
A system is said to be linear if and only if the functions g(·) and f(·) are
both linear
ẋ(t) = A(t)x(t) + B(t)u(t)
y(t) = C(t)x(t) + D(t)u(t)

Sample Paths of Dynamic Systems
Consider the state equation:
ẋ(t) = f(x(t), u(t), t)
Suppose we select a particular input function u(t) = u1(t) and some initial condition at time
t = 0, x(0) = x0
The solution to this differential equation is a particular function x1(t)
By changing u(t) and forming a family of input functions u1(t), u2(t), · · · , we obtain a
family of solutions x1(t), x2(t), · · ·
I Any member of this family of functions is referred to as a sample path
or state trajectory
Figure:

Sample Paths of Dynamic Systems
All possible values that the n-dimensional state vector x(t) can take
define an n-dimensional space
As t varies, different points are visited, thus defining a trajectory

State Spaces
State variables
I Real numbers (or interval)
I Integer values
I Values from a discrete set
{ON, OFF}, {HIGH, MEDIUM, LOW}, or {GREEN,RED, BLUE}
Continuous-state models
I The state space X is a continuum of all n-dimensional vectors of real
numbers
Discrete-state models
I The state space is a discrete set
Hybrid model
I Some state variables are discrete and some are continuous

State Spaces
Example 5: the mass spring system
We defined the 2-dimensional state vector
x =
h
x1, x2
iT
The state space X is the the set of R2
I continuous-state linear system

State Spaces
Example 6: A warehouse system
warehouse of manufactured products in a factory
we are interested in the level of inventory
x(t): number of products at time t
Output equation y(t) = x(t)
X is the set of non-negative integers {0, 1, 2, · · · }
warehouse is very large
loading a truck takes zero time
truck take away a single product at a time
no t such that u1(t) = u2(t) = 1
u1(t) =
n
1 if a product arrives at time t
0 otherwise
u2(t) =
n
1 if a truck arrives at time t
0 otherwise
x(t
+
) =
x(t) + 1 if(u1(t) = 1, u2(t) = 0)
x(t) − 1 if(u1(t) = 0, u2(t) = 1, x(t) 0)
x(t) otherwise

Deterministic and Stochastic Systems
Recall: state ⇒the ability to predict at t = t0 all future behavior of a
system
I Assuming knowledge of the input u(t) for all t ≥ t0
This assumption is not always reasonable
I precludes the unpredictable effect of nature on the input
Example 1: V (t) may be subject to random “noise”
Example 6: depends almost exclusively on the exact knowledge of the
product arrival input function u1(t) and the truck arrival input function
u2(t).
Randomness affect the state in ways that only probabilistic
mechanisms can adequately capture
A system is stochastic if at least one of its output variables is a
random variable.
Otherwise, deterministic.

The Concept of Control
So far we’ve been limited to the basic issue:
I What happens to the system output for a given input?
A system contains the idea of performing a particular function
I needs to be controlled
select the right input to achieve a desired behavior (eg. driving a car)
The input to a system is often viewed as a control signal
I aimed at achieving a desired behavior.
I we represent this desired behavior by a reference signal r(t)
The control input to the actual system (control law) u(t) = γ(r(t), t)

The Concept of Control
Example 7: Consider a linear time-invariant system
ẋ(t) = u(t), x(0) = 0
y(t) = x(t)
The desired behavior is to bring the state to the value x(t) = 1 and keep it there forever.
We define a reference signal
r(t) = 1 for all t ≥ 0
The control input is limited to operate in the range [−1, 1].
A simple control that works is :
u1(t) =

1 for t ≤ 1
0 for t 1
Not the only possible control law
u2(t) =
1
2
for t ≤ 2
0 for t 2

The Concept of Feedback
Feedback: Use any available information about the system behavior
in order to continuously adjust the control input
I In a conversation, we speak when the other party is silent
I In driving, we monitor the car’s position and speed in
I order to continuously make adjustments
I In heating a house, we use a thermostat
The key property of feedback is in making corrections, especially in
the presence of unexpected disturbances
Mathematically, using feedback implies that we should extend the
control law to include the state x(t)
u(t) = γ(r(t), x(t), t)

Open-Loop and Closed-Loop Systems

Example 8:
Initial condition:
I x(0) = 0
I Outflow rate is µ(t) = 0 for
all t ≥ 0
Task:
I maintain the fluid level at its
maximum
How:
I We will control the inflow λ(t)
so as to achieve the desired
behavior
I r(t) = K for all t ≥ 0

State and output equations
ẋ(t) =

λ(t) if x(t) K
0 otherwise
y(t) = x(t)
With the assumption of maximum rate ρ, We can choose for an open-loop design
λOL(t) =

ρ for t ≤ ρ/K
0 for t ρ/K
The solution with this control gives
x(t) = ρt
x(t) = K reached at time t = K/ρ
Drawbacks of this open-loop control law:
The slightest error in measuring or unexpected disturbance
I prevent from meeting the desired behavior

Closed-loop control system

Advantages to the use of feedback
Desired behavior less sensitive to unexpected disturbances.
Desired behavior less sensitive to possible errors in the parameter
values
The output y(t) can automatically follow or track a desired reference
signal r(t)
I continuously seeking to minimize the difference (y(t) − r(t))
Feedback cost
Sensors or other potentially complex equipment may be required to
monitor the output and provide the necessary information to the
controller.
Feedback requires some effort, which may affect the system
performance.

Discrete-Time Systems
We assumed that time is a continuous variable
I Corresponds to notion of time in the physical world
I Allows to develop models based on differential equations
Input and output variables of a system at discrete time instants
I Discrete-time system
Reasons to adopt such an approach
I Any computer we might use as a component of a system operates in
discrete-time fashion (clock ticks)
I Many differential equations can only be solved numerically (by
computer)
computer-generated solutions are discrete-time functions.
I Digital control techniques, based on discrete-time models, provide
flexibility, speed, and low cost.
I Some systems are inherently discrete-time
Economic models based on data that is recorded only at regular
discrete intervals

In discrete-time models
time line is thought of as a sequence of intervals defined by a
sequence of points t0 t1 · · · tk · · ·
It is assumed that all intervals are of equal length T
I T is the sampling interval
Discretization of time does not imply discretization of the state space

Thus, the state-based model becomes
x(k + 1) = f(x(k), u(k), k), x(0) = x0
y(k) = g(x(k), u(k), k)
For the case of linear discrete-time systems
x(k + 1) = A(k)x(k) + B(k)u(k)
y(k) = C(k)x(k) + D(k)u(k)

Discrete Event Systems
State space described by a discrete set like 0, 1, 2, · · ·
State transitions are only observed at discrete points in time
I Associate state transistion with events

The Concept of Event
An event
Occur instantaneously and causing transitions from one state value to
another.
May be identified with a specific action taken
May be spontaneous
May be the result of several conditions
We denote:
An event with the symbol e
The discrete set of events with E

The Concept of Event
Example 9: (A random walk)
A random walk is a useful model for several
interesting processes
I games of chance
In two dimensions:
I may be visualized as a particle moving one unit of distance
(a step) at a time in one of four directions: north, south,
west, or east
The direction is chosen at random and independent
of the present position
The state of this system is the position of the
particle, (x1, x2)
I x1, x2 taking integer values
State space is the discrete set
I X = {(i, j) : i, j = · · · , −1, 0, 1, · · · }
Event set is
I E = {N, S, W, E}

Time-Driven and Event-Driven Systems
In continuous-state systems
state generally changes as time changes
In discrete-time models
I The clock is what drives a typical sample path.
Refered as time-driven systems
In discrete-state systems
State changes only at certain points in time
The timing mechanism:
I At every clock tick an event e (null in case of no event)
State transition is synchronized by the clock
I At various time instants (not necessarily known in advance and not
necessarily coinciding with clock ticks)
Every event defines a distinct process
State transitions are the result of combining these asynchronous and
concurrent event processes
I Referred as event-driven systems

Time-Driven and Event-Driven Systems
Example 10: (Event-driven random walk)
The random walk described in Example 9 is a time-driven system
I with every clock tick there is a single “player” who moves the particle
However, there is an alternative view of the random walk
I suppose there are four different players:
each one responsible for moving the particle in a single direction
each player acts by occasionally issuing a signal to move the particle in
his direction
I this results in an event-driven system
defined by these four asynchronously acting players

Characteristic Properties of Discrete Event Systems
Most of the successes in system and control engineering relied on
Differential-equation-based models
I or its difference equation analog
To use these models two key properties that systems must satisfy
Continuous-state systems
I Continuous-Variable Dynamic Systems (CVDS)
Time-driven

Definition
A Discrete Event System (DES) is a discrete-state, event-driven system,
that is, its state evolution depends entirely on the occurrence of
asynchronous discrete events over time.
Examples of discrete-state systems:
State of a machine may be {ON, OFF} or {BUSY, IDLE, DOWN}
A computer running a program may be {WAITING FOR INPUT,
RUNNING, DOWN}

Comparison of sample paths for Continuous-Variable Dynamic Systems (CVDS) and Discrete
Event Systems (DES). In a CVDS, x(t) is generally the solution of a differential equation
ẋ(t) = f(x(t), u(t), t). In a DES, x(t) is a piecewise constant function, as the state jumps from
one discrete value to another whenever an event takes place.

Levels of Abstraction in the Study of DES
It is often convenient to simply write the timed sequence of events
(e1, t1), (e2, t2), (e3, t3), (e4, t4), (e5, t5), (e6, t6)
I Contains the same information as a sample path
The set of all timed sequences of events that a given system can ever
execute is called timed langage
I E: alphabet,
I sequences of events : words
Assuming that probability distribution functions are available about
the lifetime of each event
I Stochastic timed langage: a timed language together with associated
probability distribution functions

Levels of Abstraction in the Study of DES
Stochastic timed language modeling is the most detailed
If we omit the statistical information, then the corresponding timed
language enumerates all the possible sample paths of the DES
I if we delete the timing information from a timed language we obtain an
untimed langage
Three levels of abstraction at which DES are modeled and studied
Languages
I Interested in the “logical behavior
I Ensuring that a precise ordering of events takes place which satisfies a
given set of specifications
Timed langages
I Assessing the performance
I Answer questions such as:
How much time does the system spend at a particular state?
Can this sequence of events be completed by a particular deadline?
Stochastic timed languages

Examples of Discrete Event Systems

Queueing Systems
Queueing
Using a resource
I Wait
products wait for truck resource
tasks wait for CPU resouce
Queueing system
Entities that do the waiting ⇒ customers.
Resources waited for ⇒ servers.
Space where the waiting is done ⇒ queue.
Study of queueing systems
Resources are not unlimited
Customer needs are adequately satisfied
Resource access is provided in fair and efficient ways
The cost of designing and operating the system is maintained at
acceptable levels.

Queueing Systems
Capacity of the queue
Maximum number of customers
I Somtimes assumed infinite
Queueing discipline
Rule for servicing the next customer
I eg., FIFO
Viewed as a DES
event set E = {a, d} (arrivals, departures)
state variable: number of customers in queue ⇒ queue length State
space : X = {0, 1, 2, · · · }

Queueing Systems
A queuing network example.

Computer Systems
A computer systems can be
viewed as a DES
Customers:
I jobs, tasks, or
transactions
Servers:
I CPU, peripheral
devices
E = {a, d, r1, r2, d1, d2}
X = {(xC PU, x1, x2) : xC PU, x1, x2 ≥ 0}

Communication Systems
Customers
I messages, packets, or calls
Servers
I intermediate points
switching equipment , communication media
Control mechanisms (Protocols)
I ensure that access to the servers is granted
I objectives of the communication process are met

Communication Systems
Example:
A and B, both sharing a common communication medium (channel)
Channel serving one user at a time
Channel states:
I I : idle
I T : transmitting one message
I C : transmitting two or more messages (collision)
User states:
I I : idle
I T : transmitting
I W : waiting to transmit an existing message.
Two problems
I each user does not know the state of the other
I users may not know the state of the channel (may only be able to
detect one state)
States Events
X = {(xC H, xA, xB) : xC H ∈ {I, T, C}, xA ∈ {I, T, W}, xB ∈ {I, T, W}}
E = {aA, aB, τA, τB, τCH }

Hybrid Systems
Systems that combine time-driven
with event-driven dynamics
proliferation due to embedding of
microprocessors (event-driven mode)
in complex automated environments
with time-driven dynamics
I automobiles, aircraft, chemical
processes, heating, ventilation
System and associated continuous-
variable controllers abstracted as a
DES for the higher levels

Hybrid Systems

Summary of System Classifications
Static and Dynamic Systems.
I In static systems the output is always independent of past values of the
input.
I In dynamic systems, the output does depend on past values of the
input.
Differential or difference equations are generally required to describe
the behavior of dynamic systems.
Time-varying and Time-invariant Systems.
I The behavior of time-invariant systems does not change with time.
I This property, also called stationarity, implies that we can apply a
specific input to a system and expect it to always respond in the same
way.

Linear and Nonlinear Systems.
I A linear system satisfies the condition
g(a1u1 + a2u2) = a1g(u1) + a2g(u2), where u1 , u2 are two input
vectors, a1 , a2 are two real numbers, and g(·) is the resulting output.
Continuous-State and Discrete-State Systems.
I In continuous-state systems, the state variables can generally take on
any real (or complex) value.
I In discrete- state systems, the state variables are elements of a discrete
set (e.g., the non-negative integers).

Time-driven and Event-driven Systems.
I In time-driven systems, the state continuously changes as time changes.
I In event-driven systems, it is only the occurrence of asynchronously
generated discrete events that forces instantaneous state transitions.
In between event occurrences the state remains unaffected.
Deterministic and Stochastic Systems.
I A system becomes stochastic whenever one or more of its output
variables is a random variable.
I In this case, the state of the system is described by a stochastic process
I A probabilistic framework is required to characterize the system
behavior.
Discrete-time and Continuous-time Systems.
I A continuous-time system is one where all input, state, and output
variables are defined for all possible values of time.
I In discrete-time systems, one or more of these variables are defined at
discrete points in time only, usually as the result of some sampling
process.

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