Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 1: Processes and Systems
1.0 context and direction
Process control is an application area of chemical engineering - an
identifiable specialty for the ChE. It combines chemical process
knowledge (how physics, chemistry, and biology work in operating
equipment) and an understanding of dynamic systems, a topic important to
many fields of engineering. Thus study of process control allows
chemical engineers to span their own field, as well as form a useful
acquaintance with allied fields. Practitioners of process control find their
skills useful in design, operation, and troubleshooting - major categories of
chemical engineering practice.

Process control, like any coherent topic, is an integrated body of
knowledge - it hangs together on a multidimensional framework, and
practitioners draw from many parts of the framework in doing their work.
Yet in learning, we must receive information in sequence - following a
path through multidimensional space. It is like entering a large building
with unlighted rooms, holding a dim flashlight and clutching a vague map
that omits some of the stairways and passages. How best to learn one’s
way around?

In these lessons we will attempt to move through a significant portion of
the structure - say, half a textbook - in about two weeks. Then we will
repeat the journey several times, each time inspecting the rooms more
thoroughly. By this means we hope to gain, from the start, a sense of
doing an entire process control job, as well as approach each new topic in
the context of a familiar path.

1.1 the job we will do, over and over

may vary with time due to supply pressure
fluctuations and valve manipulations by the operator. The liquid density
depends on the temperature, but will usually not vary significantly with
time during the course of filling. Thus (1.2-1) becomes

)t(F
d
t
dh
A
i
=
(1.2-2)

We integrate (1.2-2) to find the liquid level as a function of time.

∫
+=
t
0
i
dt)t(F
A
1
)0(hh
(1.2-3)

1.3 planning a control scheme
Clearly the liquid level h is important, and we will call it the controlled
variable. Our control objective is to bring h quickly to its target value h

directed from a control room. These are indeed improvements. However,
the timing scheme abandons a crucial virtue of the existing scheme: by
measuring the controlled variable, the operators can react to unexpected
disturbances, such as changes in the filling rate. Using knowledge of the
controlled variable to motivate changes to the manipulated variable is a
fundamental control structure, known as feedback control
. The proposed
timing scheme has no feedback mechanism, and thus cannot accommodate
changes to h(0) and F
i
(t) in (1.2-3).

An alternative is to build on the feedback already inherent in the two-
operator scheme, but to improve its operation. We propose an automatic
controller that behaves according to the following controller algorithm:

near i max
r
near i max
r near
hh F F
hh
hh FF
hh
<=
−
>=
−
(1.3-1)


max
does not vary during filling due to
pressure fluctuations, etc. For h less than h
near
,

t
A
F
)0(hh
known)0(hF
dt
dh
A
max
max
+=
==
(1.5-1)

Equation (1.5-1) can be used to calculate t
near
, the time at which h reaches
h
near
. For h greater than h
near
,

()


Lesson 1: Processes and Systems

where the parameter t
fill
is the time required for the level to reach h
r
at
flow F
max
, starting from an empty tank.

r
fill
max
Ah
t
F
=
(1.5-3)

The plot shows the filling profile from h(0) = 0.10h
r
with several values of
h
near
/h
r
. Certainly the filling goes faster if the flow can go instantaneously
from F

0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
t/t
fill
h/hr
h
near
/h
r
= 0.95
0.75
0.501.6 defining ‘system’
In Section 1.2, we introduced a process - a tank with feed piping - whose
inventory varied in time. We thought of the process as a collection of
equipment and other material, marked off by a boundary in space,
communicating with its environment by energy and material streams.

'Process' is a good notion, important to chemical engineers. Another
useful notion is that of 'system'. A system is some collection of equipment
and operations, usually with a boundary, communicating with its
environment by a set of input and output signals. By these definitions, a
process is a type of system, but system is more abstract and general. For
example, the system boundary is often tenuous: suppose that our system

influences the liquid level, just as does F
i
.

The point of all this is to look at a single schematic and know how to view
it as a process, and as a system. View it as a process (F
o
as an outlet
stream) to write the material balance and make fluid mechanics
calculations. View it as a system (F
o
as an input) to analyze the dynamic
behavior implied by that material balance and make control calculations.

System dynamics is an engineering science useful to mechanical,
electrical, and chemical engineers, as well as others. This is because
transient behavior, for all the variety of systems in nature and technology,
can be described by a very few elements. To do our job well, we must
understand more about system dynamics how systems behave in time.
That is, we must be able to describe how important output variables react
to arbitrary disturbances.

1.7 systems within systems
We call something a system and identify its inputs and outputs as a first
step toward understanding, predicting, and influencing its behavior. In
some cases it may help to determine some of the structure within the
system boundaries; that is, if we identify some
component systems. Each
of these, of course, would have inputs and outputs, too.


final control
element
sensor
set point
manipulated
variable
other inputs other
outputs
controlled
variable
system
process
controller
final control
element
sensor
set point
manipulated
variable
other inputs other
outputs
controlled
variable
system

Figure 1.8-1 The single-loop feedback control system and its
subsystems

We will see this structure repeatedly. Inside the block called "process" is
the physical process, whatever it might be, and the block is the boundary

Lesson 2: Mathematics Review
2.0 context and direction
Imagine a system that varies in time; we might plot its output vs. time. A
plot might imply an equation, and the equation is usually an ODE
(ordinary differential equation). Therefore, we will review the math of the
first-order ODE while emphasizing how it can represent a dynamic
system. We examine how the system is affected by its initial condition
and by disturbances, where the disturbances may be non-smooth, multiple,
or delayed.

2.1 first-order, linear, variable-coefficient ODE
The dependent variable y(t) depends on its first derivative and forcing
function x(t). When the independent variable t is t
0
, y is y
0
. 00
y)t(y)t(Kx)t(y
d
t
dy
)t(a ==+
(2.1-1)

In writing (2.1-1) we have arranged a coefficient of +1 for y. Therefore
a(t) must have dimensions of independent variable t, and K has
dimensions of y/x. We solve (2.1-1) by defining the integrating factor p(t)

) and the forcing
function Kx(t). These are known as the homogeneous (as if the right-hand
side were zero) and particular (depends on the right-hand side) solutions.
In the language of dynamic systems, we can think of y(t) as the response
of the system to input disturbances Kx(t) and y(t
0
).

2.2 first-order ODE, special case for process control applications
The independent variable t will represent time. For many process control
applications, a(t) in (2.1-1) will be a positive constant; we call it the time
constant τ.

00
y)t(y)t(Kx)t(y
dt
dy
==+τ
(2.2-1)

The integrating factor (2.1-2) is

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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
τ
=
τ
=

larger time constants represent slower decay. If not further disturbed by
some x(t), the first order system reaches equilibrium at zero.

However, most practical systems are disturbed. K is a property of the
system, called the gain. By its magnitude and sign, the gain influences
how strongly y responds to x. The form of the response depends on the
nature of the disturbance.

Example: suppose x is a unit step function at time t
1
. Before we proceed
formally, let us think intuitively. From (2.2-3) we expect the response y to
decay toward zero from IC y
0
. At time t
1
, the system will respond to being
hit with a step disturbance. After a long time, there will be no memory of
the initial condition, and the system will respond only to the disturbance
input. Because this is constant after the step, we guess that the response
will also become constant.

Now the math: from (2.2-3)

()
()
()
⎟
⎠
⎞

e1)tt(KUey
dt)tt(Uee
K
ey)t(y
(2.2-4)

Figure 2.2-1 shows the solution. Notice that the particular solution makes
no contribution before time t
1
. The initial condition decays, and with no
disturbance would continue to zero. At t
1
, however, the system responds
to the step disturbance, approaching constant value K as time becomes
large. This immediate response, followed by asymptotic approach to the
new steady state, is characteristic of first-order systems. Because the
response does not track the step input faithfully, the response is said to lag

behind the input; the first-order system is sometimes called a first-order
lag.

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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
0
0.5
1
012345
disturbance

time
response
t
0
t
1
y
0
K

Figure 2.2-1 first-order response to initial condition and step
disturbance

2.3 piecewise integration of non-smooth disturbances
The solution (2.2-3) is applied over succeeding time intervals, each
featuring an initial condition (from the preceding interval) and disturbance
input.

()
()
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨

tt
1
10
t
t
tt
tt
0
1
1
0
0
(2.3-1)

Example: suppose

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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
()
⎪
⎩
⎪
⎨
⎧
<
<<−
<<
=

1t00
)t(y
2t1
1t
(2.3-3)

With a zero initial condition and no disturbance, the system remains at
equilibrium until the ramp disturbance begins at t = 1. Then the output
immediately rises in response, lagging behind the linear ramp. At t = 2,
the disturbance ceases, and the output decays back toward equilibrium.

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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
0
0.5
1
1.5
2
2.5
012345
disturbance
6
0
0.1
0.2
0.3
0.4
0.5

two disturbance functions.

002211
y)t(y)t(xK)t(xK)t(y
dt
dy
=+=+τ
(2.4-1)

Applying (2.2-3) and distributing the integral across the disturbances, we
find that the effects of the disturbances on y are additive.

()
dt)t(xee
K
dt)t(xee
K
ey)t(y
2
t
t
tt
2
1
t
t
tt
1
tt
0

0)t(y)t(xK)t(y
d
t
dy
0)t(y)t(xK)t(y
dt
dy
y)t(y0)t(y
dt
dy
02222
2
01111
1
00HH
H
==+τ
==+τ
==+τ
(2.4-4)

Equations and initial conditions (2.4-4) can be summed to recover the
original problem specification (2.4-1). The solutions are

()
dt)t(xee
K
)t(y
dt)t(xee
K

=
=
(2.4-5)

and of course these solutions can be added to recover original solution
(2.4-2). Thus we can view the problem of multiple disturbances as a
system responding to the sum of the disturbances, or as the sum of
responses from several identical systems, each responding to a single
disturbance.

Example: consider

2)0(y)3t(U)1t(U
4
3
y
4
1
dt
dy
4
1
=−−−=+
(2.4-6)

We first place the equation in standard form, in which the coefficient of y
is +1.

2)0(y)3t(U4)1t(U3y
d

8
-5
-4
-3
-2
-1
0
1
2
3
4
0246
time
response
8
0
0.5
1
0246
disturbances
8
-5
-4
-3
-2
-1
0
1
2
3


This notation at least implies that two of the exponential functions have
delayed onsets. However, further correct-but-inappropriate rearrangement
makes things even worse.

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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
()
t31
t3t1t
)3t()1t(t
ee4e321
ee4ee3e21
e4e3e21y
−
−−−
−−−−−
+−+−=
+−+−=
+−+−=
(do not do this!) (2.4-10)

The incorrect solutions are plotted with (2.4-8) in Figure 2.4-2. Equation
(2.4-9) has become discontinuous - the response takes non-physical leaps
at the onset of each new disturbance. Equation (2.4-10) has lost all
dependence on the disturbances and decays from a non-physical initial
condition. Even with the mistakes, both incorrect solutions lead to the
correct long-term condition.

dt)t(xee
K
ey)t(y
t
t
tt
tt
0
0
0
θ−
τ
+=
∫
ττ
−
τ
−−
(2.5-2)

We must integrate the disturbance considering the time delay. Take as an
example a disturbance x(t) occurring at time t
1
. The plot shows the
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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
disturbance, as well as the disturbance as the system experiences it, which
begins at time t

x(t) x(t - θ) = x
1
(t)
x(ξ)
t
0
t
1
t
1
+ θ
t
0
- θ
t
1
ξ
t
t
1
- θ
disturbance
as it occurs
disturbance as
experienced by
system
x(t) x(t - θ) = x
1
(t)
x(ξ)

()
ξξ
τ
+=
∫
ξ
θ−
τ
ξ
τ
θ
τ
−
τ
−−
d)(xeee
K
ey)t(y
0
0
t
t
tt
0
(2.5-5)

Example: consider a step disturbance at time t = 2 that affects the system
3 time units later.

)2t(U)t(x

e)2(Uee
d)2(Ued)2(Ueee
d)2(Ueeey
−−
+−+−−
−−
ξ
ξ−
ξ
ξ
−
ξ−
ξ
−
ξ−
−−=
−−=
−−−=
⎥
⎦
⎤
⎢
⎣
⎡
−ξ=
⎥
⎥
⎦
⎤
⎢

1
024681
disturbances
0
0
0.2
0.4
0.6
0.8
1
024681
time
response
0

Figure 2.5-1 step response of first order system with dead time

The time delay in responding to a disturbance is often called dead time
.
Dead time is different from lag. Lag occurs because of the combination of
y and its derivative on the left-hand side of the equation. Dead time
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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 2: Mathematics Review
occurs because of a time delay in processing a disturbance on the right-
hand side.

2.6 conclusion
Please become comfortable with handling ODEs. View them as systems;

Ai
F, C
Ao
volume VOur concern is the time-varying behavior of the process, so we should
treat our process as a dynamic system. To describe the system, we begin
by writing a component material balance over the solute.

AoAiAo
FCFCVC
dt
d
−=
(3.1-1)

In writing (3.1-1) we have recognized that the tank operates in overflow:
the volume is constant, so that changes in the inlet flow are quickly
duplicated in the outlet flow. Hence both streams are written in terms of a
single volumetric flow F. Furthermore, for now we will regard the flow as
constant in time.

Balance (3.1-1) also represents the concentration of the outlet stream, C
Ao
,
as the same as the average concentration in the tank. That is, the tank is a
perfect mixer: the inlet stream is quickly dispersed throughout the tank
volume. Putting (3.1-1) into standard form,
revised 2005 Jan 13 1

Before solving (3.1-2) we specify a reference condition: we prefer that C
Ao

be at a particular value C
Ao,r
. For steady operation in the desired state,
there is no accumulation of solute in the tank. r,Aor,Ai
r
Ao
CC0
dt
dC
F
V
−==
(3.1-3)

Thus, as expected, steady outlet conditions require a steady inlet at the
same concentration; call it C
A,r
. Let us take this reference condition as an
initial condition in solving (3.1-2). The solution is

dt)t(Ce
e
eC)t(C
Ai


3.2 response of system to steady input
Suppose inlet concentration remains steady at C
A,r
. Then from (3.1-4)

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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 3: The Blending Tank
r,A
tt
r,A
t
r,A
t
0
t
r,A
t
t
r,AAo
C1eeCeC
eC
e
eCC
=
⎟
⎠
⎞

to
value C
A1
. Until t
1
the outlet concentration is given by (3.2-1). From the
step at t
1
, the outlet concentration begins to respond.

⎟
⎠
⎞
⎜
⎝
⎛
−+=
⎟
⎠
⎞
⎜
⎝
⎛
−+=
>τ
τ
+=
τ
−−
τ

1
e1CeC
eeeCeC
tteC
e
eCC
(3.3-1)

In Figure 3.3-1, C
A,r
= 1 and C
A1
= 0.8 in arbitrary units; t
1
has been set
equal to τ. At sufficiently long time, the initial condition has no influence
and the outlet concentration becomes equal to the new inlet concentration.
After time equal to three time constants has elapsed, the response is about
95% complete – this is typical of first-order systems.

In Section 3.1, we suggested that the tank would mitigate the effect of
changes in the inlet composition. Here we see that the tank will not
eliminate a step disturbance, but it does soften its arrival.
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Spring 2006 Process Dynamics, Operations, and Control 10.450

Lesson 3: The Blending Tank
0.7
0.8
0.9


⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
<
⎟
⎠
⎞
⎜
⎝
⎛
−+
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛

)tt()tt(
1A
)tt(
r,A
21
)tt(
1A
)tt(
r,A
1r,A
Ao
221212
11
(3.4-1)

In Figure 3.4-1, C
A,r
= 0.6 and C
A1
= 1 in arbitrary units; t
1
has been set
equal to τ and t
2
to 2.5τ. We see that the tank has softened the pulse and
reduced its peak value. A pulse is a sequence of two counteracting step
changes. If the pulse duration is long (compared to the time constant τ),
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Spring 2006 Process Dynamics, Operations, and Control 10.450



From (3.1-4),

()
(
)
ωτ−+ω
τω+
+
τω+
ωτ
−=
−
τ
−
1
22
t
22
r,AAo
tantsin
1
A
e
1
A
CC
(3.5-2)

revised 2005 Jan 13 5

t/τ
input and response
8
input decaying part continuing part solution

Figure 3.5-1 first-order response to sine disturbance

3.6 frequency response and the Bode plot
The long-term response to a sine input is the most important part of the
solution; we call it the frequency response
of the system. We will
examine the frequency response for an abstract first order system.
(Because we wish to focus on the oscillatory response, we will write (3.6-
1) so that x and y vary about zero. The effect of a non-zero bias term can
be seen in (3.5-1) and (3.5-2).)

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