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Spring 2006
Process Dynamics, Operations, and Control
10.450
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
firstorder 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 nonsmooth, multiple,
or delayed.
2.1
firstorder, linear, variablecoefficient 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
.
0
0
y
)
t
(
y
)
t
(
Kx
)
t
(
y
dt
dy
)
t
(
a
=
=
+
(2.11)
In writing (2.11) 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 (2.11) by defining the integrating factor p(t)
∫
=
)
(
exp
)
(
t
a
dt
t
p
(2.12)
Notice that p(t) is dimensionless, as is the quotient under the integral.
The
solution
∫
+
=
t
t
0
0
0
dt
)
t
(
a
)
t
(
x
)
t
(
p
)
t
(
p
K
)
t
(
p
)
t
(
y
)
t
(
p
)
t
(
y
(2.13)
comprises contributions from the initial condition y(t
0
) and the forcing
function Kx(t).
These are known as the homogeneous (as if the righthand
side were zero) and particular (depends on the righthand 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
firstorder ODE, special case for process control applications
The independent variable t will represent time.
For many process control
applications, a(t) in (2.11) will be a positive constant; we call it the
time
constant
τ
.
0
0
y
)
t
(
y
)
t
(
Kx
)
t
(
y
dt
dy
=
=
+
τ
(2.21)
The integrating factor (2.12) is
revised 2005 Jan 11
1
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View Full DocumentSpring 2006
Process Dynamics, Operations, and Control
10.450
Lesson 2: Mathematics Review
τ
=
τ
=
∫
t
e
dt
exp
)
t
(
p
(2.22)
and the solution (2.13) becomes
()
dt
)
t
(
x
e
e
K
e
y
)
t
(
y
t
t
t
t
t
t
0
0
0
∫
τ
τ
−
τ
−
−
τ
+
=
(2.23)
The initial condition affects the system response from the beginning, but
its effect decays to zero according to the magnitude of the time constant 
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 (2.23) 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.
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 Spring '03
 RogerD.Kamm

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