Spring 2006
Process Dynamics, Operations, and Control
10.450
Lesson 5: Operability of Processes
5.0
context and direction
In Lesson 4, we encountered instability.
We think of stability as a
mathematical property of our linear system models.
Now we will embed
this mathematical notion within the practical context of process
operability.
That is, we must not forget that our system models help us
operate processes.
Along the way, we will encounter a special category of
instability/inoperability: the non-self-regulating process.
DYNAMIC SYSTEM BEHAVIOR
5.1
remember the stability criterion for linear systems
In Section 4.9, we introduced a stability criterion for a linear system: non-
negative poles in the transfer function (5.1-1) indicate that the system
output y(t) will not remain stable in response to a system input x(t).
1
s
a
s
a
s
a
1
)
s
(
G
)
s
(
x
)
s
(
y
1
1
n
1
n
n
n
'
'
+
+
+
+
=
=
−
−
L
(5.1-1)
As a simple example, consider a first-order system:
( )
dy
y
Kx
y
0
0
dt
′
′
′
′
τ
+
=
=
(5.1-2)
We know that the Laplace transform representation is completely
equivalent.
( )
( )
K
y
s
x
s
s
1
′
=
τ +
′
(5.1-3)
The transfer function in (5.1-3) has a single pole at -
τ
-1
.
If the time
constant
τ
is a positive quantity (as in our tank), the pole is negative and
the response is stable (as we have seen in Lesson 3).
If the time constant were a negative quantity, however, the pole would be
positive.
As we saw in Section 4.9, the response would be unstable
because of the exponential term in the solution of (5.1-2)
t
y (t)
e
−
τ
′
±
(5.1-4)
This unbounded response could be in a positive or negative direction,
depending on the sign of the gain K.
We will address on another occasion
what sort of system might have a negative time constant; for now we
recognize that encountering an unstable linear system should cause us to
look carefully at the process whose behavior it represents.
revised 2006 Feb 1
1

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