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Process Modelling and Analysis
Week 4: Nonlinear Analysis  Part One
Session 1, 2012
Lecturer: A/Prof. J. Bao
Contents
1 Stability Analysis for Linear Systems
1
2 Multiplicity
3
2.1
Output multiplicity .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Input multiplicity .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3 Stability Analysis
4
4 Phaseplane Analysis
7
4.1
Second Order Linear Systems .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1
Stability Analysis for Linear Systems
When we have a process model, we may ask the question, if the states are perturbed from the
steadystate values, what will be the dynamic behaviour of this system? Will the state converge
to a some steadystate or will it go to inﬁnity? This is stability analysis concerned with. If
the state converges to its original equilibrium or to a new steadystate, this system is said to
be stable, otherwise, it is said to be unstable. If we have bounded input
u
s
,
we redeﬁne input
variable as
u
′
=
u
−
u
s
= 0
.
x
′
=
x
−
x
s
.
A new model will be obtained with the new coordinate.
As long as the input is bounded, such variable transform will not change the boundness of the
model. Therefore we can deﬁne stability based on the zeroinput form and for the equilibrium
of zero. Mathematically, for a system with zero input
˙x
=
f
(
x
,
0
)
,
x
(
t
0
) =
x
0
(1)
Then this system is (asymptotically) stable with the equilibrium of zero if
lim
t
→∞
x
(
t
) =
0
(2)
Otherwise it is unstable. There are several stability deﬁnitions. Here we just adopt one common
deﬁnition. We will learn other stability deﬁnitions such as BIBO stability in next session’s
process control subject.
1
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View Full Document An unstable system may be very sensitive to disturbances. If we ﬁnd a process is unstable,
we need take extra care, because any disturbance may cause this process to run away. If the
state variable is temperature, unstable system implies the temperature may go to inﬁnity, at
least theoretically, it may cause ﬁre or exposition. So it is very important to analyse the stability
of models. However, it is hard to determine the stability of the process from the deﬁnition. So
we need stability criteria.
For a single variable linear system
˙
x
=
ax
(3)
has the solution:
x
(
t
) =
e
at
x
(0)
(4)
When
t
approaches inﬁnity,
x
(
t
) will converge to 0 if
a <
0;
x
(
t
) will stay at
x
(0) if
a
= 0; and
be unbounded if
a >
0
.
Therefore the system is stable if and only if
a <
0
.
In a similar fashion, for a linear system
˙x
′
=
A
x
′
+
B
u
′
(5)
with
x
(0) =
x
0
.
The “zeroinput” form of the state space model is then:
˙x
′
=
A
x
′
.
(6)
Recall that matrix
A
can be diagonalized by using
S
−
1
AS
= Λ =
λ
1
0
0
0
0
λ
2
0
0
0
0
.
.
.
.
.
.
0
0
...
λ
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This note was uploaded on 03/26/2012 for the course CHEM ENG CEIC at University of New South Wales.
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