Stability analysis in an example system

1
Stability analysis in an example system

• Direct substitution and Dead Time

2
System diagram
Consider a system that can be represented by a
first order process with dead time. An example
of this type of system might be a fertigation
system.
3
Block diagram
The system with a proportional controller might
be represented as
4
Method of direct substitution
A transfer function relating the outlet
concentration to inflow changes would be
What would the ultimate gain for the controller
be?
In order to simplify the equation, we can use a
series approximation for the exponential.
5
Finding the characteristic equation
Pades approximation
Let tp 3, t0 2, KL 1, KP 1,
6
Direct substitution
To find the ultimate gain, substitute iw for s
and solve for Kc and w
For the real part
And the imaginary part
From the imaginary part
From the real part
Note
7
Controller tuning parameters

• For a PID controller and quarter amplitude decay
  Ziegler-Nichols tuning

Table 7-1.1
8
Zigler-Nichols

• Zigler-Nichols, Quarter amplitude decay tuning

Controller Kc Ti tD
P Ku /2
PI Ku /2.2 Pu /1.2
PID Ku /1.7 Pu /2 Pu /8