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PI PID

Table 1.3 Cohen–Coon tuning rules with a reaction curve.

P
PI
PID

There is another set of tuning rules that are derived based on the reaction curve, termed Cohen and Coon tuning rules. Table 1.3 gives the PID controller parameters calculated from Cohen and Coon tuning rules.

      For the estimation of time delay , , and when using MATLAB, it is a fairly straight forward procedure to draw the lines and pinpoint the data points. The MATLAB command for finding the point on a graph is called ginput. For example, by typing

      [a,b]=ginput(1)

      a cross hair will appear on the MATLAB figure and a double click on the point of interest will yield the exact values we need. This graphic procedure will be demonstrated in the example section (see Section 1.5).

      1.3.3 Food for Thought

      1 Can you apply Ziegler-Nichols oscillation tuning method to a first order system? Why?

      2 Can you apply the reaction curve based tuning rules to unstable systems? Why?

      3 How do we decide the sign of the proportional feedback controller gain when using the Ziegler-Nichols oscillation method?

      4 Can you envisage any potential danger when using Ziegler- Nichols oscillation method?

      5 How do you design a step response experiment?

      6 What information will the step response experiment provide?

      7 How do you determine steady-state gain, parameter and time delay from a reaction curve?

      8 What are your observations when comparing Ziegler-Nichols and Cohen-Coon tuning rules, in terms of signs and values of , and ?

      9 Is there any desired closed-loop performance specification among the tuning rules?

1.4 Model Based PID Controller Tuning Rules

      This section will discuss the PID controller tuning rules that are derived based on a first order plus delay model. These tuning rules worked well in applications.

      

      1.4.1 IMC-PID Controller Tuning Rules

      The internal model control (IMC)-PID tuning rules (Rivera et al. (1986)) are proposed on the basis of a first order plus delay model:

      When using the IMC-PID tuning rules, a desired closed-loop response is specified by the transfer function from the reference signal to the output:

      where is the desired time constant chosen by the user. The PI controller parameters are related to the first order plus delay model and the desired closed-loop time constant , which are given as:

      (1.46)

      If the system has a second order transfer function with time delay in the following form:

      then a PID controller is recommended. Assuming that , then the PID controller parameters are calculated as

(1.47)

Later on, it was realized that

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