Liuping Wang - PID Control System Design and Automatic Tuning using MATLAB/Simulink

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 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?

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.

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 the choice of basically led to a pole-zero cancellation in the control system. Such a pole-zero cancellation will limit, as discussed in the next chapter, the control system performance in the disturbance rejection, particularly when is large 4. In Skogestad (2003), the IMC-PID tuning rules are modified to reduce the integral time constant as

(1.48)

while and are unchanged from (1.47).

The IMC-PID controller tuning rules are also extended to integrating systems in Skogestad (2003). Although the system has an integrator as part of its dynamics, integral control is still required for disturbance rejection (see Chapter 2).

Assuming that the system has the integrator with delay model:

(1.49)

then a PI controller is recommended with the following parameters:

(1.50)

If the transfer function for the integrating system has the form:

(1.51)

then a PID controller is recommended to have the following parameters:

(1.52)

If the system has a double integrator with the transfer function

(1.53)

then a PID controller is recommended with the following parameters:

(1.54)

The IMC-PID controller tuning rules will be studied in Examples 2.1and 2.2.

Several sets of tuning rules were introduced in Padula and Visioli (2011) and Padula and Visioli (2012). These tuning rules are based on the first order plus delay model:

Table 1.4 Padula and Visioli tuning rules (PI controller).