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

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Figure 1.13 Comparison of closed-loop output response using Ziegler–Nichols rules (Example 1.5). Key: line (1) PI control; line (2) PID control.

      1.3.2 Tuning Rules based on the First Order Plus Delay Model

      The majority of tuning rules existing in the literature are based on a first order plus delay model, which has the following transfer function:

(1.44)

      where is the steady-state gain of the system, is the time delay, and is the time constant. This is mainly because the primary applications of tuning rules are for process control where typically the process is stable with time delay. There are many methods available for obtaining a first order plus delay model. Among them is an incredibly simple procedure that is called fitting a reaction curve. This reaction curve is the so-called step response test.

The reaction curve is obtained by performing a plant step response test in open-loop operation, therefore we assume that the plant is stable, although it could be applied to integrating plant with caution. When performing this test, the plant input signal takes a step change from an initial constant value to a normal operation value, ; the measurement of the plant output signal in response to the step input change gives us the plant step response test data or the reaction curve. The response test completes when the value of the output signal reaches a constant or the signal fluctuates around a constant value due to noise and disturbances. Figure 1.14 illustrates a typical set of step response testing data, with the step change occurring at time . Figure 1.14(a) shows that the input takes a step change from the initial value to the final value and Figure 1.14(b) shows the actual output response from the steady-state output position to the steady-state output position .

The plant information obtained from the test includes the steady-state gain, defined as

      (1.45)