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

where is typically chosen to be 0.1 (10%). is chosen to be larger if the measurement noise is severe.

With the derivative filter , a typical PD controller output is calculated as

(1.12)

Figure 1.4 PD controller structure in implementation.

where is the filtered output response using (1.11). In a Laplace transform, the control signal is expressed as

(1.13)

Figure 1.4shows the block diagram used for implementation of a PD controller with a filter.

If the derivative filter was not considered in the design, there is a certain degree of performance uncertainty due to the introduction of the filter. This may not be ideal for many applications. Designing a PD controller with the filter included will be discussed in Section 3.4.1.

A proportional plus integral (PI) controller is the most widely used controller among PID controllers. With the integral action, the steady-state error that had existed with the proportional controller alone (see Example 1.1) is completely eliminated. The output of the controller is the sum of two terms, one from the proportional function and the other from the integral action, having the form,

(1.14)

where is the error signal between the reference signal and the output , is the proportional gain, and is the integral time constant. The parameter is always positive, and its value is inversely proportional to the effect of the integral action taken by the PI controller. A smaller will result in a stronger effect of the integral action.

The Laplace transform of the controller output is

(1.15)

with being the Laplace transform of the error signal . With this, the Laplace transfer function of the PI controller is expressed as

(1.16)

Figure 1.5shows a block diagram of the PI control system.

The example below is used to illustrate closed-loop control with a PI controller. For comparison purpose, we use the same plant as that used in Example 1.1.

Figure 1.5 PI control system.

Assume that the plant is a first order system with the transfer function:

(1.17)

the PI controller has the proportional gain , and the integral time constant and 0.5 respectively. Examine the locations of the closed-loop poles. With the reference signal as a unit step signal, find the steady-state value of the closed-loop output .

Solution. We calculate the closed-loop transfer function between the reference and output signals:

(1.18)

With given in (1.16) and in (1.17), we have

(1.19)

The closed-loop poles of this system are determined by the solutions of the closed-loop characteristic equation,

(1.20)

which are

(1.21)

If the quantity

then there are two identical real poles located at

If the quantity