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

(1.6)

Figure 1.2Closed-loop step response of a proportional control system ( Example 1.1).

For any value of , , i.e. not equal to the desired value at the steady state response. Figure 1.2 shows the closed-loop step response with the proportional controller and , respectively. It is seen that with the increased proportional gain , the closed-loop response speed increases and the steady-state value becomes closer to the desired value 1.

In many applications, a proportional controller is not sufficient to achieve a particular control objective such as stabilization or producing adequate damping for the closed-loop system. For instance, for a double integrator system with the transfer function:

(1.7)

the closed-loop control system with a proportional controller has a transfer function

which has a pair of closed-loop poles determined by the solutions of the polynomial equation:

These poles are at . Thus, no matter what choice we make for , the system still behaves in a sustained oscillatory manner because the pair of closed-loop poles are on the imaginary axis of the complex plane.

Now, assuming that we will additionally take the derivative of the feedback error signal into the control signal calculation, this leads to

(1.8)

where is the derivative control gain.

The Laplace transfer function of (1.8)is calculated as

Figure 1.3 Proportional plus derivative feedback control system ( ).

This is what we called a proportional plus derivative (PD) controller.

The closed-loop feedback control configuration for a PD controller is shown in Figure 1.3. For the double integrator system (1.7), with the derivative term included in the controller, then the closed-loop transfer function becomes

(1.9)

(1.10)

The closed-loop poles are determined by the solutions of the characteristic polynomial equation as

which are

Clearly, we can choose the values of and to achieve the desired closed-loop performance.

It is worthwhile emphasizing that almost without exception, the derivative term is different from the original form in the implementation. This is because the derivative term is not practically implementable and the differentiation of the output signal leads to amplification of the measurement noise. Thus, there are a few modifications with regard to the derivative terms. Firstly, in order to avoid the problem caused by a step reference trajectory change (the so-called derivative kick problem (Hägglund (2012))), the derivative action is only implemented on the output signal . Secondly, in order to avoid amplification of the measurement noise, a derivative filter is in a companionship with the derivative term .

A commonly used derivative filter is a first order filter and has its time constant linked as a percentage to the actual derivative gain in the form:

(1.11)