PID Control System Design and Automatic Tuning using MATLAB/Simulink. Liuping Wang
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examples will be used to illustrate closed-loop performance.
This chapter is suitable for those who want to understand the very basics of PID control systems. By utilization of the tuning rules, it is possible to have an application of a PID control system without further exploration.
1.2 PID Controller Structure
There are four types of controllers that belong to the family of PID controllers: the proportional (P) controller, the proportional plus integral (PI) controller, the proportional plus derivative (PD) controller and the proportional plus integral plus derivative (PID) controller. To understand the roles of the controllers, in this section we will discuss each of the structures and the PID controller parameters. From the discussions, we will establish some basic knowledge about how to use these controllers in various applications.
1.2.1 Proportional Controller
The simplest controller is the proportional controller. With this term proportional, the feedback control signal
(1.1)
where
Figure 1.1 Proportional feedback control system.
Because of its simplicity, the proportional controller is often used in the cases when little information about the system is available and the required control performance in steady-state operation is not demanding. As the controller only involves one parameter to be determined, it is possible to choose
One of the limitations of a simple proportional controller is that the steady-state error of the closed-loop control system will not be completely eliminated. We illustrate this point with the following example.
Suppose that the plant is a first order system with the following transfer function,
(1.2)
with proportional controller (). Suppose that the reference signal is a step signal with amplitude 1 and its Laplace transform is . Find the steady-state value of the output with respect to the reference signal.
Solution. From Figure 1.1, the closed-loop control system from the reference signal to the plant output has the transfer function,
(1.3)
With any positive , the closed-loop system is stable where its pole is determined by the solution of the polynomial equation 1,
(1.4)
which is .
The Laplace transform of the output, , is
(1.5)
where . Applying final value theorem to the stable closed-loop system, we calculate
(1.6)
Figure 1.2 Closed-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.
1.2.2 Proportional Plus Derivative Controller
In many applications, a proportional controller
(1.7)
the closed-loop control system with a proportional controller
which has a pair of closed-loop poles determined by the solutions of the polynomial equation:
These poles are at
Now, assuming that we will