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

This book is accompanied by a companion website:

www.wiley.com/go/Wang/PIDcontrolsystem

The website includes:

Lecture slides

MATLAB/Simulink programs

Scan this QR code to visit the companion website.

This chapter will introduce the basic ideas of PID control systems. It starts with an introduction to the roles of proportional control, integral control, and derivative control, followed by an introduction to the various tuning rules developed over the past several decades. These tuning rules are mainly developed for first order plus delay systems and are simple to use. However, they do not, in general, guarantee satisfactory control system performance. Simulation 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.

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.

The simplest controller is the proportional controller. With this term proportional, the feedback control signal is computed in proportion to the feedback error with the formula,

(1.1)

where is the proportional gain and the feedback error is the difference between the reference signal and the output signal ( ). The block diagram for the closed-loop feedback control configuration is shown in Figure 1.1where , , , and are the Laplace transforms of the reference signal, feedback error, control signal, and output signal, respectively. represents the Laplace transfer function of the plant.

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 without detailed information about the plant.

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