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PID Control System Design and Automatic Tuning using MATLAB/Simulink. Liuping WangЧитать онлайн книгу.

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.

      

      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

is computed in proportion to the feedback error
with the formula,

      (1.1)

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 1.1 where
,
,
, 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.

      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,

       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,

       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)

Graph depicting Time (sec) on the horizontal axis, Output on the vertical axis, and two curves plotted for Kc 8, 80.

       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

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

equation

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

equation

      These poles are at images. Thus, no matter what choice we make for images, 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


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