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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.

Example 1.1

       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 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

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