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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.