Modern Control System Theory and Design, 2nd Edition

5.7.  ZERO-ERROR SYSTEMS

The transfer function of the general feedback system, containing a forward transfer function G(s) and a feedback transfer function H(s), is given by the general expression

From Eq. (5.26b) of Section 5.4, the error for this system can be expressed as^*

It can be shown [2] that 1/(1 + K_p) is a function of B_m+1 − A_n+1, and it is necessary that B_m+1 = A_n+1 for zero steady-state error when the input is a step function. In addition, this implies that the forward transfer function contains at least one pure integrator as discussed previously in Section 5.4. From the general system transfer function of Eq. (5.50), it can be seen that there are many possible forms of C(s)/R(s) that will yield zero steady-state error with a step input. When the numerator consists of the constant B_m+1 the system is called a zero steady-state step error system [2]:

We can illustrate such a system with an example of a control system we have thoroughly analyzed in Chapter 4, the second-order control system, shown in Figure 4.1 which contains a single pure integrator. We also know from Table 5.1 that this control system has zero steady-state error to a unit step input (since G(s) has one ...