and a2 = 1. Then, from (4-38), we get e1 = 3 so that slm-a = -1 < 0 and the system NCT is locally asymptotically stable.
Various simulations have been obtained using a Runge-Kutta algorithm [19] to numerically integrate the closed-loop nonlinear system. In some of these simulations, an internal model system has been used which does not contain dynamics corresponding to the second harmonic of the reference signal. Such a design results when the
internal model system is chosen in accordance with linear servomechanism theory. To give a fair comparison between the two control schemes, the closed-loop eigenvalues of the design based on linear theory are identical to those given above, except that the eigenvalues at -4 t j2 are no longer needed due to a reduction in system order.
Figure 4-1 shows the responses obtained for both design approaches when only a reference signal is applied. The proposed control design works well (see Fig. 4-1(a)) and the tracking error is completely eliminated in steady-state. The design based on linear servomechanism theory, however, results in a steady-state tracking error which is sinusoidal with a frequency twice that of the reference signal.
Figure 4-2 again shows the responses obtained using both design approaches. Here, however, a constant disturbance has been introduced in addition to the sinusoidal reference signal.
In Figure 4.3 the amplitude of the disturbance has been increased and as a result, the transient response in the system designed by linear theory is very poor. Note that in order to show the complete response, the scaling of the plot in Figure 4-3(b) is different from the scaling used in previous figures.