7.8.  COMPENSATION AND DESIGN USING THE NICHOLS CHART

The Nichols-chart method has been developed in Section 6.10. We demonstrated in that section how one could obtain the closed-loop frequency response of a feedback control system by superimposing the open-loop gain-phase characteristics onto the Nichols chart. Specifically, we obtained the closed-loop frequency response of the system shown in Figure 6.49. The intersections of the open-loop gain–phase characteristics and the Nichols closed-loop gain characteristics were shown in Figure 6.51a. The resulting closed-loop frequency response was illustrated in Figure 6.51b. It indicated a maximum value of peaking M_p of 6.758 dB (2.2) and the frequency at which it occurred, ω_p, was 8.989 rad/sec. We indicated in Sections 6.10 and 6.12 that a value of M_p = 6.758 dB (2.2) does not represent a good design. This section demonstrates how the control engineer may use the Nichols chart in order to achieve a specified performance.

Let us assume for this problem that an acceptable value of M_p is 1.4 (2.92 dB). This may be achieved by adding a phase-lag or phase-lead network in cascade with the forward-loop transfer function G(s). A phase-lag network is used for this problem although a solution can be found as easily using a phase-lead network. We shall demonstrate that for an M_p of 2.92 dB the object is to modify the gain–phase characteristics on the Nichols chart so that it is just tangent to the 2.92 dB locus and does not enter it. By restricting the gain-phase characteristics to areas external to the M = 2.92 dB locus, we will have limited M_p to 2.92 dB, because the interior of this locus represents values of M greater than 2.92 dB.

Studying the characteristics of Figure 6.51a, we see that relatively large magnitudes of G(jω) exist for ω < 3 rad/sec. Therefore, it is not desirable to shift these magnitudes inside the M_p = 2.92 dB curve. In addition, it is desirable to attenuate G(jω) by a factor of about 3 in the range of frequencies of ω = 5 − 12 rad/sec. A phase-lag network, (1 + Ts)/(1 + αTs), whose factor α equals 3 will achieve this if is chosen at about 1 rad/sec. Solving for αT and T, we get αT = 1.74 and T = 0.58.

In order to obtain the gain-phase characteristics of the open-loop system, the Bode diagram is first drawn as indicated in Figure 7.36. Then for each value of ω the magnitude and phase of the open-loop compensated characteristics are plotted into a Nichols chart as shown in Figure 7.37. Because the open-loop gain-phase characteristics are just tangent to the constant-magnitude locus corresponding to M = 2.976 dB (1.4), we have achieved our goal. Notice that we have shifted ω = 1 rad/sec by about −35°, but this does not increase M_p. Figures 7.36 and 7.37 were obtained using MATLAB, and are contained in the M-files that are part of my MCSTD Toolbox. It is important to note that the use of MATLAB and the MCSTD Toolbox permits one to obtain the Nichols chart directly without having to first obtain the Bode diagram.