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Modern Control System Theory and Design, 2nd Edition by Stanley M. Shinners

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6.10.  NICHOLS CHART

The Nichols chart [21] is a very useful technique for determining stability and the closed-loop frequency response of a feedback system. Stability is determined from a plot of the open-loop gain versus phase characteristics. At the same time, the closed-loop frequency response of the system is determined by utilizing contours of constant closed-loop amplitude and phase shift which are overlaid on the gain-phase plot.

In order to derive the basic Nichols chart relationships, let us consider the unity-feedback system illustrated in Figure 6.45. The closed-loop transfer function is given by

Image

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Figure 6.45   Block diagram of a simple feedback control system.

or

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where M(ω) represents the amplitude component of the transfer function and α(ω) the phase component of the transfer function. The radian frequency at which the maximum value of C()/R() occurs is called the resonant frequency of the system, ωp, and the maximum value of C()/R() is denoted by Mp. For the system illustrated in Figure 6.45, we would expect a typical closed-loop frequency response to have the general form shown in Figure 6.46.

From Section 4.2 we know that a small margin of stability would mean ...

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