The techniques necessary to construct and analyze the open-loop frequency response of a feedback control system utilizing the Bode-diagram approach were presented in Section 6.7. This section illustrates how the Bode diagram can be used for designing a feedback-control system in order to meet certain specifications regarding relative stability, transient response, and accuracy. It is important to emphasize that the Bode-diagram approach is used very frequently by the practicing control engineer. Its use is due to the fact that the anticipated theoretical results may be relatively simply checked with actual performance in the laboratory just by opening the feedback loop and obtaining an open-loop frequency response of the system.

Bode’s primary contribution to the control art is summarized in two theorems [7]. We introduce the concepts embodied in these theorems first in a qualitative manner, and then the mathematical statements are given.

*Bode’s first theorem* essentially state that the slopes of the asymptotic amplitude–log-frequency curve implies a certain corresponding phase shift. For example, in Section 6.7 it was shown that a slope of 20*n* dB/decade (or 6*n* dB/octave) corresponded to a phase shift of 90*n*° for *n* = 0, ±1, ±2, …. Furthermore, this theorem states that the slope at crossover (where the attenuation–log-frequency curve crosses the 0-dB line) is weighted more heavily toward determining system ...

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