12.6: Implications of Bode’s Integral

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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The loop transfer function \(PC\) cannot roll off too rapidly in the crossover region, and this limits how ”dramatic” can be the loop shapes that we create to achieve robustness, nominal performance, or both. The simple reason is that a steep slope induces a large phase loss, which in turn degrades the phase margin. To see this requires a short foray into Bode’s integral. For a transfer function \(H(s)\), the crucial relation is

\[ \angle H(j \omega_0) \, = \, \frac{1}{\pi} \int\limits_{-\infty}^{\infty} \frac{d}{d \nu} [ \log(|H(j \omega)|) \cdot \log( \coth (|\nu|/2))] \, d \nu, \]where \(\nu = \log(\omega / \omega_0)\), and \(\coth()\) is the hyperbolic cotangent. The integral is hence taken over the log of a frequency normalized with \(\omega_0\). It is not hard to see how the integral controls the angle: the function \(\log( \coth( |\nu|/2)) \) is nonzero only near \(\nu = 0\), implying that the angle depends only on the local slope \(d( \log |H|)/ d\nu\). Thus, if the slope is large, the angle is large.

Example \(\PageIndex{1}\)

Suppose \(H(s) = \omega_0^n / s^n\), i.e., it is a simple function with \(n\) poles at the origin, and no zeros; \(\omega_0\) is a fixed constant. It follows that \(|H| = \omega_0^n / \omega^n\), and \(\log |H| = -n \log (\omega / \omega_0)\), so that \(d(\log |H|) / d\nu = -n\). Then we have just

\[ \angle H \, = \, -\frac{n}{\pi} \int\limits_{-\infty}^{\infty} \log (\coth (|\nu|/2)) \, d\nu \, = \, -\frac{n \pi}{2}. \]

This integral is easy to look up or compute. Each pole at the origin induces \(90°\) of phase loss. In the general case, each pole not at the origin induces \(90°\) of phase loss for frequencies above the pole. Each zero at the origin adds \(90°\) phase lead, while zeros not at the origin add \(90°\) of phase lead for frequencies above the zero. In the immediate neighborhood of these poles and zeros, the phase may vary significantly with frequency.

The Nyquist loci are clearly susceptible to these variations in phase, and the phase margin can be easily lost if the slope of \(PC\) at crossover (where the magnitude is unity) is too steep. The slope can safely be first-order (\(-20dB/decade)\), equivalent to a single pole), and may be second-order (\(-40dB/decade)\) if an adequate phase angle can be maintained near crossover.