Bode: Gain, Phase and What Feedback Costs

Nyquist's criterion answers a yes-or-no question about a loop you have already built. It does not tell you what to change, and at Bell Labs in the 1930s that left designers measuring amplifier after amplifier without a way to steer the next one. Hendrik Bode's answer, worked out through the 1930s and published as a book in 1945, is that for an ordinary minimum-phase network the phase is not free. It is determined by how fast the gain is falling: a slope of twenty decibels per decade goes with ninety degrees of lag, forty with a hundred and eighty. The stability question therefore becomes a question about a SLOPE, and the design rule that runs all of classical control falls straight out - cross through unity gain at twenty decibels per decade, because forty means a hundred and eighty degrees, and a hundred and eighty degrees at unity gain is an oscillator. The deeper result is the one that has no workaround. For a stable loop with no right-half-plane poles that falls off faster than one over frequency, the integral of the log of the sensitivity over all frequencies is exactly zero. Sensitivity below one is where feedback is helping you; the integral says every decibel of help at one frequency is paid for by a decibel of harm at another. Engineers call it the waterbed: press it down here and it comes up there. That is why the design decisions in this blueprint are all about WHERE, never how much. Turning the gain down, adding a dominant pole, adding a lead network and buying a faster plant all reach the same phase margin and cost completely different things - accuracy everywhere, speed, noise at the actuator, or money.