In Nyquist's seminal paper, he applied the idea of sinusoidal analysis to dynamic systems, but in very arcane mathematical terms, given the education of fellow engineers in his day. Nine years later, Bode presented his equally significant paper. He taught us that by expressing the amplitude of a sinusoidal frequency response in decibels, it becomes a simple matter of adding the amplitude and phase responses of one device (e.g. a valve) to the amplitude and phase responses of another device (e.g. the load and actuator subsystem) to obtain the composite phase and amplitude responses for a system. In those days - when the computational aid of choice was a slide rule - such a simple process was indeed welcome. In spite of the revolution in computers and the ease with which they perform complex calculations, the well-known Bode plot remains a favorite tool of system designers.

Bode and Nyquist (both were research engineers at Bell Laboratories) were intrigued by the idea of determining why some audio amplifiers would break into oscillation, while others would not. Today, designers of motion-control and feedback-control systems still struggle with the problems of oscillation in machines. If allowed to continue, this oscillation can be self-destructive. Frequency response methods allow us to make a reasonable estimate of the limits of electronic tuning that will produce a stable, oscillation-free servomechanism.

Bode's methods require that we test and study an open-loop system, then use analytical techniques to ask "what if" the loop is closed. For example, a complete positional servo mechanism, Figure 3, can be tested in its open-loop configuration to find the conditions necessary to make the system go unstable. Note that we do not use sinusoidal test data (frequency response characteristics) to determine how well the system will behave with sinusoidal inputs, but, rather, how much servo-loop gain we can get before the system oscillates. Such are the subtleties of frequency response methods.

To illustrate, consider the system of Figure 3. It will be first tested with the feedback switch in the open-loop position. As frequency increases, we look for any frequency that causes a 180° phase lag between command input and the open-loop feedback signal. (In hydromechanical systems, it is essentially certain that such a frequency exists.) At that frequency, if the output (open-loop feedback signal) has an amplitude equal to or greater than the input command amplitude, then the feedback switch can be closed. The 180° phase shift then undergoes another 180° of phase shift through the negative feedback process. The result is that the sinusoidal command input stimulus can be removed, and the closed-loop system will be in a state of sustained oscillation. This is an unstable system and is impractical if the oscillations cannot be stopped. Reducing servo-loop gain is the normal procedure to stop the oscillations. This is accomplished by changing the gain setting of the servo amplifier.

The foregoing paragraph presents the classical criterion for servo-loop stability in non-mathematical terms. It can be simplified a bit by changing the search for the 180° phase shift frequency. Consider this: Suppose in the search, the critical, 180° phase shift frequency is found, but the amplitude of the open-loop feedback signal is less than the command input amplitude. The open-loop gain at this frequency is less than one, which is less than zero on the decibel scale.

We now ask ourselves, "How can we increase the gain so that the feedback signal amplitude equals the command signal amplitude (zero dB of servo loop gain)?" We need only to increase the servo amplifier gain, and the system will break into oscillation. Therefore, anytime there is a 180° phase shift frequency, simply increasing the servo amplifier gain achieves sustained oscillation. A 180° phase-shift frequency exists in every electrohydraulic system, so we can always adjust such a system to the point of instability.

Of course, we do not want the system to oscillate. The purpose of tuning to the point of instability is to find the ultimate gain that will produce it. The gain is then reduced to about one-half the value that causes steady oscillations and is left there. This 50% reduction in gain is about the same as a 5-dB gain margin. That is, the gain is set 5-dB below the point of instability. This is sufficient for many electrohydraulic motion control systems. Using frequency response methods during the design process, we can predict what the gain will be for instability. Therefore, we can estimate the errors from instability to be expected in the servo system.

### Resonant and valve frequencies

The hydromechanical resonant frequency (HMRF) is inversely related to the volume of fluid and the load mass: the greater the volume of fluid under compression, and the greater the load mass, the lower the HMRF. The lower the HMRF, the more difficult it is to achieve snappy, responsive control of the servo system. Instead, the system becomes slow and springy. Indeed, there are those who characterize the compressibility of fluid as equivalent to a spring. The analogy has some value.

This springiness can be a system bottleneck when the HMRF is too low. I have seen systems with an HMRF as low as 0.5 Hz, as high as 700 Hz, and all values in between. Low HMRF is a characteristic of large masses connected to small cylinders. Increasing cylinder area always has the effect of raising the HMRF. HMRF becomes the system bottleneck when it is less than the valve's frequency. The valve frequency, fv, is the frequency that produces a 90° phase lag according to frequency-response test data published by the valve's manufacturer.

Now we have a direct basis of comparing one frequency with another, which enables us to draw important conclusions. It is true that when it is less than the valve frequency, HMRF limits system response. However, the valve becomes the limiting device when its frequency is less than the HMRF. The rule is easy: The dynamic bottleneck is the lesser of fv and fn.

More often than not, a system is more difficult to design for crisp response when its HMRF is less than its valve frequency. Unfortunately, a system's HMRF usually is lower than its valve frequency. This means, therefore, that HMRF usually is dominant, which represents the most challenging design scenario. In other words, the worst-case scenario is the most common scenario.

Furthermore, when the valve frequency is about twice that of the HMRF, increasing the valve frequency produces negligible affect on system performance, because performance will be influenced almost totally by the HMRF. It should be clear that the closed-loop bandwidth must always be less than the lesser of fv and fn. The only question remaining: How much less?