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?
Limitations to closed-loop bandwidth
The maximum closed-loop bandwidth (frequency response) must be less by an amount called the separation ratio, which always has a value less than one. In mathematical terms:
fmax < ps × (lesser of fv or fn), where
fmax is the maximum system closed bandwidth
ps is the separation ratio, and
fv and fn are the valve and hydromechanical resonant frequencies in Hz.
When HMRF dominates (< fv), the separation ratio is controlled entirely by the damping ratio of the hydromechanical system:
ps = 2Zn
where Zn is the damping ratio, a measure of the tendency for an oscillation to subside.
Two conditions contribute to damping — internal leakage from one side of the actuator to the other (whether from within the actuator or the control valve) and friction (whether from the actuator or its load). Because manufacturers strive to reduce internal leakage and friction, it should come as no surprise that the degree of damping in most hydromechanical systems can be very low. In fact, when the load can be moved with negligible friction (as when supported by a recirculating linear ball bearing), the damping ratio may be as low as 0.03 or 0.05. Admittedly, the system friction and the damping ratio are the most elusive quantities to evaluate in a system. Nonetheless, they, along with the frequencies, absolutely govern the performance limits of the system.
Sample calculations
Consider an example to demonstrate this discussion. Suppose that a system's hydromechanical resonant frequency has been calculated and found to be 18 Hz. Further suppose that its servovalve has a 90° phase lag frequency of 65 Hz, and we estimate the hydromechanical damping ratio, due to both friction and internal valve leakage, to be about 0.05. We can calculate the maximum possible closed-loop system bandwidth:
fmax < ps x (lesser of fv or fn)
fmax < 2 x 0.05 x 18
fmax < 1.8 Hz
The maximum closed-loop bandwidth, fmax, is only 1.8 Hz, which is only a tenth of the HMRF! At startup time, we increase the system bandwidth by increasing the servo amplifier gain. If we increase the gain until we have 1.8 Hz of bandwidth, and then attempt further increases, the servo loop will break into sustained oscillation, rendering it worthless. The gain must be decreased to re-establish stability.
System bandwidth is important because a direct inverse relationship exists between it and positioning accuracy, or, more correctly, the positioning error and the following error. It has been shown that:
∆xp = (∆IT × Gsp)/(2 ’ fsys).
Where ∆xp is the expected steady state positioning error (in.)
∆IT is the total expected valve current variation (amperes) caused by eight known external disturbances,
Gsp is the speed gain [(in.x A) / sec] at the highest expected speed and load, and
fsys is the actual closed-loop system bandwidth (Hz).
The output position is never where we want it to be — only close. There are eight known disturbances in the electrohydraulic positional servo mechanism that cause imperfect positioning:
• valve temperature changes
• supply pressure variations
• tank port pressure variations
• breakaway friction
• load variations
• valve hysteresis
• valve threshold, and
• valve dead zone.
All of these must be resolved into an equivalent valve current, then added together to yield the total expected valve current, delta IT. In general, evaluating these eight "error contributors" for a given system is more than a trivial process. Experience tells us, however, that for zero-lapped valves with "typical" servovalve performance, delta IT is about 2% or 3% of rated valve current. If the valve is proportional and has substantial overlap, then we usually use the overlap only and ignore the other seven contributors.
Technically, speed gain must be calculated - using the characteristics of the selected control valve - for the worst-case loading condition. If the designer uses good engineering practice for selecting the control valve (if the valve is selected to provide maximum power transfer at the worst-case load-and-speed combination), then speed gain Gsp will equal approximately the target design actuator speed divided by about 2/3 of valve rated current. Armed with this information, we can now estimate the expected system "accuracy."
Suppose the system we are designing must propel a load at 21 in./sec under worst-case conditions using a servovalve with essentially zero overlap. The numerator of the equation at the upper-left portion of this page can be evaluated first:
∆IT × Gsp = (0.02 × IR x 21) ÷ (0.67 × IR), where
IR is the rated valve current, which cancels out of the equation.
Now, ∆IT Gsp = (0.02 × 21) × 0.67
∆ITG sp = 0.63 in./sec.
If we assume that the servoloop has been tuned to the maximum allowed before instability occurs, then fsys is set to fmax, so the error can be estimated:
∆xp = ∆I T Gsp ÷ 2 ’ fsys.
But because the numerator has already been evaluated:
∆xp = 0.63 ÷ (2 ’ × 1.8),
∆xp = ±0.055 in.
We could expect, then, that the long-term positioning capability of this system would be about 0.055 in.
Jack L. Johnson is a contributing editor for Hydraulics & Pneumatics and president of IDAS Engineering, Milwaukee.
