The Art of the Helicopter John Watkinson - Karatunov.net

Technical background 57
the inertia of the system that determine the maximum slew rate. Figure 2.23 showed
the effect of different damping factors on system response. In the case of a servo the
amount of velocity feedback determines the damping factor. With little or no velocity
feedback the slew rate will be high, but hunting will be excessive and the step response
will be a decaying oscillation. With too much velocity feedback the response will have
no overshoot, but will be very slow. In between these is the condition known as critical
damping. This is defined as the amount of damping which gives the fastest response
without any overshoot. The critical damping condition is simply a definable condition,
and it is not necessarily the optimum response. In most cases the response speed can
be increased significantly with the penalty of a very small overshoot.
In an unpowered system, if the pilot experiences resistance to a control input, he will
automatically apply more force to obtain the desired control position. In a fully powered
system, the pilot (or the autopilot) does not feel this resistance and cannot compensate.
The feedback system must be designed to overcome resistance automatically. In a simple
servo, the load will stop at a position where the force from the motor is balancing the
disturbing force. Clearly in a simple feedback system, the motor can only produce a
force if there is a position error. The result is that the external force pushes the load
out of position until the position error is large enough to oppose further motion. The
ability of a servo to oppose external force in this way is called stiffness.
The stiffness of a servo can be improved by increasing the gain given to the position
error. Then a given resistance to an external force will be obtained with a smaller position
error. Increasing the gain may also speed up the step response, but not beyond the
slew rate limit. If a control input exceeds the slew rate limit, for a period of time the
position error will be large and the system is not feedback controlled. In this state
the system is said to be in an open loop condition.
Clearly high gain is desirable in a servo because it increases response speed and
stiffness. However, high gain can also cause instability. During servo development, a
control input is created which is sinusoidal and the frequency is swept upwards from a
very low value. The amplitude and phase response are plotted against the frequency.
As the frequency rises, the servo has to work harder to follow the rapid movement and
so the position error may increase. This causes the load to lag the input. At some high
frequency, the phase lag may reach 180 ◦ .
With a 180 ◦ phase shift, a negative feedback system has turned into a positive feedback
system. In other words the sense of the correcting action is wrong. If the loop
gain is above unity at the frequency where the phase response has gone to 180 ◦ , the
system will oscillate at that frequency. This may be spontaneous upon applying power
or result from a small control input. Clearly this would be catastrophic in any control
system. In practice feedback systems must contain a filter to reduce the loop gain at
high frequencies to prevent oscillation.
Another possibility that allows higher loop gain is to have a signal processor containing
an inverse model of the phase characteristics of the servo loop. When the servo
loop lags, the processor will introduce a phase lead to balance out the lag. The use of
these filters and processors in a servo loop is called compensation.
If high gain cannot be used for stability reasons, it is possible to improve the accuracy
of a servo in the long term by integrating the error. Figure 2.36 shows a feedback system
with integral control. A small error at the input to the integrator will become larger
as the integrator operates until the loop acts to cancel it. Integral control is useful for
overcoming friction in mechanisms.
Feedback loops can be nested which means that one operates inside another. For
example, Figure 2.37 shows an automatic navigation system. The flight director error
loop provides an error signal to the autopilot in order to keep the machine on track.