The PD controller from subsection 4.4.2 has the form
which can be written as
where s is given by
and
The two gains to be adjusted are thus _{ }
and_{ }.
For constant values of _{ },
_{ }
behaves like the proportional gain. Due to the composite gain
structure, increases in _{ }
not only increases the derivative gain, but also results in an
increase in effective proportional gain. To increase the derivative
gain without increasing the effective proportional gain, _{
}
must be reduced as _{ }
is increased.
Increase in effective proportional gain initially reduces regulation
errors, however if this gain is increased beyond a certain boundary,
limit cycling results. The limit cycling is most pronounced about the
vehicle roll axis, which is the axis with the lowest moment of
inertia. A possible cause for this limit cycling is interaction
between the controller and the attitude estimator. Rotations about
the roll axis create angular accelerations which perturb the
measurement of the gravity vector which the vehicle assumes to point
down (Subsections 4.2.1 and 4.3.3). This causes errors in the
estimation of the vehicle attitude which in turn produces control
errors. None of the gain sets tested resulted in unstable operation
in attitude hold, however very large gains produced severe limit
cycling.
If the effective derivative gain is too large, the vehicle begins to
experience high frequency thrust perturbations. Examination of the
signals within the controller has shown that this thruster noise is
caused by increased sensitivity to noise in the angular velocity
estimate. This fact limits the level of damping that can be added to
the system.
The goal of the PD tuning was determine a set of gains that produce
relatively small steady state regulation errors with no significant
limit cycling. To this end, the vehicle was commanded to hold
attitude constant aligned with the tank axes using the quaternion
based PD controller. First, a value for _{ }
was chosen. With the controller using this value for the derivative
gain, the proportional gain _{ }
was then increased until limit cycling was observed. This process was
repeated for several values of _{ }.
The results are shown in Figure 53. Initially, increase in _{
}
caused a reduction in regulation error. However, if _{ }
is increased beyond a certain boundary, limit cycling results. Thus
for each value of _{ }
chosen, a maximum value for _{ }
may be determined, and for that pair of gains, the average regulation
error may be observed.
Figure 53 Maximum _{ } and the corresponding average attitude error as a function of _{ }
Notice that the minimum regulation error occurs when _{
}
= 2.2. At this point, the maximum value for _{ }
is 35, with an average attitude error of 0.7°.
The angular error could be reduced even further if the proportional
gain could be increased. Unfortunately, increasing _{ }
beyond the boundary indicated by the dashed line in Figure 53
results in limit cycles due to interaction between the estimator and
controller. One possible approach is to use a new gain strategy which
reduces the proportional gain as the magnitude of the angular error
increases, thus "softening" the response away from equilibrium enough
to reduce the susceptibility to limit cycling. The gain remains
unmodified near the desired attitude thus maintaining low regulation
error. The gain modification creates a new proportional gain _{
}
as a function of the attitude error.

_{} 
(5.1) 
The algorithm then tests to see if _{ }
is less than _{ }/5.
If so, it sets _{ }=_{}/5,
thus _{ }
is reduced to a minimum of_{ }/5.
This function is shown in Figure 54. The minimum occurs when the
desired attitude is greater than approximately 5° from the
desired attitude.
Figure 54 Gain modification strategy intended to reduce susceptibility to limit cycling
To test this modified gain, the previous experiments were
repeated: choosing a value for _{ },
and increasing _{ }
until limit cycling occurred. The results are shown in Figure 55.
The new feedback strategy appears to allow higher gains without limit
cycling, and moreover the regulation error is reduced to about
.3° or a factor of 2.3 improvement. The gain set producing this
new minimum regulation error is _{ }
= 68, _{ }
= 1. This gain modification strategy was used for all of the
remaining data in this chapter.
Figure 55 Comparison of maximum _{ } and the corresponding average attitude error as a function of _{ } with and without _{ } gain modification strategy. (the data points on the solid lines were recorded using the gain values designated by x's on the limit cycle boundary)
Figure 56 Attitude regulation error vs. time using_{ } gain modification strategy (_{ }=68, _{ }=1)