Relation between structural compliance and allowable friction in a servomechanism

In a great many control systems, error is primarily caused by transient load torques, often the result of static friction. To keep the error due to transient load torques small a high output inertia, and high values of gain crossover frequencies are required in the control loops. Integral networks that increase the static stiffness of the control system generally have negligible effect against transient load torques. The maximum error produced by a step of torque is approximately as follows for a rate feedback system:
\theta_{e} = \frac{T}{J \omega_{c}\omega_{r}}
(1) where
T
is the magnitude of the step of torque,
J
is the output intertia, ω
c
, is the gain-crossover frequency of the position loop, and
\omega{r}
is the gain-crossover frequency of the rate loop. Thus, if the error is to be kept within a given bound
\theta_{e}
, the maximum allowable friction torque T
f
is given by
T_{f} \leq \theta_{e}J\omega_{c}\omega_{r}
(2) Structural compliance is a severe limitation upon the allowable values of gain-crossover frequency. The maximum value of rate loop gain-crossover frequency ω
r
in practice is usually about half the natural frequency of the structure ω
n
. The position loop gain-cross-over frequency is usually at least a factor of three below that of the rate loop. Therefore one can assume that
\omega_{r} \leq \frac{\omega_{n}}{2}
(3)
\omega_{c} \leq \frac{\omega_{r}}{3} \leq \frac{\omega_{n}}{6}
(4) Substituting (3) and (4) into (2) gives
T_{f} \leq (1/12)J\omega_{n}^{2}\theta_{e}
(5) Equation (5) shows that regardless of the type of control system used, there is a maximum possible value of friction torque that is related only to the allowable error, the output inertia, and the structural natural frequency. Although this limitation is derived in terms of a specific servo configuration, it also holds approximately for other configurations, and cannot be improved by integral networks.