Fundamental design tension
Consider a system with a PD controller, actuator, sensor, and plant with three inputs, the reference input, the external
disturbance, and the measurement noise. Our design objective is to determine the PD gains such that the system is
robustly stable to plant uncertainties, rejects external disturbances, attenuates the measurement noise, and meets
given performance specifications (such as settling time and percent overshoot). In the case of the self-balancing robot,
we have significant measurement noise stemming from the noisy gyro and accelerometer. If we analyze the tracking
error, E(s), we find that a fundamental design tension, that is, to reduce the sensitivity to plant changes (robustness)
and to reject the external disturbances, we want to make the controller gains ‘large’. On the other hand, to attenuate
the measurement noise, we want to select the controller gains to be ‘small’. This fundamental tension does not stem
from using a PD controller, but is a fundamental design challenge no matter what controller structure is employed.
Fortunately, there is often a separation of frequencies over which the external disturbances and measurement noise
exist. Measurement noise is typically high frequency and external disturbances are typically low frequency. If this is not
the case in a particular situation, the design problem becomes significantly more difficult. The classic control design
approach is for the designer to design a controller with high gain at low frequencies and low gain at high frequencies!
Consider the PD controller for the selfbalancing robot—increase the gains and observe the improved performance to
an external gentle push. As you continue to increase the gains you should see the feedback control system response
begin to deteriorate as it begins to amplify the measurement noise. The trade-off results in a set of controller gains
that provide stable acceptable performance in the presence of plant changes (adding a mass at the top of the
robot) and external disturbances (gentle pushes), while attenuating the measurement noise (from noisy gyro and
accelerometer).
Self Balancing Robot Assembly 157
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