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Instability

Consider the case where $P_D>R$. This implies that the net damping $\hat{R}=R-P_D < 0$, which corresponds to a scenario in which the magnitude of the vibration increases exponentially over time. At first this might seem to violate energy conservation since the kinetic and potential energy of the system increase over time. However, the power supply for the amplifier connected to the motor serves as the source of additional power.

Occasionally, certain kinds of instability may be desirable, but usually the goal is to quickly drive a quantity to zero. As a result, much of feedback control involves proving that controlled systems are stable. More sophisticated proofs revolve around proving how fast system quantities decay to zero.


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Download pidcontrol.pdf

``PID Control of a Plucked String'', by Edgar Berdahl, and Julius O. Smith III,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2007-04-09 under the Creative Commons License (Attribution 2.5), by Edgar Berdahl, and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA