Sensitivity of the zeros of flexible structures to sensor and actuator location

In practice, flexible structures will often have to be controlled by sensors and actuators not physically collocated. Unfortunately, the transmission zeros of noncollocated structures are known to be potentially far more sensitive to system perturbations, for instance, small variations in the sensor or actuator positions, than the zeros of collocated structures. The paper defines two new transmission zero condition numbers that quantify this sensitivity precisely and lead to a clear geometrical interpretation of the circumstances under which a noncollocated structure will possess ill-conditioned zeros. Furthermore, the condition number associated with a particular zero is shown to provide a measure of the sensitivity of the high-gain, closed-loop pole that approaches this zero asymptotically. Finally, the results obtained are used to suggest a new approach to the difficult sensor/actuator placement problem.