Sensor/Actuator Selection and Placement for Control of Elastic Continua

Abstract General controllability and observability conditions are developed for a flexible body whose dynamics are modelled by partial differential equations. These conditions reveal the types, number and location of actuators and sensors necessary to ensure controllability and observability of all modes of vibration. It is proved that the number and types of sensors/actuators required are directly related to the order of the partial differential equation. Controllability and observability can be guaranteed with the sensors/actuators located at an arbitrary point on the structure, although the degree of modal excitation/detection depends on their particular position. The implementation of the above principles relies on co-location of all the sensors, and co-location of all the actuators. In general, it also requires devices which may not be easily mechanized. Fortunately, the general controllability and observability criteria are quite conservative and it is shown that, under realistic boundary conditions, they are satisfied with just two types of conventional sensors and actuators. Moreover, in most cases, complete controllability and observability of the flexible-body modes can be achieved with only one sensor/actuator pair located at a free boundary. For illustration, the above principles are applied to an undamped flexible beam with arbitrary boundary conditions. Extensions to damped and gyroscopic systems are pointed out. The case of distributed sensors and ‘Distributed Identity’ and ‘Distributed Slope’ operators. It is verified analytically that the distributed action of these devices leads to an attenuated detection/excitation of higher-frequency vibrations. Although the structure was idealized by using a continuum modelling approach with a theoretically infinite number of modes, the above results can be applied to the identification and control of more complex structures modelled by other techniques, such as the finite element method.