Using properties of multivariable flexible structures. it is shown that the Hankel singular values can be approximately decomposed to the sum of Hankel sigular values of individual sensor and actuator combination. The formula for the approximate Hankel singular value is simple and is found to be sufficiently accurate for actuator and sensor placement problems considered so far. This decomposition allows one to evaluate and characterize each actuator and sensor pair in terms of their joint controllability and observability. For multimodal systems, placement metrics such as the trace or determinant of Hankel singular value matrix can be formulated and sovled. It is shown that for the special case where the trace of the Hankel singular value matrix is used as the placement metric, the actuator placement problem becomes trivial. For illustration, two examples of actuator placement problem for fiexed sensors are given.
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