Uniqueness of Solutions for the Identification of Linear Reduced Order Structural Systems

The problem of identification of structural systems is an inverse problem that uses input (say force excitation) and output information (accelerations, for instance) to obtain an optimal model to describe the system’s behavior. Since a full instrumentation setup is expensive, situations usually arise where only partial measurements are available. Uniqueness of the solution in these circumstances might not be guaranteed. This paper analyzes the minimum number of measurements required to ensure that only one solution exists for the identification problem of mass, damping and stiffness distributions of shear-type N degrees of freedom linear structures. Three typical configurations of measurements are studied with two distinct theoretical approaches, one based on classical polynomial theory, the other based on reduced order model theory. Both these approaches lead to the conclusion that only one input and one or two output measurements are sufficient to guarantee uniqueness of identification, depending on the selected location of the input measurement. Additionally, the identification of a 3DOF system is carried out analytically with the usage of Sylvester’s Dyalitic Elimination to show that fewer measurements than the ones proposed lead to non-unique identification. This fact is also illustrated with the usage of a recently developed optimization technique, with which convergence to the different solutions is observed depending on the initial estimate used.