Second-order structural identification procedure via state-space-based system identification

Abstract

A theory is presented for transforming the system-theory-based realization models into the corresponding physical-coordinate-based structural models. The theory has been implemented into a computational procedure and applied to several example problems. The results show that the present transformation theory yields an objective model basis possessing a unique set of structural parameters from an infinite set of equivalent system realization models. For proportionally damped systems, the transformation directly and systematically yields the normal modes and modal damping. When nonproportional damping is present, the relative magnitude and phase of the damped mode shapes are separately characterized, and a corrective transformation is then employed to capture the undamped normal modes and nondiagonal modal damping matrix. namics equations has hampered the application (and acceptance) of the system-theory-based structural identification techniques and can eventually curtail the progress of hybrid experimental/analyti- cal modeling and design efforts. The present paper offers a theory for transforming the system- theory-based realization models into the corresponding physical- coordinate-based structural models. Since a key idea employed in the development of the present theory is an objective common basis normalization, it is designated as a common basis-normal- ized structural identification (CBSI) procedure. The resultant model is unique for a given sequence of Markov parameters, which in turn are uniquely determined for a linear structure with given inputs and outputs. Therefore, the realized model, after transformation, has a one-to-one correspondence with the physical parameters of the system and can either yield data for finite ele- ment model correlation or alternatively for direct calculation of mass, stiffness, and damping matrices of the original structure. Specifically, we begin with the so-called McMillan transformation employed by Longman and Juang9 and show that the McMillan transformation does not in general yield the desired structural nor- mal modes. To arrive at an objective transformation basis of gener- alized coordinates, we invoke two invariance properties. The first is the output invariance property, viz., the outputs are invariant with respect to any choice of generalized coordinates. The second is the normal mode identity, viz., for proportionally damped cases the mode shapes for the displacement, velocity, and acceleration vectors are the same. There are several byproducts that the present theory provides, primarily due to the common basis normalization employed in the theory. First, the present transformation theory allows the integra- tion of different realized models with varying actuator and sensor locations if they arise from the same structure. Second, each sensor and actuator or groups of sensors and actuators can be processed in parallel and combined concurrently or sequentially for the con- struction of a global model. Third, it can facilitate a decentralized real-time control implementation. From a structural dynamics point of view, the transformation to an objective basis provides a state- space model coinciding with the canonical form of the second- order equations of motion, thus extracting the classical real-valued parameters of interest to modal testing, mass-normalized normal modes, and the general modal damping matrix, while maintaining the system equivalence properties of the state-space form.