Constrained Eigensystem Realization Algorithm for Lightly Damped Distributed Structures

The temporal correlation method for modal identification of lightly damped distributed structures is extended to include system realization and is shown to be a constrained version of the eigensystem realization algorithm. The method relies on the temporal and spatial orthogonality of the modes of vibration, which are properties of lightly damped or nearly self-adjoint distributed systems. Using these properties, the system realization and modal identification can be performed in the configuration space, as opposed to the state space. In this manner, the computational requirements of the eigensystem realization algorithm are decreased, in addition to constrain- ing the algorithm to take advantage of the lightly damped behavior of the system so that the technique may be less sensitive to sensor noise in the system measurements and provide improved realization capabilities. Analytical and experimental examples illustrate and verify the method. The results are compared with the standard version of the eigensystem realization algorithm. HE eigensystem realization algorithm (ERA) has been gaining popularity for use as a tool for identifying modal parameters of structures. Developed and tested by J.-N. Juang and R. S. Pappa at NASA Langley Research Center, the ERA has been used successfully for minimum-order system realiza- tion and modal identification of structures. The ERA was first introduced as an extension of the Ho-Kalman algorithm for constructing a discrete-time state-space representation of a linear system from noisy sensor measurements.1 In the time domain, the ERA uses the discrete-time state equations x(k +1) = Ax(k) + Bu(k) and the output equation y = Cx(k) and consists of constructing the coefficient matrices A, B, and C using the impulse response. The method is particularly well suited for systems with closely spaced or repeated eigenvalues, as it allows more than a single set of output data generated by different inputs to be used in the technique. This makes it possible to identify eigenfunctions belonging to repeated ei- genvalues. The minimum-order system realization consists of determining the system order and then constructing the mini- mum-order discrete-time state-transition matrix A. The solu- tion of the eigenvalue problem associated withvl renders the identified modal parameters. Several recent works have been published by Juang and Pappa1-6 and Longman and Juang.7 In addition, Pappa has developed a very useful public domain ERA code for implementation on VAX/VMS systems. In many ongoing structures/controls experiments, the struc- ture possesses light levels of damping and demands active and/or passive control for vibration suppression. Usually, many sensors are used for system identification and state estimation of the structure to ensure adequate performance of the control system. The degree of noise contamination in the sensors is generally unknown and can make the system realiza- tion process especially difficult. Modal survey tests typically use a large number of sensors to adequately define mode shapes, and they can be greater in number than the number of modes to be identified. Using a large number of measurement stations enhances the noise filtering capabilities of the identifi- cation as the sensor measurements can use spatial filters (e.g.,