A method, called the Eigensystem Realization Algorithm (ERA), is developed for modal parameter identification and model reduction of dynamic systems from test data. A new approach is introduced in conjunction with the singular value decomposition technique to derive the basic formulation of minimum order realization which is an extended version of the Ho-Kalman algorithm. The basic formulation is then transformed into modal space for modal parameter identification. Two accuracy indicators are developed to quantitatively identify the system modes and noise modes. For illustration of the algorithm, examples are shown using simulation data and experimental data for a rectangular grid structure.

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An eigensystem realization algorithm for modal parameter identification and model reduction

Staff members at Los Alamos National Laboratory (LANL) produced a summary of the structural health monitoring literature in 1995. This presentation will summarize the outcome of an updated review covering the years 1996 2001. The updated review follows the LANL statistical pattern recognition paradigm for SHM, which addresses four topics: 1. Operational Evaluation; 2. Data Acquisition and Cleansing; 3. Feature Extraction; and 4. Statistical Modeling for Feature Discrimination. The literature has been reviewed based on how a particular study addresses these four topics. A significant observation from this review is that although there are many more SHM studies being reported, the investigators, in general, have not yet fully embraced the well-developed tools from statistical pattern recognition. As such, the discrimination procedures employed are often lacking the appropriate rigor necessary for this technology to evolve beyond demonstration problems carried out in laboratory setting.

A review of structural health monitoring literature 1996-2001

In this paper a new frequency domain technique is introduced for the modal identification of output-only systems, i.e. in the case where the modal parameters must be estimated without knowing the input exciting the system. By its user friendliness the technique is closely related to the classical approach where the modal parameters are estimated by simple peak picking. However, by introducing a decomposition of the spectral density function matrix, the response spectra can be separated into a set of single degree of freedom systems, each corresponding to an individual mode. By using this decomposition technique close modes can be identified with high accuracy even in the case of strong noise contamination of the signals. Also, the technique clearly indicates harmonic components in the response signals.

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Modal identification of output-only systems using frequency domain decomposition

LEAST SQUARES APPROXIMATION A Curve Fitting Example Linear Batch Estimation Linear Least Squares Weighted Least Squares Constrained Least Squares Linear Sequential Estimation Nonlinear Least Squares Estimation Basis Functions Advanced Topics Matrix Decompositions in Least Squares Kronecker Factorization and Least Squares Levenberg-Marquardt Method Projections in Least Squares Summary PROBABILITY CONCEPTS IN LEAST SQUARES Minimum Variance Estimation Estimation without a Prior State Estimates Estimation with a Prior State Estimates Unbiased Estimates Maximum Likelihood Estimation Cramer-Rao Inequality Nonuniqueness of the Weight Matrix Bayesian Estimation Advanced Topics Analysis of Covariance Errors Ridge Estimation Total Least Squares Summary REVIEW OF DYNAMICAL SYSTEMS Linear System Theory The State Space Approach Homogeneous Linear Dynamical Systems Forced Linear Dynamical Systems Linear State Variable Transformations Nonlinear Dynamical Systems Parametric Differentiation Observability Discrete-Time Systems Stability of Linear and Nonlinear Systems Attitude Kinematics and Rigid Body Dynamics Attitude Kinematics Rigid Body Dynamics Spacecraft Dynamics and Orbital Mechanics Spacecraft Dynamics Orbital Mechanics Aircraft Flight Dynamics Vibration Summary PARAMETER ESTIMATION: APPLICATIONS Global Positioning System Navigation Attitude Determination Vector Measurement Models Maximum Likelihood Estimation Optimal Quaternion Solution Information Matrix Analysis Orbit Determination Aircraft Parameter Identification Eigen-system Realization Algorithm Summary SEQUENTIAL STATE ESTIMATION A Simple First-Order Filter Example Full-Order Estimators Discrete-Time Estimators The Discrete-Time Kalman Filter Kalman Filter Derivation Stability and Joseph's Form Information Filter and Sequential Processing Steady-State Kalman Filter Correlated Measurement and Process Noise Orthogonality Principle The Continuous-Time Kalman Filter Kalman Filter Derivation in Continuous Time Kalman Filter Derivation from Discrete Time Stability Steady-State Kalman Filter Correlated Measurement and Process Noise The Continuous-Discrete Kalman Filter Extended Kalman Filter Advanced Topics Factorization Methods Colored-Noise Kalman Filtering Consistency of the Kalman Filter Adaptive Filtering Error Analysis Unscented Filtering Robust Filtering Summary BATCH STATE ESTIMATION Fixed-Interval Smoothing Discrete-Time Formulation Continuous-Time Formulation Nonlinear Smoothing Fixed-Point Smoothing Discrete-Time Formulation Continuous-Time Formulation Fixed-Lag Smoothing Discrete-Time Formulation Continuous-Time Formulation Advanced Topics Estimation/Control Duality Innovations Process Summary ESTIMATION OF DYNAMIC SYSTEMS: APPLICATIONS GPS Position Estimation GPS Coordinate Transformations Extended Kalman Filter Application to GPS Attitude Estimation Multiplicative Quaternion Formulation Discrete-Time Attitude Estimation Murrell's Version Farrenkopf's Steady-State Analysis Orbit Estimation Target Tracking of Aircraft The a-b Filter The a-b-g Filter Aircraft Parameter Estimation Smoothing with the Eigen-system Realization Algorithm Summary OPTIMAL CONTROL AND ESTIMATION THEORY Calculus of Variations Optimization with Differential Equation Constraints Pontryagin's Optimal Control Necessary Conditions Discrete-Time Control Linear Regulator Problems Continuous-Time Formulation Discrete-Time Formulation Linear Quadratic-Gaussian Controllers Continuous-Time Formulation Discrete-Time Formulation Loop Transfer Recovery Spacecraft Control Design Summary APPENDIX A MATRIX PROPERTIES Basic Definitions of Matrices Vectors Matrix Norms and Definiteness Matrix Decompositions Matrix Calculus APPENDIX B BASIC PROBABILITY CONCEPTS Functions of a Single Discrete-Valued Random Variable Functions of Discrete-Valued Random Variables Functions of Continuous Random Variables Gaussian Random Variables Chi-Square Random Variables Propagation of Functions through Various Models Linear Matrix Models Nonlinear Models APPENDIX C PARAMETER OPTIMIZATION METHODS C.1 Unconstrained Extrema C.2 Equality Constrained Extrema C.3 Nonlinear Unconstrained Optimization C.3.1 Some Geometrical Insights C.3.2 Methods of Gradients C.3.3 Second-Order (Gauss-Newton) Algorithm APPENDIX D COMPUTER SOFTWARE Index

Optimal Estimation of Dynamic Systems

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems.
 However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken.
 We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator.
 This generalizes DMD to a larger class of datasets, including nonsequential time series.
 We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively.
 We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples.
 Such computations are not considered in the existing literature but can be understood using our more general framework.
 In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory.
 It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science.
 We show that under certain conditions, DMD is equivalent to LIM.

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On dynamic mode decomposition: Theory and applications

The basic concept of the Eigensystem Realization Algorithm for modal parameter identification and model reduction is extended to minimize the distortion of the identified parameters caused by noise. The mathematical foundation for the properties of accuracy indicators, such as the singular values of the data matrix and modal amplitude coherence, is provided, based on knowledge of the noise characteristics. These indicators quantitatively discriminate noise from system information and are used to reduce the realized system model to a better approximation of the true model. Monte Carlo Simulations are included to support the analytical studies.

Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algorithm

A new method is described for assessing the consistency of structural modal parameters identified with the Eigensystem Realization Algorithm. Identification results show varying consistency in practice due to many sources including high modal density, nonlinearity, and inadequate excitation. Consistency is considered to be a reliable indicator of accuracy. The new method is the culmination of many years of experience in developing a practical implementation of the Eigensystem Realization Algorithm. The effectiveness of the method is illustrated using data from NASA Langley's Controls-Structures-Interaction Evolutionary Model.

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A consistent-mode indicator for the eigensystem realization algorithm

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Consistent-mode indicator for the eigensystem realization algorithm

The technique of using hundreds or thousands of projected dots of light as targets for photogrammetry and videogrammetry of gossamer space structures is documented. Photogrammetry calculates the three-dimensional coordinates of each target on the structure, and videogrammetry tracks the coordinates vs time. Gossamer structures characteristically contain large areas of delicate, thin-film membranes. Examples include solar sails, large antennas, inflatable solar arrays, solar-power concentrators and transmitters, sun shields, and planetary balloons and habitats. Using projected-dot targets avoids the unwanted mass, stiffness, and installation costs of traditional retroreflective adhesive targets. Four laboratory applications are covered that demonstrate the practical effectiveness of white-light dot projection for both static-shape and dynamic measurement of reflective and diffuse surfaces, respectively. Comparisons are made between dot-projection videogrammetry and traditional laser vibrometry for membrane vibration measurements. A promising extension of existing techniques using a novel laser-induced fluorescence approach is introduced.

Richard S. Pappa

Papers

An eigensystem realization algorithm for modal parameter identification and model reduction

Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algorithm

A consistent-mode indicator for the eigensystem realization algorithm

Consistent-mode indicator for the eigensystem realization algorithm

Dot-Projection Photogrammetry and Videogrammetry of Gossamer Space Structures