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Applied system identification
01 Jan 1994-
TL;DR: In this paper, the authors introduce the concept of Frequency Domain System ID (FDSI) and Frequency Response Functions (FRF) for time-domain models, as well as Frequency-Domain Models with Random Variables and Kalman Filter.
Abstract: 1. Introduction. 2. Time-Domain Models. 3. Frequency-Domain Models. 4. Frequency Response Functions. 5. System Realization. 6. Observer Identification. 7. Frequency Domain System ID. 8. Observer/Controller ID. 9. Recursive Techniques. Appendix A: Fundamental Matrix Algebra. Appendix B: Random Variables and Kalman Filter. Appendix C: Data Acquisition.
Citations
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TL;DR: In this paper, a novel approach of stochastic subspace identification is presented that incorporates the idea of the reference sensors already in the identification step: the row space of future outputs is projected into the rowspace of past reference outputs.
1,236 citations
TL;DR: In this paper, the authors presented a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator, which requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a black box integrator.
Abstract: The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005) Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions
1,146 citations
TL;DR: In this article, a review of stochastic system identification methods that have been used to estimate the modal parameters of vibrating structures in operational conditions is presented. But it is not shown that many of these methods have an output-only counterpart.
Abstract: This paper reviews stochastic system identification methods that have been used to estimate the modal parameters of vibrating structures in operational conditions. It is found that many classical input-output methods havean output-only counterpart. For instance, the Complex Mode Indication Function (CMIF) can be applied both to Frequency Response Functions and output power and cross spectra. The Polyreference Time Domain (PTD) method applied to impulse responses is similar to the Instrumental Variable (IV) method applied to output covariances. The Eigensystem Realization Algorithm (ERA) is equivalent to stochastic subspace identification.
849 citations
TL;DR: This approach is an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes, and if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation.
Abstract: The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.
849 citations
Cites background from "Applied system identification"
...[50]), one often constructs deterministic governing equations from inherently stochastic data (either due to measurement or process noise)....
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TL;DR: This work develops a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models.
Abstract: We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations---only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. W...
665 citations
Cites methods from "Applied system identification"
...We discuss these particular methods in more detail in this section due to the historical connection between data-driven methods like ERA [24, 23] and modal decomposition methods such as Balanced Proper Orthogonal Decomposition (BPOD) [40], which produce balanced input-output models [35]....
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...ERA constructs an input-output model using impulse response data, and OKID is typically used in conjunction when convenient impulse response data is unavailable [25, 37, 36, 23]....
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...The observer/Kalman filter identification method allows minimal realization algorithms such as ERA to be generalized from impulse response data to data that is driven by rich input signals [25, 37, 36, 23]....
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