Empirical Differential Balancing for Nonlinear Systems

Kawano, Yu; Scherpen, Jacquelien M. A.

doi:10.1016/j.ifacol.2017.08.920

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…where F T ∈ R n . Note that (30) represents a natural extension of the relation imposed in (17). There, fitting a linear structure is instead enforced.…”

Section: The General Proceduresmentioning

confidence: 99%

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Toward fitting structured nonlinear systems by means of dynamic mode decomposition

Gosea¹,

Duff²

2020

Preprint

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The dynamic mode decomposition (DMD) is a data-driven method used for identifying the dynamics of complex nonlinear systems. It extracts important characteristics of the underlying dynamics by means of measured time-domain data produced either by means of experiments or by numerical simulations. In the original methodology, the measurements are assumed to be approximately related by a linear operator. Hence, a linear discrete-time system is fitted to the given data. However, often, nonlinear systems modeling physical phenomena have a particular known structure. In this contribution, we propose an identification and reduction method based on the classical DMD approach allowing to fit a structured nonlinear system to the measured data. We mainly focus on two types of nonlinearities: bilinear and quadratic-bilinear. By enforcing this additional structure, more insight into extracting the nonlinear behavior of the original process is gained. Finally, we demonstrate the proposed methodology for different examples, such as the Burgers' equation and the coupled van der Pol oscillators.

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“…where F T ∈ R n . Note that (30) represents a natural extension of the relation imposed in (17). There, fitting a linear structure is instead enforced.…”

Section: The General Proceduresmentioning

confidence: 99%

“…Consider the coupled van der Pol oscillators along a limit cycle example given in [17]. The dynamics are characterized by the following six differential equations with linear and nonlinear Singular value decay of the data matrices The approximation error (cubic) terms:…”

Section: Coupled Van Der Pol Oscillatorsmentioning

confidence: 99%

Toward fitting structured nonlinear systems by means of dynamic mode decomposition

Gosea¹,

Duff²

2020

Preprint

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“…The empirical Gramian framework -emgr -implements empirical Gramian computation for system input-output coherence and parameter identifiability evaluation. Possible future extensions of emgr may include Koopman Gramians [98], empirical Riccati covariance matrices [99], or empirical differential balancing [100]. Finally, further examples and applications can be found at the emgr project website: http://gramian.de.…”

Section: Concluding Remarkmentioning

confidence: 99%

emgr—The Empirical Gramian Framework

Himpe

2018

Algorithms

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System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramians are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework -emgrimplements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction. Keywords

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“…There are only a few papers [7]- [9] working towards model reduction of nonlinear control systems based on simulations or experiments, and several papers attempt to use reducedorder models based on the Koopman operator for control problems [10]- [13]. However, [8]- [13] have not studied the properties of the achieved reduced-order models.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Model Reduction of Monotone Systems by Nonlinear DC Gains

Kawano

Besselink

Scherpen

et al. 2020

IEEE Trans. Automat. Contr.

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In this paper, we develop data-driven model reduction methods for monotone nonlinear control systems based on a nonlinear version of the DC gain. The nonlinear DC gain is a function of the amplitude of the input and can be used to evaluate the importance of each state variable. In fact, the nonlinear DC gain is directly related to the infinity-induced norm of the system as well as a notion of output reachability. Given the DC gain, model reduction is performed by either truncating not-so-important state variables or aggregating state variables having similar importances. Under such truncation and clustering, monotonicity and boundedness of the nonlinear DC gain are preserved; moreover, these two operations can be approximately performed based on simulation or experimental data alone. This empirical model reduction approach is illustrated by an example of a gene regulatory network.

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Empirical Differential Balancing for Nonlinear Systems

Cited by 9 publications

References 15 publications

Toward fitting structured nonlinear systems by means of dynamic mode decomposition

Toward fitting structured nonlinear systems by means of dynamic mode decomposition

emgr—The Empirical Gramian Framework

Data-Driven Model Reduction of Monotone Systems by Nonlinear DC Gains

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