# Linear Matrix Inequality Approaches to Koopman Operator Approximation

@article{Dahdah2021LinearMI, title={Linear Matrix Inequality Approaches to Koopman Operator Approximation}, author={Steven Dahdah and James Richard Forbes}, journal={ArXiv}, year={2021}, volume={abs/2102.03613} }

Koopman operator theory [1–4] provides a means to globally represent a nonlinear system as a linear system by transforming its states into an infinite-dimensional space of lifted states. The Koopman operator advances the current lifted state of the system to the next lifted state, much like the state transition matrix of a linear system. While originally proposed by B. O. Koopman in 1931 [1], modern computational resources, along with recent theoretical developments [2–4], have led to a…

## One Citation

System Norm Regularization Methods for Koopman Operator Approximation

- Computer ScienceArXiv
- 2021

DMD and DMD with control are reformulated as convex optimization problems with linear matrix inequality constraints and hard asymptotic stability constraints and system norm regularizers are considered as methods to improve the numerical conditioning of the approximate Koopman operator.

## References

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- Computer ScienceArXiv
- 2021

The results show that the Koopman operator, the associated Hilbert space of observables and a suitable dictionary can be obtained by solving two rank-constrained semi-definite programs (SDP).

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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.

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The authors exploit the Koopman operator to develop a systematic, data-driven approach for constructing a linear representation in terms of higher order derivatives of the underlying nonlinear dynamics, which enables fast control synthesis of nonlinear systems.

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The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice, and the paper highlights its strengths in applied and numerical contexts.

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A spectral expansion for general linear autonomous dynamical systems with analytic observables, and the notion of generalized eigenfunctions of the associated Koopman operator is defined, and isostables for a general class of nonlinear systems are defined.

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This thesis proposes a choice of the regularization parameter based on the statistical properties of fictitious validation data which can be used to avoid computationally costly techniques such as cross-validation, where the problem is solved multiple times to find a suitable parameter value.

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This work describes this Koopman-based system identification method and its application to model predictive controller design, which yields an explicit control-oriented linear model rather than just a "black-box" input-output mapping.

LMI Properties and Applications in Systems, Stability, and Control Theory

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The equivalency of some of the LMIs in this document may be straightforward to more experienced readers, but the authors believe that some readers may benefit from the presentation of multiple equivalent LMIs.