Learning Stable Koopman Embeddings

Fan, Fletcher; Rye, David; Shi, Guodong; Manchester, Ian R.

doi:10.48550/arxiv.2110.06509

Cited by 3 publications

(3 citation statements)

References 26 publications

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“…In Section 5.6.1 we show that in our case the eigenvector r T for the dominant eigenvalue λ 1 of the Koopman operator, restricted to the transition set (called A F above) also encodes the escape times from F . This implies that other approaches to time series analysis that approximate the dominant mode of the Koopman operator should also be able to determine in which parts of the reconstructed phase space the system are most susceptible to freezing (escape from stepping) [5].…”

Section: Preferred Transition Statesmentioning

confidence: 99%

Time series analysis and modelling of the freezing of gait phenomenon

Ai¹,

Sieber²,

Young³

et al. 2022

Preprint

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Freezing of Gait (FOG) is one of the most debilitating symptoms of Parkinson's Disease and is associated with falls and loss of independence. The patho-physiological mechanisms underpinning FOG are currently poorly understood. In this paper we combine time series analysis and mathematical modelling to study the FOG phenomenon's dynamics. We focus on the transition from stepping in place into freezing and treat this phenomenon in the context of an escape from an oscillatory attractor into an equilibrium attractor state. We extract a discrete-time discretespace Markov chain from experimental data and divide its state space into communicating classes to identify the transition into freezing. This allows us to develop a methodology for computationally estimating the time to freezing as well as the phase along the oscillatory (stepping) cycle of a patient experiencing Freezing Episodes (FE). The developed methodology is general and could be applied to any time series featuring transitions between different dynamic regimes including time series data from forward walking in people with FOG .

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Section: Preferred Transition Statesmentioning

confidence: 99%

Time series analysis and modelling of the freezing of gait phenomenon

Ai¹,

Sieber²,

Young³

et al. 2022

Preprint

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“…However, the learned Koopman operator is not guaranteed to be stable. [24] proposes a method to guarantee stability by learning a stable Koopman operator by utilizing a particular operator parameterization that ensures that the computed Koopman operator is Schur stable. Further, [25,26,27] propose data-driven approaches for identification of Koopman invariant subspaces whose applicability is, however, limited to uncontrolled nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Neural Koopman Lyapunov Control

Zinage¹,

Bakolas²

2022

Preprint

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Learning and synthesizing stabilizing controllers for unknown nonlinear systems is a challenging problem for real-world and industrial applications. Koopman operator theory allow one to analyze nonlinear systems through the lens of linear systems and nonlinear control systems through the lens of bilinear control systems. The key idea of these methods, lies in the transformation of the coordinates of the nonlinear system into the Koopman observables, which are coordinates that allow the representation of the original system (control system) as a higher dimensional linear (bilinear control) system. However, for nonlinear control systems, the bilinear control model obtained by applying Koopman operator based learning methods is not necessarily stabilizable and therefore, the existence of a stabilizing feedback control is not guaranteed which is crucial for many real world applications. Simultaneous identification of these stabilizable Koopman based bilinear control systems as well as the associated Koopman observables is still an open problem. In this paper, we propose a framework to identify and construct these stabilizable bilinear models and its associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown nonlinear control system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier. Our proposed approach thereby provides provable guarantees of global asymptotic stability for the nonlinear control systems with unknown dynamics. Numerical simulations are provided to validate the efficacy of our proposed class of stabilizing feedback controllers for unknown nonlinear systems.

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“…The works in [16], [17] use methods based on neural networks for this task, while [18] directly learns the Koopman eigenfunctions spanning invariant subspaces. Moreover, the works in [19], [20] approximate finitedimensional Koopman models relying on knowledge about the system's attractors and their stability. In our previous work, we have provided efficient algebraic algorithms to identify exact Koopman-invariant subspaces [21], [22] or approximate them with tunable predefined accuracy [23].…”

Section: Introductionmentioning

confidence: 99%

Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

Haseli¹,

Cortés²

2022

Preprint

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Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.

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Learning Stable Koopman Embeddings

Cited by 3 publications

References 26 publications

Time series analysis and modelling of the freezing of gait phenomenon

Time series analysis and modelling of the freezing of gait phenomenon

Neural Koopman Lyapunov Control

Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

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