Singular Time Delayed System Stability Theory in the sense of Lyapunov: A Quite New Approach

Debeljković, Dragutin Lj.; Stojanović, Sreten B.; Višnjić, Nemanja S.; Milinković, S.A.

doi:10.1109/acc.2007.4282161

Cited by 9 publications

(1 citation statement)

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“…However, effectively modeling states present in highly nonlinear dynamical systems while also accurately quantifying uncertainty is still a challenging task [1]. Research on nonlinear dynamic systems has a long history, while the parameters of the complex system may not be known precisely; thus, we can only obtain approximate results and require stability analysis (such as the Lyapunov stability or structural stability) [2]. In recent years, fully data-driven machine learning methods, in particular deep neural networks, appear to be a viable alterna-tive, which have a remarkable ability to learn complex patterns from training samples.…”

Section: Introductionmentioning

confidence: 99%

A Convex Relaxation Approach for Learning the Robust Koopman Operator

Jin

Peng

et al. 2022

Wireless Communications and Mobile Computing

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Although data-driven models, especially deep learning, have achieved astonishing results on many prediction tasks for nonlinear sequences, challenges still remain in finding an appropriate way to embed prior knowledge of physical dynamics in these models. In this work, we introduce a convex relaxation approach for learning robust Koopman operators of nonlinear dynamical systems, which are intended to construct approximate space spanned by eigenfunctions of the Koopman operator. Different from the classical dynamic mode decomposition, we use the layer weights of neural networks as eigenfunctions of the Koopman operator, providing intrinsic coordinates that globally linearize the dynamics. We find that the approximation of space can be regarded as an orthogonal Procrustes problem on the Stiefel manifold, which is highly sensitive to noise. The key contribution of this paper is to demonstrate that strict orthogonal constraint can be replaced by its convex relaxation, and the performance of the model can be improved without increasing the complexity when dealing with both clean and noisy data. After that, the overall model can be optimized via backpropagation in an end-to-end manner. The comparisons of the proposed method against several state-of-the-art competitors are shown on nonlinear oscillators and the lid-driven cavity flow.

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Section: Introductionmentioning

confidence: 99%

A Convex Relaxation Approach for Learning the Robust Koopman Operator

Jin

Peng

et al. 2022

Wireless Communications and Mobile Computing

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Separation principle of delay perturbed singular systems

Ben Mrad,

Ellouze

2023

IMA Journal of Mathematical Control and Information

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In this paper, we establish a separation principle for a class of time-varying delay perturbed singular systems. Furthermore, we propose a singular observer to estimate the system states. Based on the Lyapunov–Krasovskii functionals, the practical stability of the proposed singular observer is achieved. These results are applied to show that a separation principle for perturbed singular systems can be obtained. Eventually, we provide a numerical example to verify the validity of the proposed results.

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