Local Asymptotic Stability Analysis and Region of Attraction Estimation with Gaussian Processes

Lederer, Armin; Hirche, Sandra

doi:10.1109/cdc40024.2019.9029489

Cited by 15 publications

(9 citation statements)

References 15 publications

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“…Since we do not have any target values for the supervised learning, we cannot directly apply the GP regression approach. Therefore, we approximate the infinite horizon cost Ṽ∞ (x) = ∞ k=1 l(f k (x)), where f k (•) denotes the k-times application of the dynamics f (•) and l : R d → R + is a chosen stage cost, by transforming the Bellman equation at training points into a regression problem as proposed in (Lederer and Hirche, 2019). This is formalized in the following lemma.…”

Section: Learning Nonparametric Control Lyapunov Functionsmentioning

confidence: 99%

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Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Xiao

Lederer

Hirche

2020

Preprint

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Modelling real world systems involving humans such as biological processes for disease treatment or human behavior for robotic rehabilitation is a challenging problem because labeled training data is sparse and expensive, while high prediction accuracy is required from models of these dynamical systems. Due to the high nonlinearity of problems in this area, datadriven approaches gain increasing attention for identifying nonparametric models. In order to increase the prediction performance of these models, abstract prior knowledge such as stability should be included in the learning approach. One of the key challenges is to ensure sufficient flexibility of the models, which is typically limited by the usage of parametric Lyapunov functions to guarantee stability. Therefore, we derive an approach to learn a nonparametric Lyapunov function based on Gaussian process regression from data. Furthermore, we learn a nonparametric Gaussian process state space model from the data and show that it is capable of reproducing observed data exactly. We prove that stabilization of the nominal model based on the nonparametric control Lyapunov function does not modify the behavior of the nominal model at training samples. The flexibility and efficiency of our approach is demonstrated on the benchmark problem of learning handwriting motions from a real world dataset, where our approach achieves almost exact reproduction of the training data.

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Section: Learning Nonparametric Control Lyapunov Functionsmentioning

confidence: 99%

“…Due to (Lederer and Hirche, 2019), a function satisfying this equation on a finite set of pairs (x, f (x)) can be obtained through noiseless GP regression with the kernel k…”

Section: Learning Nonparametric Control Lyapunov Functionsmentioning

confidence: 99%

Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Xiao

Lederer

Hirche

2020

Preprint

Self Cite

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“…The Lyapunov Neural Network [21] can incrementally adapt the RoA's shape given an initial safe set. As an alternative, GPs can obtain a Lyapunov-like function [22], or an LF can be synthesized to provide guarantee's on a controller's stability while training [23].…”

Section: Introductionmentioning

confidence: 99%

Data-Efficient Characterization of the Global Dynamics of Robot Controllers with Confidence Guarantees

Vieira¹,

Sivaramakrishnan²,

Ye³

et al. 2022

Preprint

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This paper proposes an integration of surrogate modeling and topology to significantly reduce the amount of data required to describe the underlying global dynamics of robot controllers, including closed-box ones. A Gaussian Process (GP), trained with randomized short trajectories over the statespace, acts as a surrogate model for the underlying dynamical system. Then, a combinatorial representation is built and used to describe the dynamics in the form of a directed acyclic graph, known as Morse graph. The Morse graph is able to describe the system's attractors and their corresponding regions of attraction (RoA). Furthermore, a pointwise confidence level of the global dynamics estimation over the entire state space is provided. In contrast to alternatives, the framework does not require estimation of Lyapunov functions, alleviating the need for high prediction accuracy of the GP. The framework is suitable for data-driven controllers that do not expose an analytical model as long as Lipschitz-continuity is satisfied. The method is compared against established analytical and recent machine learning alternatives for estimating RoAs, outperforming them in data efficiency without sacrificing accuracy. Link to code: https://go.rutgers.edu/49hy35en

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“…The complexity of systems we encounter prompts a shift from classical parametric techniques in favor of more flexible machine learning techniques (e.g. neural networks or Gaussian processes) for prediction [1,2], model-based control [3,4] and analysis [5,6,7]. Traditionally, representations of systems are in the immediate state-space concerned with "dynamics of states".…”

Section: Introductionmentioning

confidence: 99%

Koopman Operator Dynamical Models: Learning, Analysis and Control

Bevanda,

Sosnowski,

Hirche

2021

Preprint

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The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional -necessitating finite approximations -for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigm's relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.

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Local Asymptotic Stability Analysis and Region of Attraction Estimation with Gaussian Processes

Cited by 15 publications

References 15 publications

Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Data-Efficient Characterization of the Global Dynamics of Robot Controllers with Confidence Guarantees

Koopman Operator Dynamical Models: Learning, Analysis and Control

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