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Combining Trajectory Data with Analytical Lyapunov Functions for Improved Region of Attraction Estimation [article]

Lucas Lugnani, Morgan Jones, Luís F. C. Alberto, Mathew Peet, Daniel Dotta
2022 arXiv   pre-print
This paper proposes a novel methodology for solving the problem of estimating the Region of Attraction (ROA) of a nonlinear system by combining classical model based methods with modern data driven methods  ...  The method is carried out by using analytical Lyapunov functions, such as energy functions, in combination with data that is used to fit a converse Lyapunov function.  ...  CONCLUSION This work proposes a novel methodology for ROA estimation using an approximated converse Lyapunov function, derived from trajectory data, together with an analytical Lyapunov function.  ... 
arXiv:2111.09382v3 fatcat:wjpmbkmcb5fcpesw4ki4gybyti

Region-of-convergence estimation for learning-based adaptive controllers

John F. Quindlen, Ufuk Topcu, Girish Chowdhary, Jonathan P. How
2016 2016 American Control Conference (ACC)  
These level sets can be combined with the a priori known Lyapunov function for such systems to provide barrier certificates, verifying the safety of the system to maximum allowable error limits.  ...  These conditions, coupled with the known Lyapunov functions describing the adaptation, are used to form an optimization procedure to construct verifiable level sets for the system response.  ...  Paired with the fact that the known Lyapunov function for such systems is bounded, this information forms a maximum invariant level set on safe trajectories.  ... 
doi:10.1109/acc.2016.7525292 dblp:conf/amcc/QuindlenTCH16 fatcat:wfculqi4zrhrdp7dbcxe7jll44

Nonlinear Trajectory-Based Region of Attraction Estimation for Aircraft Dynamics Analysis [article]

Brian Lai, Torbjørn Cunis, Laurent Burlion
2021 arXiv   pre-print
First, this paper presents improvements to a method for estimating the region of attraction (ROA) of nonlinear systems governed by ordinary differential equations (ODEs) based only on trajectory measurements  ...  A piecewise polynomial model of the GTM is used to simulate trajectories and the developed analysis tools are used to estimate the ROA around a trim condition based only on this trajectory data.  ...  This set is denoted the region of attraction (ROA), and defined as := { ∈ R : lim →∞ ( , ) = 0}. (3) where lim →∞ ( , ) ∞ for divergent trajectories. To estimate , Lyapunov functions are used.  ... 
arXiv:2106.08850v1 fatcat:dpfn2lwfjjc5fc2djzrcjrbcre

Learning Region of Attraction for Nonlinear Systems [article]

Shaoru Chen, Mahyar Fazlyab, Manfred Morari, George J. Pappas, Victor M. Preciado
2021 arXiv   pre-print
Estimating the region of attraction (ROA) of general nonlinear autonomous systems remains a challenging problem and requires a case-by-case analysis.  ...  Specifically, our method searches for robust Lyapunov functions using counterexamples, i.e., the states at which the Lyapunov conditions fail.  ...  Of more practical use is estimating the region of attraction of the origin. Definition 1 (Region of attraction).  ... 
arXiv:2110.00731v1 fatcat:f5ssglplwvdrfk6a4gksxgdtfi

Learning Dynamical Systems using Local Stability Priors [article]

Arash Mehrjou, Andrea Iannelli, Bernhard Schölkopf
2020 arXiv   pre-print
a Lyapunov function of the system as a regularization term.  ...  A coupled computational approach to simultaneously learn a vector field and the region of attraction of an equilibrium point from generated trajectories of the system is proposed.  ...  This is the case, for example, for the region of attraction (ROA) of equilibria [4] and region of contraction of limit cycles [5] .  ... 
arXiv:2008.10053v1 fatcat:toxtm6wqobeenjxbxsofprojeq

Stabilizing Neural Control Using Self-Learned Almost Lyapunov Critics [article]

Ya-Chien Chang, Sicun Gao
2021 arXiv   pre-print
We use sample-based approaches and the Almost Lyapunov function conditions to estimate the region of attraction and invariance properties through the learned Lyapunov critic functions.  ...  We develop new methods for learning neural control policies and neural Lyapunov critic functions in the model-free reinforcement learning (RL) setting.  ...  We showed that the learned Lyapunov critics can be used to estimate regions of attraction for the controllers based on Almost Lyapunov conditions.  ... 
arXiv:2107.04989v1 fatcat:aq6rpkuf65bk3k4gdci2uzoara

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

Armin Lederer, Sandra Hirche
2019 2019 IEEE 58th Conference on Decision and Control (CDC)  
We develop a novel approach to characterize the region of attraction using a Lyapunov-like function, which is analyzed with a sampling-based interval analysis algorithm.  ...  Determining the region of attraction of nonlinear systems is a difficult problem, which is typically approached by means of Lyapunov theory.  ...  Although V ∞ is smaller than the true region of attraction V num due to regression errors, it is still a large improvement over the set V 4 obtained with the method in [9] . V.  ... 
doi:10.1109/cdc40024.2019.9029489 dblp:conf/cdc/LedererH19 fatcat:wgdnq73ex5hnrheihgsjos7tmi

Stability of Power Grids: State-of-the-art and Future Trends

Nikita Tomin, Daniil Panasetsky, Alexey Iskakov, C. Rehtanz, N. Voropai
2019 EPJ Web of Conferences  
Function; estimation of the ROA of IEEE 39-bus system using Gaussian process and Converse Lyapunov function.  ...  Transient stability concepts are illustrated with simple examples; in particular, we consider two machine learning-based methods for computing region of attraction: ROA produced by Neural Network Lyapunov  ...  Acknowledgment This work was supported by the Russian Scientific Foundation (No. 19-19-00673) under the project "Development of new methods for stability assessment and control in complex electric power  ... 
doi:10.1051/epjconf/201921701017 fatcat:dwtqr3lcabby3jecdvc2m3luay

Using neural networks to estimate regions of stability

E.D. Ferreira, B.H. Krogh
1997 Proceedings of the 1997 American Control Conference (Cat. No.97CH36041)  
This paper presents a new method to estimate the region of stability of an asymptotically stable equilibrium point of an autonomous nonlinear system using a neural network.  ...  The neural network results are compared with estimates obtained by previously proposed methods for some sample two dimensional problems and for an inverted pendulum.  ...  It is necessary to perform experiments to estimate the region of stability, but actual data from the system operation can be used as training data for the network to improve its estimation.  ... 
doi:10.1109/acc.1997.611036 fatcat:qb4xmeyoufhjrhvphlab5f3u4q

Active Sampling-based Binary Verification of Dynamical Systems [article]

John F. Quindlen, Ufuk Topcu, Girish Chowdhary, Jonathan P. How
2018 arXiv   pre-print
Various case studies demonstrate the closed-loop verification procedure and highlight improvements in prediction error over both existing analytical and statistical verification techniques.  ...  This work presents a data-driven statistical verification procedure that instead constructs statistical learning models from simulated training data to separate the set of possible perturbations into "  ...  The region of θ values for which the trajectory converges, Θ saf e , is also known as the region-of-attraction (ROA).  ... 
arXiv:1706.04268v2 fatcat:bpee2b7f4revnpi7t6bw3fbe5q

Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents [article]

Tatyana A. Alexeeva, and Nikolay V. Kuznetsov, Timur N. Mokaev
2021 arXiv   pre-print
We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically.  ...  We use an analytical approach for localization of a global attractor and study limiting dynamics of the model.  ...  /) and Financial Research Institute of the Ministry of Finance of the Russian Federation, a number of whose employees the authors thank for for helpful suggestions and comments.  ... 
arXiv:2107.13907v2 fatcat:5sx54egx3vetvehzoqbmwufery

Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents

Tatyana A. Alexeeva, Nikolay V. Kuznetsov, Timur N. Mokaev
2021 Chaos, Solitons & Fractals  
We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically.  ...  We use an analytical approach for localization of a global attractor and study limiting dynamics of the model.  ...  Acknowledgments This paper was prepared with the support by the Leading Scientific Schools of Russia: project NSh-2624.2020.  ... 
doi:10.1016/j.chaos.2021.111365 fatcat:uvqcgkqs3nhsdoolpcgmgzp27i

Data-Driven Robust Stabilization with Robust DOA Enlargement for Nonlinear Systems [article]

Chaolun Lu, Yongqiang Li, Zhongsheng Hou, Yuanjing Feng, Yu Feng, Ronghu Chi, Xuhui Bu
2019 arXiv   pre-print
For non-affine nonlinear system, Li et al. (2019) proposes a new nonlinear control method to solve the robust stabilization problem with estimation of the robust closed-loop DOA (Domain of attraction).  ...  However, Li et al. (2019) assumes that the Lyapunov function is given and does not consider the problem of finding a good Lyapunov function to enlarge the estimate of the robust closed-loop DOA.  ...  The analytical expression of the volume of the estimate of the closed-loop DOA is hard to be derived, but it is easy to evaluate its value for a given positive-definite function.  ... 
arXiv:1912.11480v1 fatcat:uc5aquuwnbehriqyhvv4n75ppq

A Review of Safe Online Learning for Nonlinear Control Systems

Matthew Osborne, Hyo-Sang Shin, Antonios Tsourdos
2021 2021 International Conference on Unmanned Aircraft Systems (ICUAS)  
This paper highlights some of the main approaches for safe online learning of stabilisable nonlinear control systems with a focus on safety certification for stability.  ...  Learning for autonomous dynamic control systems that can adapt to unforeseen environmental changes are of great interest but the realisation of a practical and safe online learning algorithm is incredibly  ...  The authors would also like to thank the following researchers for their kind assistance. Sumeet Singh, Ian Manchester and Johan Löfberg.  ... 
doi:10.1109/icuas51884.2021.9476765 fatcat:6uf4ad73ynccbgtbo7sqkn3qty

Bayesian Safe Learning and Control with Sum-of-Squares Analysis and Polynomial Kernels [article]

Alex Devonport, He Yin, Murat Arcak
2020 arXiv   pre-print
The method maintains safety by ensuring that the system state stays within the region of attraction (ROA) of a stabilizing control policy while collecting data.  ...  We propose an iterative method to safely learn the unmodeled dynamics of a nonlinear system using Bayesian Gaussian process (GP) models with polynomial kernel functions.  ...  ESTIMATING THE REGION OF ATTRACTION For safe learning with a GP model we must ensure that there is a region of state space which we are confident can be explored safely.  ... 
arXiv:2004.00662v1 fatcat:cfvcm6ymcvathop4zoqnqarulu
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