A Convex Data-Driven Approach for Nonlinear Control Synthesis

Choi, Hyungjin; Vaidya, Umesh; Chen, Yongxin

doi:10.3390/math9192445

A Convex Data-Driven Approach for Nonlinear Control Synthesis

Journal Article · Fri Oct 01 00:00:00 EDT 2021 · Mathematics

DOI:https://doi.org/10.3390/math9192445· OSTI ID:1825330

; Vaidya, Umesh; Chen, Yongxin

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

View Journal Article

Sponsoring Organization:: USDOE

Grant/Contract Number:: OE0000876

OSTI ID:: 1825330

Alternate ID(s):: OSTI ID: 1834090
OSTI ID: 1834124

Journal Information:: Mathematics, Journal Name: Mathematics Journal Issue: 19 Vol. 9; ISSN MBSAAR; ISSN 2227-7390

Publisher:: MDPI AGCopyright Statement

Country of Publication:: Switzerland

Language:: English

References (21)

A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition Williams, Matthew O.; Kevrekidis, Ioannis G.; Rowley, Clarence W. Journal of Nonlinear Science, Vol. 25, Issue 6 https://doi.org/10.1007/s00332-015-9258-5	journal	June 2015
Semidefinite programming relaxations for semialgebraic problems Parrilo, Pablo A. Mathematical Programming, Vol. 96, Issue 2 https://doi.org/10.1007/s10107-003-0387-5	journal	May 2003
A dual to Lyapunov's stability theorem Rantzer, Anders Systems & Control Letters, Vol. 42, Issue 3 https://doi.org/10.1016/S0167-6911(00)00087-6	journal	March 2001
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control Korda, Milan; Mezić, Igor Automatica, Vol. 93 https://doi.org/10.1016/j.automatica.2018.03.046	journal	July 2018
Stability in the almost everywhere sense: A linear transfer operator approach Rajaram, R.; Vaidya, U.; Fardad, M. Journal of Mathematical Analysis and Applications, Vol. 368, Issue 1 https://doi.org/10.1016/j.jmaa.2010.02.032	journal	August 2010
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control Klus, Stefan; Nüske, Feliks; Peitz, Sebastian Physica D: Nonlinear Phenomena, Vol. 406 https://doi.org/10.1016/j.physd.2020.132416	journal	May 2020
Data-Driven Optimal Control Using Transfer Operators Das, Apurba Kumar; Huang, Bowen; Vaidya, Umesh 2018 IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC.2018.8619057	conference	December 2018
Feedback Stabilization Using Koopman Operator Huang, Bowen; Ma, Xu; Vaidya, Umesh 2018 IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC.2018.8619727	conference	December 2018
Learning control for polynomial systems using sum of squares relaxations Guo, Meichen; De Persis, Claudio; Tesi, Pietro 2020 59th IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC42340.2020.9303924	conference	December 2020
A Semi-Algebraic Optimization Approach to Data-Driven Control of Continuous-Time Nonlinear Systems Dai, T.; Sznaier, M. IEEE Control Systems Letters, Vol. 5, Issue 2 https://doi.org/10.1109/LCSYS.2020.3003505	journal	April 2021
Data-driven Identification and Prediction of Power System Dynamics Using Linear Operators Sharma, Pranav; Huang, Bowen; Ajjarapu, Venkatramana 2019 IEEE Power & Energy Society General Meeting (PESGM) https://doi.org/10.1109/PESGM40551.2019.8973724	conference	August 2019
Nonlinear Control Synthesis by Convex Optimization Prajna, S.; Parrilo, P. A.; Rantzer, A. IEEE Transactions on Automatic Control, Vol. 49, Issue 2 https://doi.org/10.1109/TAC.2003.823000	journal	February 2004
Lyapunov Measure for Almost Everywhere Stability Vaidya, Umesh; Mehta, Prashant G. IEEE Transactions on Automatic Control, Vol. 53, Issue 1 https://doi.org/10.1109/TAC.2007.914955	journal	February 2008
Optimal Stabilization Using Lyapunov Measures Raghunathan, Arvind; Vaidya, Umesh IEEE Transactions on Automatic Control, Vol. 59, Issue 5 https://doi.org/10.1109/TAC.2013.2289707	journal	May 2014
Global Stability Analysis Using the Eigenfunctions of the Koopman Operator Mauroy, Alexandre; Mezic, Igor IEEE Transactions on Automatic Control, Vol. 61, Issue 11 https://doi.org/10.1109/TAC.2016.2518918	journal	November 2016
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator Mezić, Igor Annual Review of Fluid Mechanics, Vol. 45, Issue 1 https://doi.org/10.1146/annurev-fluid-011212-140652	journal	January 2013
Applied Koopman operator theory for power systems technology Susuki, Yoshihiko; Mezic, Igor; Raak, Fredrik Nonlinear Theory and Its Applications, IEICE, Vol. 7, Issue 4 https://doi.org/10.1587/nolta.7.430	journal	January 2016
Control synthesis for nonlinear optimal control via convex relaxations Zhao, Pengcheng; Mohan, Shankar; Vasudevan, Ram 2017 American Control Conference (ACC) https://doi.org/10.23919/ACC.2017.7963353	conference	May 2017
Optimal Quadratic Regulation of Nonlinear System Using Koopman Operator Ma, Xu; Huang, Bowen; Vaidya, Umesh 2019 American Control Conference (ACC) https://doi.org/10.23919/ACC.2019.8814903	conference	July 2019
Sample Complexity for Nonlinear Stochastic Dynamics Chen, Yongxin; Vaidya, Umesh 2019 American Control Conference (ACC) https://doi.org/10.23919/ACC.2019.8815138	conference	July 2019
Numerical Differentiation of Noisy, Nonsmooth Data Chartrand, Rick ISRN Applied Mathematics, Vol. 2011 https://doi.org/10.5402/2011/164564	journal	January 2011

Similar Records

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

Other · Fri Sep 04 00:00:00 EDT 2020 · OSTI ID:1693451

A Convex Data-Driven Approach for Nonlinear Control Synthesis

Citation Formats

References (21)

Similar Records

Related Subjects