A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles

Eva Ahbe; Andrea Iannelli; Roy S. Smith

doi:10.3934/jcd.2022016

Article Contents

2023, Volume 10, Issue 1: 75-104. Doi: 10.3934/jcd.2022016

This issue Previous Article A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems Next Article Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions

A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles

1.
Automatic Control Laboratory, ETH Zürich, Physikstrasse 3, 8092 Zürich, Switzerland

^* Corresponding author: iannelli@control.ee.ethz.ch
^* Corresponding author: iannelli@control.ee.ethz.ch

Received: October 2021

Revised: March 2022

Early access: August 2022

Published: January 2023

This work is supported by the Swiss National Science Foundation under grant no. 200021_178890

Abstract / Introduction Full Text(HTML) Figure(12) / Table(2) Related Papers Cited by

Abstract

The paper proposes a Lyapunov theory-based method to compute inner estimates of the region of attraction (ROA) of stable limit cycles. The approach is based on a transformation of the system to transverse coordinates, defined on a moving orthonormal coordinate system (MOC) for which a novel construction is presented. The proposed center point MOC (cp-MOC) is associated with a user-defined center point and provides flexibility to the construction of the transverse coordinates. In particular, compared to the standard approach based on hyperplanes orthogonal to the flow, the new construction allows the analyst to obtain larger regions of the state space where the well-definedness property of the transformation is satisfied. This has important benefits when using transverse coordinates to compute inner estimates of the ROA. To demonstrate these improvements, a sum-of-squares optimization-based formulation is proposed for computing inner estimates of the ROA of limit cycles for polynomial dynamics described in transverse coordinates. Different algorithmic options are explored, taking into account computational and accuracy aspects. Results are shown for three different systems exhibiting increasing complexity. The presented algorithms are extensively compared, and the newly cp-MOC is shown to markedly outperform existing approaches.

Keywords:

Mathematics Subject Classification: Primary: 93D05, 70K05; Secondary: 90C23.

Citation:

FullText(HTML)

Figure 1. Illustration of the orthogonal projection of a notional 3-dimensional periodic orbit with periodic solution $ \omega_0 $

Download: Full-size image PowerPoint slide

Figure 2. Left plot: Illustration of the hyperplanes $ \mathcal{S}( \tau) $ given by the class-MOC for the limit cycle of the Van der Pol oscillator. Right plot: Illustration of the hyperplanes $ \mathcal{S}( \tau) $ given by the cp-MOC with $ x{}_c = [0,0]^ T $ for the limit cycle of the Van der Pol oscillator

Download: Full-size image PowerPoint slide

Figure 3. Comparison of $ \mathcal{R} $ sizes obtained from Algorithm 1 for the algorithmic choice $ {\rm{EE-}}\partial(r)_m $ where both quadratic and quartic Lyapunov functions were used for class-MOC and cp-MOC

Download: Full-size image PowerPoint slide

Figure 4. Illustration of the computed ROA estimates shown in Figure 3 for the Van der Pol oscillator (59). The results for the quartic Lyapunov function are overlaid with those for the quadratic. Left plot: ROA estimates for the class-MOC. Right plot: ROA estimates for the cp-MOC

Download: Full-size image PowerPoint slide

Figure 5. Phase portrait of the dual-orbit system (60) in the neighborhood of the attractive and unstable LCs. The green lines show examples of converging trajectories while the red lines represent diverging ones

Download: Full-size image PowerPoint slide

Figure 6. ROA estimates as a function of $ \tau $, obtained from the different algorithmic choices presented in 4.2 for the class-MOC

Download: Full-size image PowerPoint slide

Figure 7. ROA estimates as a function of $ \tau $, obtained from the different algorithmic choices presented in 4.2 for the cp-MOC. For comparison, the $ {\rm{EE-}}\partial(4)_m $ result obtained for the class-MOC is included

Download: Full-size image PowerPoint slide

Figure 8. Illustration of selected ROA estimates shown in Figure 7 and Figure 6 for the dual-orbit system (60). The results for the quartic Lyapunov function are overlaid with those for the quadratic. Left plot: ROA estimates for the class-MOC. Right plot: ROA estimates for the cp-MOC

Download: Full-size image PowerPoint slide

Figure 9. Illustration of the hyperplanes in the cp-MOC from two different angles. In the right plot, the red dashed line indicates the $ 1 $-dimensional subspace given by the intersection of the hyperplanes. In the left plot, this collpases to a point

Download: Full-size image PowerPoint slide

Figure 10. Results showing the volume of $ \mathcal{R} $ on each of the $ 49 $ selected distinct hyperplanes $ \mathcal{S}( \tau_i), i = 1,...,49 $, obtained from the different algorithmic options in Section 4.2 and from both presented options of MOC in Section 2.3. A quadratic and a quartic (where applicable) Lyapunov function were used

Download: Full-size image PowerPoint slide

Figure 11. Results showing $ \mathcal{R} $ obtained from $ {\rm{EE-}}\partial(4)_s $, plotted on each corresponding hyperplane. The $ \phi $-$ \theta $ plane of the $ 3 $-dimensional plot is shown. Left plot: Results for the class-MOC. Right plot: Results for the cp-MOC

Download: Full-size image PowerPoint slide

Figure 12. Rotated view of the plots in Figure 11. Left plot: Results for the class-MOC. Right plot: Results for the cp-MOC

Download: Full-size image PowerPoint slide

Table 1. Comparison of iteration numbers obtained for the dual orbit example for each algorithmic option and MOC

Algorithmic	class-MOC		cp-MOC
option	$ \, \partial( V) =2\, $	$ \, \partial( V) =4\, $	$ \, \partial( V) =2\, $	$ \, \partial( V) =4\, $
$ {\rm{EE-}}\partial(r)_m $	23	51	20	49
$ {\rm{EE-}}\partial(r)_s $	10	9	8	8
$ {\rm{SE-}}\partial(r) $	39	32	39	33
$ {\rm{VSS-}}\partial(2) $	12	-	12	-
$ {\rm{VSS-}}\partial_{\rm{lin}} $	10	-	6	-

| Show Table

DownLoad: CSV

Table 2. Comparison of iteration numbers obtained for the deterministic kite example for each algorithmic option and MOC

Algorithmic	class-MOC		cp-MOC
option	$ \, \partial( V) =2\, $	$ \, \partial( V) =4\, $	$ \, \partial( V) =2\, $	$ \, \partial( V) =4\, $
$ {\rm{EE-}}\partial(r)_m $	19	44	11	10
$ {\rm{EE-}}\partial(r)_s $	10	18	15	19
$ {\rm{SE-}}\partial(r) $	54	27	18	15
$ {\rm{VSS-}}\partial(2) $	8	-	12	-
$ {\rm{VSS-}}\partial_{\rm{lin}} $	7	-	14	-

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	Supplemental material to the paper "A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles", ETH Research Collection, 2021. doi: 10.3929/ethz-b-000492932.
[2]	E. Ahbe, P. Listov, A. Iannelli and R. S. Smith, Feedback control design maximizing the region of attraction of stochastic systems using Polynomial Chaos Expansion, in IFAC World Congress, 2020.
[3]	E. Ahbe, T. A. Wood and R. S. Smith, Stability verification for periodic trajectories of autonomous kite power systems, in IEEE European Control Conference, 2018, 46-51. doi: 10.23919/ECC.2018.8550252.
[4]	E. Ahbe, T. A. Wood and R. S. Smith, Transverse contraction-based stability analysis for periodic trajectories of controlled power kites with model uncertainty, in IEEE Conference on Decision and Control, 2018, 6501-6506. doi: 10.1109/CDC.2018.8619106.
[5]	A. A. Ahmadi and A. Majumdar, DSOS and SDSOS optimization: More tractable alternatives to sum of squares and semidefinite optimization, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 193-230. doi: 10.1137/18M118935X.
[6]	U. Ahrens, M. Diehl and R. Schmehl, Airborne Wind Energy, Springer, Berlin, 2013. doi: 10.1007/978-3-642-39965-7.
[7]	S. Bittanti, P. Colaneri and G. D. Nicolao, The periodic Riccati equation, in The Riccati Equation (eds. S. Bittanti, A. Laub and J. Williams), Springer, Berlin, 1991,127-162. doi: 10.1007/978-3-642-58223-3_6.
[8]	S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.
[9]	A. Chakraborty, P. Seiler and G. J. Balas, Nonlinear region of attraction analysis for flight control verification and validation, Control Engineering Practice, 19 (2011), 335-345. doi: 10.1016/j.conengprac.2010.12.001.
[10]	G. Chesi, Estimating the domain of attraction for non-polynomial systems via LMI optimizations, Automatica, 45 (2009), 1536-1541. doi: 10.1016/j.automatica.2009.02.011.
[11]	G. Chesi, Domain of Attraction Analysis and Control via SOS Programming, London, U.K., Springer, 2011. doi: 10.1007/978-0-85729-959-8.
[12]	C. C. Chung and J. Hauser, Nonlinear control of a swinging pendulum, Automatica, 31 (1995), 851-862. doi: 10.1016/0005-1098(94)00148-C.
[13]	L. Fagiano, A. U. Zgraggen, M. Morari and M. Khammash, Automatic crosswind flight of tethered wings for airborne wind energy: Modeling, control design and experimental results, IEEE Transactions on Control Systems Technology, 22 (2014), 1433-1447. doi: 10.1109/TCST.2013.2279592.
[14]	L. B. Freidovich and A. S. Shiriaev, Transverse linearization for underactuated nonholonomic mechanical systems with application to orbital stabilization, in Distributed Decision Making and Control, Springer-Verlag London Limited, 2012, chapter 11,245-258. doi: 10.1007/978-1-4471-2265-4_11.
[15]	A. Gasull and H. Giacomini, A new criterion for controlling the number of limit cycles of some generalized liénard equations, Journal of Differential Equations, 185 (2002), 54-73. doi: 10.1006/jdeq.2002.4172.
[16]	P. Giesl, On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, Journal of Mathematical Analysis and Applications, 354 (2009), 606-618. doi: 10.1016/j.jmaa.2009.01.027.
[17]	J. K. Hale, Ordinary Differential Equations, R. E. Krierger Pub. Co., New York, 1980.
[18]	J. Hauser and C. C. Chung, Converse Lyapunov functions for exponentially stable periodic orbits, System & Control Letters, 23 (1994), 27-34. doi: 10.1016/0167-6911(94)90078-7.
[19]	A. Iannelli, A. Marcos and M. Lowenberg, Nonlinear robust approaches to study stability and postcritical behavior of an aeroelastic plant, IEEE Transactions on Control Systems Technology, 27 (2019), 703-716. doi: 10.1109/TCST.2017.2779110.
[20]	A. Iannelli, A. Marcos and M. Lowenberg, Robust estimations of the Region of Attraction using invariant sets, Journal of the Franklin Institute, 356 (2019), 4622-4647. doi: 10.1016/j.jfranklin.2019.02.013.
[21]	A. Iannelli, P. Seiler and A. Marcos, Region of attraction analysis with integral quadratic constraints, Automatica, 109 (2019), 108543. doi: 10.1016/j.automatica.2019.108543.
[22]	Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun and A. Packard, Controls applications of sum of squares programming, in Positive Polynomials in Control, Springer, Berlin, Heidelberg, 2005, 3-22. doi: 10.1007/10997703_1.
[23]	Z. W. Jarvis-Wloszek, Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems Using Sum-of-Squares Optimization, PhD thesis, University of California, Berkeley, 2003.
[24]	M. Jones and M. M. Peet, Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions, in Proceedings of the American Control Conference, American Automatic Control Council, 2019, 2084-2091. doi: 10.23919/ACC.2019.8814848.
[25]	G. A. Leonov, Generalization of the Andronov-Vitt theorem, Regular and Chaotic Dynamics, 11 (2006), 281-289. doi: 10.1070/RD2006v011n02ABEH000351.
[26]	N. Levinson and O. K. Smith, A general equation for relaxation oscillations, Duke mathmatical Journal, 9 (1942), 382-403. doi: 10.1215/S0012-7094-42-00928-1.
[27]	J. Loefberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, 2004. doi: 10.1109/CACSD.2004.1393890.
[28]	J. Löfberg, Pre- and post-processing sum-of-squares programs in practice, IEEE Transactions on Automatic Control, 54 (2009), 1007-1011. doi: 10.1109/TAC.2009.2017144.
[29]	A. Majumdar and R. Tedrake, Funnel libraries for real-time robust feedback motion planning, The International Journal of Robotics Research, 36 (2017), 947-982. doi: 10.1177/0278364917712421.
[30]	I. R. Manchester, Transverse dynamics and regions of stability for nonlinear hybrid limit cycles, in IFAC Proceedings Volumes, IFAC, 44 (2011), 6285-6290. doi: 10.3182/20110828-6-IT-1002.03063.
[31]	I. R. Manchester, M. M. Tobenkin, M. Levashov and R. Tedrake, Regions of attraction for hybrid limit cycles of walking robots, in IFAC Proceedings Volumes, IFAC, 44 (2011), 5801-5806. doi: 10.3182/20110828-6-IT-1002.03069.
[32]	J. Moore, R. Cory and R. Tedrake, Robust post-stall perching with a simple fixed-wing glider using LQR-trees, Bioinspiration & Biomimetics, 9 (2014), 1-15. doi: 10.1088/1748-3182/9/2/025013.
[33]	A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler, P. Parrilo, M. M. Peet and D. Jagt, SOSTOOLS: Sum of Squares Optimization Toolbox For MATLAB, http://arXiv.org/abs/1310.4716, 2021, Available from https://github.com/oxfordcontrol/SOSTOOLS.
[34]	A. Papachristodoulou and S. Prajna, A tutorial on sum of squares techniques for systems analysis, in Proceedings of the American Control Conference, 2005, 2686-2700. doi: 10.1109/ACC.2005.1470374.
[35]	A. Papachristodoulou and S. Prajna, Analysis of non-polynomial systems using the sum of squares decomposition, in Positive Polynomials in Control, 2005, 23-43. doi: 10.1007/10997703_2.
[36]	P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD thesis, California Institute of Technology, 2000.
[37]	S. Prajna, A. Papachristodoulou and F. Wu, Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach, in 5th Asian Control Conference, IEEE, 2004,157-165.
[38]	P. B. Reddy and I. A. Hiskens, Limit-induced stable limit cycles in power systems, IEEE Russia Power Tech, 1-5. doi: 10.1109/PTC.2005.4524706.
[39]	G. Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry, Math. Ann, 207 (1974), 87-97. doi: 10.1007/BF01362149.
[40]	W. Tan and A. Packard, Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming, IEEE Transactions on Automatic Control, 53 (2008), 565-571. doi: 10.1109/TAC.2007.914221.
[41]	J. Z. Tang, A. M. Boudali and I. R. Manchester, Invariant funnels for underactuated dynamic walking robots: New phase variable and experimental validation, in IEEE International Conference on Robotics and Automation, 2017, 3497-3504. doi: 10.1109/ICRA.2017.7989400.
[42]	R. Tedrake, I. R. Manchester, M. Tobenkin and J. W. Roberts, LQR-trees: Feedback motion planning via sums-of-squares verification, The International Journal of Robotics Research, 29 (2010), 1038-1052. doi: 10.1177/0278364910369189.
[43]	M. M. Tobenkin, I. R. Manchester and R. Tedrake, Invariant funnels around trajectories using sum-of-squares programming, in IFAC Proceedings Volumes, IFAC, 44 (2011), 9218-9223. doi: 10.3182/20110828-6-IT-1002.03098.
[44]	U. Topcu, A. Packard and P. Seiler, Local stability analysis using simulations and sum-of-squares programming, Automatica, 44 (2008), 2669-2675. doi: 10.1016/j.automatica.2008.03.010.
[45]	U. Topcu, A. Packard, P. Seiler and G. Balas, Help on sos [ask the experts], IEEE Control Systems Magazine, 30 (2010), 18-23. doi: 10.1109/MCS.2010.937045.
[46]	M. Urabe, Nonlinear Autonomous Oscillations: Analytical Theory, Academic Press, 1967.
[47]	G. Valmorbida and J. Anderson, Region of attraction estimation using invariant sets and rational Lyapunov functions, Automatica, 75 (2017), 37-45. doi: 10.1016/j.automatica.2016.09.003.
[48]	S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 53, Springer-Verlag New York, 2003.
[49]	T. A. Wood, H. Hesse, A. U. Zgraggen and R. S. Smith, Model-based flight path planning and tracking for tethered wings, in IEEE Conference on Decision and Control, 2015, 6712-6717. doi: 10.1109/CDC.2015.7403276.
[50]	M. Wu, Z. Yang and W. Lin, Domain-of-attraction estimation for uncertain non-polynomial systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3044-3052. doi: 10.1016/j.cnsns.2013.12.001.
[51]	Y. Zheng, G. Fantuzzi and A. Papachristodoulou, Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization, Annual Reviews in Control, 52 (2021), 243-279. doi: 10.1016/j.arcontrol.2021.09.001.