# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021213
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## Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component

 1 School of Information and Automation Engineering, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy 2 Dipartimento di Ingegneria dell'Informazione, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

* Corresponding author: Simone Fiori

Received  March 2021 Revised  June 2021 Early access August 2021

The present paper builds on the previous contribution by the second author, S. Fiori, Synchronization of first-order autonomous oscillators on Riemannian manifolds, Discrete and Continuous Dynamical Systems – Series B, Vol. 24, No. 4, pp. 1725 – 1741, April 2019. The aim of the present paper is to optimize a previously-developed control law to achieve synchronization of first-order non-linear oscillators whose state evolves on a Riemannian manifold. The optimization of such control law has been achieved by introducing a transverse control field, which guarantees reduced control effort without affecting the synchronization speed of the oscillators. The developed non-linear control theory has been analyzed from a theoretical point of view as well as through a comprehensive series of numerical experiments.

Citation: Adolfo Damiano Cafaro, Simone Fiori. Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021213
##### References:
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Gravdahl, Leader/follower synchronization of satellite attitude without angular velocity measurements, in Proceedings of the 44th IEEE Conference on Decision and Control, 2005, 7270–7277. [6] A. A. Castrejón-Pita and P. L. Read, Synchronization in a pair of thermally coupled rotating baroclinic annuli: Understanding atmospheric teleconnections in the laboratory, Physical Review Letters, 104 (2010), 204501. doi: 10.1103/PhysRevLett.104.204501. [7] I. Chueshov, P. E. Kloeden and M. Yang, Synchronization in coupled stochastic sine-Gordon wave model, Discrete & Continuous Dynamical Systems - B, 21 (2016), 2969-2990.  doi: 10.3934/dcdsb.2016082. [8] D. R. Creveling, P. E. Gill and H. D. I. Abarbanel, State and parameter estimation in nonlinear systems as an optimal tracking problem, Physics Letters A, 372 (2008), 2640-2644.  doi: 10.1016/j.physleta.2007.12.051. [9] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing, 40 (1993), 626-633.  doi: 10.1109/82.246163. [10] K. Ding and Q.-L. Han, Master-slave synchronization of nonautonomous chaotic systems and its application to rotating pendulums, International Journal of Bifurcation and Chaos, 22 (2012), 1250147. doi: 10.1142/S0218127412501477. [11] F. Dörfler, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010.  doi: 10.1073/pnas.1212134110. [12] M. D. Duong, C. Teraoka, T. Imamura, T. Miyoshi and K. Terashima, Master-slave system with teleoperation for rehabilitation, IFAC Proceedings Volumes, 38 (2005), 48-53.  doi: 10.3182/20050703-6-CZ-1902.01410. [13] R. Femat and G. Solís-Perales, On the chaos synchronization phenomena, Physics Letters A, 262 (1999), 50-60.  doi: 10.1016/S0375-9601(99)00667-2. [14] S. Fiori, Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 3077-3100.  doi: 10.1007/s11071-018-4546-x. [15] S. Fiori, Synchronization of first-order autonomous oscillators on Riemannian manifolds, Discrete & Continuous Dynamical Systems - B, 24 (2019), 1725-1741.  doi: 10.3934/dcdsb.2018233. [16] S. Fiori, I. Cervigni, M. Ippoliti and C. Menotta, Extension of a PID control theory to Lie groups applied to synchronising satellites and drones, IET Control Theory & Applications, 14 (2020), 2628-2642.  doi: 10.1049/iet-cta.2020.0226. [17] I. Fischer, Y. Liu and P. Davis, Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication, Physical Review A, 62 (2000), 011801. doi: 10.1103/PhysRevA.62.011801. [18] J. M. Gonzalez-Miranda, Synchronization and Control of Chaos: An Introduction for Scientists and Engineers, World Scientific Publishing Company, 2004. doi: 10.1142/p352. [19] S. Guo, S. Zhang, Z. Song and M. Pang, Development of a human upper limb-like robot for master-slave rehabilitation, in Proceedings of the 2013 ICME International Conference on Complex Medical Engineering, 2013,693–696. [20] Z. Guo, S. Gong, S. Yang and T. Huang, Global exponential synchronization of multiple coupled inertial memristive neural networks with time-varying delay via nonlinear coupling, Neural Networks, 108 (2018), 260-271.  doi: 10.1016/j.neunet.2018.08.020. [21] F. C. Hoppensteadt and E. M. Izhikevich, Synchronization of MEMS resonators and mechanical neurocomputing, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 48 (2001), 133-138.  doi: 10.1109/81.904877. [22] A.-S. Hu and S. D. Servetto, Asymptotically optimal time synchronization in dense sensor networks, in Proceedings of the 2nd ACM International Conference on Wireless Sensor Networks and Applications, WSNA'03, New York, NY, USA, 2003, Association for Computing Machinery, 1–10. [23] A. Khan and S. Kumar, Measure of chaos and adaptive synchronization of chaotic satellite systems, International Journal of Dynamics and Control, 7 (2019), 536-546.  doi: 10.1007/s40435-018-0481-4. [24] J.-S. Li, I. Dasanayake and J. Ruths, Control and synchronization of neuron ensembles, IEEE Transactions on Automatic Control, 58 (2013), 1919-1930.  doi: 10.1109/TAC.2013.2250112. [25] X. Li and R. Rakkiyappan, Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1515-1523.  doi: 10.1016/j.cnsns.2012.08.032. [26] G. M. Mahmoud and E. E. Mahmoud, Complete synchronization of chaotic complex nonlinear systems with uncertain parameters, Nonlinear Dynamics, 62 (2010), 875-882.  doi: 10.1007/s11071-010-9770-y. [27] J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer New York, 2012. doi: 10.1007/978-1-4614-1806-1_59. [28] M. Mitsuishi, A. Morita, N. Sugita, S. Sora, R. Mochizuki, K. Tanimoto, Y. M. Baek, H. Takahashi and K. Harada, Master-slave robotic platform and its feasibility study for micro-neurosurgery, The International Journal of Medical Robotics and Computer Assisted Surgery, 9 (2013), 180-189.  doi: 10.1002/rcs.1434. [29] T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S. Williams, E. Ott and R. Roy, Complex dynamics and synchronization of delayed-feedback nonlinear oscillators, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368 (2010), 343-366.  doi: 10.1098/rsta.2009.0225. [30] D. Sadaoui, A. Boukabou, N. Merabtine and M. Benslama, Predictive synchronization of chaotic satellites systems, Expert Systems with Applications, 38 (2011), 9041-9045.  doi: 10.1016/j.eswa.2011.01.117. [31] A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76.  doi: 10.1137/060673400. [32] S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano and W. L. Ditto, Controlling chaos in the brain, Nature, 370 (1994), 615-620.  doi: 10.1038/370615a0. [33] F. Sorrentino and E. Ott, Using synchronism of chaos for adaptive learning of time-evolving network topology, Physical Review E, 79 (2009), 016201. doi: 10.1103/PhysRevE.79.016201. [34] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276.  doi: 10.1038/35065725. [35] P. J. Uhlhaas and W. Singer, Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology, Neuron, 52 (2006), 155-168.  doi: 10.1016/j.neuron.2006.09.020. [36] A. Vaccaro, V. Loia, G. Formato, P. Wall and V. Terzija, A self-organizing architecture for decentralized smart microgrids synchronization, control, and monitoring, IEEE Transactions on Industrial Informatics, 11 (2015), 289-298.  doi: 10.1109/TII.2014.2342876. [37] N. Wanichnukhrox, T. Maneewarn and S. Songschon, Master-slave control for walking rehabilitation robot, in Proceedings of the 6th International Conference on Rehabilitation Engineering & Assistive Technology, i-CREATe'12, Midview City, SGP, 2012, Singapore Therapeutic, Assistive & Rehabilitative Technologies (START) Centre. [38] C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific Publishing Company, 2007. [39] X. Wu, C. Xu and J. Feng, Complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems and coupling time delays, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1004-1014.  doi: 10.1016/j.cnsns.2014.07.003. [40] J.-P. Yeh and K.-L. Wu, A simple method to synchronize chaotic systems and its application to secure communications, Mathematical and Computer Modelling, 47 (2008), 894-902.  doi: 10.1016/j.mcm.2007.06.021.

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##### References:
 [1] R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002), 47-97.  doi: 10.1103/RevModPhys.74.47. [2] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002. [3] Y. M. Baek, Y. Kozuka, N. Sugita, A. Morita, S. Sora, R. Mochizuki and M. Mitsuishi, Highly precise master-slave robot system for super micro surgery, in Proceedings of the 2010 IEEE International Conference on Biomedical Robotics and Biomechatronics, 2010,740–745. doi: 10.1109/BIOROB.2010.5625946. [4] S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares and C. S. Zhou, The synchronization of chaotic systems, Physics Reports, 366 (2002), 1-101.  doi: 10.1016/S0370-1573(02)00137-0. [5] A. K. Bondhus, K. Y. Pettersen and J. T. Gravdahl, Leader/follower synchronization of satellite attitude without angular velocity measurements, in Proceedings of the 44th IEEE Conference on Decision and Control, 2005, 7270–7277. [6] A. A. Castrejón-Pita and P. L. Read, Synchronization in a pair of thermally coupled rotating baroclinic annuli: Understanding atmospheric teleconnections in the laboratory, Physical Review Letters, 104 (2010), 204501. doi: 10.1103/PhysRevLett.104.204501. [7] I. Chueshov, P. E. Kloeden and M. Yang, Synchronization in coupled stochastic sine-Gordon wave model, Discrete & Continuous Dynamical Systems - B, 21 (2016), 2969-2990.  doi: 10.3934/dcdsb.2016082. [8] D. R. Creveling, P. E. Gill and H. D. I. Abarbanel, State and parameter estimation in nonlinear systems as an optimal tracking problem, Physics Letters A, 372 (2008), 2640-2644.  doi: 10.1016/j.physleta.2007.12.051. [9] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing, 40 (1993), 626-633.  doi: 10.1109/82.246163. [10] K. Ding and Q.-L. Han, Master-slave synchronization of nonautonomous chaotic systems and its application to rotating pendulums, International Journal of Bifurcation and Chaos, 22 (2012), 1250147. doi: 10.1142/S0218127412501477. [11] F. Dörfler, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010.  doi: 10.1073/pnas.1212134110. [12] M. D. Duong, C. Teraoka, T. Imamura, T. Miyoshi and K. Terashima, Master-slave system with teleoperation for rehabilitation, IFAC Proceedings Volumes, 38 (2005), 48-53.  doi: 10.3182/20050703-6-CZ-1902.01410. [13] R. Femat and G. Solís-Perales, On the chaos synchronization phenomena, Physics Letters A, 262 (1999), 50-60.  doi: 10.1016/S0375-9601(99)00667-2. [14] S. Fiori, Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 3077-3100.  doi: 10.1007/s11071-018-4546-x. [15] S. Fiori, Synchronization of first-order autonomous oscillators on Riemannian manifolds, Discrete & Continuous Dynamical Systems - B, 24 (2019), 1725-1741.  doi: 10.3934/dcdsb.2018233. [16] S. Fiori, I. Cervigni, M. Ippoliti and C. Menotta, Extension of a PID control theory to Lie groups applied to synchronising satellites and drones, IET Control Theory & Applications, 14 (2020), 2628-2642.  doi: 10.1049/iet-cta.2020.0226. [17] I. Fischer, Y. Liu and P. Davis, Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication, Physical Review A, 62 (2000), 011801. doi: 10.1103/PhysRevA.62.011801. [18] J. M. Gonzalez-Miranda, Synchronization and Control of Chaos: An Introduction for Scientists and Engineers, World Scientific Publishing Company, 2004. doi: 10.1142/p352. [19] S. Guo, S. Zhang, Z. Song and M. Pang, Development of a human upper limb-like robot for master-slave rehabilitation, in Proceedings of the 2013 ICME International Conference on Complex Medical Engineering, 2013,693–696. [20] Z. Guo, S. Gong, S. Yang and T. Huang, Global exponential synchronization of multiple coupled inertial memristive neural networks with time-varying delay via nonlinear coupling, Neural Networks, 108 (2018), 260-271.  doi: 10.1016/j.neunet.2018.08.020. [21] F. C. Hoppensteadt and E. M. Izhikevich, Synchronization of MEMS resonators and mechanical neurocomputing, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 48 (2001), 133-138.  doi: 10.1109/81.904877. [22] A.-S. Hu and S. D. Servetto, Asymptotically optimal time synchronization in dense sensor networks, in Proceedings of the 2nd ACM International Conference on Wireless Sensor Networks and Applications, WSNA'03, New York, NY, USA, 2003, Association for Computing Machinery, 1–10. [23] A. Khan and S. Kumar, Measure of chaos and adaptive synchronization of chaotic satellite systems, International Journal of Dynamics and Control, 7 (2019), 536-546.  doi: 10.1007/s40435-018-0481-4. [24] J.-S. Li, I. Dasanayake and J. Ruths, Control and synchronization of neuron ensembles, IEEE Transactions on Automatic Control, 58 (2013), 1919-1930.  doi: 10.1109/TAC.2013.2250112. [25] X. Li and R. Rakkiyappan, Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1515-1523.  doi: 10.1016/j.cnsns.2012.08.032. [26] G. M. Mahmoud and E. E. Mahmoud, Complete synchronization of chaotic complex nonlinear systems with uncertain parameters, Nonlinear Dynamics, 62 (2010), 875-882.  doi: 10.1007/s11071-010-9770-y. [27] J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer New York, 2012. doi: 10.1007/978-1-4614-1806-1_59. [28] M. Mitsuishi, A. Morita, N. Sugita, S. Sora, R. Mochizuki, K. Tanimoto, Y. M. Baek, H. Takahashi and K. Harada, Master-slave robotic platform and its feasibility study for micro-neurosurgery, The International Journal of Medical Robotics and Computer Assisted Surgery, 9 (2013), 180-189.  doi: 10.1002/rcs.1434. [29] T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S. Williams, E. Ott and R. Roy, Complex dynamics and synchronization of delayed-feedback nonlinear oscillators, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368 (2010), 343-366.  doi: 10.1098/rsta.2009.0225. [30] D. Sadaoui, A. Boukabou, N. Merabtine and M. Benslama, Predictive synchronization of chaotic satellites systems, Expert Systems with Applications, 38 (2011), 9041-9045.  doi: 10.1016/j.eswa.2011.01.117. [31] A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76.  doi: 10.1137/060673400. [32] S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano and W. L. Ditto, Controlling chaos in the brain, Nature, 370 (1994), 615-620.  doi: 10.1038/370615a0. [33] F. Sorrentino and E. Ott, Using synchronism of chaos for adaptive learning of time-evolving network topology, Physical Review E, 79 (2009), 016201. doi: 10.1103/PhysRevE.79.016201. [34] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276.  doi: 10.1038/35065725. [35] P. J. Uhlhaas and W. Singer, Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology, Neuron, 52 (2006), 155-168.  doi: 10.1016/j.neuron.2006.09.020. [36] A. Vaccaro, V. Loia, G. Formato, P. Wall and V. Terzija, A self-organizing architecture for decentralized smart microgrids synchronization, control, and monitoring, IEEE Transactions on Industrial Informatics, 11 (2015), 289-298.  doi: 10.1109/TII.2014.2342876. [37] N. Wanichnukhrox, T. Maneewarn and S. Songschon, Master-slave control for walking rehabilitation robot, in Proceedings of the 6th International Conference on Rehabilitation Engineering & Assistive Technology, i-CREATe'12, Midview City, SGP, 2012, Singapore Therapeutic, Assistive & Rehabilitative Technologies (START) Centre. [38] C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific Publishing Company, 2007. [39] X. Wu, C. Xu and J. Feng, Complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems and coupling time delays, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1004-1014.  doi: 10.1016/j.cnsns.2014.07.003. [40] J.-P. Yeh and K.-L. Wu, A simple method to synchronize chaotic systems and its application to secure communications, Mathematical and Computer Modelling, 47 (2008), 894-902.  doi: 10.1016/j.mcm.2007.06.021.
Simulation of the evolution of the master system (in green color), controlled systems $\Sigma_L$ (26) (in red color) and $\Sigma_G$ (27) (in blue color) on the sphere $\mathbb{S}^2$, displayed in terms of state components, together with the values taken by the Lyapunov function (7) during evolution (in blue color for $\Sigma_G$ and in red color for $\Sigma_L$)
Simulation of the evolution of the controlled systems $\Sigma_L$ (26) (in red color) and $\Sigma_G$ (27) (in blue color) on the sphere $\mathbb{S}^2$, displayed in terms of transverse control field components, together with the values taken by the control effort during evolution
Synchronization of a master/slave pair oscillators on the sphere $\mathbb{S}^7$ illustrated in terms of state components as well as kinetic energy (in green color for the master oscillator, red color for the system $\Sigma_L$ and blue color for the system $\Sigma_G$)
Synchronization of a master/slave pair oscillators on the sphere $\mathbb{S}^7$ – with and without the transverse field $\tau_G$ – illustrated in terms of control efforts and Lypunov function values (in red color for the system without transverse component and blue color for the system with transverse component). The left-bottom panes shows the course of the difference $\|u\|_{z_s}^2-\|u+\tau_G\|_{z_s}^2$ which takes non-negative values
Synchronization of a master/slave pair oscillators on the sphere $\mathbb{S}^7$ illustrated in terms of transverse field components as well as control efforts and Lypunov function values (in red color for the system $\Sigma_L$ and blue color for the system $\Sigma_G$)
Synchronization of two master/slave oscillators on $\mathbb{SO}(3)$ by the control field (4). In the top panel, the evolution of the squared Riemannian distance $d^2(z^s, z^m)$ is represented versus time. In the bottom panel, the evolution of the squared control effort related to the control law $u$ is represented over times
Synchronization of two master/slave oscillators on $\mathbb{SO}(3)$ by the control field $\tilde{u}$ with (48) as transverse control field. In the top panel, the evolution of the squared Riemannian distance $d^2(z^s, z^m)$ is represented versus time. In the bottom panel, the evolution of the squared control effort associated to the control field $\tilde{u}$ is represented
Synchronization of two master/slave oscillators on $\mathbb{SO}(3)$: Comparison of the squared control effort resulting from the application of the control laws $\tilde{u}(t)$ (in blue color) and $u(t)$ (in red color)
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