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Synchronization of dynamical systems on Riemannian manifolds by an extended PIDtype control theory: Numerical evaluation
1.  Dipartimento di Ingegneria dell'Informazione, Università Politecnica delle Marche, Via Brecce Bianche, Ancona, 60131, Italy 
2.  Graduate School of Information and Automation Engineering, Università Politecnica delle Marche, Via Brecce Bianche, Ancona, 60131, Italy 
The present document outlines a nonlinear control theory, based on the PID regulation scheme, to synchronize two secondorder dynamical systems insisting on a Riemannian manifold. The devised extended PID scheme, referred to as MPID, includes an unconventional component, termed 'canceling component', whose purpose is to cancel the natural dynamics of a system and to replace it with a desired dynamics. In addition, this document presents numerical recipes to implement such systems, as well as the devised control scheme, on a computing platform and a large number of numerical simulation results focused on the synchronization of Duffinglike nonlinear oscillators on the unit sphere. Detailed numerical evaluations show that the canceling contribution of the MPID control scheme is not critical to the synchronization of two oscillators, however, it possesses the beneficial effect of speeding up their synchronization. Simulation results obtained in nonideal conditions, namely in the presence of additive disturbances and delays, reveal that the devised synchronization scheme is robust against highfrequency additive disturbances as well as against observation delays.
References:
[1] 
D. P. Atherton, Almost six decades in control engineering, IEEE Control Systems Magazine, 34 (2014), 103110. 
[2] 
A. M. Bloch, An Introduction to Aspects of Geometric Control Theory, in Nonholonomic Mechanics and Control (eds. P. Krishnaprasad and R. Murray), vol. 24 of Interdisciplinary Applied Mathematics, Springer, New York, NY, 2015. 
[3] 
F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, vol. 49 of Texts in Applied Mathematics, Springer Verlag, New YorkHeidelbergBerlin, 2004. 
[4] 
J. C. Butcher, RungeKutta Methods, chapter 3, John Wiley & Sons, Ltd, 2016. doi: 10.1002/9781119121534.ch3. 
[5] 
G. Chen and X. Yu, Chaos Control – Theory and Applications, Lecture Notes in Control and Information Sciences, Springer, 2003. doi: 10.1007/b79666. 
[6] 
L. Cong, J. Mu, Q. Liu, H. Wang, L. Wang, Y. Li and C. Qiao, Thermal noise decoupling of microNewton thrust measured in a torsion balance, Symmetry, 13 (2021), 1357. doi: 10.3390/sym13081357. 
[7] 
D. N. Das, R. Sewani, J. Wang and M. K. Tiwari, Synchronized truck and drone routing in package delivery logistics, IEEE Transactions on Intelligent Transportation Systems, 1–11. 
[8] 
P. Deng, G. Amirjamshidi and M. Roorda, A vehicle routing problem with movement synchronization of drones, sidewalk robots, or footwalkers, Transportation Research Procedia, 46 (2020), 2936. doi: 10.1016/j.trpro.2020.03.160. 
[9] 
R. Dhelika, A. F. Hadi and P. A. Yusuf, Development of a motorized hospital bed with swerve drive modules for holonomic mobility, Applied Sciences, 11 (2021), 11356. doi: 10.3390/app112311356. 
[10] 
S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207–222, URL http://www.sciencedirect.com/science/article/pii/S1007570416304932. doi: 10.1016/j.cnsns.2016.11.025. 
[11] 
S. Fiori, Nondelayed synchronization of nonautonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 30773100. doi: 10.1007/s110710184546x. 
[12] 
S. Fiori, Extension of a PID control theory to Lie groups applied to synchronising satellites and drones, IET Control Theory & Applications, 14 (2020), 26282642. doi: 10.1049/ietcta.2020.0226. 
[13] 
S. Fiori, Manifold calculus in system theory and control–Fundamentals and firstorder systems, Symmetry, 13 (2021), 2092. doi: 10.3390/sym13112092. 
[14] 
R. Fuentes, G. P. Hicks and J. M. Osborne, The spring paradigm in tracking control of simple mechanical systems, Automatica, 47 (2011), 9931000. doi: 10.1016/j.automatica.2011.01.046. 
[15] 
S. Gajbhiye and R. N. Banavar, The EulerPoincaré equations for a spherical robot actuated by a pendulum, IFAC Proceedings Volumes, 45 (2012), 72–77, URL http://www.sciencedirect.com/science/article/pii/S1474667015337459, 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control. doi: 10.3182/201208293IT4022.00011. 
[16] 
V. Ghaffari and F. Shabaninia, Synchronization of nonlinear dynamical systems using extended Kalman filter and its application in some wellknown chaotic systems, Nonlinear Studies, 25 (2018), 273286. 
[17] 
O. Golevych, O. Pyvovar and P. Dumenko, Synchronization of nonlinear dynamic systems under the conditions of noise action in the channel, Latvian Journal of Physics and Technical Sciences, 55 (2018), 7076. doi: 10.2478/lpts20180023. 
[18] 
I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley & Sons, Ltd., Chichester, 2011. doi: 10.1002/9780470977859. 
[19] 
Y. Li, L. Li and C. Zhang, AMT starting control as a soft starter for belt conveyors using a datadriven method, Symmetry, 13 (2021), 1808. doi: 10.3390/sym13101808. 
[20] 
M. A. Magdy and T. S. Ng, Regulation and control effort in selftuning controllers, IEE Proceedings D – Control Theory and Applications, 133 (1986), 289292. doi: 10.1049/ipd.1986.0046. 
[21] 
J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Transactions on Automatic Control, 66 (2021), 43114318. doi: 10.1109/TAC.2020.3030849. 
[22] 
A. Návrat and P. Vašík, On geometric control models of a robotic snake, Note di Matematica, 37 (2017), 120129. doi: 10.1285/i15900932v37suppl1p119. 
[23]  M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. 
[24] 
K. Ojo, S. Ogunjo and A. Olagundoye, Projective synchronization via active control of identical chaotic oscillators with parametric and external excitation, International Journal of Nonlinear Science, 24 (2017), 7683. 
[25] 
J. M. Osborne and G. P. Hicks, The geodesic spring on the Euclidean sphere with paralleltransportbased damping, Notices of the AMS, 60 (2013), 544556. doi: 10.1090/noti997. 
[26] 
Y.s. Reddy and S.h. Hur, Comparison of optimal control designs for a 5 MW wind turbine, Applied Sciences, 11 (2021), 8774. doi: 10.3390/app11188774. 
[27] 
L. Righetti, Control and Synchronization with Nonlinear Dynamical Systems for an Application to Humanoid Robotics, Ecole Polytechnique Fédérale de Lausanne, 2004, URL https://nyuscholars.nyu.edu/en/publications/controlandsynchronizationwithnonlineardynamicalsystemsfor. 
[28] 
R. W. H. Sargent, Optimal control, Computational and Applied Mathematics, 124 (2000), 361371. doi: 10.1016/S03770427(00)004180. 
[29] 
M. Shiino and K. Okumura, Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena, Discrete and Continuous Dynamical Systems  Series S, 2013 (2013), 685694. doi: 10.3934/proc.2013.2013.685. 
[30] 
K. Sreenath, T. Lee and V. Kumar, Geometric control and differential flatness of a quadrotor UAV with a cablesuspended load, in 52nd IEEE Conference on Decision and Control, 2013, 2269–2274. 
[31] 
A. Varga, G. Eigner, I. Rudas and J. K. Tar, Experimental and simulationbased performance analysis of a computed torque control (CTC) method running on a double rotor aeromechanical testbed, Electronics, 10 (2021), 1745. doi: 10.3390/electronics10141745. 
[32] 
Y. Wang, Y. Lu and R. Xiao, Application of nonlinear adaptive control in temperature of Chinese solar greenhouses, Electronics, 10 (2021), 1582. doi: 10.1109/CCDC52312.2021.9601368. 
[33] 
C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific Publishing Co Pte Ltd, Singapore, 2007. doi: 10.1142/6570. 
[34] 
M. Zarei, A. Kalhor and M. Masouleh, An experimental oscillation damping impedance control for the Novint Falcon haptic device based on the phase trajectory length function concept, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233 (2019), 26632672. doi: 10.1177/0954406218799779. 
[35] 
Z. Zhang, J. Cheng and Y. Guo, PDbased optimal ADRC with improved linear extended state observer, Entropy, 23 (2021), Paper No. 888, 15 pp. doi: 10.3390/e23070888. 
[36] 
Z. Zhong, M. Xu, J. Xiao and H. Lu, Design and control of an omnidirectional mobile wallclimbing robot, Applied Sciences, 11 (2021), 11065. doi: 10.3390/app112211065. 
show all references
References:
[1] 
D. P. Atherton, Almost six decades in control engineering, IEEE Control Systems Magazine, 34 (2014), 103110. 
[2] 
A. M. Bloch, An Introduction to Aspects of Geometric Control Theory, in Nonholonomic Mechanics and Control (eds. P. Krishnaprasad and R. Murray), vol. 24 of Interdisciplinary Applied Mathematics, Springer, New York, NY, 2015. 
[3] 
F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, vol. 49 of Texts in Applied Mathematics, Springer Verlag, New YorkHeidelbergBerlin, 2004. 
[4] 
J. C. Butcher, RungeKutta Methods, chapter 3, John Wiley & Sons, Ltd, 2016. doi: 10.1002/9781119121534.ch3. 
[5] 
G. Chen and X. Yu, Chaos Control – Theory and Applications, Lecture Notes in Control and Information Sciences, Springer, 2003. doi: 10.1007/b79666. 
[6] 
L. Cong, J. Mu, Q. Liu, H. Wang, L. Wang, Y. Li and C. Qiao, Thermal noise decoupling of microNewton thrust measured in a torsion balance, Symmetry, 13 (2021), 1357. doi: 10.3390/sym13081357. 
[7] 
D. N. Das, R. Sewani, J. Wang and M. K. Tiwari, Synchronized truck and drone routing in package delivery logistics, IEEE Transactions on Intelligent Transportation Systems, 1–11. 
[8] 
P. Deng, G. Amirjamshidi and M. Roorda, A vehicle routing problem with movement synchronization of drones, sidewalk robots, or footwalkers, Transportation Research Procedia, 46 (2020), 2936. doi: 10.1016/j.trpro.2020.03.160. 
[9] 
R. Dhelika, A. F. Hadi and P. A. Yusuf, Development of a motorized hospital bed with swerve drive modules for holonomic mobility, Applied Sciences, 11 (2021), 11356. doi: 10.3390/app112311356. 
[10] 
S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207–222, URL http://www.sciencedirect.com/science/article/pii/S1007570416304932. doi: 10.1016/j.cnsns.2016.11.025. 
[11] 
S. Fiori, Nondelayed synchronization of nonautonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 30773100. doi: 10.1007/s110710184546x. 
[12] 
S. Fiori, Extension of a PID control theory to Lie groups applied to synchronising satellites and drones, IET Control Theory & Applications, 14 (2020), 26282642. doi: 10.1049/ietcta.2020.0226. 
[13] 
S. Fiori, Manifold calculus in system theory and control–Fundamentals and firstorder systems, Symmetry, 13 (2021), 2092. doi: 10.3390/sym13112092. 
[14] 
R. Fuentes, G. P. Hicks and J. M. Osborne, The spring paradigm in tracking control of simple mechanical systems, Automatica, 47 (2011), 9931000. doi: 10.1016/j.automatica.2011.01.046. 
[15] 
S. Gajbhiye and R. N. Banavar, The EulerPoincaré equations for a spherical robot actuated by a pendulum, IFAC Proceedings Volumes, 45 (2012), 72–77, URL http://www.sciencedirect.com/science/article/pii/S1474667015337459, 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control. doi: 10.3182/201208293IT4022.00011. 
[16] 
V. Ghaffari and F. Shabaninia, Synchronization of nonlinear dynamical systems using extended Kalman filter and its application in some wellknown chaotic systems, Nonlinear Studies, 25 (2018), 273286. 
[17] 
O. Golevych, O. Pyvovar and P. Dumenko, Synchronization of nonlinear dynamic systems under the conditions of noise action in the channel, Latvian Journal of Physics and Technical Sciences, 55 (2018), 7076. doi: 10.2478/lpts20180023. 
[18] 
I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley & Sons, Ltd., Chichester, 2011. doi: 10.1002/9780470977859. 
[19] 
Y. Li, L. Li and C. Zhang, AMT starting control as a soft starter for belt conveyors using a datadriven method, Symmetry, 13 (2021), 1808. doi: 10.3390/sym13101808. 
[20] 
M. A. Magdy and T. S. Ng, Regulation and control effort in selftuning controllers, IEE Proceedings D – Control Theory and Applications, 133 (1986), 289292. doi: 10.1049/ipd.1986.0046. 
[21] 
J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Transactions on Automatic Control, 66 (2021), 43114318. doi: 10.1109/TAC.2020.3030849. 
[22] 
A. Návrat and P. Vašík, On geometric control models of a robotic snake, Note di Matematica, 37 (2017), 120129. doi: 10.1285/i15900932v37suppl1p119. 
[23]  M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. 
[24] 
K. Ojo, S. Ogunjo and A. Olagundoye, Projective synchronization via active control of identical chaotic oscillators with parametric and external excitation, International Journal of Nonlinear Science, 24 (2017), 7683. 
[25] 
J. M. Osborne and G. P. Hicks, The geodesic spring on the Euclidean sphere with paralleltransportbased damping, Notices of the AMS, 60 (2013), 544556. doi: 10.1090/noti997. 
[26] 
Y.s. Reddy and S.h. Hur, Comparison of optimal control designs for a 5 MW wind turbine, Applied Sciences, 11 (2021), 8774. doi: 10.3390/app11188774. 
[27] 
L. Righetti, Control and Synchronization with Nonlinear Dynamical Systems for an Application to Humanoid Robotics, Ecole Polytechnique Fédérale de Lausanne, 2004, URL https://nyuscholars.nyu.edu/en/publications/controlandsynchronizationwithnonlineardynamicalsystemsfor. 
[28] 
R. W. H. Sargent, Optimal control, Computational and Applied Mathematics, 124 (2000), 361371. doi: 10.1016/S03770427(00)004180. 
[29] 
M. Shiino and K. Okumura, Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena, Discrete and Continuous Dynamical Systems  Series S, 2013 (2013), 685694. doi: 10.3934/proc.2013.2013.685. 
[30] 
K. Sreenath, T. Lee and V. Kumar, Geometric control and differential flatness of a quadrotor UAV with a cablesuspended load, in 52nd IEEE Conference on Decision and Control, 2013, 2269–2274. 
[31] 
A. Varga, G. Eigner, I. Rudas and J. K. Tar, Experimental and simulationbased performance analysis of a computed torque control (CTC) method running on a double rotor aeromechanical testbed, Electronics, 10 (2021), 1745. doi: 10.3390/electronics10141745. 
[32] 
Y. Wang, Y. Lu and R. Xiao, Application of nonlinear adaptive control in temperature of Chinese solar greenhouses, Electronics, 10 (2021), 1582. doi: 10.1109/CCDC52312.2021.9601368. 
[33] 
C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific Publishing Co Pte Ltd, Singapore, 2007. doi: 10.1142/6570. 
[34] 
M. Zarei, A. Kalhor and M. Masouleh, An experimental oscillation damping impedance control for the Novint Falcon haptic device based on the phase trajectory length function concept, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233 (2019), 26632672. doi: 10.1177/0954406218799779. 
[35] 
Z. Zhang, J. Cheng and Y. Guo, PDbased optimal ADRC with improved linear extended state observer, Entropy, 23 (2021), Paper No. 888, 15 pp. doi: 10.3390/e23070888. 
[36] 
Z. Zhong, M. Xu, J. Xiao and H. Lu, Design and control of an omnidirectional mobile wallclimbing robot, Applied Sciences, 11 (2021), 11065. doi: 10.3390/app112211065. 
Experiment type  Control method  Results  Figures 
Syncing of two identical oscillators and of two different oscillators.  MPID with 
When the systems are identical, switching off the canceling component does not hinder syncing, while different systems cannot sync without the aid of the canceling component.  2 
Syncing of two identical oscillators.  Proportional control action only.  No sync achieved.  3 
Syncing of two identical oscillators.  Proportional, integral and derivative control actions.  Sync achieved quickly and smoothly.  4 
Syncing of two identical oscillators.  Proportional and integral control actions.  Sync achieved slowly.  5 
Syncing of two identical oscillators.  Proportional and derivative control actions.  Short initial transient compared to the full MPID case.  6 
Syncing of two identical oscillators.  Full MPID with two different values of the proportional term coefficient.  Overshoot more apparent for a higher value of the proportional action coefficient.  7 
Syncing of two identical oscillators.  Full MPID with two different values of the derivative term coefficient.  Quicker convergence for higher value of the derivative action coefficient.  8 
Syncing of two identical oscillators.  Full MPID with two different values of the integral term coefficient.  Large values of this coefficient entail quicker convergence at the expense of larger oscillations around the set point.  9 
Syncing of two identical oscillators.  MPID controller including the canceling term vs. not including the canceling term. Unfavorable initial velocity.  Shorter initial transient vs. longer initial transient to convergence.  10 & 11 
Syncing of two identical oscillators.  MPID controller including the canceling term vs. not including the canceling term. Favorable initial velocity.  Longer initial transient vs. shorter initial transient to convergence.  12 & 13 
Syncing of two different oscillators of the same species.  Full MPID controller.  Synchronization achieved.  14 
Syncing of two different oscillators.  MPID controller with canceling term vs. no canceling term.  Syncronization achieved thanks to the canceling term. Absence of the canceling terms makes syncing almost to no avail.  15 & 16 
Experiment type  Control method  Results  Figures 
Syncing of two identical oscillators and of two different oscillators.  MPID with 
When the systems are identical, switching off the canceling component does not hinder syncing, while different systems cannot sync without the aid of the canceling component.  2 
Syncing of two identical oscillators.  Proportional control action only.  No sync achieved.  3 
Syncing of two identical oscillators.  Proportional, integral and derivative control actions.  Sync achieved quickly and smoothly.  4 
Syncing of two identical oscillators.  Proportional and integral control actions.  Sync achieved slowly.  5 
Syncing of two identical oscillators.  Proportional and derivative control actions.  Short initial transient compared to the full MPID case.  6 
Syncing of two identical oscillators.  Full MPID with two different values of the proportional term coefficient.  Overshoot more apparent for a higher value of the proportional action coefficient.  7 
Syncing of two identical oscillators.  Full MPID with two different values of the derivative term coefficient.  Quicker convergence for higher value of the derivative action coefficient.  8 
Syncing of two identical oscillators.  Full MPID with two different values of the integral term coefficient.  Large values of this coefficient entail quicker convergence at the expense of larger oscillations around the set point.  9 
Syncing of two identical oscillators.  MPID controller including the canceling term vs. not including the canceling term. Unfavorable initial velocity.  Shorter initial transient vs. longer initial transient to convergence.  10 & 11 
Syncing of two identical oscillators.  MPID controller including the canceling term vs. not including the canceling term. Favorable initial velocity.  Longer initial transient vs. shorter initial transient to convergence.  12 & 13 
Syncing of two different oscillators of the same species.  Full MPID controller.  Synchronization achieved.  14 
Syncing of two different oscillators.  MPID controller with canceling term vs. no canceling term.  Syncronization achieved thanks to the canceling term. Absence of the canceling terms makes syncing almost to no avail.  15 & 16 
Experiment type  Control method  Results  Figures 
Syncing of two different oscillators, either moderately or severely damped.  Full MPID controller.  Moderate damping makes syncing possible before collapsing of trajectories, while severe damping prevents synchronization.  17 & 18 
Syncing of two different oscillators, where state/velocity measurements are affected by random disturbances.  Full MPID controller.  Syncing is achieved, although a larger noise level makes the syncing process slower. Large control efforts required.  19, 20 & 21 
Syncing of two different oscillators, where state/velocity measurements are affected by a sinusoidal disturbance.  Full MPID controller.  Syncing is achieved in the presence of a very fastoscillating disturbance, where a slowoscillating disturbance disrupts the syncing process.  22 & 23 
Syncing of two different oscillators in the presence of a (known) observation delay.  Full MPID controller.  Synchronization is achieved to the delayed leader's state.  24 
Syncing of two different oscillators in the presence of a (known) observation delay and large state/velocity observation disturbance.  Full MPID controller.  Syncing is achieved to the delayed leader's state. Large control efforts required.  25 
Experiment type  Control method  Results  Figures 
Syncing of two different oscillators, either moderately or severely damped.  Full MPID controller.  Moderate damping makes syncing possible before collapsing of trajectories, while severe damping prevents synchronization.  17 & 18 
Syncing of two different oscillators, where state/velocity measurements are affected by random disturbances.  Full MPID controller.  Syncing is achieved, although a larger noise level makes the syncing process slower. Large control efforts required.  19, 20 & 21 
Syncing of two different oscillators, where state/velocity measurements are affected by a sinusoidal disturbance.  Full MPID controller.  Syncing is achieved in the presence of a very fastoscillating disturbance, where a slowoscillating disturbance disrupts the syncing process.  22 & 23 
Syncing of two different oscillators in the presence of a (known) observation delay.  Full MPID controller.  Synchronization is achieved to the delayed leader's state.  24 
Syncing of two different oscillators in the presence of a (known) observation delay and large state/velocity observation disturbance.  Full MPID controller.  Syncing is achieved to the delayed leader's state. Large control efforts required.  25 
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