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doi: 10.3934/dcdsb.2022047
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## Synchronization of dynamical systems on Riemannian manifolds by an extended PID-type 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

*Corresponding author: Simone Fiori

Received  September 2021 Revised  December 2022 Early access March 2022

The present document outlines a non-linear control theory, based on the PID regulation scheme, to synchronize two second-order dynamical systems insisting on a Riemannian manifold. The devised extended PID scheme, referred to as M-PID, 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 Duffing-like non-linear oscillators on the unit sphere. Detailed numerical evaluations show that the canceling contribution of the M-PID 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 non-ideal conditions, namely in the presence of additive disturbances and delays, reveal that the devised synchronization scheme is robust against high-frequency additive disturbances as well as against observation delays.

Citation: Simone Fiori, Italo Cervigni, Mattia Ippoliti, Claudio Menotta. Synchronization of dynamical systems on Riemannian manifolds by an extended PID-type control theory: Numerical evaluation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022047
##### References:
 [1] D. P. Atherton, Almost six decades in control engineering, IEEE Control Systems Magazine, 34 (2014), 103-110. [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 York-Heidelberg-Berlin, 2004. [4] J. C. Butcher, Runge-Kutta 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 micro-Newton 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 foot-walkers, Transportation Research Procedia, 46 (2020), 29-36.  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, 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. [12] S. Fiori, 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. [13] S. Fiori, Manifold calculus in system theory and control–Fundamentals and first-order 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), 993-1000.  doi: 10.1016/j.automatica.2011.01.046. [15] S. Gajbhiye and R. N. Banavar, The Euler-Poincaré 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: 10.3182/20120829-3-IT-4022.00011. [16] V. Ghaffari and F. Shabaninia, Synchronization of nonlinear dynamical systems using extended Kalman filter and its application in some well-known chaotic systems, Nonlinear Studies, 25 (2018), 273-286. [17] O. Golevych, O. Pyvovar and P. Dumenko, Synchronization of non-linear dynamic systems under the conditions of noise action in the channel, Latvian Journal of Physics and Technical Sciences, 55 (2018), 70-76.  doi: 10.2478/lpts-2018-0023. [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 data-driven method, Symmetry, 13 (2021), 1808.  doi: 10.3390/sym13101808. [20] M. A. Magdy and T. S. Ng, Regulation and control effort in self-tuning controllers, IEE Proceedings D – Control Theory and Applications, 133 (1986), 289-292.  doi: 10.1049/ip-d.1986.0046. [21] J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Transactions on Automatic Control, 66 (2021), 4311-4318.  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), 120-129.  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), 76-83. [25] J. M. Osborne and G. P. Hicks, The geodesic spring on the Euclidean sphere with parallel-transport-based damping, Notices of the AMS, 60 (2013), 544-556.  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/control-and-synchronization-with-nonlinear-dynamical-systems-for-. [28] R. W. H. Sargent, Optimal control, Computational and Applied Mathematics, 124 (2000), 361-371.  doi: 10.1016/S0377-0427(00)00418-0. [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), 685-694.  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 cable-suspended 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 simulation-based 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), 2663-2672.  doi: 10.1177/0954406218799779. [35] Z. Zhang, J. Cheng and Y. Guo, PD-based 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 wall-climbing 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), 103-110. [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 York-Heidelberg-Berlin, 2004. [4] J. C. Butcher, Runge-Kutta 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 micro-Newton 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 foot-walkers, Transportation Research Procedia, 46 (2020), 29-36.  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, 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. [12] S. Fiori, 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. [13] S. Fiori, Manifold calculus in system theory and control–Fundamentals and first-order 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), 993-1000.  doi: 10.1016/j.automatica.2011.01.046. [15] S. Gajbhiye and R. N. Banavar, The Euler-Poincaré 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: 10.3182/20120829-3-IT-4022.00011. [16] V. Ghaffari and F. Shabaninia, Synchronization of nonlinear dynamical systems using extended Kalman filter and its application in some well-known chaotic systems, Nonlinear Studies, 25 (2018), 273-286. [17] O. Golevych, O. Pyvovar and P. Dumenko, Synchronization of non-linear dynamic systems under the conditions of noise action in the channel, Latvian Journal of Physics and Technical Sciences, 55 (2018), 70-76.  doi: 10.2478/lpts-2018-0023. [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 data-driven method, Symmetry, 13 (2021), 1808.  doi: 10.3390/sym13101808. [20] M. A. Magdy and T. S. Ng, Regulation and control effort in self-tuning controllers, IEE Proceedings D – Control Theory and Applications, 133 (1986), 289-292.  doi: 10.1049/ip-d.1986.0046. [21] J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Transactions on Automatic Control, 66 (2021), 4311-4318.  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), 120-129.  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), 76-83. [25] J. M. Osborne and G. P. Hicks, The geodesic spring on the Euclidean sphere with parallel-transport-based damping, Notices of the AMS, 60 (2013), 544-556.  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/control-and-synchronization-with-nonlinear-dynamical-systems-for-. [28] R. W. H. Sargent, Optimal control, Computational and Applied Mathematics, 124 (2000), 361-371.  doi: 10.1016/S0377-0427(00)00418-0. [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), 685-694.  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 cable-suspended 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 simulation-based 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), 2663-2672.  doi: 10.1177/0954406218799779. [35] Z. Zhang, J. Cheng and Y. Guo, PD-based 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 wall-climbing robot, Applied Sciences, 11 (2021), 11065.  doi: 10.3390/app112211065.
Feedback control scheme, where $y_\mathrm{sp}$ denotes the set point, $y_\mathrm{m}$ denotes a measure of the controlled variable $y$, $u$ denotes a control signal and $e$ denotes an error signal, which quantifies the discrepancy between the measured controlled variable and the set point
Synchronization of two Duffing-type oscillators by a M-PID controller. The left-hand panel shows the values of the distance $d(z, x)$ on the top and of the control effort $\sigma$ on the bottom, taken by a hard Duffing oscillator and a soft Duffing oscillator. Instead, the right-hand panel shows the values of the distance $d(z, x)$ on the top and of the control effort $\sigma$ on the bottom, taken by two hard Duffing oscillators. In both cases, the follower is controlled by an M-PID controller with $u_\mathrm{C}\neq 0$ during the first $10$ seconds of the simulation and with $u_\mathrm{C}$ set to zero from $t = 10$ outward. The proportional control coefficient was set to $\kappa_\mathrm{P} = 10$, the integral control coefficient was set to $\kappa_\mathrm{I} = 5$ and the derivative control coefficient was set to $\kappa_\mathrm{D} = 10$
Synchronization of two hard Duffing oscillators where the follower is controlled by a P-controller (a M-PID with only the proportional term). In both panels the black lines are about the follower dynamics while the red lines are referred to the leader dynamics. The left-hand panel shows the trajectories on the sphere. The follower's initial velocity is represented by a green arrow. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 0.01$
Synchronization of two hard Duffing oscillators where the follower is controlled by a full M-PID controller. Values and graphic elements are as in the Figure 3. The integral control coefficient value was set to $\kappa_\mathrm{I} = 5$, the derivative control coefficient value was set to $\kappa_\mathrm{D} = 10$ and the proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PI-controller, with only the proportional and integral terms. Values and graphic elements are as in the Figure 3. The integral coefficient value was set to $\kappa_\mathrm{I} = 2.5$ and the proportional term coefficient value was set to $\kappa_\mathrm{P} = 10$
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PD-controller, with only the proportional and derivative terms. Values and graphic elements are as in the Figure 3. The derivative control coefficient value was set to $\kappa_\mathrm{D} = 10$ and the proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PID controller: comparison of the effects of two different values of proportional term coefficient. In both panels, the black lines denote the follower's dynamics, while the red lines denote the leader's dynamic. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory, when the proportional coefficient value was set to $\kappa_\mathrm{P} = 5$. The right-hand panel shows the same quantities when the proportional coefficient value was set to $\kappa_\mathrm{P} = 150$. In both cases, the value of the integral coefficient was set to $\kappa_\mathrm{I} = 5$ and the value of the derivative coefficient was set to $\kappa_\mathrm{D} = 5$
Synchronization of two hard Duffing oscillators where the follower is controlled by M-PID controller: comparison with two different values of derivative term coefficient. Values and graphic elements are as in the Figure 7. The left-hand panel shows results when the derivative coefficient was set to $\kappa_\mathrm{D} = 0.5$, while the right-hand panel shows results when the derivative coefficient was set to $\kappa_\mathrm{D} = 3$. In both cases, the value of the integral coefficient was set to $\kappa_\mathrm{I} = 25$ and of the proportional coefficient was set to $\kappa_\mathrm{P} = 5$
Synchronization of two hard Duffing oscillators where the follower is controlled by M-PID controller: comparison with two different values of integral term coefficient. Values and graphic elements are as in the Figure 7. The left-hand panel shows results when the integral coefficient was set to $\kappa_\mathrm{I} = 3$, while the right-hand panel shows results when the value of the integral coefficient value was set to $\kappa_\mathrm{I} = 25$. In both cases, the derivative coefficient value was set to $\kappa_\mathrm{D} = 5$ and the proportional coefficient was set to $\kappa_\mathrm{P} = 5$
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PID controller that includes the $u_\mathrm{C}$ term and the initial velocity direction of the follower is not favorable for synchronization (i.e., it is opposed to the leader trajectory). In both panels the black lines indicate the follower's dynamics, while the red lines indicate the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The follower's initial velocity is represented by a green arrow and its initial state by a green open circle, while the leader's initial state is represented by a white open circle. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. The proportional control coefficient value is $\kappa_\mathrm{P} = 10$, the integral control coefficient value is $\kappa_\mathrm{I} = 5$ and the derivative control coefficient value is $\kappa_\mathrm{D} = 10$
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PID controller that does not include the $u_\mathrm{C}$ term and the initial velocity direction of the follower is not favorable for synchronization. Values and graphic elements are as in the Figure 10
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PID controller endowed with the $u_\mathrm{C}$ term when the initial velocity direction of the follower is favorable for synchronization. In both panels the black lines denote the follower's dynamics, while the red lines denote the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The follower's initial velocity is represented by a green arrow and its initial state by a green open circle, while the leader's initial state is represented by a white open circle. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. In particular, the right-bottom panel shows the total control effort $\sigma$ in blue, the PID contribution $\sigma_\mathrm{PID}$ in magenta and the contribution deriving from $u_\mathrm{C}$ term in green color. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$, the integral control coefficient value was set to $\kappa_\mathrm{I} = 5$ and the derivative control coefficient value was set to $\kappa_\mathrm{D} = 10$
Synchronization of two hard Duffing oscillators where the follower is controlled by a M-PID controller without the $u_\mathrm{C}$ term when the initial velocity direction of the follower is favorable for synchronization. Values and graphic elements are as in the Figure 12
Synchronization of two hard Duffing oscillators endowed with different reference points, where the follower is controlled by a M-PID controller. In both panels the black lines denote the follower's dynamics, while the red lines denote the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The follower's initial velocity is represented by a green arrow, while the leader's initial velocity is represented by a white arrow. The two different reference points have been represented as two colored open circles on the sphere (green for the leader reference point and white for the follower's one). The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. In this case, the leader's and follower's starting points, initial velocities and reference points are taken randomly. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$, the coefficient of the integral control term value was set to $\kappa_\mathrm{I} = 5$ and the derivative coefficient value was set to $\kappa_\mathrm{D} = 10$
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller endowed with the presence of the $u_\mathrm{C}$ term. In both panels the black lines represent the follower's dynamic, while the red ones represent the leader's dynamic. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The follower's initial velocity is represented by a green arrow. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. In particular, in the right-bottom panel, the total control effort $\sigma$ is represented in blue, the PID contribution $\sigma_\mathrm{PID}$ in magenta and the contribution deriving from $u_\mathrm{C}$ term in green color. The value of the proportional control coefficient was set to $\kappa_\mathrm{P} = 10$, the value of the integral control coefficient was set to $\kappa_\mathrm{I} = 5$ and the value of the derivative control coefficient was set to $\kappa_\mathrm{D} = 10$
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller without the $u_\mathrm{C}$ term. Values and graphic elements are as in the Figure 15
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller and both the follower and the leader are damped. In both panels, the black lines correspond to the follower's dynamics, while the red lines correspond to the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The follower's initial velocity is represented by a green arrow. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$, the integral control coefficient value was set to $\kappa_\mathrm{I} = 5$ and the derivative coefficient value was set to $\kappa_\mathrm{D} = 10$. In this case, the value of the damping coefficient was set to $\mu = 0.1$
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller and both the follower and the leader are damped. Values and graphic elements are as in the Figure 17. In this case the value of the damping coefficient was set to $\mu = 0.9$
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller, the state/velocity measurements are affected by random disturbances. In both panels the black lines denote the follower's dynamics, while the red lines denote the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on a sphere over time. The follower's initial velocity is represented by a green arrow. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$, the integral control coefficient value was set to $\kappa_\mathrm{I} = 5$ and the derivative coefficient value was set to $\kappa_\mathrm{D} = 10$. In this case, the value of the coefficient $b_z$ was set to $50$ and the value of the coefficient $b_w$ was set to $0.01$
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller and the state/velocity measurements are affected by random disturbances. Values and graphic elements are as in the Figure 19 except that, in this case, the value of the coefficient $b_z$ was set to $300$
Synchronization of an hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller and the state/velocity measurements are affected by random disturbances. Values and graphic elements are as in the Figure 19 except that, in this case, the value of the coefficient $b_z$ was set to $300$ and the value of the coefficient $b_w$ was set to $0.1$
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller and the state/velocity measurements are affected by sinusoidal disturbances. In both panels the black lines denote the follower's dynamics, while the red lines denote the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The follower's initial velocity is represented by a green arrow. The right-hand panel shows the values taken by the kinetic energy, the potential energy, the total energy, the control effort and the state-to-state distance over the generated trajectory. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$, the integral control coefficient value was set to $\kappa_\mathrm{I} = 5$ and the derivative coefficient value was set to $\kappa_\mathrm{D} = 10$. In this simulation, the noise coefficients were set to $b_z = 300$ and $b_w = 0.01$ and the sinusoidal disturbance's angular frequency was set to $\Omega = 1,000$ rad/s
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller and the state/velocity measurements are affected by sinusoidal disturbances. Values and graphic elements are as in the Figure 22 except that the sinusoidal disturbance's angular frequency was set to $\Omega = 5$ rad/s
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller. In both panels the black lines denote the follower's dynamics, while the red lines denote the leader's dynamics. The left-hand panel shows the trajectories of the two oscillators on the sphere over time. The right-hand panel shows the values taken by the kinetic energy, the potential energy and the state-to-state distances. The proportional control coefficient value was set to $\kappa_\mathrm{P} = 10$, the integral control coefficient value was set to $\kappa_\mathrm{I} = 5$ and the derivative coefficient value was set to $\kappa_\mathrm{D} = 10$. In this case, the time delay was set to $l = 0.4$ seconds and $h = 0.0002$ was the chosen value of the stepsize
Synchronization of a hard Duffing oscillator and a soft Duffing oscillator, where the follower is controlled by a M-PID controller in the presence of both time-delay and additive noise. Values and graphic elements are as in the Figure 24 and $b_z = 300$ and $b_w = 0.01$. As it can be noticed the total energy of the leader oscillator is constant over time, because it is supposed not to be subjected to external forces. In fact leader's total energy is preserved over time, while follower's total energy is not constant being subjected to the control action of the PID system
Numerical experiments under ideal conditions
 Experiment type Control method Results Figures Syncing of two identical oscillators and of two different oscillators. M-PID with $u_\mathrm{C}\neq 0$ only in the first half of time. 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 M-PID case. 6 Syncing of two identical oscillators. Full M-PID 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 M-PID 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 M-PID 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. M-PID 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. M-PID 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 M-PID controller. Synchronization achieved. 14 Syncing of two different oscillators. M-PID 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. M-PID with $u_\mathrm{C}\neq 0$ only in the first half of time. 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 M-PID case. 6 Syncing of two identical oscillators. Full M-PID 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 M-PID 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 M-PID 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. M-PID 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. M-PID 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 M-PID controller. Synchronization achieved. 14 Syncing of two different oscillators. M-PID 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
Numerical experiments under ideal conditions
 Experiment type Control method Results Figures Syncing of two different oscillators, either moderately or severely damped. Full M-PID 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 M-PID 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 M-PID controller. Syncing is achieved in the presence of a very fast-oscillating disturbance, where a slow-oscillating disturbance disrupts the syncing process. 22 & 23 Syncing of two different oscillators in the presence of a (known) observation delay. Full M-PID 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 M-PID 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 M-PID 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 M-PID 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 M-PID controller. Syncing is achieved in the presence of a very fast-oscillating disturbance, where a slow-oscillating disturbance disrupts the syncing process. 22 & 23 Syncing of two different oscillators in the presence of a (known) observation delay. Full M-PID 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 M-PID controller. Syncing is achieved to the delayed leader's state. Large control efforts required. 25
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