December  2018, 23(10): 4541-4555. doi: 10.3934/dcdsb.2018175

Identification of generic stable dynamical systems taking a nonlinear differential approach

1. 

School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, P.O. Box: 14155-6455, Tehran, Iran

2. 

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

* Corresponding author's e-mail address: salehani@ut.ac.ir (M. Khajeh Salehani)

Received  September 2017 Revised  December 2017 Published  May 2018

Fund Project: This work was supported in part by a grant from the Institute for Research in Fundamental Sciences (IPM) [No. 95510037]

Identifying new stable dynamical systems, such as generic stable mechanical or electrical control systems, requires questing for the desired systems parameters that introduce such systems. In this paper, a systematic approach to construct generic stable dynamical systems is proposed. In fact, our approach is based on a simple identification method in which we intervene directly with the dynamics of our system by considering a continuous $1$-parameter family of system parameters, being parametrized by a positive real variable $\ell$, and then identify the desired parameters that introduce a generic stable dynamical system by analyzing the solutions of a special system of nonlinear functional-differential equations associated with the $\ell$-varying parameters. We have also investigated the reliability and capability of our proposed approach.

To illustrate the utility of our result and as some applications of the nonlinear differential approach proposed in this paper, we conclude with considering a class of coupled spring-mass-dashpot systems, as well as the compartmental systems - the latter of which provide a mathematical model for many complex biological and physical processes having several distinct but interacting phases.

Citation: Mahdi Khajeh Salehani. Identification of generic stable dynamical systems taking a nonlinear differential approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4541-4555. doi: 10.3934/dcdsb.2018175
References:
[1]

K. Alexis, G. Nikolakopoulos and A. Tzes, Design and experimental verification of a constrained finite time optimal control scheme for the attitude control of a quadrotor helicopter subject to wind gusts, In: Proc. IEEE Int. Conf. Robot. Autom., Anchorage, Alaska, USA, (2010), 1636-1641.

[2]

H. Bilharz, Bemerkung zu einem Satze von Hurwitz, Zeitschrift f${\rm{\ddot{u}}}$r Angewandte Mathematik und Mechanik, 24 (1944), 77-82. doi: 10.1002/zamm.19440240205.

[3]

D. Cabecinhas, R. Naldi, L. Marconi, C. Silvestre and R. Cunha, Robust take-off and landing for a quadrotor vehicle, In: Proc. IEEE Int. Conf. Robot. Autom., Anchorage, Alaska, USA, (2010), 1630-1635. doi: 10.1109/ROBOT.2010.5509430.

[4]

F. Calogero, Nonlinear differential algorithm to compute all the zeros of a generic polynomial, J. Math. Physics, 57 (2016), 083508, 3pp. doi: 10.1063/1.4960821.

[5]

F. Calogero, Comment on "Nonlinear differential algorithm to compute all the zeros of a generic polynomial", J. Math. Physics, 57 (2016), 104101, 4pp. doi: 10.1063/1.4965441.

[6]

A. Cauchy, Calcul des indices des fonctions, Calcul des indices des fonctions, (2011), 416-466. doi: 10.1017/CBO9780511702501.013.

[7]

H. Cremer, Über den Zusammenhang zwischen den Routhschen und Hurwitzschen Stabilitätskriterien, Zeitschrift f${\rm{\ddot{u}}}$r Angewandte Mathematik und Mechanik, 27 (1947), 160-161. doi: 10.1002/zamm.19470250525.

[8]

H. Cremer and F. H. Effertz, Über die algebraischen Kriterien f${\rm{\ddot{u}}}$r die Stabilität von Regelsystemen, Mathematische Annalen, 137 (1959), 328-350. doi: 10.1007/BF01360969.

[9]

G. Frobenius, Ueber das Trägheitsgesetz der quadratischen Formen, J. f${\rm{\ddot{u}}}$ die reine und angewandte Mathematik, 114 (1895), 187-230. doi: 10.1515/crll.1895.114.187.

[10]

F. R. Gantmacher, Matrizentheorie, [Russion original, Moscow, 1968], Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-71243-2.

[11]

S. D. Hanford, L. N. Long and J. F. Horn, A small semiautonomous rotary-wing unmanned air vehicle (UAV), In: Proc. AIAA Infotech at Aerospace Conf., Washington DC, USA, 2005.

[12]

E. G. Hardy, On the zeros of a class of integral functions, Messenger of Mathematics, 34 (1904), 97-101.

[13]

B. HerisseT. HamelR. Mahony and F. X. Russotto, Landing a VTOL unmanned aerial vehicle on a moving platform using optical flow, IEEE Trans. Robot., 28 (2012), 77-89. doi: 10.1109/TRO.2011.2163435.

[14]

C. Hermite, Extrait d'une lettre de Mr. Ch. Hermite de Paris à Mr. Borchardt de Berlin sur le nombre des racines d'une équation algébrique comprises entre des limites données, J. f${\rm{\ddot{u}}}$r die reine und angewandte Mathematik, 52 (1856), 39-51. doi: 10.1515/crll.1856.52.39.

[15]

F. Hoffmann, N. Goddemeier and T. Bertram, Attitude estimation and control of a quadrocopter, In: Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Taipei, Taiwan, (2010), 1072-1077. doi: 10.1109/IROS.2010.5649111.

[16]

A. Hurwitz, Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Mathematische Annalen, 46 (1895), 273-284. doi: 10.1007/BF01446812.

[17]

J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc., 25 (1923), 325-332. doi: 10.1090/S0002-9947-1923-1501248-1.

[18]

K. G. J. Jacobi, Über eine elementare Transformation eines in Bezug auf jedes von zwei Variablen-Systemen linearen homogenen Ausdrucks, J. fü die reine und angewandte Mathematik, 53 (1857), 265-270 [see: Gesammelte Werke, pp. 583-590. Chelsea Publishing Co., New York (1969)]. doi: MR1579002.

[19]

S. H. Lehnigk, Liapunov's direct method and the number of roots with positive real parts of a polynomial with constant complex coefficients, SIAM J. on Control, 5 (1967), 234-244. doi: 10.1137/0305016.

[20]

A. Liénard and M. H. Chipart, Sur le signe de la partie réelle des racines d'une équation algébrique, J. de Mathématiques Pures et Appliquées, 10 (1914), 291-346.

[21]

D. MellingerM. ShominN. Michael and V. Kumar, Cooperative grasping and transport using multiple quadrotors, Distributed Auton. Syst., 83 (2013), 545-558. doi: 10.1007/978-3-642-32723-0_39.

[22]

N. MichaelJ. Fink and V. Kumar, Cooperative manipulation and transportation with aerial robots, Auton. Robot., 30 (2011), 73-86. doi: 10.15607/RSS.2009.V.001.

[23]

A. N. Michel, L. Hou and D. Liu, Stability of Dynamical Systems. On the Role of Monotonic and Non-monotonic Lyapunov Functions, 2$^{nd}$ edition, Systems & Control: Foundations & Applications. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-15275-2.

[24]

W. Michiels and S.-I. Niculescu, Stability, Control, and Computation for Time-delay Systems. An Eigenvalue-based Approach, 2$^{nd}$ edition, Advances in Design and Control, 27. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014. doi: 10.1137/1.9781611973631.

[25]

N. Michael and V. Kumar, Control of ensembles of aerial robots, Proc. IEEE, 99 (2011), 1587-1620. doi: 10.1109/JPROC.2011.2157275.

[26]

B. Michini, J. Redding, N. K. Ure, M. Cutler and J. P. How, Design and flight testing of an autonomous variable-pitch quadrotor, In: Proc. IEEE Int. Conf. Robot. Autom., Shanghai, China, (2011), 2978-2979. doi: 10.1109/ICRA.2011.5980561.

[27]

J. Moser, Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics, Princeton University Press, Princeton, NJ, 2001. doi: 10.1515/9781400882694.

[28]

L. Orlando, Sul problema di Hurwitz relative alle parti reali delle radici di un'equazione algebrica, Math. Ann., 71 (1911), 233-245. doi: 10.1007/BF01456650.

[29]

P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov, Proceedings of the Cambridge Philosophical Society, 58 (1962), 694-702. doi: 10.1017/S030500410004072X.

[30]

P. PoundsR. Mahony and P. Corke, Modelling and control of a large quadrotor robot, Control Eng. Practice, 18 (2010), 691-699. doi: 10.1016/j.conengprac.2010.02.008.

[31]

E. J. Routh, A Treatise on the Stability of a given State of Motion, Macmillan, London, 1877. [Reprinted in: A. T. Fuller, Stability of Motion, Taylor & Francis, London (1975), 19-138.]

[32]

C. Sturm, Autres démonstrations du même théorème, J. de Mathématiques Pures et Appliquées, 1 (1836), 290–308. [English translation in: Stability of Motion, Taylor & Francis, London (1975), 189–207.]

[33]

C. Sturm and A. Liouville, Démonstration d'un théorème de M. Cauchy, relatif aux racines imaginaires des équations, J. de Mathématiques Pures et Appliquées, 1 (1836), 278-289.

[34]

A. Tayebi and S. McGilvray, Attitude stabilization of a VTOL quadrotor aircraft, IEEE Trans. Control Syst. Technol., 14 (2006), 562-571. doi: 10.1109/TCST.2006.872519.

[35]

S. Trapp and M. Matthies, Chemodynamics and Environmental Modeling. An Introduction, Springer Berlin, Heidelberg, 1998.

[36]

N. I. Vitzilaios and N. C. Tsourveloudis, An experimental test bed for small unmanned helicopters, J. Intell. Robot. Syst., 54 (2000), 769-794. doi: 10.1007/s10846-008-9284-8.

[37]

X. Wu, Y. Liu and J. J. Zhu, Design and real time testing of a trajectory linearization flight controller for the quanserUFO, In: Proc. Amer. Control Conf., Athens, OH, USA, (2003), 3913-3918.

show all references

References:
[1]

K. Alexis, G. Nikolakopoulos and A. Tzes, Design and experimental verification of a constrained finite time optimal control scheme for the attitude control of a quadrotor helicopter subject to wind gusts, In: Proc. IEEE Int. Conf. Robot. Autom., Anchorage, Alaska, USA, (2010), 1636-1641.

[2]

H. Bilharz, Bemerkung zu einem Satze von Hurwitz, Zeitschrift f${\rm{\ddot{u}}}$r Angewandte Mathematik und Mechanik, 24 (1944), 77-82. doi: 10.1002/zamm.19440240205.

[3]

D. Cabecinhas, R. Naldi, L. Marconi, C. Silvestre and R. Cunha, Robust take-off and landing for a quadrotor vehicle, In: Proc. IEEE Int. Conf. Robot. Autom., Anchorage, Alaska, USA, (2010), 1630-1635. doi: 10.1109/ROBOT.2010.5509430.

[4]

F. Calogero, Nonlinear differential algorithm to compute all the zeros of a generic polynomial, J. Math. Physics, 57 (2016), 083508, 3pp. doi: 10.1063/1.4960821.

[5]

F. Calogero, Comment on "Nonlinear differential algorithm to compute all the zeros of a generic polynomial", J. Math. Physics, 57 (2016), 104101, 4pp. doi: 10.1063/1.4965441.

[6]

A. Cauchy, Calcul des indices des fonctions, Calcul des indices des fonctions, (2011), 416-466. doi: 10.1017/CBO9780511702501.013.

[7]

H. Cremer, Über den Zusammenhang zwischen den Routhschen und Hurwitzschen Stabilitätskriterien, Zeitschrift f${\rm{\ddot{u}}}$r Angewandte Mathematik und Mechanik, 27 (1947), 160-161. doi: 10.1002/zamm.19470250525.

[8]

H. Cremer and F. H. Effertz, Über die algebraischen Kriterien f${\rm{\ddot{u}}}$r die Stabilität von Regelsystemen, Mathematische Annalen, 137 (1959), 328-350. doi: 10.1007/BF01360969.

[9]

G. Frobenius, Ueber das Trägheitsgesetz der quadratischen Formen, J. f${\rm{\ddot{u}}}$ die reine und angewandte Mathematik, 114 (1895), 187-230. doi: 10.1515/crll.1895.114.187.

[10]

F. R. Gantmacher, Matrizentheorie, [Russion original, Moscow, 1968], Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-71243-2.

[11]

S. D. Hanford, L. N. Long and J. F. Horn, A small semiautonomous rotary-wing unmanned air vehicle (UAV), In: Proc. AIAA Infotech at Aerospace Conf., Washington DC, USA, 2005.

[12]

E. G. Hardy, On the zeros of a class of integral functions, Messenger of Mathematics, 34 (1904), 97-101.

[13]

B. HerisseT. HamelR. Mahony and F. X. Russotto, Landing a VTOL unmanned aerial vehicle on a moving platform using optical flow, IEEE Trans. Robot., 28 (2012), 77-89. doi: 10.1109/TRO.2011.2163435.

[14]

C. Hermite, Extrait d'une lettre de Mr. Ch. Hermite de Paris à Mr. Borchardt de Berlin sur le nombre des racines d'une équation algébrique comprises entre des limites données, J. f${\rm{\ddot{u}}}$r die reine und angewandte Mathematik, 52 (1856), 39-51. doi: 10.1515/crll.1856.52.39.

[15]

F. Hoffmann, N. Goddemeier and T. Bertram, Attitude estimation and control of a quadrocopter, In: Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Taipei, Taiwan, (2010), 1072-1077. doi: 10.1109/IROS.2010.5649111.

[16]

A. Hurwitz, Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Mathematische Annalen, 46 (1895), 273-284. doi: 10.1007/BF01446812.

[17]

J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc., 25 (1923), 325-332. doi: 10.1090/S0002-9947-1923-1501248-1.

[18]

K. G. J. Jacobi, Über eine elementare Transformation eines in Bezug auf jedes von zwei Variablen-Systemen linearen homogenen Ausdrucks, J. fü die reine und angewandte Mathematik, 53 (1857), 265-270 [see: Gesammelte Werke, pp. 583-590. Chelsea Publishing Co., New York (1969)]. doi: MR1579002.

[19]

S. H. Lehnigk, Liapunov's direct method and the number of roots with positive real parts of a polynomial with constant complex coefficients, SIAM J. on Control, 5 (1967), 234-244. doi: 10.1137/0305016.

[20]

A. Liénard and M. H. Chipart, Sur le signe de la partie réelle des racines d'une équation algébrique, J. de Mathématiques Pures et Appliquées, 10 (1914), 291-346.

[21]

D. MellingerM. ShominN. Michael and V. Kumar, Cooperative grasping and transport using multiple quadrotors, Distributed Auton. Syst., 83 (2013), 545-558. doi: 10.1007/978-3-642-32723-0_39.

[22]

N. MichaelJ. Fink and V. Kumar, Cooperative manipulation and transportation with aerial robots, Auton. Robot., 30 (2011), 73-86. doi: 10.15607/RSS.2009.V.001.

[23]

A. N. Michel, L. Hou and D. Liu, Stability of Dynamical Systems. On the Role of Monotonic and Non-monotonic Lyapunov Functions, 2$^{nd}$ edition, Systems & Control: Foundations & Applications. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-15275-2.

[24]

W. Michiels and S.-I. Niculescu, Stability, Control, and Computation for Time-delay Systems. An Eigenvalue-based Approach, 2$^{nd}$ edition, Advances in Design and Control, 27. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014. doi: 10.1137/1.9781611973631.

[25]

N. Michael and V. Kumar, Control of ensembles of aerial robots, Proc. IEEE, 99 (2011), 1587-1620. doi: 10.1109/JPROC.2011.2157275.

[26]

B. Michini, J. Redding, N. K. Ure, M. Cutler and J. P. How, Design and flight testing of an autonomous variable-pitch quadrotor, In: Proc. IEEE Int. Conf. Robot. Autom., Shanghai, China, (2011), 2978-2979. doi: 10.1109/ICRA.2011.5980561.

[27]

J. Moser, Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics, Princeton University Press, Princeton, NJ, 2001. doi: 10.1515/9781400882694.

[28]

L. Orlando, Sul problema di Hurwitz relative alle parti reali delle radici di un'equazione algebrica, Math. Ann., 71 (1911), 233-245. doi: 10.1007/BF01456650.

[29]

P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov, Proceedings of the Cambridge Philosophical Society, 58 (1962), 694-702. doi: 10.1017/S030500410004072X.

[30]

P. PoundsR. Mahony and P. Corke, Modelling and control of a large quadrotor robot, Control Eng. Practice, 18 (2010), 691-699. doi: 10.1016/j.conengprac.2010.02.008.

[31]

E. J. Routh, A Treatise on the Stability of a given State of Motion, Macmillan, London, 1877. [Reprinted in: A. T. Fuller, Stability of Motion, Taylor & Francis, London (1975), 19-138.]

[32]

C. Sturm, Autres démonstrations du même théorème, J. de Mathématiques Pures et Appliquées, 1 (1836), 290–308. [English translation in: Stability of Motion, Taylor & Francis, London (1975), 189–207.]

[33]

C. Sturm and A. Liouville, Démonstration d'un théorème de M. Cauchy, relatif aux racines imaginaires des équations, J. de Mathématiques Pures et Appliquées, 1 (1836), 278-289.

[34]

A. Tayebi and S. McGilvray, Attitude stabilization of a VTOL quadrotor aircraft, IEEE Trans. Control Syst. Technol., 14 (2006), 562-571. doi: 10.1109/TCST.2006.872519.

[35]

S. Trapp and M. Matthies, Chemodynamics and Environmental Modeling. An Introduction, Springer Berlin, Heidelberg, 1998.

[36]

N. I. Vitzilaios and N. C. Tsourveloudis, An experimental test bed for small unmanned helicopters, J. Intell. Robot. Syst., 54 (2000), 769-794. doi: 10.1007/s10846-008-9284-8.

[37]

X. Wu, Y. Liu and J. J. Zhu, Design and real time testing of a trajectory linearization flight controller for the quanserUFO, In: Proc. Amer. Control Conf., Athens, OH, USA, (2003), 3913-3918.

Figure 1.  Schematic of a coupled spring-mass-dashpot system
Figure 2.  Schematic of an open compartmental system
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