2014, 4(2): 93-101. doi: 10.3934/naco.2014.4.93

A sufficient condition of Euclidean rings given by polynomial optimization over a box

1. 

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China

Received  June 2013 Revised  December 2013 Published  May 2014

A sufficient condition of Euclidean rings is given by polynomial optimization. Then, through computation, we give all norm-Euclidean square number fields, four examples of norm-Euclidean cubic number fields and two examples of norm-Euclidean cyclotomic fields, with the absolute of a norm less than 1 over the corresponding box, respectively.
Citation: Shenggui Zhang. A sufficient condition of Euclidean rings given by polynomial optimization over a box. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 93-101. doi: 10.3934/naco.2014.4.93
References:
[1]

K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control,, IEEE Control Systems Magazine, 25 (2005), 26.  doi: 10.1109/MCS.2005.1499389.  Google Scholar

[2]

C. K. Chen and T. K. Dao, Speed-adaptive roll-angle-tracking control of an unmanned bicycle using fuzzy logic,, Vehicle System Dynamics, 48 (2010), 133.   Google Scholar

[3]

C. Cornejo and L. Alvarez-Icaza, Passivity based control of under-actuated mechanical systems with nonlinear dynamic friction,, J. Vibration and Control, 18 (2012), 1025.  doi: 10.1177/1077546311408469.  Google Scholar

[4]

M. L. Fair and S. L. Campbell, Active incipient fault detection in continuous time systems with multiple simultaneous faults,, Numerical Algebra, 1 (2011), 211.  doi: 10.3934/naco.2011.1.211.  Google Scholar

[5]

L. Feng, Robust Control Design: An Optimal Control Approach,, Wayne State University, (2007).   Google Scholar

[6]

N. H. Getz, Dynamic Inversion of Nonlinear Maps with Applications to Nonlinear Control and Robotics,, Ph.D. Dissertation, (1995).   Google Scholar

[7]

Y. Harata, Y. Banno and K. Taji, Parametric excitation based bipedal walking: Control method and optimization,, Numerical Algebra, 1 (2011), 171.  doi: 10.3934/naco.2011.1.171.  Google Scholar

[8]

C. L. Hwang, H. M. Wu and C. L. Shih, Fuzzy sliding-mode underactuated control for autonomous dynamic balance of an electrical bicycle,, IEEE Trans. Control Systems Technology, 17 (2009), 658.   Google Scholar

[9]

N. H. K. Iuchi, H. Niki and T. Murakami, Attitude control of bicycle motion by steering angle and variable COG control,, Proc. 31st Annual Conference of IEEE Industrial Electronics Society, (2005), 16.   Google Scholar

[10]

R. N. Jazar, Mathematical theory of auto-driver for autonomous vehicles,, J. Vibration and Control, 16 (2010), 253.  doi: 10.1177/1077546309104467.  Google Scholar

[11]

R. Khaled and N. G. Chalhoub, A dynamic model and a robust controller for a fully-actuated marine surface vessel,, J. Vibration and Control, 17 (2011), 801.   Google Scholar

[12]

L. Lujng, System Identification Theory for User,, Linkopping University, ().   Google Scholar

[13]

M. S. Mahmoud, Computer-Operated Systems Control,, Marcel Dekker Inc., (1991).   Google Scholar

[14]

M. S. Mahmoud, Robust control of blood gases during extracorporeal circulation,, IET Control Theory and Applications, 5 (2011), 1577.  doi: 10.1049/iet-cta.2010.0665.  Google Scholar

[15]

M. S. Mahmoud, Resilient $\begin{eqation*}\frac{L_2}{L_\infty} \end{equation*}$ filtering of polytopic systems with state delays,, IET Control Theory And Applications, 1 (2007), 141.  doi: 10.1049/iet-cta:20045281.  Google Scholar

[16]

M. S. Mahmoud and A. Y. Al-Rayyah., Efficient parameterisation to stability and feedback synthesis of linear time-delay systems,, IET control theory and applications, 3 (2009), 1107.  doi: 10.1049/iet-cta.2008.0152.  Google Scholar

[17]

M. S. Mahmoud and Yuanqing Xia, Robust filter design for piecewise discrete-time systems with time-varying delays,, International Journal of Robust and Nonlinear Control, 20 (2010), 544.  doi: 10.1002/rnc.1447.  Google Scholar

[18]

M. S. Mahmoud and M. M. Hussain, Design of linear systems with saturating actuators: A survey,, Int. J. Numerical Algebra, 2 (2012), 413.  doi: 10.3934/naco.2012.2.413.  Google Scholar

[19]

J. Meijaard, J. Papadopoulos, A. Ruina and A. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review,, Proc. the Royal Society A: Mathematical, 463 (2007).  doi: 10.1098/rspa.2007.1857.  Google Scholar

[20]

K. Mendrok and Tadeusz Uhl, Load identification using a modified modal filter technique,, J. Vibration and Control, 16 (2010), 89.  doi: 10.1177/1077546309103274.  Google Scholar

[21]

G. T. Michaltsos, Bouncing of a vehicle on an irregularity: A mathematical model,, J. Vibration and Control, 16 (2010), 181.  doi: 10.1177/1077546309104878.  Google Scholar

[22]

H. Moradi, M. R. Movahhedy, and G. Vossoughi, Sliding mode control of machining chatter in the presence of tool wear and parametric uncertainties,, J. Vibration and Control, 16 (2010), 231.   Google Scholar

[23]

U. Nenner, R. Linker and P. Gutman, Robust feedback stabilization of an unmanned motorcycle,, Control Engineering Practice, (2010).   Google Scholar

[24]

Omar S. Al-Buraiki and El Ferik, Sami, Adaptive control of autonomous bicycle kinematics,, Proc. 13th Automation and Systems (ICCAS), (2013), 20.   Google Scholar

[25]

M. C. Pai, Sliding mode control of vibration in uncertain time-delay systems,, J. Vibration and Control, 16 (2010), 2131.  doi: 10.1177/1077546309350865.  Google Scholar

[26]

H. Schttler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Int. J. Numerical Algebra, 2 (2012), 631.  doi: 10.3934/naco.2012.2.631.  Google Scholar

[27]

R. Sharp and D. Limebeer, A motorcycle model for stability and control analysis,, Multi-body System Dynamics, 6 (2001), 123.   Google Scholar

[28]

R. Sharp, Optimal preview speed-tracking control for motorcycles,, Multi-body System Dynamics, 18 (2007), 397.   Google Scholar

[29]

S. Sivrioglu, H control for suppressing acoustic modes of a distributed structure using cluster sensing and actuation,, J. Vibration and Control, 16 (2010), 439.   Google Scholar

[30]

N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle,, Proc. IEEE International Conference Robotics and Biometrics, (2006), 1652.   Google Scholar

[31]

T. Yamaguchi, T. Shibata and T. Murakami, Self-sustaining approach of electric bicycle by acceleration control based backstepping,, Proc. 33rd Annual Conference of the IEEE Industrial Electronics Society, (2007), 2610.   Google Scholar

[32]

K. Zhou and J. C. Doyle, Essentials of Robust Control,, NJ: Prentice Hall, (1998).   Google Scholar

show all references

References:
[1]

K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control,, IEEE Control Systems Magazine, 25 (2005), 26.  doi: 10.1109/MCS.2005.1499389.  Google Scholar

[2]

C. K. Chen and T. K. Dao, Speed-adaptive roll-angle-tracking control of an unmanned bicycle using fuzzy logic,, Vehicle System Dynamics, 48 (2010), 133.   Google Scholar

[3]

C. Cornejo and L. Alvarez-Icaza, Passivity based control of under-actuated mechanical systems with nonlinear dynamic friction,, J. Vibration and Control, 18 (2012), 1025.  doi: 10.1177/1077546311408469.  Google Scholar

[4]

M. L. Fair and S. L. Campbell, Active incipient fault detection in continuous time systems with multiple simultaneous faults,, Numerical Algebra, 1 (2011), 211.  doi: 10.3934/naco.2011.1.211.  Google Scholar

[5]

L. Feng, Robust Control Design: An Optimal Control Approach,, Wayne State University, (2007).   Google Scholar

[6]

N. H. Getz, Dynamic Inversion of Nonlinear Maps with Applications to Nonlinear Control and Robotics,, Ph.D. Dissertation, (1995).   Google Scholar

[7]

Y. Harata, Y. Banno and K. Taji, Parametric excitation based bipedal walking: Control method and optimization,, Numerical Algebra, 1 (2011), 171.  doi: 10.3934/naco.2011.1.171.  Google Scholar

[8]

C. L. Hwang, H. M. Wu and C. L. Shih, Fuzzy sliding-mode underactuated control for autonomous dynamic balance of an electrical bicycle,, IEEE Trans. Control Systems Technology, 17 (2009), 658.   Google Scholar

[9]

N. H. K. Iuchi, H. Niki and T. Murakami, Attitude control of bicycle motion by steering angle and variable COG control,, Proc. 31st Annual Conference of IEEE Industrial Electronics Society, (2005), 16.   Google Scholar

[10]

R. N. Jazar, Mathematical theory of auto-driver for autonomous vehicles,, J. Vibration and Control, 16 (2010), 253.  doi: 10.1177/1077546309104467.  Google Scholar

[11]

R. Khaled and N. G. Chalhoub, A dynamic model and a robust controller for a fully-actuated marine surface vessel,, J. Vibration and Control, 17 (2011), 801.   Google Scholar

[12]

L. Lujng, System Identification Theory for User,, Linkopping University, ().   Google Scholar

[13]

M. S. Mahmoud, Computer-Operated Systems Control,, Marcel Dekker Inc., (1991).   Google Scholar

[14]

M. S. Mahmoud, Robust control of blood gases during extracorporeal circulation,, IET Control Theory and Applications, 5 (2011), 1577.  doi: 10.1049/iet-cta.2010.0665.  Google Scholar

[15]

M. S. Mahmoud, Resilient $\begin{eqation*}\frac{L_2}{L_\infty} \end{equation*}$ filtering of polytopic systems with state delays,, IET Control Theory And Applications, 1 (2007), 141.  doi: 10.1049/iet-cta:20045281.  Google Scholar

[16]

M. S. Mahmoud and A. Y. Al-Rayyah., Efficient parameterisation to stability and feedback synthesis of linear time-delay systems,, IET control theory and applications, 3 (2009), 1107.  doi: 10.1049/iet-cta.2008.0152.  Google Scholar

[17]

M. S. Mahmoud and Yuanqing Xia, Robust filter design for piecewise discrete-time systems with time-varying delays,, International Journal of Robust and Nonlinear Control, 20 (2010), 544.  doi: 10.1002/rnc.1447.  Google Scholar

[18]

M. S. Mahmoud and M. M. Hussain, Design of linear systems with saturating actuators: A survey,, Int. J. Numerical Algebra, 2 (2012), 413.  doi: 10.3934/naco.2012.2.413.  Google Scholar

[19]

J. Meijaard, J. Papadopoulos, A. Ruina and A. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review,, Proc. the Royal Society A: Mathematical, 463 (2007).  doi: 10.1098/rspa.2007.1857.  Google Scholar

[20]

K. Mendrok and Tadeusz Uhl, Load identification using a modified modal filter technique,, J. Vibration and Control, 16 (2010), 89.  doi: 10.1177/1077546309103274.  Google Scholar

[21]

G. T. Michaltsos, Bouncing of a vehicle on an irregularity: A mathematical model,, J. Vibration and Control, 16 (2010), 181.  doi: 10.1177/1077546309104878.  Google Scholar

[22]

H. Moradi, M. R. Movahhedy, and G. Vossoughi, Sliding mode control of machining chatter in the presence of tool wear and parametric uncertainties,, J. Vibration and Control, 16 (2010), 231.   Google Scholar

[23]

U. Nenner, R. Linker and P. Gutman, Robust feedback stabilization of an unmanned motorcycle,, Control Engineering Practice, (2010).   Google Scholar

[24]

Omar S. Al-Buraiki and El Ferik, Sami, Adaptive control of autonomous bicycle kinematics,, Proc. 13th Automation and Systems (ICCAS), (2013), 20.   Google Scholar

[25]

M. C. Pai, Sliding mode control of vibration in uncertain time-delay systems,, J. Vibration and Control, 16 (2010), 2131.  doi: 10.1177/1077546309350865.  Google Scholar

[26]

H. Schttler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Int. J. Numerical Algebra, 2 (2012), 631.  doi: 10.3934/naco.2012.2.631.  Google Scholar

[27]

R. Sharp and D. Limebeer, A motorcycle model for stability and control analysis,, Multi-body System Dynamics, 6 (2001), 123.   Google Scholar

[28]

R. Sharp, Optimal preview speed-tracking control for motorcycles,, Multi-body System Dynamics, 18 (2007), 397.   Google Scholar

[29]

S. Sivrioglu, H control for suppressing acoustic modes of a distributed structure using cluster sensing and actuation,, J. Vibration and Control, 16 (2010), 439.   Google Scholar

[30]

N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle,, Proc. IEEE International Conference Robotics and Biometrics, (2006), 1652.   Google Scholar

[31]

T. Yamaguchi, T. Shibata and T. Murakami, Self-sustaining approach of electric bicycle by acceleration control based backstepping,, Proc. 33rd Annual Conference of the IEEE Industrial Electronics Society, (2007), 2610.   Google Scholar

[32]

K. Zhou and J. C. Doyle, Essentials of Robust Control,, NJ: Prentice Hall, (1998).   Google Scholar

[1]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[2]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[3]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[4]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[5]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[6]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010

[7]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[8]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[9]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021005

[10]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 681-695. doi: 10.3934/cpaa.2020285

[11]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[12]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003

[13]

Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021003

[14]

Sugata Gangopadhyay, Constanza Riera, Pantelimon Stănică. Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations. Advances in Mathematics of Communications, 2021, 15 (2) : 241-256. doi: 10.3934/amc.2020056

[15]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[16]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[17]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[18]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[19]

Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005

[20]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176

 Impact Factor: 

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]