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]

Aihua Li. An algebraic approach to building interpolating polynomial. Conference Publications, 2005, 2005 (Special) : 597-604. doi: 10.3934/proc.2005.2005.597

[2]

Pieter C. Allaart. An algebraic approach to entropy plateaus in non-integer base expansions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6507-6522. doi: 10.3934/dcds.2019282

[3]

Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070

[4]

Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67

[5]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019102

[6]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755

[7]

David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 31-54. doi: 10.3934/dcds.2003.9.31

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm. Communications on Pure & Applied Analysis, 2003, 2 (1) : 33-50. doi: 10.3934/cpaa.2003.2.33

[10]

Palash Sarkar, Shashank Singh. A unified polynomial selection method for the (tower) number field sieve algorithm. Advances in Mathematics of Communications, 2019, 13 (3) : 435-455. doi: 10.3934/amc.2019028

[11]

Reza Kamyar, Matthew M. Peet. Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2383-2417. doi: 10.3934/dcdsb.2015.20.2383

[12]

Brigitte Vallée. Euclidean dynamics. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 281-352. doi: 10.3934/dcds.2006.15.281

[13]

Qifeng Cheng, Xue Han, Tingting Zhao, V S Sarma Yadavalli. Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter. Journal of Industrial & Management Optimization, 2019, 15 (1) : 177-198. doi: 10.3934/jimo.2018038

[14]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557

[15]

Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349

[16]

Donglei Du, Tianping Shuai. Errata to:''Optimal preemptive online scheduling to minimize $l_{p}$ norm on two processors''[Journal of Industrial and Management Optimization, 1(3) (2005), 345-351.]. Journal of Industrial & Management Optimization, 2008, 4 (2) : 339-341. doi: 10.3934/jimo.2008.4.339

[17]

Yi An, Zhuohan Li, Changzhi Wu, Huosheng Hu, Cheng Shao, Bo Li. Earth pressure field modeling for tunnel face stability evaluation of EPB shield machines based on optimization solution. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020101

[18]

Songting Luo, Leonidas J. Guibas, Hong-Kai Zhao. Euclidean skeletons using closest points. Inverse Problems & Imaging, 2011, 5 (1) : 95-113. doi: 10.3934/ipi.2011.5.95

[19]

Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002

[20]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

 Impact Factor: 

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]