Linearization - Advantages and Shortcomings Toward Control System Design
Keywords:Jacobian, Linear system, nonlinear system, equilibrium point, Linearization
This brief manuscript discusses the necessity to linearize nonlinear systems. Thorough review on nonlinear phenomena in dynamical and numerical system is presented. The methodology to linearize nonlinear system in Jacobian approach is shown in didactic manner. Numerical and dynamical example of nonlinear system is provided to enhance understanding. Afterward, the comparison between both linearized and non-linearized system is literally discussed. The outcomes concluded that linearization process is a linear approximation of a nonlinear system that is only valid in a small region around an operating point.
F. Chiou, J. P. Gentle and T. R. McJunkin, Dispatchable renewable energy model for microgrid power system. IEEE Conference on Technologies for Sustainability, pp.195-199, 2016.
Z. Cao, F. O'Rourke and W. Lyons, Performance modelling of a small-scale wind and solar energy hybrid system, 28th Irish Signals and Systems Conference, pp.1-6, 2017.
M. I. Ahmed, M. Y. Hazlina and M. M. Rashid, Mathematical Modeling and Control of Active Suspension System for a Quarter Car Railway Vehicle, Malaysian Journal of Mathematical Sciences, Vol. 10(S), pp.227-241, 2016.
Jiwang Du, Qichang He and Xiumin Fan, Automating generation of the assembly line models in aircraft manufacturing simulation, IEEE International Symposium on Assembly and Manufacturing, pp.155-159, 2013.
Wen-Chyuan Chiang, Timothy L. Urban and Chunyong Luo, Balancing stochastic two-sided assembly lines, International Journal of Production Research, Vol. 54 (Issue 20), pp.6232-6250, 2016.
H. M. Al-Qahtani, Amin A. Mohammed and M. Sunar, Dynamics and Control of a Robotic Arm Having Four Links, Arabian Journal for Science and Engineering, Vol. 42 (Issue 5), pp.1841-1852, 2017.
A. Kirecci, M. Topalbekiroglu and I. Eker, Experimental evaluation of a model reference adaptive control for a hydraulic robot: A case study, Robotica, Vol.21 (Issue 1): pp.71-78, 2003.
S. Roggia, F. Cupertino, C. Gerada and M. Galea, Axial Position Estimation of Conical Shaped Motors for Aerospace Traction Applications. IEEE Transactions on Industry Applications, Vol. 53 (Issue 6): pp.5405-5414, 2017.
A. Witkowska, M. Tomera and R. mierzchalski, A Backstepping Approach to Ship Course Control, International Journal of Applied Mathematics and Computer Science, Vol. 17 (Issue 1): pp.73-85, 2007.
Yanjun Huang, Amir Khajepour, Farshid Bagheri and Majid Bahrami, Modelling and optimal energy-saving control of automotive air-conditioning and refrigeration systems, Journal of Automobile Engineering, Vol. 231 (Issue 3): pp.291-309, 2016.
Kis Andras, Physical models for micro and nano systems, Cole polytechnique fdrale de Lausanne, Course books 2017-2018.
Muhammad Nizam Kamarudin, Abdul Rashid Husain and Mohamed Noh Ahmad, Robust bounded control for uncertain nonlinear systems: application to a nonlinear strict feedback wind turbine model with explicit wind speed dynamics, Journal of Theoretical and Applied Information Technology, Vol. 63 (Issue 3): pp.718-732, 2014.
Nor Syaza Farhana Mohamad Murad, Muhammad Nizam Kamarudin, Sahazati Md Rozali and Mohd Hendra Hairi, Achieving optimum tip-speed-ratio of a two-mass wind turbine system, IEEE International Conference on Power and Energy (PECon), pp.757-762, 2016.
Nouby M. Ghazaly, Ali A. Kamel and M. O. Mousa, Influence of Misalignment and Backlash on Spur Gear using FEM, International Journal of Mechanical and Production Engineering, Vol. 2 (Issue 12), pp.2320-2092, 2014.
Muhammad Nizam Kamarudin, Abdul Rashid Husain and Mohamed Noh Ahmad, Variable Speed Wind Turbine with External Stiffness and Rotor Deviation Observer, Applied Mechanics and Materials, Vol. 661, pp.154-159, 2014.
Zheng Wang, Xiaoping Liu, Kefu Liu, Shuai Li and Huanqing Wang, Backstepping-Based Lyapunov Function Construction using Approximate Dynamic Programming and Sum of Square Techniques, IEEE Transactions On Cybernetics, Vol. 47 (Issue 10), pp.3393- 3403, 2017.
Tushar Kanti Roy, Md. Apel Mahmud, Weixiang Shen, and Amanullah Maung Than Oo, Nonlinear Adaptive Excitation Controller Design for Multimachine Power Systems with Unknown Stability Sensitive Parameters, IEEE Transactions On Control Systems Technology, Vol. 25 (Issue 6), pp.2060-2072, 2017.
Muhammad Nizam Kamarudin, Abdul Rashid Husain and Mohamed Noh Ahmad, Control of Uncertain Nonlinear Systems using Mixed Nonlinear Damping Function and Backstepping Techniques, IEEE International Conference on Control System, Computing and Engineering, Penang, pp.105-109, 2012.
M. R. James, I. R. Petersen and V. Ugrinovskii, A Popov stability condition for uncertain linear quantum systems, American Control Conference, pp. 2551-2555, 2013.
E. Jackson and J. Aggarwal, Popov and Hurwitz stability criterions: A comparison, IEEE Transactions on Automatic Control, Vol. 11 (Issue 3), pp.623-24, 1996.
Xinye Lia, Huabiao Zhang and Lijuan Zhang, Response of the Duffng-van der Pol oscillator under position feedback control with two time delays, Shock and Vibration, Vol. 18, pp. 377386, 2011.
Marios Tsatsos, Theoretical and Numerical Study of the Van der Pol equation, Ph.D Thesis Dissertation, Department of Physics,School of Sciences, Aristotle University of Thessaloniki, July 2016.
Nikhil M. Kriplani, Stephen Bowyer, Jennifer Huckaby and Michael B. Steer, Modelling of an Esaki Tunnel Diode in a Circuit Simulator, Active and Passive Electronic Components, Vol. 2011, pp.1-8, 2011.
Volker Mehrmann and Hongguo Xu, An analysis of the pole placement problem: The single-input case, Electronic Transactions on Numerical Analysis, Vol.4, pp.89-105, 1996.
Yuan Gong Sun, LongWang, Guangming Xie and Mei Yu, Improved overshoot estimation in pole placements and its application in observer-based stabilization for switched systems, IEEE Transactions on Automatic Control, vol. 51 (Issue 12), pp.1962-1966, 2006.
Kaiyang Yang and Robert Orsi, Generalized pole placement via static output feedback: A methodology based on projections, Automatica, Vol. 42 (Issue1), pp.2143-2150, 2006.
Muhammad Nizam Kamarudin, Abdul Rashid Husain and Sahazati Md.Rozali, Observer-based output feedback control with linear quadratic performance, Procedia Engineering, Vol.53 (), pp.233-240, 2013.
Jos_e B Mare and Jos_e A De Don, Solution of the input-constrained LQR problem using dynamic programming, Systems & Control Letters, Vol. 56 (Issue 5), pp.342-348, 2007.
Alberto Bemporad, Manfred Morari, Vivek Dua and Efstratios N Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica, Vol. 38 (Issue 1), :3-20, 2002.
Y. Eun, C. Gokcek, P. T. Kabamba and S. M. Merkov, An LQG approach to systems with saturating actuators and anti-windup implementation, Mathematical Problems in Engineering, Vol.8 (Issue 4-5):311-321, 2002.
M. N. Kamarudin, S. M. Rozali, M. H. Hairi, F. Hanaffi, M. S. M. Aras and M. K. M. Zambri, Realization of Real-Time Hardware-in-the-Loop for a Liquid Level with Open-loop Ziegler Nichols Technique, International Journal of Electrical Engineering and Applied Sciences (IJEEAS), Vol.1(2), pp.47-52, 2018.
Kiam Heong Ang, Gregory Chong and Yun Li, PID control system analysis, design, and technology, IEEE Transactions on Control Systems Technology, Vol.13 (Issue 4), pp.559-576, 2005.
Guillermo J. Silva, Aniruddha Datta and S. P. Bhattacharyya, New results on the synthesis of PID controllers, IEEE Transactions on Automatic Control, Vol. 47 (Issue 2), pp.241-252, 2002.
Zhendong Sun and Xiaohua Xia, The explicit linear quadratic regulator for constrained systems. Automatica, Vol. 33 (Issue 7) pp.1339-1344, 1997.
C. Daizhan, H. Xiaoming and W. Yuzhen, Non-regular feedback linearization of nonlinear systems via a normal form algorithm, Automatica, Vol. 40, pp.439-447, 2004.
M. Ababneh, M. Salah and K. Alwidyan, Non-regular feedback linearization of nonlinear systems via a normal form algorithm, Jordan Journal of Mechanical and Industrial Engineering, Vol. 5 (Issue 6), pp.567-571, 2011.
How to Cite
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).