A Novel High-Order Fuzzy Systems with Decomposition in to Zero-and First-Order Fuzzy Structures in Nonlinear Dynamic Systems
الموضوعات :
Zohreh Zeighami
1
,
Mohammad Reza Jahed Motlag
2
,
Ali Moarefianpour
3
1 - Department of Electrical Engineering, Sirjan Branch, Islamic Azad University, Sirjan, Iran
2 - Department of Computer Engineering, Iran University of Science and Technology, Tehran, Iran
3 - Department of Electrical Engineering, Science and Research Branch,Islamic Azad University,Tehran,Iran
الکلمات المفتاحية: Stability, Approximation, Taylor series expansion, Nonlinear dynamic system modeling, high-order fuzzy systems,
ملخص المقالة :
Fuzzy modeling is a relatively new system modeling method with a proven efficiency record in various fields. Although zero- and first-order fuzzy systems are common due to their simplicity, their linear structure faces challenges when modeling nonlinear systems with state-variable interaction. These challenges include an increase in the number of rules and the inability to stabilize highly nonlinear systems. One solution is to use high-order fuzzy systems, which have a nonlinear structure and can represent model input interactions. In previous research, high-order fuzzy modeling has been investigated for static and nonlinear systems based on data, but such modeling has not been applied to dynamic systems with nonlinear nature which is a model of industrial processes. The present paper proposes a novel fuzzy structure inspired by the Taylor series expansion for dynamic systems with nonlinear state-space equations. This structure has a high degree of freedom in modeling complex nonlinear processes and can be adapted to the state-space equations of the system. The main novelty of this method is the conversion of a nonlinear high-order fuzzy structure into a set of first-order fuzzy structures. Another advantage is the ability to calculate the coefficients of the high-order fuzzy system from the Taylor series coefficients of the dynamic system’s model. Fuzzy systems have made various applications possible in the field of approximation. The present paper also proves the approximation ability and convergence of the proposed structure and determines its convergence criteria.
[1] Y. C.Shin, and C. X u,, Intelligent Systems: Modeling, Optimization, and Control, CRC Press, 2017.
[2] S. Jafarzadeh, Stability Analysis and Control Design of TSK Systems with Applications in Power Systems, PhD Dissertation, University of Nevada, Reno, 2012.
[3] Ogata, Modern Control Engineering, Prentice-Hall,5th, 2010.
[4] Y. Liu, L. Yan and L. Fang, "An adaptive neurocomplex- fuzzy-inferential modeling mechanism for generating higher order TSK models," Neurocomputing, vol. 365, pp. 94-101, 2019.
[5] B. Qin, Y. Nojima, H. Ishibuchi and S. Wang, "Realizing Deep High-Order TSK Fuzzy Classifier by Ensembling Interpretable Zero-Order TSK Fuzzy Subclassifiers," IEEE Transaction on Fuzzy Systems, vol. 19, no. 11, pp. 3441-3455, 2020.
[6] A. Kalhor, B. N. Araabi and C. Lucasi, "A new highorder takagi-sugeno fuzzy model based on deformed
linear models," Amirkabir International Journal of Modeling, Identification, Simulation & Control, vol. 42, pp. 43-54, 2010.
[7] Y.-L. Kuo and I. E. C. Resmi, "Model Predictive Control Based on a Takagi–Sugeno Fuzzy Model,"
International Journal of Fuzzy Systems, vol. 21, pp. 556-570, 2019. [8] R. E. Bellman, Adaptive Control Processes, A Guide Tour. Princeton University Press, Princeton, NJ, 2016.
[9] G. A. Heydari, A. Gharaveisi and M. Vali, "Chaotic time series prediction via artificialneural square fuzzy
inference system," Expert Systems With Applications, vol. 55, pp. 461-468, 2016.
[10] Q. Ren, "High Order Type-2 Takagi-sugeno-kang Fuzzy Logic System,PHD thesis,Ecole Polytechnique
De Montreal," 2007.
[11] E. F. Camacho and C. Bordons Alba, Model Predictive Control, 2 ed., Springer, 2007.
[12] A. Cavallo, "High-order fuzzy sliding manifold control," Fuzzy Sets and Systems, vol. 156, no. 2, pp. 149-266, 2005.
[13] K. Demirli and P. Muthukumaran, "Higher order fuzzy system identification using subtractive clustering,"
Journal of Intelligent and Fuzzy Systems, vol. 9, no. 3- 4, pp. 129-158, 2000.
[14] G. A. Heydari, "Optimal Design of Nonlinear Predictive Controller Based on High-Order Fuzzy Systems( in persian)," Shahid Bahonar University of Kerman, Iran,Kerman, 2015.
[15] L. j. Herrera, H. Pomares, I. Rojas and O. Valenzuela, "TaSe,a Taylor series-based fuzzy system model that combines interpretability and accuracy," Fuzzy Sets and Systems, vol. 153, no. 3, pp. 403-427, 2005.
[16] N. K. Kasabov and Q. Song, "DENFIS:Dynamic evolving neural-fuzzy inference system and its
application for time-series prediction," IEEE Trans. on Fuzzy Systems, vol. 10, p. 144–154, 2002.
[17] C. R. Ramaker and B. L. Cutler, "Dynamic Matrix Control - A computer control Algorithm," in Proceedings of the 1980 Joint Automatic Control Conference, San Francisco: American Automatic Control Council, 1980.
[18] K. Wiktorowicz and T. Krzeszowski, "Training High- Order Takagi-Sugeno Fuzzy Systems Using Batch Least Squares and Particle Swarm Optimization," International Journal of Fuzzy Systems, vol. 22, pp. 22-34, 2019.
[19] Karaboga, Dervis, Kaya and Ebubekir, "Adaptive network based fuzzy inference system (ANFIS)
training approaches: a comprehensive survey," Artificial Intelligence Review,Springer, vol. 52, pp. 2263-2293, 2018.
[20] M. V. Bobyr and S. G. Emelyanov, "A nonlinear method of learning neuro-fuzzy models for dynamic
control systems," Applied Soft Copmputing, vol. 88, 2020.
[21] G. A. Heydari, A. A. Gharaveisi and M. A. Vali, "New formulation for representing higher order tsk fuzzy systems," IEEE Trans. on Fuzzy Systems, vol. 24, pp. 854-864, 2016.
[22] S.-M. Chen, "Forcasting Enrollments Based On High- Order Fuzzy Time Series," Cybernetics and Systems, Taylor & Francis, vol. 33, no. 1, pp. 1-16, 2002.
[23] N. K. Kasabov, "DENFIS: Dynamic Evolving Neural- Fuzzy Inference System and Its Application for Time- Series Prediction," in IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2002.
[24] E. Navid Sadjadi, M. Ebrahimi and Z. Gachloo, "Discussion on Accuracy of Approximation with
Smooth Fuzzy Models," IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), 2020.
[25] G. A. Heydari, A. A. Gharaveisi and M. A. Vali, "Photovoltaic cells modeling via artificial neural square fuzzy inference system," Modares Journal of Electrical Engineering, vol. 4, no. 11, pp. 11-19, 2012.
[26] N.Ebrahimi, Gh.A Heydari and A.A Gharaveisi, "Optimum Design Of Fuzzy Supervisor Controller For Maximum Power Point Tracking in Sollar Cells," in third annual clean energy conference, kerman, 2013.
[27] S. Eslahi Tatafi, G. Heydari and A. Gharaveisi, "Short- Term Electricity Price Forecasting Using Optimal TSK
Fuzzy Systems," Journal of mathematics and computer science, pp. 238-246, 2014.
[28] I. Boulkaibet, K. .Belarbi, S. .Bououden and T. .Marwala, "A new T-S fuzzy model predictive control
for nonlinear processes," Expert Systems With Applications, vol. 88, pp. 132-151, 2017.
[29] I. Boulkaibet, S.Belarbi, S.Bououden, M.Chadli and T.Marwala, "An adaptive fuzzy predictive control of
nonlinear processes based onMulti- kernel least squares support vector regression,," Applied Soft Computing Journal,Elsevier, vol. 73, pp. 572-590, 2018.
[30] N.Vafamand, “Nonlinear system identification based on Takagi-Sugeno fuzzy modeling and unscented
Kalman filter,” ISI Transactions, vol. 74, pp. 134-143, 2018.
[31] S. Dadashi Arani and A. Moarefianpoor, "A sum of Tracking Control Design for Discrete-time Polynomial Systemss: A Sum-of-Squares approach," Modares Journal Of Electrical Engineering, vol. 16, no. 1, pp. 1-7, 2016.
[32] R. Salimi Tari and A. Ali Moarefianpour, "Observer- Based Tracking Control Design for a Class of Fuzzy
Polynomial Systems," Journal of Control, vol. 12, no. 1, pp. 1-11, 2018.
[33] R. Salimi Tari and A. Moarefianpour, "Observer‐based tracking controller design for a class of polynomial fuzzy systems with disturbance," Asian Journal of Control, vol. 23, no. 4, pp. 1982-1991, 2020.
[34] H. K. Lam, "Polynomial fuzzy-model-based control systems: Stability analysis via piecewise-linear membership functions," IEEE Transactions on Fuzzy Systems, vol. 19, no. 3, pp. 588-593, 2011.
[35] N. Behmanesh Fard, P. Kohan Sedgh, A. Khayatian, M.H. Asemani, “Design of a TS fuzzy controller for the simultaneous stabilization of nonlinear systems,” Computational Interlligence in Electrical Engineering, 2021.
[36] k. Tanaka, H. Yoshida, H. Ohtake and h. O-Wang, "A Sum-of-Squares Approach to Modeling and Control of Nonlinear Dynamical Systems With Polynomial Fuzzy Systems," IEEE Transaction on Fuzzy Systems, vol. 17, no. 4, pp. 911 922, 2009.
[37] T. Kazuo and H. O.Wang, "Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach," John Wiley & Sons, 2004.
[38] S. Prajna, A. Papachristodoulou and W. Fen, "Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach," in Control Conference, Melbourne, VIC, Australia, 2004.
[39] Z.Zeighami,M.R.jahed- Motlagh,A.Moarefianpour,G.Heydari, "A new stability criterion for high-order dynamic fuzzy systems," Iranian Journal of Fuzzy Systems, vol. 19, no. 2, pp. 187-203, 2022.
[40] J. R. Castro, O. Castillo, M. A. Sanchez, O. Mendoza, A. Rodríguez-Diaz and P. Melin, "Method for Higher
Order polynomial Sugeno Fuzzy Inference Systems," Information Sciences, vol. 351, pp. 76-89, 2016.
[41] Wang, L.X., A Course In Fuzzy Systems And Control, Prentice-Hall International,Inc, 1977.
[42] G. A. Heydari, A. Gharaveisi and M. Vali, "Chaotic time series prediction via artificial neural square fuzzy inference system," Expert Systems With Applications, vol. 55, pp. 461-468, 2016.
[43] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc, 1976.
[44] K. Koenigsberger, Analysis 2, 5 ed., Springer, 2004.
[45] I. J. P. Crowe-Wright, "Control Theory: The Double Pendulum Inverted on a Cart,Master Thesis of Science
Mathematics," New Mexico, 2018.
[46] A. Bogdanov, "Optimal Control of a Double Inverted Pendulum on a Cart,Technical Report," Department of
Computer Science & Electrical Engineering OHSU,OGI School of Science and Engineering ,reserchgate, Portland, 2004.
