Study of the TP transformation via the inverted pendulum example

Andrea Wéber; Miklós Kuczmann

doi:10.14513/actatechjaur.00562

Authors

Andrea Wéber Széchenyi István University, Department of Automation, Egyetem tér 1, 9026 Győr, Hungary
Miklós Kuczmann Széchenyi István University, Department of Automation, Egyetem tér 1, 9026 Győr, Hungary

DOI:

https://doi.org/10.14513/actatechjaur.00562

Keywords:

tensor product based control, nonlinear dynamic systems, linear matrix inequality, inverted pendulum

Abstract

This study decribes the Tensor Product (TP) based model construction through the nonlinear dynamic system of the inverted pendulum. It presents the steps of TP modeling, various weighting functions and the Linear Matrix Inequality (LMI) based approach. LMI control has been used to stabilize the nonlinear system. This study shows the quasi-Linear Parameter-Varying (qLPV) state-space modeling and the Higher-Order Singular Value Decomposition (HOSVD) based TP model transformation. Some research in this issue already exists, but only the furuta, rotary, single and parallel-type double pendulum have been examined. In this paper the TP model transformation of the inverted pendulum is analyzed in terms of stability.

Downloads

Download data is not yet available.

References

S. Iles, F. Kolonic, J. Matusko, Linear Matrix Inequalities Based H∞ Control of Gantry Crane using Tensor Product Transformation, Proceedings of the 18th International Conference on ProcessControl, Tatranská Lomnica, Slovakia, 2011, pp. 92–99.

D. Guang-Ren, Y. Hai-Hua, LMIs in Control Systems: Analysis, Design and Applications, 1st Edition, CRC Press, Taylor and Francis Group, Boca Raton,2013. doi: https://doi.org/10.1201/b15060

X. Liu, X. Xin, Z. Li, Z. Chen, Near Optimal Control Based on the Tensor-Product Technique, IEEE Transactions on Circuits and Systems II: Express Briefs 64 (5) (2017) pp. 560–564. doi: https://doi.org/10.1109/TCSII.2016.2592986

A. Szollosi, P. Baranyi, Influence of the Tensor Product model representation of qLPV models on the feasibility of Linear Matrix Inequality, Asian Journal of Control 18 (4) (2015) pp. 1328–1342.

B. Lantos, Theory and Design of Control Systems III., Academic Press, Budapest,2018.

Sz. Nagy, Z. Petres, P. Baranyi, TP tool - A MATLAB toolbox for TP model transformation, 8th International Symposium of Hungarian Researchers on Computational Intelligence and Informatics, CINTI (2007), Budapest, Hungary pp.483–495.

S. Iles, J. Matusko, F. Kolonic, Furuta Pendulum - a Tensor Product Model-based Design Approach Case Study, 2011 Proceedings of the 34th International Convention MIPRO, Opatija, Croatia, 2011.

P. Grof, Y. Yam, TP transformation based control of rotary pendulumy, IEEE International Conference on Systems, Man, and Cybernetics (2015) pp. 2620–2625. doi: https://doi.org/10.1109/SMC.2015.458

J. Matusko, V. Lesic, F. Kolonic, S. Iles, Tensor product based control of the Single Pendulum Gantry process with stable neural network based friction compensation, Advanced Intelligent Mechatronics (AIM), 2011 IEEE/ASME International Conference, Budapest. doi: https://doi.org/10.1109/AIM.2011.6027152

F. Kolonic, A. Poljugan, Experimental Control Design by TP Model Transformation, 2006 IEEE International Conference on Mechatronics, Budapest, Hungary, 2006. doi: https://doi.org/10.1109/ICMECH.2006.252605

Sz. Nagy, Z. Petres, P. Baranyi, TP model transformation based controller designfor the parallel-type double inverted pendulum, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence),Hong Kong, China, 2008. doi: https://doi.org/10.1109/FUZZY.2008.4630551

P. Baranyi, TP- Model Transformation Based Control Design Frameworks, Control Engineering, Springer Book, 2016.

P. Baranyi, Y. Yam, P. V ́arlaki, Tensor Product Model Transformation in Polytopic Model Based Control, 1st Edition, Automation and Control Engineering, CRC Press, Taylor and Francis Group, 2018. doi: https://doi.org/10.1201/9781315218045

P. Baranyi, Extracting LPV and qLPV Structures from State-Space Functions: A TP Model Transformation Based Framework, IEEE Transactions on FuzzySystems 28 (3) (2017) pp. 499–509. doi: https://doi.org/10.1109/TFUZZ.2019.2908770

P. Korondi, Tensor Product Model Transformation-based Sliding Surface Design, Acta Polytechnica Hungarica 3 (4) (2006) pp. 23–35.

F. Kolonic, A. Poljugan, I. Petrovic, Tensor Product Model Transformation-based Controller Design for Gantry Crane Control System-an Application Approach, Acta Polytechnica Hungarica 3 (4) (2006) pp. 95–112.

P. Varkonyi, D. Tikk, P. Korondi, P. Baranyi, A New Algorithm for RNO-INO Type Tensor Product Model Representation, 2005 IEEE International Conferenceon Intelligent Engineering Systems, Cruising on the Mediterranean Sea, Spain, 2005, pp. 263–266. doi: https://doi.org/10.1109/INES.2005.1555170

Y. Yam, P. Baranyi, C.T. Yamg, Reduction of Fuzzy Rule Base via Singular Value Decomposition, IEEE Transactions on Fuzzy Systems 7 (2) (1999) pp.120–132. doi: https://doi.org/10.1109/91.755394

G. Bergqvist, E. Larsson, The Higher Order Singular Value Decomposition: Theory and an Application,IEEE Signal Processing Magazine 27 (3) (2010) pp.151–154. doi: https://doi.org/10.1109/MSP.2010.936030

P. Baranyi, P. Szeidl, P. Várlaki, Y. Yam, Definition of the HOSVD Based Canonical Form of Polytopic Dynamic Models, Proceedings of the 2006 IEEE International Conference on Mechatronics, Budapest, 2006 pp. 660–665. doi: https://doi.org/10.1109/ICMECH.2006.252604

P. Szeidl, P. Várlaki, HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems, Journal of Advanced Computational Intelligence and Intelli-gent Informatics 13 (1) (2009) pp. 52–60.

M. Kuczmann, Signals and Systems, Universitas- Győr Nonprofit Kft., Győr, 2010.

V. C. S. Campos, F. O. Souza, L. A. B. Torres, R. M. Palhares, New Stability Conditions Based on Piecewise Fuzzy Lyapunov Functions and Tensor Product Transformations, IEEE Transactions on Fuzzy Systems21 (4) (2013) pp. 784–760. doi: https://doi.org/10.1109/TFUZZ.2012.2230178

H. O. Wang, K. Tanaka, M. F. Griffin, Parallel Distributed Compensation of Nonlinear Systems by Takagi-Sugeno Fuzzy Model, Proceedings of 1995 IEEE International Conference on Fuzzy Systems, Yokohama, Japan, 1995 pp. 531–538. doi: https://doi.org/10.1109/FUZZY.1995.409737

B. P. Lal, T. Barjeev, O. G. Hari, Optimal Control of Nonlinear Inverted Pendulum System using PID Controller and LQR: Performance Analysis withoutand with Disturbance Input, International Journal of Automation and Computing (2014) pp. 661–670.

M. Kuczmann, Comprehensive Survey of PID Controller Design for the Inverted Pendulum, Acta Technica Jaurinensis 12 (1) (2019) pp. 55–81. doi: https://doi.org/10.14513/actatechjaur.v12.n1.492

G. Zhao, Z. Wang, Z. Song, A Novel Tensor Product Model Transformation-based Adaptive Variable Universe of Discourse Controller, Journal of the FranklinInstitute 353 (17) (2016) pp. 4471–4499. doi: https://doi.org/10.1016/j.jfranklin.2016.08.026

S. Kuntanapreeda, Tensor Product Model Transformation Based Control and Synchronization of a Class of Fractional-Order Chaotic Systems, Asian Journalof Control 17 (2) (2015) pp. 371–380. doi: https://doi.org/10.1002/asjc.839

B. Lantos, Theory and Design of Control Systems I., Academic Press, Budapest,2009.

Z. Petres, B. Reskó, P. Baranyi, TP Model Transformation Based Control of the TORA System, Production Systems and Information Engineering (2004) pp.159–175.

K. Tanaka, H. O. Wang, Fuzzy Control Systems Design and Analysis: A linear Matrix Inequality Approach, John Wiley and Sons, 2001. doi: https://doi.org//10.1002/0471224596.ch2

M. Kuczmann, State Space Based Linear Controller Design for the Inverted Pendulum, Acta Technica Jaurinensis12 (2) (2019) pp. 130–147. doi: https://doi.org/10.14513/actatechjaur.v12.n2.499