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Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework

  • Wojciech P. Hunek EMAIL logo and Tomasz Feliks
Published/Copyright: August 23, 2021

Abstract

The advanced analytical study in the field of fractional-order non-full rank inverse model control design is presented in the paper. Following the recent results in this matter it is certain, that the inverse model control-oriented perfect control law can be established for the non-full rank integer-order systems being under the discrete-time state-space reference with zero value. It is shown here, that the perfect control paradigm can be extended to cover the multivariable non-full rank plants governed by the more general Grünwald-Letnikov discrete-time state-space model. Indeed, the postulated approach significantly reduces both iterative and non-iterative computational effort, mainly derived from the approximation of the Moore-Penrose inverse of the non-full rank matrices to finally be inverted. A prevention provided by the new method excludes the detrimental matrix behavior in the form of singularity, often avoided due to the observed ill-conditioned sensitivity. Thus, the new defined robust fractional-order non-full rank instance of such control strategy, supported by the pole-free mechanism, gives rise to the introduction of the general unified non-full rank perfect control-originated theory. Numerical algorithms with simulation investigation clearly confirm the innovative peculiarities provided by the manuscript.

References

[1] A.A. Ahmed, B.K. Koh, Y.I. Lee, A comparison of finite control set and continuous control set model predictive control schemes for speed control of induction motors. IEEE Trans. on Industrial Informatics 14, No 4 (2018), 1334–1346; DOI: 10.1109/TII.2017.2758393.10.1109/TII.2017.2758393Search in Google Scholar

[2] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications. Springer, New York (2006).Search in Google Scholar

[3] R. Bhattarai, N. Gurung, S. Kamalasadan, Minimum variance controller based adaptive control for Doubly fed induction generator. In: Proc. of the 2016 North American Power Symposium, Denver, CO, USA (2016), 1–6; DOI: 10.1109/NAPS.2016.7747913.10.1109/NAPS.2016.7747913Search in Google Scholar

[4] R. Bisht, S. Subramaniam, R. Bhattarai, S. Kamalasadan, Adaptive minimum variance control of grid connected single phase inverters in synchronously rotating dq reference frame. In: Proc. of the 2018 IEEE Industry Applications Society Annual Meeting, Portland, OR, USA (2018), 1–10; DOI: 10.1109/IAS.2018.8544589.10.1109/IAS.2018.8544589Search in Google Scholar

[5] Z. Chen, M. Yin, Y. Zou, K. Meng, Z. Dong, Maximum wind energy extraction for variable speed wind turbines with slow dynamic behavior. IEEE Trans. on Power Systems 32, No 4 (2017), 3321–3322; DOI: 10.1109/TPWRS.2016.2623981.10.1109/TPWRS.2016.2623981Search in Google Scholar

[6] R. Cioć, Physical and geometrical interpretation of Grünwald-Letnikov differintegrals: Measurement of path and acceleration. Fract. Calc. Appl. Anal. 19, No 1 (2016), 161–172; DOI: 10.1515/fca-2016-0009; https://www.degruyter.com/journal/key/FCA/19/1/html.10.1515/fca-2016-0009Search in Google Scholar

[7] C. Cubukcuoglu, A. Kirimtat, B. Ekici, F. Tasgetiren, P.N. Suganthan, Evolutionary Computation for Theatre Hall Acoustics. Springer International Publishing, Cham (2019)10.1007/978-3-030-01641-8_4Search in Google Scholar

[8] T. Feliks and W.P. Hunek, A new non-full rank algorithm for the imc-derived d-step mimo structures in the pole-free state space. IEEE Access 8 (2020), 121357–121365; DOI: 10.1109/ACCESS.2020.3006806.10.1109/ACCESS.2020.3006806Search in Google Scholar

[9] A. Goldenshluger, L. Mirkin, On minimum-variance event-triggered control. IEEE Control Systems Letters 1, No 1 (2017), 32–37; DOI: 10.1109/LCSYS.2017.2700620.10.1109/LCSYS.2017.2700620Search in Google Scholar

[10] M.J. Gomes Silva, C. Silva Araujo, S.T. Marques Bezerra, C. Rocha Souto, S. Arnaud Silva, H. Pimentel Gomes, Generalized minimum variance control for water distribution system. IEEE Latin America Transactions 13, No 3 (2015), 651–658; DOI: 10.1109/TLA.2015.7069088.10.1109/TLA.2015.7069088Search in Google Scholar

[11] Y. Gui, C.H. Kim, C.C. Chung, Y. Kang, Intra-day unit commitment for wind farm using model predictive control method. In: 2013 IEEE Power Energy Society General Meeting, Vancouver, BC, Canada (2013), 1–5; DOI: 10.1109/PESMG.2013.6672813.10.1109/PESMG.2013.6672813Search in Google Scholar

[12] W.P. Hunek, Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation. Fract. Calc. Appl. Anal. 22, No 2 (2019), 424–443; DOI: 10.1515/fca-2019-0026; https://www.degruyter.com/journal/key/FCA/22/2/html.10.1515/fca-2019-0026Search in Google Scholar

[13] W.P. Hunek, Towards a General Theory of Control Zeros for LTI MIMO Systems. Opole University of Technology Press, Opole, Poland (2011).Search in Google Scholar

[14] W.P. Hunek, T. Feliks, A geometric-based approach to the maximum-speed state and output variables for some class of IMC structures. In: Proc. of the 6th IEEE Intern. Conf. on Control, Decision and Information Technologies, Paris, France (2019), 1385–1389; DOI: 10.1109/CoDIT.2019.8820536.10.1109/CoDIT.2019.8820536Search in Google Scholar

[15] W.P. Hunek, T. Feliks, A new extension of inverse model control design to non-full rank state-space plants. In: Proc. of the 2020 European Control Conference, St. Petersburg, Russia (2020), 1783-1788; DOI: 10.23919/ECC51009.2020.9143669.10.23919/ECC51009.2020.9143669Search in Google Scholar

[16] W.P. Hunek, T. Feliks, A new geometric-oriented minimum-energy perfect control design in the IMC-based state-space domain. IEEE Access 8 (2020), 41733–41739; DOI: 10.1109/ACCESS.2020.2977278.10.1109/ACCESS.2020.2977278Search in Google Scholar

[17] W.P. Hunek, P. Majewski, Perfect reconstruction of signal – a new polynomial matrix inverse approach. EURASIP J. on Wireless Commun.and Networking 2018, No 1 (2018), 107; DOI: 10.1186/s13638-018-1122-5.10.1186/s13638-018-1122-5Search in Google Scholar

[18] W.P. Hunek, L.Wach, A new stability theory for grünwald–letnikov inverse model control in the multivariable lti fractional-order framework. Symmetry 11, No 10 (2019), # 1322; DOI: 10.3390/sym11101322.10.3390/sym11101322Search in Google Scholar

[19] A. Inoue, M. Deng, A. Yanou, T. Henmi, Multi-variable generalized minimum variance control with time-delay using interactor matrix. In: Proc. of the 2019 IEEE Intern. Conf. on Advanced Mechatronic Systems, Kusatsu, Japan (2019), 81–86; DOI: 10.1109/ICAMechS.2019.8861635.10.1109/ICAMechS.2019.8861635Search in Google Scholar

[20] V. Kaminskas, E. Ščiglinskas, Minimum variance control of human emotion as reactions to a dynamic virtual 3D face. In: Proc.of the 4th Workshop on Advances in Information, Electr. and Electr. Engineering, Vilnius, Lithuania (2016), 1–5; DOI: 10.1109/AIEEE.2016.7821810.10.1109/AIEEE.2016.7821810Search in Google Scholar

[21] V. Kiryakova, Generalized Fractional Calculus and Applications. Long-man Scientific & Technical, Harlow; and John Wiley & Sons, Inc., New York (1993).Search in Google Scholar

[22] M. Kishida, R.D. Braatz, Inversion-based output regulation of chemotaxis using a constrained influx of chemical signaling molecules. In: 2013 American Control Conference, Washington, DC, USA (2013), 3443–3448; DOI: 10.1109/ACC.2013.6580363.10.1109/ACC.2013.6580363Search in Google Scholar

[23] S. Lee, C.C. Chung, Reference redesigned perfect tracking control, with application to servo control system. In: Proc. of the 53rd IEEE Conf. on Decision and Control, Los Angeles, CA, USA (2014), 4542–4547; DOI: 10.1109/CDC.2014.7040098.10.1109/CDC.2014.7040098Search in Google Scholar

[24] J. Li, M. Gan, A novel robust perfect tracking control method for nonlinear servo systems. In: Proc. of the 37th IEEE Chinese Control Conf., Wuhan, China (2018), 3790–3795; DOI: 10.23919/ChiCC.2018.8482835.10.23919/ChiCC.2018.8482835Search in Google Scholar

[25] H. Ma, N. Li, P. Stanimirović, V.N. Katsikis, Perturbation theory for Moore–Penrose inverse of tensor via Einstein product. Computat. and Appl. Math. 38, No 111 (2019), 1–24; DOI: 10.1007/s40314-019-0893-6.10.1007/s40314-019-0893-6Search in Google Scholar

[26] A. Mystkowski, A. Zolotas, PLC-based discrete fractional-order control design for an industrial-oriented water tank volume system with input delay. Fract. Calc. Appl. Anal. 21, No 4 (2018), 1005–1026; DOI: 10.1515/fca-2018-0055; https://www.degruyter.com/journal/key/FCA/21/4/html.10.1515/fca-2018-0055Search in Google Scholar

[27] S. Okada, S. Masuda, Data-driven minimum variance control using regulatory closed-loop data based on the FRIT method. In: Proc. of the 56th Annual Conf. of the Soc. of Instrument and Control Engineers of Japan, Kanazawa, Japan (2017), 253–254; DOI: 10.23919/SICE.2017.8105758.10.23919/SICE.2017.8105758Search in Google Scholar

[28] L. Qida, T. Shubin, Y. Yongkuan, Performance evaluation of generalized minimum variance multi-disturbance control system. In: Proc. of the 2019 IEEE Chinese Control And Decision Conf., Nanchang, China (2019), 3286–3290; DOI: 10.1109/CCDC.2019.8833345.10.1109/CCDC.2019.8833345Search in Google Scholar

[29] M.-B. Radac, R.-E. Precup, Data-driven MIMO model-free reference tracking control with nonlinear state-feedback and fractional order controllers. Applied Soft Computing 73 (2018), 992–1003; DOI: 10.1016/j.asoc.2018.09.035.10.1016/j.asoc.2018.09.035Search in Google Scholar

[30] P.N. Suganthan, Letter: On non-iterative learning algorithms with closed-form solution. Applied Soft Computing 70 (2018), 1078–1082; DOI: 10.1016/j.asoc.2018.07.013.10.1016/j.asoc.2018.07.013Search in Google Scholar

[31] F. Vanhoenshoven, G. Nápoles, W. Froelich, J.L. Salmeron, K. Vanhoof, Pseudoinverse learning of Fuzzy Cognitive Maps for multivariate time series forecasting. Applied Soft Computing 95 (2020), # 106461; DOI: 10.1016/j.asoc.2020.106461.10.1016/j.asoc.2020.106461Search in Google Scholar

[32] D. Wang, A. Wang, Y. Fu, J. Xiao, H. Liu, Robust nonlinear perfect control for semiconductor refrigeration device. In: Proc. of the 10th IEEE Intern. Conf. on Software, Knowledge, Information Management Applications, Chengdu, China (2016), 413–417; DOI: 10.1109/SKIMA.2016.7916257.10.1109/SKIMA.2016.7916257Search in Google Scholar

[33] P. Yang, X. Zhou, H. Guo, Y. Zhou, Research on self-turning minimum variance control algorithm in non-minimum phase system. In: Proc. of the 15th IEEE Intern. Conf. on Networking, Sensing and Control, Zhuhai, China (2018), 1–5; DOI: 10.1109/ICNSC.2018.8361273.10.1109/ICNSC.2018.8361273Search in Google Scholar

[34] R. Yokoyama, S. Masuda, Data-driven generalized minimum variance regulatory control with constrained controller structure. In: Proc. of the 2016 Intern. Conf. on Advanced Mechatronic Systems, Melbourne, VIC, Australia (2016), 17–22; DOI: 10.1109/ICAMechS.2016.7813414.10.1109/ICAMechS.2016.7813414Search in Google Scholar

[35] W. Yunjie, W. Junfeng, L. Xiaodong, T. Dapeng, A new control method with perfect tracking control for flight simulator. In: Proc. of the 2011 IEEE Chinese Control and Decision Conf., Mianyang, China (2011), 2571–2576; DOI: 10.1109/CCDC.2011.5968644.10.1109/CCDC.2011.5968644Search in Google Scholar

Received: 2020-06-12
Revised: 2021-07-22
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

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