Robust stability testing function for a complex interval family of fractional-order polynomials. (English) Zbl 1501.93112
Summary: The robust stability of a family of interval fractional-order systems with complex coefficients is investigated in this study. The concept of “a family of interval fractional-order systems with complex coefficients” means that the characteristic function of a control system can be of both commensurate and non-commensurate orders, the coefficients of the characteristic function can be uncertain parameters, and may be complex numbers. At first, a simple graphical procedure is presented for robust stability analysis. The “robust stability testing function” is then extended to look at the robust conditions. To the best of the authors’ knowledge, no auxiliary function for analyzing the robust stability of the systems under investigation has been introduced until now. Moreover, lower and upper frequency bounds are provided which are useful to improve the computational efficiency of testing the robust stability conditions. Eventually, to verify the results, analytical examples and numerical simulations are provided.
References:
| [1] | Doye, I. N.; Salama, K. N.; Laleg-Kirati, T.-M., Robust fractional-order proportional-integral observer for synchronization of chaotic fractional-order systems, IEEE/CAA J. Autom. Sin., 6, 1, 268-277 (2018) |
| [2] | Huang, J.; Chen, Y.; Li, H.; Shi, X., Fractional order modeling of human operator behavior with second order controlled plant and experiment research, IEEE/CAA J. Autom. Sin., 3, 3, 271-280 (2016) |
| [3] | Victor, S.; Malti, R.; Garnier, H.; Oustaloup, A., Parameter and differentiation order estimation in fractional models, Automatica, 49, 4, 926-935 (2013) · Zbl 1284.93228 |
| [4] | Mujumdar, A.; Tamhane, B.; Kurode, S., Observer-based sliding mode control for a class of noncommensurate fractional-order systems, IEEE/ASME Trans. Mechatron., 20, 5, 2504-2512 (2015) |
| [5] | Mohyud-Din, S. T.; Ali, A.; Bin-Mohsin, B., On biological population model of fractional order, Int. J. Biomath., 9, 05, 1650070 (2016) · Zbl 1342.92194 |
| [6] | Sun, G.; Wu, L.; Kuang, Z.; Ma, Z.; Liu, J., Practical tracking control of linear motor via fractional-order sliding mode, Automatica, 94, 221-235 (2018) · Zbl 1401.93064 |
| [7] | Sun, G.; Ma, Z., Practical tracking control of linear motor with adaptive fractional order terminal sliding mode control, IEEE/ASME Trans. Mechatron., 22, 6, 2643-2653 (2017) |
| [8] | Tavazoei, M. S.; Tavakoli-Kakhki, M., Compensation by fractional-order phase-lead/lag compensators, IET Control Theory Appl., 8, 5, 319-329 (2014) |
| [9] | Tepljakov, A.; Petlenkov, E.; Belikov, J.; Astapov, S., Tuning and digital implementation of a fractional-order pd controller for a position servo, Int. J. Microelectron. Comput. Sci., 4, 3, 116-123 (2013) |
| [10] | Tepljakov, A.; Petlenkov, E.; Belikov, J., FOPID controller tuning for fractional FOPDT plants subject to design specifications in the frequency domain, 2015 European Control Conference (ECC), 3502-3507 (2015) |
| [11] | Zheng, W.; Luo, Y.; Chen, Y.; Wang, X., Synthesis of fractional order robust controller based on bodes ideas, ISA Trans., 111, 290-301 (2021) |
| [12] | Arya, P. P.; Chakrabarty, S., A robust internal model-based fractional order controller for fractional order plus time delay processes, IEEE Control Syst. Lett., 4, 4, 862-867 (2020) |
| [13] | Ghorbani, M.; Tavakoli-Kakhki, M.; Tepljakov, A.; Petlenkov, E.; Farnam, A.; Crevecoeur, G., Robust stability analysis of interval fractional-order plants with interval time delay and general form of fractional-order controllers, IEEE Control Syst. Lett., 6, 1268-1273 (2021) |
| [14] | Zheng, S.; Liang, B.; Liu, F.; Yang, Z.; Xie, Y., Robust stability of fractional order system with polynomial uncertainties based on sum-of-squares approach, J. Frankl. Inst., 357, 12, 8035-8058 (2020) · Zbl 1447.93276 |
| [15] | Zhang, Q.; Lu, J., Robust stability of output feedback controlled fractional-order systems with structured uncertainties in all system coefficient matrices, ISA Trans., 105, 51-62 (2020) |
| [16] | Aghayan, Z. S.; Alfi, A.; Machado, J. T., Robust stability of uncertain fractional order systems of neutral type with distributed delays and control input saturation, ISA Trans., 111, 144-155 (2021) |
| [17] | Moornani, K. A.; Haeri, M., On robust stability of LTI fractional-order delay systems of retarded and neutral type, Automatica, 46, 2, 362-368 (2010) · Zbl 1205.93118 |
| [18] | Moornani, K. A.; Haeri, M., On robust stability of linear time invariant fractional-order systems with real parametric uncertainties, ISA Trans., 48, 4, 484-490 (2009) |
| [19] | Moornani, K. A.; Haeri, M., Robust stability testing function and Kharitonov-like theorem for fractional order interval systems, IET Control Theory Appl., 4, 10, 2097-2108 (2010) |
| [20] | Ghorbani, M.; Tavakoli-Kakhki, M.; Estarami, A. A., Robust FOPID stabilization of retarded type fractional order plants with interval uncertainties and interval time delay, J. Frankl. Inst., 356, 16, 9302-9329 (2019) · Zbl 1423.93292 |
| [21] | Ghorbani, M.; Tavakoli-Kakhki, M., Robust stabilizability of fractional order proportional integral controllers for fractional order plants with uncertain parameters: a new value set based approach, J. Vib. Control, 26, 11-12, 965-975 (2020) |
| [22] | Ghorbani, M., Robust stability analysis of interval fractional-order plants by fractional-order controllers: an approach to reduce additional calculation, Int. J. Gen. Syst., 50, 1, 1-25 (2021) |
| [23] | Ghorbani, M.; Tavakoli-Kakhki, M., Robust stability analysis of uncertain incommensurate fractional order quasi-polynomials in the presence of interval fractional orders and interval coefficients, Trans. Inst. Meas. Control, 43, 5, 1117-1125 (2021) |
| [24] | Wei, J.; Liu, K.; Radice, G., Study on stability and rotating speed stable region of magnetically suspended rigid rotors using extended Nyquist criterion and gain-stable region theory, ISA Trans., 66, 154-163 (2017) |
| [25] | Ren, Y.; Fang, J., Complex-coefficient frequency domain stability analysis method for a class of cross-coupled antisymmetrical systems and its extension in MSR systems, Math. Probl. Eng. (2014) · Zbl 1407.93315 |
| [26] | Jha, M.-S.; Dauphin-Tanguy, G.; Ould-Bouamama, B., Robust fault detection with interval valued uncertainties in bond graph framework, Control Eng. Pract., 71, 61-78 (2018) |
| [27] | S. P. Bhattacharyya, H. C.; Keel, L. H., Robust Control: The Parametric Approach (1995), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ · Zbl 0838.93008 |
| [28] | Minnichelli, R. J.; Anagnost, J. J.; Desoer, C. A., An elementary proof of Kharitonov’s stability theorem with extensions, IEEE Trans. Autom. Control, 34, 9, 995-998 (1989) · Zbl 0693.93068 |
| [29] | Janssens, F. L.; van der Ha, J. C., Stability of spinning satellite under axial thrust, internal mass motion, and damping, J. Guid., Control, Dyn., 38, 4, 761-771 (2015) |
| [30] | Mohsenipour, R.; Liu, X., Robust D-stability test of LTI general fractional order control systems, IEEE/CAA J. Autom. Sin., 7, 3, 853-864 (2020) |
| [31] | Tan, N.; Özgüven, Ö. F.; Özyetkin, M. M., Robust stability analysis of fractional order interval polynomials, ISA Trans., 48, 2, 166-172 (2009) |
| [32] | Tan, N., Computation of the frequency response of multilinear affine systems, IEEE Trans. Autom. Control, 47, 10, 1691-1696 (2002) · Zbl 1364.93496 |
| [33] | Shao, X.; Sun, G.; Yao, W.; Liu, J.; Wu, L., Adaptive sliding mode control for quadrotor UAVs with input saturation, IEEE/ASME Trans. Mechatron. (2021) |
| [34] | Závacká, J.; Bakošová, M.; Vaneková, K., Design of robust PI controllers for control of an exothermic chemical reactor, Proceedings of the 14th WSEAS International Conference on Systems (2010) |
| [35] | Matusu, R.; Senol, B.; Pekar, L., Robust PI control of interval plants with gain and phase margin specifications: application to a continuous stirred tank reactor, IEEE Access, 8, 145372-145380 (2020) |
| [36] | Tu, Y.-W.; Ho, M.-T., Robust second-order controller synthesis for model matching of interval plants and its application to servo motor control, IEEE Trans. Control Syst. Technol., 20, 2, 530-537 (2011) |
| [37] | Simfukwe, D. D.; Pal, B. C., Robust and low order power oscillation damper design through polynomial control, IEEE Trans. Power Syst., 28, 2, 1599-1608 (2012) |
| [38] | Marden, M., Geometry of Polynomials, vol. 3 (1949), American Mathematical Soc. · Zbl 0038.15303 |
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