Robust FOPID stabilization of retarded type fractional order plants with interval uncertainties and interval time delay. (English) Zbl 1423.93292
J. Franklin Inst. 356, No. 16, 9302-9329 (2019); corrigendum ibid 358, No. 2, 1692 (2021).
Summary: This study investigates the robust stability of the retarded type of interval fractional order plants with an interval time delay. To this end, the characteristic quasi-polynomial is divided into two terms. The first term is simply the denominator interval polynomial of the open loop system and the second term is the multiplication of the interval delay term in the numerator of the open loop system which is an interval polynomial. Each of these two terms of the characteristic quasi-polynomial makes their own value sets in the complex plane for a given frequency. In this paper, based on these two value sets and by using the zero exclusion principle, the robust stability of the closed loop system by applying a FOPID controller is analyzed. Finally, two numerical examples and an experimental verification are provided to demonstrate the effectiveness of the proposed method in the robust stabilization of fractional order plants with interval uncertainties and interval time delay.
MSC:
| 93D09 | Robust stability |
| 93D21 | Adaptive or robust stabilization |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Efe, M. O., Fractional order systems in industrial automation a survey, IEEE Trans. Ind. Inf., 7, 4, 582-591 (2011) |
| [2] | Bonnet, C.; Partington, J. R., Coprime factorizations and stability of fractional differential systems, Syst. Control Lett., 41, 3, 167-174 (2000) · Zbl 0985.93048 |
| [3] | Busłowicz, M., Stability of linear continuous time fractional order systems with delays of the retarded type, Bull. Polish Acad. Sci. Tech. Sci., 56, 4, 319-324 (2008) |
| [4] | Merrikh-Bayat, F.; Karimi-Ghartemani, M., An efficient numerical algorithm for stability testing of fractional-delay systems, ISA Trans., 48, 1, 32-37 (2009) |
| [5] | Gao, Z., A graphic stability criterion for non-commensurate fractional-order time-delay systems, Nonlinear Dyn., 78, 3, 2101-2111 (2014) · Zbl 1345.34123 |
| [6] | Shi, M.; Wang, Z., An effective analytical criterion for stability testing of fractional-delay systems, Automatica, 47, 9, 2001-2005 (2011) · Zbl 1227.93055 |
| [7] | Hamamci, S. E., Stabilization using fractional order PI and PID controllers, Nonlinear Dyn., 51, 1-2, 329-343 (2008) · Zbl 1170.93023 |
| [8] | Hamamci, S. E.; Koksal, M., Calculation of all stabilizing fractional order PD controllers for integrating time delay systems, Comput. Math. Appl., 59, 5, 1621-1629 (2010) · Zbl 1189.93125 |
| [9] | Gao, Z.; Yan, M.; Wei, J., Robust stabilizing regions of fractional order \(PD^μ\) controllers of time delay fractional order systems, J. Process Control, 24, 1, 37-47 (2014) |
| [10] | Moornani, K. A.; Haeri, M., Robustness in fractional proportional-integral-derivative-based closed-loop systems, IET Control Theory Appl., 4, 10, 1933-1944 (2010) |
| [11] | Zheng, S.; Li, W., Stabilizing region of \(PD^μ\) controller for fractional order system with general interval uncertainties and an interval delay, J. Frankl. Inst., 355, 3, 1107-1138 (2018) · Zbl 1393.93111 |
| [12] | Chen, L.; Wu, R.; He, Y.; Yin, L., Robust stability and stabilization of fractional-order linear systems with polytopic uncertainties, Appl. Math. Comput., 257, 274-284 (2015) · Zbl 1338.93293 |
| [13] | Li, C.; Wang, J., Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0 <α <1 case, J. Frankl. Inst., 349, 7, 2406-2419 (2012) · Zbl 1287.93063 |
| [14] | Ma, Y.; Lu, J.; Chen, W., Robust stability and stabilization of fractional order linear systems with positive real uncertainty, ISA Trans., 53, 2, 199-209 (2014) |
| [15] | Kharitonov, V. L., Asymptotic stability of an equilibrium position of a family of systems of differential equations, Differ. Equ., 14, 1483 (1978) · Zbl 0409.34043 |
| [16] | Dasgupta, S., Kharitonov’s theorem revisited, Syst. Control Lett., 11, 5, 381-384 (1988) · Zbl 0673.93071 |
| [17] | Huang, Y. J.; Wang, Y. J., Robust PID tuning strategy for uncertain plants based on the Kharitonov theorem, ISA Trans., 39, 4, 419-431 (2000) |
| [18] | Olshevsky, A.; Olshevsky, V., Kharitonovs theorem and Bezoutians, Linear Algebra Appl., 399, 285-297 (2005) · Zbl 1066.15003 |
| [19] | Yeung, K.; Wang, S., A simple proof of Kharitonov’s theorem, IEEE Trans. Autom. Control, 32, 9, 822-823 (1987) · Zbl 0624.34052 |
| [20] | Tan, N.; Ozguven, O. F.; Ozyetkin, M. M., Robust stability analysis of fractional order interval polynomials, ISA Trans., 48, 2, 166-172 (2009) |
| [21] | 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) |
| [22] | Gao, Z., An analytical method on the stabilization of fractional-order plants with one fractional-order term and interval uncertainties using fractional-order \(PI^λ D^μ\) controllers, Trans. Inst. Measurem. Control, 40, 15, 4133-4142 (2018) |
| [23] | Gao, Z., Analytical criterion on stabilization of fractional-order plants with interval uncertainties using fractional-order \(PD^μ\) controllers with a filter, ISA Trans., 83, 25-34 (2018) |
| [24] | 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 |
| [25] | 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) |
| [26] | Gao, Z., Robust stability criterion for fractional order systems with interval uncertain coefficients and a time delay, ISA Trans., 58, 76-84 (2015) |
| [27] | Gao, Z., Robust stabilization of interval fractional order plants with one time delay by fractional order controllers, J. Frankl. Inst., 354, 2, 767-786 (2017) · Zbl 1355.93161 |
| [28] | Mohsenipour, R.; Fathi Jegarkandi, M., Robust stability analysis of fractional-order interval systems with multiple time delays, Int. J. Robust Nonlinear Control, 29, 6, 1823-1839 (2019) · Zbl 1416.93161 |
| [29] | Liang, T.; Chen, J.; Lei, C., Algorithm of robust stability region for interval plant with time delay using fractional order \(PI^λ D^μ\) controller, Commun. Nonlinear Sci. Numer. Simulat., 17, 2, 979-991 (2012) · Zbl 1239.93090 |
| [30] | Hamidian, H.; Beheshti, M. T.H., A robust fractional-order PID controller design based on active queue management for TCP network, Int. J. Syst. Sci., 49, 1, 211-216 (2018) · Zbl 1385.93022 |
| [31] | Mohsenipour, R.; Jegarkandi, M. F., A comment on algorithm of robust stability region for interval plant with time delay using fractional order \(PI^λ D^μ\) controller, Commun. Nonlinear Sci. Numer. Simulat., 63, 202-204 (2018) · Zbl 1508.93231 |
| [32] | Hamamci, S. E., An algorithm for stabilization of fractional order time delay systems using fractional order PID controllers, IEEE Trans. Autom. Control, 52, 10, 1964-1969 (2007) · Zbl 1366.93464 |
| [33] | U.N.T.G. GmbH, Technical description of RT512 Process Trainer Level, (2003), https://www.gunt.de/en/products/mechatronics/automation-and-process-control-engineering/simple-process-engineering-control-systems/level-control-trainer/080.51200/rt512/glct-1:pa-148:ca-83:pr-1178. |
| [34] | Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F. M., Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE Trans. Circu. Syst. I Fund. Theory Appl., 47, 1, 25-39 (2000) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
