An algebraic stability test for fractional order time delay systems
DOI:
https://doi.org/10.11121/ijocta.01.2020.00803Keywords:
Fractional order systems, pproximation, Time delay, StabilityAbstract
In this study, an algebraic stability test procedure is presented for fractional order time delay systems. This method is based on the principle of eliminating time delay. The stability test of fractional order systems cannot be examined directly using classical methods such as Routh-Hurwitz, because such systems do not have analytical solutions. When a system contains the square roots of s, it is seen that there is a double value function of s. In this study, a stability test procedure is applied to systems including sqrt(s) and/or different fractional degrees such as s^alpha where 0 < ? < 1, and ? include in R. For this purpose, the integer order equivalents of fractional order terms are first used and then the stability test is applied to the system by eliminating time delay. Thanks to the proposed method, it is not necessary to use approximations instead of time delay term such as Pade. Thus, the stability test procedure does not require the solution of higher order equations.
Downloads
References
Petras, I. (2011). Stability test procedure for a certain class of the fractional-order systems.Proceedings of the 2011 12th International Carpathian Control Conference, ICCC’2011, (5), 303-307.
Podlubny, I. (1999). Fractional-order systems and PIDμ-controllers. IEEE Transactions on Automatic Control, 44(1), 208–214.
Liu, M., Dassios I., Milano F. (2019). On the Stability Analysis of Systems of Neutral Delay Differential Equations. Circuits, Systems, and Signal Processing, 38(4), 1639-1653.
Milano, F., Dassios, I. (2016). Small-Signal Stability Analysis for Non-Index 1 Hessenberg Form Systems of Delay Differential-Algebraic Equations. IEEE Transactions on Circuits and Systems I: Regular Papers, 63(9), 1521-1530.
Liu, M., Dassios I., Tzounas, G., Milano F. (2019). Stability Analysis of Power Systems with Inclusion of Realistic-Modeling of WAMS Delays. IEEE Transactions on Power Systems, 34(1), 627-636.
Petras, I. (2010). Fractional-Order Memristor-Based Chua’s Circuit.IEEE Transactions on Circuits and Systems, 57(12), 975–979.
Dorcak, L., Valsa, J., Gonzalez, E., Terpak, J., Petras, I., & Pivka, L. (2013). Analogue realization of fractional-order dynamical systems. Entropy, 15(10), 4199-4214.
Shah, P., & Agashe, S. (2016). Review of fractional PID controller. Mechatronics, 38, 29– 41.
Podlubny, I., Petras, I., O’Leary, P., Dorcak, L., & Vinagre, B. M. (2002). Analogue realizations of fractional order controllers. Non- linear dynamics, 29, 281-296.
Luo, Y., & Chen, Y. (2012). Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica, 48(9), 2159–2167.
Tang, X., Shi, Y., & Wang, L. L. (2017). A new framework for solving fractional optimal control problems using fractional pseudospectral methods. Automatica, 78, 333-340.
Razminia, A., & Baleanu, D. (2013). Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics, 23(7), 873-879.
Ozturk, N. (1995). Stability Independent of distributed Lag for A Special Class of Distributed Parameter Systems. In Proceedings of the 34th IEEE Conference on Decision and Control, 13-15 December, New Orleans, LA, USA, 3245-3246).
Tan, N.,¨Ozg¨uven, ¨O. F., & ¨Ozyetkin,M. M. (2009). Robust stability analysis of fractional order interval polynomials. ISA Transactions, 48(2), 166-172.
Ozturk, N. and Uraz, A. (1985). An Analysis Stability Test for a Certain Class of Distributed Parameter Systems with Delays. IEEE Transactions on Circuits and Systems, CAS-32, 393-396.
Petras, I. (2008). Stability of Fractional- Order Systems with Rational Orders. Fractional calculus and Applied Aalysis, 12(3), 25.
Sabatier, J., Moze, M., & Farges, C. (2010). LMI stability conditions for fractional order systems. Computers and Mathematics with Applications, 59(5), 1594-1609.
Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. Computational engineering in systems applications, 963-968.
Matignon, D. (1998). Stability properties for generalized fractional differential systems. ESAIM: Proceedings Fractional Differential Systems: Models, Methods and Aplicaitons,(5), 145-158.
Lazarevic, M. P., & Spasic, A. M. (2009). Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Mathematical and Computer Modelling, 49(3-4), 475–481.
Lazarevic, M. P., & Debeljkovi´c, D. L. (2008). Finite time stability analysis of linear autonomous fractional order systems with delayed state. Asian Journal of Control, 7(4), 440-447.
Hwang, C., & Cheng, Y. C. (2006). A numerical algorithm for stability testing of fractional delay systems. Automatica, 42(5), 825-831.
Oustaloup, A., Melchior, P., Lanusse, P., Dancla, F., & Crone, E. (2000). The CRONE toolbox for Matlab.In Proceedings of the 2000 IEEE International Symposium on Computer-Aided Control System Design. Anchorage, Alaska, USA September 25-27, 2000,190-195.
Melchior, P., Orsoni, B., Oustaloup, A., Cnrs, U. M. R., & Bordeaux, U. (2001). The CRONE toolbox for Matlab: Fractional Path Planning Design in Robotics.In Proceedings 10th IEEE International Workshop on Robot and Human Interactive Communication, ROMAN 2001 (Cat. No.01TH8591), 534-540.
Costa, D. V. and J. S. da. (2004). NINTEGER: A Non-Integer Control Toolbox for MATLAB. In Proceedings of 1st IFAC Workshop on Fractional Differentation and Its Applications, Bordeaux, France, 1-6.
Tepljakov, A., Petlenkov, E., & Belikov, J. (2011). FOMCON: Fractional-order modeling and control toolbox for MATLAB.Proceedings of the 18th International Conference Mixed Design of Integrated Circuits and Systems- MIXDES 2011, (4), 684-689.
Ozyetkin, M. M. (2018). A simple tuning method of fractional order PI- PDμcontrollers for time delay systems. ISA Transactions, 74, 77–87.
Ozyetkin, M. M., Yeroˇglu, C., Tan, N., & Taˇgluk, M. E. (2010). Design of PI and PID controllers for fractional order time delay systems. IFAC Proceedings Volumes (IFAC- PapersOnline), 43, 355-360.
Krishna, B. T. (2011). Studies on fractional order differentiators and integrators: A survey. Signal Processing, 91(3), 386-426.
Chen, Y. Q., Petr´aˇs, I., & Xue, D. (2009). Fractional order control - A tutorial. Proceedings of the American Control Conference, 1397-1411.
Silva, G. J., Datta, A., & Bhattacharyya, S. P. (2005). PID Controllers for Time-Delay Systems. Birkhauser Boston.
Walton, K., & Marshall, J. E. (1987). Direct method for TDS stability analysis.IEE Proceedings D Control Theory and Applications, 134(2), 101-107.
Ozturk, N. (1988). Stability Intervals for Delay Systems.In Proceedings of the 27th Conference on Decision and Control, Austin, Texas, 2215-2216.
Downloads
Published
How to Cite
Issue
Section
License
Articles published in IJOCTA are made freely available online immediately upon publication, without subscription barriers to access. All articles published in this journal are licensed under the Creative Commons Attribution 4.0 International License (click here to read the full-text legal code). This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.
Under the Creative Commons Attribution 4.0 International License, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles in IJOCTA, so long as the original authors and source are credited.
The readers are free to:
- Share — copy and redistribute the material in any medium or format
- Adapt — remix, transform, and build upon the material
- for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
under the following terms:
- Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
This work is licensed under a Creative Commons Attribution 4.0 International License.
