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Research Papers

# Robust Stability Analysis of Uncertain Linear Fractional-Order Systems With Time-Varying Uncertainty for 0 < α < 2

[+] Author and Article Information

School of Electrical and Computer Engineering,
Shiraz University,
Shiraz, Iran

School of Electrical and Computer Engineering,
Shiraz University,
Shiraz, Iran
e-mail: asemani@shirazu.ac.ir

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received September 14, 2018; final manuscript received September 26, 2018; published online November 8, 2018. Assoc. Editor: Mazen Farhood.

J. Dyn. Sys., Meas., Control 141(3), 031005 (Nov 08, 2018) (8 pages) Paper No: DS-17-1302; doi: 10.1115/1.4041607 History: Received September 14, 2018; Revised September 26, 2018

## Abstract

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order $0<α<2$, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with $0<α<1$ and $1<α<2$ to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

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## References

Monje, C. A. , Chen, Y. Q. , Vinagre, B. M. , Xue, D. , and Feliu-Batlle, V. , 2010, Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London.
Caponetto, R. , Dongola, G. , Fortuna, L. , and Petr, I. , 2010, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Tavazoei, M. S. , and Haeri, M. , 2010, “ Stabilization of Unstable Fixed Points of Fractional-Order Systems by Fractional-Order Linear Controllers and Its Applications in Suppression of Chaotic Oscillations,” ASME J. Dyn. Syst. Meas. Control, 132(2), p. 021008.
Rajagopal, K. , Vaidyanathan, S. , Karthikeyan, A. , and Duraisamy, P. , 2017, “ Dynamic Analysis and Chaos Suppression in a Fractional-Order Brushless DC Motor,” Electr. Eng., 99(2), pp. 721–733.
Bhalekar, S. , and Daftardar-Gejji, V. , 2016, “ Chaos in Fractional Order Financial Delay System,” Comput. Math. Appl. (in press).
Chen, X. , Chen, Y. , Zhang, B. , and Qiu, D. , 2016, “ A Method of Modeling and Analysis for Fractional-Order dc-dc Converters,” IEEE Trans. Power Electron, 32(9), pp. 7034–7044.
Tavakoli-Kakhki, M. , and Haeri, M. , 2011, “ Temperature Control of a Cutting Process Using Fractional Order Proportional-Integral-Derivative Controller,” ASME J. Dyn. Syst. Meas. Control, 133(5), p. 051014.
Tang, Y. , Wang, Y. , Han, M. , and Lian, Q. , 2016, “ Adaptive Fuzzy Fractional-Order Sliding Mode Controller Design for Antilock Braking Systems,” ASME J. Dyn. Syst. Meas. Control, 138(4), p. 041008.
Nataraj, P. S. V. , and Tharewal, S. , 2006, “ On Fractional-Order QFT Controllers,” ASME J. Dyn. Syst. Meas. Control, 129(2), pp. 212–218.
Matignon, D. , 1996, “ Stability Results for Fractional Differential Equations With Applications to Control Processing,” Comput. Eng. Syst. Appl., 2, pp. 963–968.
Tavazoei, M. S. , and Haeri, M. , 2009, “ A Note on the Stability of Fractional Order Systems,” Math. Comput. Simul., 79(5), pp. 1566–1577.
Sabatier, J. , Moze, M. , and Farges, C. , 2010, “ LMI Stability Conditions for Fractional Order Systems,” Comput. Math. Appl., 59(5), pp. 1594–1609.
Dadras, S. , Dadras, S. , Malek, H. , and Chen, Y. Q. , 2017, “ A Note on the Lyapunov Stability of Fractional Order Nonlinear Systems,” ASME Paper No. DETC2017-68270.
Shen, J. , and Lam, J. , “ Stability and Performance Analysis for Positive Fractional-Order Systems With Time-Varying Delays,” IEEE Trans. Autom. Control, 61(9), pp. 2676–2681.
Chen, Y. Q. , Ahn, H. S. , and Podlubny, I. , 2006, “ Robust Stability Check of Fractional Order Linear Time Invariant Systems With Interval Uncertainties,” Signal Process., 86(10), pp. 2611–2618.
Ahn, H. S. , and Chen, Y. Q. , 2008, “ Necessary and Sufficient Stability Condition of Fractional-Order Interval Linear Systems,” Automatica, 44(11), pp. 2985–2988.
Alagoz, B. B. , Yeroglu, C. , Senol, B. , and Ates, A. , 2015, “ Probabilistic Robust Stabilization of Fractional Order Systems With Interval Uncertainty,” ISA Trans., 57, pp. 101–110. [PubMed]
Gao, Z. , 2017, “ Robust Stabilization of Interval Fractional-Order Plants With One Time-Delay by Fractional-Order Controllers,” J. Franklin Inst., 354(2), pp. 767–786.
Lan, Y.-H. , and Zhou, Y. , 2011, “ LMI-Based Robust Control of Fractional-Order Uncertain Linear Systems,” Comput. Math. Appl., 62(3), pp. 1460–1471.
Ji, Y. , and Qiu, J. , 2015, “ Stabilization of Fractional-Order Singular Uncertain Systems,” ISA Trans., 56, pp. 53–64. [PubMed]
Qian, D. , Li, C. , Agarwal, R. P. , and Wong, P. J. Y. , 2010, “ Stability Analysis of Fractional Differential System With Riemann–Liouville Derivative,” Math. Comput. Modell., 52(5–6), pp. 862–874.
Alavian-Shahri, E. , Alfi, A. , and Machado, T. , 2017, “ Robust Stability and Stabilization of Uncertain Fractional Order Systems Subject to Input Saturation,” J. Vib. Control, 24(16), pp. 3676–3683.
Wu, M. Y. , 1974, “ A Note on Stability of Linear Time-Varying Systems,” IEEE Trans. Autom. Control, 19(2), pp. 162–164.
Levin, B. J. , 1964, Distribution of Zeros of Entire Functions, Translation of Mathematical Monograph, Vol. 5, Providence, RI.
Corduneanu, C. , 1971, Principles of Differential and Integral Equations, Allyn and Bacon, Boston, MA.
Wen, X. J. , 2008, “ Stability Analysis of a Class of Nonlinear Fractional-Order Systems,” IEEE Trans. Circuits Syst. II, 55(11), pp. 1178–1182.
Bellman, R. , 1960, Introduction to Matrix Analysis, McGraw-Hill, New York.
Ait-Rami, M. , El-faiz, S. , Benzaouia, A. , and Tadeo, F. , 2009, “ Robust Exact Pole Placement Via an LMI-Based Algorithm,” IEEE Trans. Autom. Control, 54(2), pp. 394–398.
Soltan, A. , Radwan, G. A. , and Soliman, M. A. , 2015, “ Fractional Order Sallen–Key and KHN Filters: Stability and Poles Allocation,” Circuits, Syst., Signal Process., 34(5), pp. 1461–1480.

## Figures

Fig. 1

The plot of |E0.3,0.3( t∡arg(−4.064+2.623i)) | for t≥0

Fig. 2

The plot of |t∡arg(−4.064+2.623i)E0.3,0.3( t∡arg(−4.064+2.623i)) | for t≥0

Fig. 3

The non-smooth uncertainty function S(t)

Fig. 4

The response of system (40) in the presence of time-varying uncertainty S(t)

Fig. 5

The plot of |E1.4,1.4( t∡arg(−4.064+2.623i)) | for t≥0

Fig. 6

The plot of |t∡arg(−4.064+2.623i)E1.4,1.4( t∡arg(−4.064+2.623i)) | for t≥0

Fig. 7

The response of system (40) in the presence of time-varying uncertainty Δ(t)=0.6 (1−e−10 t)I3

Fig. 8

The circuit model of Sallen–Key filter [30]

Fig. 9

The plot of |E0.9,0.9( t∡arg(−0.5+1.936i)) | for t≥0

Fig. 10

The plot of |t∡arg(−0.5+1.936i)E0.9,0.9( t∡arg(−0.5+1.936i)) | for t≥0

Fig. 11

The zero-input response of system (44) in the presence of time-varying uncertainty ΔR(t)

Fig. 12

The nonsmooth uncertainty ΔR(t)

## Errata

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