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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
Mohammad Tavazoei

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

Mohammad Hassan Asemani

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

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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Figures

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
Fig. 3

The non-smooth uncertainty function S(t)

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
Fig. 8

The circuit model of Sallen–Key filter [30]

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 12

The nonsmooth uncertainty ΔR(t)

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
Fig. 10

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

Grahic Jump Location
Fig. 11

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

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