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Research Papers

Robust Stability Analysis of Distributed-Order Linear Time-Invariant Systems With Uncertain Order Weight Functions and Uncertain Dynamic Matrices

[+] Author and Article Information
Hamed Taghavian

Electrical Engineering Department,
Sharif University of Technology,
Tehran 1458889694, Iran

Mohammad Saleh Tavazoei

Electrical Engineering Department,
Sharif University of Technology,
Tehran 1458889694, Iran
e-mail: tavazoei@sharif.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 27, 2016; final manuscript received June 12, 2017; published online August 28, 2017. Assoc. Editor: Yunjun Xu.

J. Dyn. Sys., Meas., Control 139(12), 121010 (Aug 28, 2017) (9 pages) Paper No: DS-16-1330; doi: 10.1115/1.4037268 History: Received June 27, 2016; Revised June 12, 2017

Bounded-input bounded-output (BIBO) stability of distributed-order linear time-invariant (LTI) systems with uncertain order weight functions and uncertain dynamic matrices is investigated in this paper. The order weight function in these uncertain systems is assumed to be totally unknown lying between two known positive bounds. First, some properties of stability boundaries of fractional distributed-order systems with respect to location of eigenvalues of dynamic matrix are proved. Then, on the basis of these properties, it is shown that the stability boundary of distributed-order systems with the aforementioned uncertain order weight functions is located in a certain region on the complex plane defined by the upper and lower bounds of the order weight function. Thereby, sufficient conditions are obtained to ensure robust stability in distributed-order LTI systems with uncertain order weight functions and uncertain dynamic matrices. Numerical examples are presented to verify the obtained results.

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Figures

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Fig. 1

Functions hc(a) and hs(a) that cut each other at the point (0,2/π)

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Fig. 2

The segment line joining points Ei and Fi placed on the curve cl

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Fig. 3

Stability boundaries associated with sample weight functions (58), which are surrounded by the curves obtained from Theorem 4.1

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Fig. 4

The location of eigenvalues of some random matrices satisfying Eq. (48) with constant matrices (59), which are placed in a circle obtained from Theorem 4.2. Placing these eigenvalues in the left side of curve cl confirms BIBO stability of the considered uncertain distributed-order system.

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Fig. 5

Step responses of distributed-order systems with various weight functions and random dynamic matrices fulfilling assumptions considered in Example 1

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Fig. 6

Step responses of distributed-order systems in the form Eq. (60) with various constant weight functions satisfying Eq. (61) and several random parameters a and b satisfying the constraints addressed in Example 2

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