Research Papers

Numerical Stability Analysis of Linear Incommensurate Fractional Order Systems

[+] Author and Article Information
Sambit Das

Department of Mechanical Engineering,
IIT Kharagpur 721302, India
e-mail: sambitiitkgp@gmail.com

Anindya Chatterjee

Department of Mechanical Engineering,
IIT Kanpur 208016,India
e-mail: anindya100@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received November 9, 2012; final manuscript received March 5, 2013; published online xx xx, xxxx. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(4), 041012 (May 31, 2013) (6 pages) Paper No: CND-12-1196; doi: 10.1115/1.4023966 History: Received November 09, 2012; Revised March 05, 2013

We present a method for detecting right half plane (RHP) roots of fractional order polynomials. It is based on a Nyquist-like criterion with a system-dependent contour which includes all RHP roots. We numerically count the number of origin encirclements of the mapped contour to determine the number of RHP roots. The method is implemented in Matlab, and a simple code is given. For validation, we use a Galerkin based strategy, which numerically computes system eigenvalues (Matlab code is given). We discuss how, unlike integer order polynomials, fractional order polynomials can sometimes have exponentially large roots. For computing such roots we suggest using asymptotics, which provide intuition but require human inputs (several examples are given).

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Grahic Jump Location
Fig. 1

Clockwise oriented contour on complex s plane that includes all RHP zeros of f(s)

Grahic Jump Location
Fig. 2

This closed loop system gives the same output as the open loop system of Eq. (7), provided we take G(s)=1/ansαn and H(s)=a0+∑k=1n-1aksαk




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