# Stability Analysis of Fractional-Order Systems

### Summary:

Fractional calculus is a mathematical area introduced as an extension of traditional calculus in the eighteenth century. It generally deals with derivatives and integrals of both integer and non-integer orders. Over the past few decades, fractional calculus has been employed increasingly within various research disciplines to model and design dynamical systems, leading to the emergence of the field of fractional-order systems. In fact, many real world processes have proved to be of fractional nature. To name a few, fractional calculus can be applied to modeling heat conduction, finance systems, and the dynamics of viscoelastic materials, transmission lines, and electrical capacitors. A dynamical system involving fractional calculus is referred to as a fractional-order system.

Of the most important properties of any dynamical system is their stability. In this project, we exclusively deal with the stability properties of fractional-order systems. Our goal in this project is to characterize the stability criteria – the safe zone – of different fractional-order systems, based on the system parameters. Our specific contributions throughout this research are as follows:

• Stability preservation of fractional-order systems through frequency-based approximations [3, 5]: In practice, to computationally analyze fractional-order operators, we must approximate them by integer-order operators. Thus, a comprehensive comparison between the stability criteria of a fractional-order system and its integer-order approximation is crucial, that constitutes our objective in [3, 5]. To achieve our goal, we consider a general commensurate fractional-order system, with the derivative order between 0 and 1, and an approximation of it via a well-known frequency-based method. We argue that for small frequencies, i.e., those close to 0, the stability criterion of the original system is a subset of that of the approximating one, which means an unstable system may become stable through approximation. We further discuss explicitly the intermediate and high frequencies and show particularly that the two stability criteria in these cases do not yield an inclusion relation.
• Stability analysis of fractional-order Van der Pol system : It was previously observed that the fractional-order Van der Pol system could demonstrate oscillating behavior. However, the oscillations range of the fractional-order Van der Pol system remained to be determined. In , we fully address this problem by obtaining a succinct inequality, depending on the system parameters, that precisely determines the oscillation range of the system. Furthermore, we show that if the oscillations of the fractional-order Van der Pol system are of regular type, the resulting limit cycle is not unique and indeed varies according to the initial conditions. This characteristic of fractional-order Van der Pol system is of particular interest as it opposes that of its integer-order counterpart where the limit cycle is unique. We finally verified the results using numerical examples.
• The relationship between the inner dimension and the maximum number of frequencies in fractional-order systems [1, 4, 6]: For a general LTI fractional-order system in the Pseudo state space form where the fractional derivatives are rational numbers between 0 and 1, we investigate in [1, 4, 6] the maximum number of generated frequencies given the inner dimension of the system. More specifically, inspired by the restricted difference bases concept, we develop a novel multi-frequency oscillatory fractional order system that led to a non-trivial lower bound on the maximum of number of generated frequencies given the inner dimension of the system. We further establish a number of other lower and upper bounds on that number and verified the effectiveness of the results via numerical examples.

### Reference

1. M.S. Tavazoei, M. Haeri, M. Siami, S. Bolouki. “Bounds for Maximum Number of Frequencies which Can Exist in Oscillations Generated by Fractional Order LTI Systems.” IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4003-4012. August 2010.
2. M.S. Tavazoei, M. Haeri, M. Attari, S. Bolouki, M. Siami. “More Details on Analysis of Fractional-Order Van Der Pol Oscillator.” Journal of Vibration and Control, vol. 15, no. 6, pp. 803-819. February 2009.
3. M.S. Tavazoei, M. Haeri, S. Bolouki, M. Siami. “Stability Preservation Analysis for Frequency Based Methods in Numerical Simulation of Fractional Order Systems.” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 321-338. October 2008.
4. S. Bolouki, M. Haeri, M.S. Tavazoei, M. Siami. “Upper and Lower Bounds for the Maximum Number of Frequencies Exist in Oscillations Generated by Fractional Order Systems.” 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey. November 2008.
5. M.S. Tavazoei, M. Haeri, M. Siami, S. Bolouki. “Stability Preservation Problem in Methods Finding Rational Approximation of Fractional Order Systems.” 6th EUROMECH Nonlinear Dynamics Conference (ENOC 2008), St. Petersburg, Russia. June 2008.
6. S. Bolouki, M. Haeri, M.S. Tavazoei, M. Siami. “Some Bounds on Maximum Number of Frequencies Existing in Oscillations Produced by Linear Fractional Order Systems.” New Trends in Nanotechnology and Fractional Calculus Applications, 3, pp. 213-220, Springer, DOI: 10.1007/978-90-481-3293-5_17. October 2009.