[Paper Review] Linearized stability analysis of Caputo-Katugampola fractional-order nonlinear systems

D. Boucenna, Abdellatif Ben Makhlouf+3 more|arXiv (Cornell University)|Mar 10, 2018

Advanced Control Systems Design4 citations

TL;DR

This paper establishes a linearized asymptotic stability criterion for Caputo-Katugampola fractional-order nonlinear systems with order $\alpha \in (0,1)$ and shape parameter $\rho > 0$. By transforming the linear part into a nearly diagonal form and constructing a Lyapunov-Perron operator, the authors prove that if the linearized system is asymptotically stable (i.e., all eigenvalues of matrix $A$ satisfy $|\arg(\lambda)| > \alpha\pi/2$), then the nonlinear system inherits this stability. The result extends prior work on Caputo fractional systems to the more general Caputo-Katugampola framework.

ABSTRACT

In this paper, a linearized asymptotic stability result for a Caputo-Katugampola fractional-order systems is described. An application is given to demonstrate the validity of the proposed results.

Motivation & Objective

• To extend linearized stability theory from Caputo to Caputo-Katugampola fractional-order systems.
• To establish sufficient conditions under which the zero solution of a nonlinear Caputo-Katugampola system is asymptotically stable.
• To generalize existing results for $\rho = 1$ to arbitrary $\rho > 0$, broadening applicability to a wider class of fractional dynamics.
• To validate the theoretical framework via an application to a fractional-order Lorenz system with linear feedback control.

Proposed method

• Transform the system matrix $A$ into a form close to diagonal via similarity transformation to simplify stability analysis.
• Define a Lyapunov-Perron operator that maps solutions of the nonlinear system to fixed points, enabling stability analysis via fixed-point theory.
• Use the Caputo-Katugampola fractional derivative to model the system dynamics, incorporating both fractional order $\alpha$ and shape parameter $\rho$.
• Apply the semigroup property and properties of the Mittag-Leffler function to estimate solution norms and control growth.
• Establish a bound on the Lipschitz constant $\ell_f(r)$ of the nonlinear term $f(x)$ to ensure convergence to zero.
• Verify stability by showing that the limit superior of the solution norm is bounded by a contraction factor less than one.

Experimental results

Research questions

• RQ1 Under what conditions is the zero solution of a Caputo-Katugampola fractional-order nonlinear system asymptotically stable?
• RQ2 How does the inclusion of the shape parameter $\rho$ affect the stability criteria compared to the classical Caputo case ($\rho = 1$)?
• RQ3 Can the linearized stability result for Caputo systems be extended to the more general Caputo-Katugampola derivative framework?
• RQ4 What is the role of the Lyapunov-Perron operator in proving asymptotic stability for nonlinear fractional systems?
• RQ5 How can the theoretical stability condition be applied and verified in a concrete nonlinear system such as the fractional Lorenz system?

Key findings

• The zero solution of the nonlinear Caputo-Katugampola system is asymptotically stable if all eigenvalues of the matrix $A$ satisfy $|\arg(\lambda)| > \alpha\pi/2$, extending the classical stability condition to the generalized derivative.
• The theoretical framework successfully proves asymptotic stability for the closed-loop Caputo-Katugampola fractional-order Lorenz system with $\alpha = 0.9$, $\rho = 1.2$, and a linear feedback controller.
• The eigenvalues of the closed-loop system matrix $A + BK$ are $\lambda_1 = -2.8229$, $\lambda_2 = -48.1771$, and $\lambda_3 = -3$, all satisfying $|\arg(\lambda_i)| > \alpha\pi/2 \approx 1.4137$ radians.
• The Lipschitz constant $\ell_h(r)$ of the nonlinear term in the Lorenz system is sufficiently small such that $\ell_h(r)C(\alpha,\lambda) < 1$, ensuring contraction and convergence to zero.
• The application demonstrates that the proposed method is effective in stabilizing chaotic fractional-order systems through linear feedback control.
• The proof technique, based on the Lyapunov-Perron operator and norm estimation using Mittag-Leffler functions, provides a robust analytical pathway for stability in nonlinear fractional systems.