Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems

Abstract This paper presents two new lemmas related to the Caputo fractional derivatives, when α ∈ 0 , 1 , for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results.

[1]  Manuel A Duarte-Mermoud,et al.  Fractional adaptive control for an automatic voltage regulator. , 2013, ISA transactions.

[2]  Noureddine Elalami,et al.  Application of fractional adaptive High-Gain controller to a LEO (Low Earth Orbit) satellite , 2009, 2009 International Conference on Computers & Industrial Engineering.

[3]  D. C. Dodder.  R-Matrix Analysis , 1976 .

[4]  Alain Oustaloup,et al.  Fractional system identification for lead acid battery state of charge estimation , 2006, Signal Process..

[5]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[6]  Nasser Sadati,et al.  Design of a fractional order PID controller for an AVR using particle swarm optimization , 2009 .

[7]  Dumitru Baleanu,et al.  Stability of q-fractional non-autonomous systems , 2013 .

[8]  Blas M. Vinagre,et al.  A Fractional Adaptation Scheme for Lateral Control of an AGV , 2008 .

[9]  D. Baleanu,et al.  On the Stability of Some Discrete Fractional Nonautonomous Systems , 2012 .

[10]  Manuel A. Duarte-Mermoud,et al.  Lyapunov functions for fractional order systems , 2014, Commun. Nonlinear Sci. Numer. Simul..

[11]  Tianzeng Li,et al.  Stability Analysis of Fractional-Order Nonlinear Systems with Delay , 2014 .

[12]  Wei Jiang,et al.  Uniform stability of fractional neutral systems: a Lyapunov-Krasovskii functional approach , 2013, Advances in Difference Equations.

[13]  Samir Ladaci,et al.  Fractional order adaptive high-gain controllers for a class of linear systems , 2008 .

[14]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[15]  Samir Ladaci,et al.  On Fractional Adaptive Control , 2006 .

[16]  Anuradha M. Annaswamy,et al.  Stable Adaptive Systems , 1989 .

[17]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[18]  YaLi He,et al.  Application of Fractional-order Model Reference Adaptive Control on Industry Boiler Burning System , 2010, 2010 International Conference on Intelligent Computation Technology and Automation.

[19]  Y. Q. Chen,et al.  Using Fractional Order Adjustment Rules and Fractional Order Reference Models in Model-Reference Adaptive Control , 2002 .

[20]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[21]  Thierry Poinot,et al.  Fractional modelling and identification of thermal systems , 2011, Signal Process..

[22]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.