Nyquist-based stability analysis of non-commensurate fractional-order delay systems

Abstract As a generalization of the first and second order models, the elementary fractional-order models have been widely used in various engineering fields. However, most of the previous studies only focus on commensurate fractional-order models. In this paper, a general non-commensurate elementary fractional-order delay system is investigated. First, the stability of the studied fractional-order delay system is analyzed based on Nyquist theorem. Then, a series of sufficient stability conditions are presented for different combinations of parameters, including the fractional orders (α, β), time delay (τ), pseudo-damping factor (ζ), and natural frequency (ω0). Finally, three examples are given to show the effectiveness of the presented results.

[1]  R. Gorenflo,et al.  Time-fractional derivatives in relaxation processes: a tutorial survey , 2008, 0801.4914.

[2]  Dean Zhao,et al.  Finite-time stabilization for a class of high-order stochastic nonlinear systems with an output constraint , 2019, Appl. Math. Comput..

[3]  Igor Podlubny,et al.  On the regional controllability of the sub-diffusion process with Caputo fractional derivative , 2016 .

[4]  Lu Liu,et al.  Continuous fractional-order Zero Phase Error Tracking Control. , 2018, ISA transactions.

[5]  K. Moore,et al.  Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[6]  Lu Liu,et al.  Fractional-order partial pole assignment for time-delay systems based on resonance and time response criteria analysis , 2019, J. Frankl. Inst..

[7]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[8]  Ahmed S Elwakil,et al.  Fractional-order circuits and systems: An emerging interdisciplinary research area , 2010, IEEE Circuits and Systems Magazine.

[9]  Mohammad Saleh Tavazoei,et al.  Stability Preservation Analysis for Frequency-Based Methods in Numerical Simulation of Fractional Order Systems , 2008, SIAM J. Numer. Anal..

[10]  R. Armentano,et al.  Modeling the arterial wall mechanics using a novel high-order viscoelastic fractional element , 2015 .

[11]  Roberto Kawakami Harrop Galvão,et al.  Fractional Order Modeling of Large Three-Dimensional RC Networks , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[12]  Xavier Moreau,et al.  An overview of the CRONE approach in system analysis, modeling and identification, observation and control , 2008 .

[13]  X. Moreau,et al.  Stability and resonance conditions of second-order fractional systems , 2018 .

[14]  Thierry Poinot,et al.  LPV continuous fractional modeling applied to ultracapacitor impedance identification , 2015 .

[15]  Dumitru Baleanu,et al.  Lyapunov functions for Riemann-Liouville-like fractional difference equations , 2017, Appl. Math. Comput..

[16]  G. Yin,et al.  Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes , 2018 .

[17]  Junwei Lu,et al.  Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition , 2019, Appl. Math. Comput..

[18]  Xavier Moreau,et al.  Stability and resonance conditions of elementary fractional transfer functions , 2011, Autom..

[19]  Mihailo P. Lazarević,et al.  Modeling of Bioimpedance for Human Skin Based on Fractional Distributed- Order Modified Cole Model , 2014 .

[20]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[21]  Mohamed Aoun,et al.  Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind , 2015, Commun. Nonlinear Sci. Numer. Simul..

[22]  V. E. Tarasov.  Fractional hydrodynamic equations for fractal media , 2005, physics/0602096.

[23]  Serdar Ethem Hamamci.  An Algorithm for Stabilization of Fractional-Order Time Delay Systems Using Fractional-Order PID Controllers , 2007, IEEE Transactions on Automatic Control.

[24]  Qing Wang,et al.  Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions , 2016, Neurocomputing.

[25]  Lu Liu,et al.  Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment , 2019, International Journal of Robust and Nonlinear Control.

[26]  Chunhai Kou,et al.  Stability analysis by Krasnoselskii's fixed point theorem for nonlinear fractional differential equations , 2015, Appl. Math. Comput..

[27]  YangQuan Chen,et al.  Fractional-order exponential switching technique to enhance sliding mode control , 2017 .

[28]  Ivo Petras,et al.  Stability of Fractional-Order Systems , 2011 .

[29]  Yongguang Yu,et al.  Mittag-Leffler stability of fractional-order Hopfield neural networks , 2015 .

[30]  Lu Liu,et al.  Closed-loop time response analysis of irrational fractional-order systems with numerical Laplace transform technique , 2019, Appl. Math. Comput..

[31]  Dominik Sierociuk,et al.  Diffusion process modeling by using fractional-order models , 2015, Appl. Math. Comput..

[32]  Lu Liu,et al.  Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis , 2019, International Journal of Robust and Nonlinear Control.

[33]  I. Podlubny.  Fractional-order systems and PIλDμ-controllers , 1999, IEEE Trans. Autom. Control..

[34]  Igor Podlubny,et al.  Mittag-Leffler stability of fractional order nonlinear dynamic systems , 2009, Autom..

[35]  António M. Lopes,et al.  Milk Characterization Using Electrical Impedance Spectroscopy and Fractional Models , 2018, Food Analytical Methods.

[36]  S. Das,et al.  Fractional Order Modeling of a PHWR Under Step-Back Condition and Control of Its Global Power With a Robust ${\rm PI}^{\lambda} {\rm D} ^{\mu}$ Controller , 2011, IEEE Transactions on Nuclear Science.