Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems

Adaptive and non-adaptive minimal realization (MR) fractional order observers (FOO) for linear time-invariant systems (LTIS) of a possibly different derivation order (mixed order observers, MOO) are studied in this paper. Conditions on the convergence and robustness are provided using a general framework which allows observing systems defined with any type of fractional order derivative (FOD). A qualitative discussion is presented to show that the derivation orders of the observer structure and for the parameter adjustment are relevant degrees of freedom for performance optimization. A control problem is developed to illustrate the application of the proposed observers.

[1]  D. Matignon,et al.  Some Results on Controllability and Observability of Finite-dimensional Fractional Differential Systems , 1996 .

[2]  Zhiqiang Gao,et al.  On the augmentation of Luenberger Observer-based state feedback design for better robustness and disturbance rejection , 2015, 2015 American Control Conference (ACC).

[3]  Manuel A. Duarte-Mermoud,et al.  On the Lyapunov theory for fractional order systems , 2016, Appl. Math. Comput..

[4]  M. Duarte-Mermoud,et al.  Robustness and convergence of fractional systems and their applications to adaptive schemes , 2016, 1609.05544.

[5]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[6]  Hassan Hammouri,et al.  Observer-based approach to fault detection and isolation for nonlinear systems , 1999, IEEE Trans. Autom. Control..

[7]  M. Rapaić,et al.  Variable-Order Fractional Operators for Adaptive Order and Parameter Estimation , 2014, IEEE Transactions on Automatic Control.

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

[9]  Rachid Malti,et al.  A note on ℒpℒp-norms of fractional systems , 2013, Autom..

[10]  Christophe Farges,et al.  On Observability and Pseudo State Estimation of Fractional Order Systems , 2012, Eur. J. Control.

[11]  S. Shankar Sastry,et al.  Global stability proofs for continuous-time indirect adaptive control schemes , 1987 .

[12]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[13]  M. Duarte-Mermoud,et al.  Robust mixed order backstepping control of non‐linear systems , 2018, IET Control Theory & Applications.

[14]  H. Oya,et al.  Observer-based robust control giving consideration to transient behavior for linear systems with structured uncertainties , 2002, IEEE 2002 28th Annual Conference of the Industrial Electronics Society. IECON 02.

[15]  Alain Oustaloup,et al.  How to impose physically coherent initial conditions to a fractional system , 2010 .

[16]  Hiromitsu Ohmori,et al.  State and parameter estimation of lithium-ion battery by Kreisselmeier-type adaptive observer for fractional calculus system , 2015, 2015 54th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE).

[17]  Yong-Hong Lan,et al.  Observer-based robust control of a (1⩽ a ≪ 2) fractional-order uncertain systems: a linear matrix inequality approach , 2012 .

[18]  M. Duarte-Mermoud,et al.  Convergence of fractional adaptive systems using gradient approach. , 2017, ISA transactions.

[19]  J. Partington,et al.  Coprime factorizations and stability of fractional differential systems , 2000 .

[20]  H. T. Tuan,et al.  Stability of fractional‐order nonlinear systems by Lyapunov direct method , 2017, IET Control Theory & Applications.

[21]  Jingcheng Wang,et al.  Robust adaptive observer for fractional order nonlinear systems: An LMI approach , 2014, The 26th Chinese Control and Decision Conference (2014 CCDC).

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

[23]  Manuel A. Duarte-Mermoud,et al.  On fractional extensions of Barbalat Lemma , 2015, Syst. Control. Lett..

[24]  Jinde Cao,et al.  Observer Design for Tracking Consensus in Second-Order Multi-Agent Systems: Fractional Order Less Than Two , 2017, IEEE Transactions on Automatic Control.

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

[26]  D. Luenberger.  Observers for multivariable systems , 1966 .

[27]  I. Podlubny.  Fractional differential equations , 1998 .

[28]  Sara Dadras,et al.  A New Fractional Order Observer Design for Fractional Order Nonlinear Systems , 2011 .

[29]  Manuel Duarte Ortigueira,et al.  On the initial conditions in continuous-time fractional linear systems , 2003, Signal Process..

[30]  S. Sastry,et al.  Adaptive Control: Stability, Convergence and Robustness , 1989 .

[31]  Driss Boutat,et al.  Nonasymptotic Pseudo-State Estimation for a Class of Fractional Order Linear Systems , 2017, IEEE Transactions on Automatic Control.

[32]  G. Kreisselmeier.  Adaptive observers with exponential rate of convergence , 1977 .

[33]  Manuel A. Duarte-Mermoud,et al.  Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems , 2015, Commun. Nonlinear Sci. Numer. Simul..

[34]  Yiheng Wei,et al.  On fractional order adaptive observer , 2015, Int. J. Autom. Comput..