Fractional Order Unknown Inputs Fuzzy Observer for Takagi–Sugeno Systems with Unmeasurable Premise Variables

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO.

[1]  Rachid Mansouri,et al.  Multi-model identification of a fractional non linear system , 2013 .

[2]  Krzysztof J. Latawiec,et al.  Modeling of discrete-time fractional-order state space systems using the balanced truncation method , 2017, J. Frankl. Inst..

[3]  Jinde Cao,et al.  Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control , 2019, Mathematics.

[4]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[5]  Hui Li,et al.  State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems , 2016, Signal Process..

[6]  X. Liao,et al.  Observer-based fuzzy control for nonlinear fractional-order systems via fuzzy T-S models: The 1 < α < 2 case , 2014 .

[7]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[8]  Saïd Mammar,et al.  How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: Dynamic extension approach , 2018, Eng. Appl. Artif. Intell..

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

[10]  Yiming Fang,et al.  Fractional-disturbance-observer-based Sliding Mode Control for Fractional Order System with Matched and Mismatched Disturbances , 2019, International Journal of Control, Automation and Systems.

[11]  Mehrdad Saif,et al.  Estimation and Fault Diagnosis of Lithium-Ion Batteries: A Fractional-Order System Approach , 2018, Mathematical Problems in Engineering.

[12]  Liping Wen,et al.  Rational Function Approximation for Fractional Order Differential and Integral Operators , 2011 .

[13]  Dumitru Baleanu,et al.  Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach , 2014, Commun. Nonlinear Sci. Numer. Simul..

[14]  Wenqiang Ji,et al.  Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems , 2013 .

[15]  Chun-Lai Li,et al.  Robust control for fractional-order four-wing hyperchaotic system using LMI , 2013 .

[16]  Salim Ibrir,et al.  New sufficient conditions for observer-based control of fractional-order uncertain systems , 2015, Autom..

[17]  Vicente Feliu,et al.  On Fractional PID Controllers: A Frequency Domain Approach , 2000 .

[18]  Eric A. Butcher,et al.  Optimal observer-based feedback control for linear fractional-order systems with periodic coefficients , 2019, Journal of Vibration and Control.

[19]  Xiaojian Li,et al.  Adaptive fault estimation for T-S fuzzy systems with unmeasurable premise variables , 2018 .

[20]  Tao Yu,et al.  Passivity-based fractional-order sliding-mode control design and implementation of grid-connected photovoltaic systems , 2018 .

[21]  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.

[22]  Mohamed-Faouzi Harkat,et al.  A new approach for state estimation of uncertain multiple model with unknown inputs. Application to sensor fault diagnosis , 2016 .

[23]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[24]  Piotr Ostalczyk,et al.  Discrete Fractional Calculus: Applications In Control And Image Processing , 2015 .

[25]  Yong Wang,et al.  Frequency domain subspace identification of commensurate fractional order input time delay systems , 2011 .

[26]  Ioannis K. Argyros,et al.  Intelligent Numerical Methods: Applications to Fractional Calculus , 2015, Studies in Computational Intelligence.

[27]  J. Ragot,et al.  Estimation of State and Unknown Inputs of a Nonlinear System Represented by a Multiple Model , 2004 .

[28]  Dorota Mozyrska,et al.  The -Transform Method and Delta Type Fractional Difference Operators , 2015 .

[29]  Yun Zhang,et al.  A New State of Charge Estimation Method for Lithium-Ion Battery Based on the Fractional Order Model , 2019, IEEE Access.

[30]  Ammar A. Aldair,et al.  Design of Fractional Order controller based on Evolutionary Algorithm for a full vehicle nonlinear active suspension systems , 2010 .

[31]  Jang-Hyun Park,et al.  Approximation-Free Output-Feedback Non-Backstepping Controller for Uncertain SISO Nonautonomous Nonlinear Pure-Feedback Systems , 2019 .

[32]  Li Junmin,et al.  Robust Stability and Stabilization of Fractional Order Systems Based on Uncertain Takagi-Sugeno Fuzzy Model With the Fractional Order 1≤v < 2 , 2013 .

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

[34]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[35]  Jun-Guo Lu,et al.  Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order $\alpha$: The $0≪\alpha≪1$ Case , 2010, IEEE Transactions on Automatic Control.

[36]  M. Bettayeb,et al.  Observation and sliding mode observer for nonlinear fractional-order system with unknown input. , 2016, ISA transactions.

[37]  Mohamed Darouach,et al.  Robust stabilization of uncertain descriptor fractional-order systems , 2013, Autom..

[38]  S. Ibrir.  Robust state estimation with q-integral observers , 2004, Proceedings of the 2004 American Control Conference.

[39]  Hamid Reza Karimi,et al.  Robust Observer Design for Unknown Inputs Takagi–Sugeno Models , 2013, IEEE Transactions on Fuzzy Systems.

[40]  Feng Ding,et al.  A Hierarchical Approach for Joint Parameter and State Estimation of a Bilinear System with Autoregressive Noise , 2019, Mathematics.

[41]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

[42]  Behrooz Safarinejadian,et al.  Simultaneous state estimation and parameter identification in linear fractional order systems using coloured measurement noise , 2016, Int. J. Control.

[43]  Saeid Nahavandi,et al.  Observer design for positive fractional-order interconnected time-delay systems , 2019, Trans. Inst. Meas. Control.

[44]  Mathieu Moze,et al.  Pseudo state feedback stabilization of commensurate fractional order systems , 2009, 2009 European Control Conference (ECC).

[45]  Mohammed Chadli,et al.  LMI Solution for Robust Static Output Feedback Control of Discrete Takagi–Sugeno Fuzzy Models , 2012, IEEE Transactions on Fuzzy Systems.

[46]  Bing Liu,et al.  Observer design for a class of nonlinear fractional-order systems with unknown input , 2017, J. Frankl. Inst..

[48]  Alain Oustaloup,et al.  A Lyapunov approach to the stability of fractional differential equations , 2009, Signal Process..

[49]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[50]  Thierry-Marie Guerra,et al.  Nonquadratic Stabilization Conditions for a Class of Uncertain Nonlinear Discrete Time TS Fuzzy Models: A New Approach , 2008, IEEE Transactions on Automatic Control.

[51]  Mohamed Darouach,et al.  Design of unknown input fractional-order observers for fractional-order systems , 2013, Int. J. Appl. Math. Comput. Sci..

[52]  Peng Yang,et al.  Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form , 2006, Autom..