Mixed order robust adaptive control for general linear time invariant systems

Abstract We provide a solution to the adaptive control problem of an unknown linear system of a given derivation order, using a reference model or desired poles defined in a possibly different derivation order and employing continuous adjustment of parameters ruled by possibly another different derivation order. To this purpose, we present an extension for the fractional settings of the Bezout’s lemma and gradient steepest descent adjustment. We analyze both the direct and indirect approaches to adaptive control. We discuss some robustness advantages/disadvantages of the fractional adjustment of parameters in comparison with the integer one and, through simulations, the possibility to define optimal derivation order controllers.

[1]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[2]  姚建勇,et al.  Robust Adaptive Asymptotic Tracking Control of A Class of Nonlinear Systems with Unknown Input Dead-Zone , 2015 .

[3]  Samir Ladaci,et al.  Direct fractional adaptive pole placement control for minimal phase systems , 2015, 2015 16th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA).

[5]  Ma Qian,et al.  Factional order model reference adaptive control based on Lyapunov stability theory , 2016, 2016 35th Chinese Control Conference (CCC).

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

[7]  W. T. Tutte,et al.  Encyclopedia of Mathematics and its Applications , 2001 .

[8]  Huijun Gao,et al.  Disturbance Observer-Based Adaptive Tracking Control With Actuator Saturation and Its Application , 2016, IEEE Transactions on Automation Science and Engineering.

[9]  K. Narendra,et al.  Robust adaptive control in the presence of bounded disturbances , 1986 .

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

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

[12]  Yang Yi,et al.  Adaptive control of output feedback nonlinear systems with unmodeled dynamics and output constraint , 2017, J. Frankl. Inst..

[13]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[14]  Maria Adler,et al.  Stable Adaptive Systems , 2016 .

[15]  Baris Baykant Alagoz,et al.  Implementation of Model Reference Adaptive Controller with Fractional Order Adjustment Rules for Coaxial Rotor Control Test System , 2016 .

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

[17]  Yongping Pan,et al.  Adaptive Fuzzy Backstepping Control of Fractional-Order Nonlinear Systems , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[18]  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.

[19]  Bao Shi,et al.  On Fractional Model Reference Adaptive Control , 2014, TheScientificWorldJournal.

[20]  Jianbin Qiu,et al.  Adaptive Neural Control of Stochastic Nonlinear Time-Delay Systems With Multiple Constraints , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[21]  Xin-Ping Guan,et al.  Stability analysis for fractional-order PD controlled delayed systems , 2016, J. Frankl. Inst..

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

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

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

[25]  A. Alikhanov.  A priori estimates for solutions of boundary value problems for fractional-order equations , 2010, 1105.4592.

[26]  Samir Ladaci,et al.  Indirect fractional order pole assignment based adaptive control , 2016 .

[27]  N. D. Cong,et al.  Asymptotic Stability of Linear Fractional Systems with Constant Coefficients and Small Time-Dependent Perturbations , 2016, 1601.06538.

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

[29]  Zhe Gao,et al.  Robust stabilization criterion of fractional-order controllers for interval fractional-order plants , 2015, Autom..

[30]  Chao Yang,et al.  Distributed containment control of fractional-order uncertain multi-agent systems , 2016, J. Frankl. Inst..

[31]  Jianbin Qiu,et al.  Recent Advances on Fuzzy-Model-Based Nonlinear Networked Control Systems: A Survey , 2016, IEEE Transactions on Industrial Electronics.

[32]  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..

[33]  Rufus Isaacs,et al.  Differential Games , 1965 .

[34]  R. Cristi,et al.  Global stability of adaptive pole placement algorithms , 1985 .

[35]  Manuel A. Duarte-Mermoud,et al.  Boundedness and convergence on fractional order systems , 2016, J. Comput. Appl. Math..

[36]  Andrea L'Afflitto.  Differential games, partial-state stabilization, and model reference adaptive control , 2017, J. Frankl. Inst..

[37]  Maria Letizia Corradini,et al.  FO sliding surface for the robust control of integer-order LTI plants , 2014 .

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

[39]  Farshad Merrikh-Bayat,et al.  Fractional-order unstable pole-zero cancellation in linear feedback systems☆ , 2012, 1207.6962.