Generalized fractional controller for singular systems of differential equations

Abstract In this article we consider a class of singular linear systems of first order, and introduce a generalized fractional order feedback controller of Caputo type. The closed loop system in question is a singular system of differential equations having both first, and fractional order derivatives. We provide a comprehensive theory for the existence and uniqueness of solutions, as well as for the stability of the system with inclusion of the fractional order controller. An example of a singular system with a fractional order proportional integral controller, as well as an example on a 3-bus power system with inclusion of a fractional order damping controller, is given to illustrate our theory.

[1]  Ioannis K. Dassios,et al.  Geometric relation between two different types of initial conditions of singular systems of fractional nabla difference equations , 2017 .

[2]  W. Rugh.  Linear System Theory , 1992 .

[3]  Margarita Rivero,et al.  On systems of linear fractional differential equations with constant coefficients , 2007, Appl. Math. Comput..

[4]  Bo Tang,et al.  Quasilinearization technique for solving nonlinear Riemann-Liouville fractional-order problems , 2020, Appl. Math. Comput..

[5]  Dumitru Baleanu,et al.  On a singular system of fractional nabla difference equations with boundary conditions , 2013 .

[6]  D. Baleanu,et al.  Numerical solution of some fractional dynamical systems in medicine involving non-singular kernel with vector order , 2019 .

[7]  Federico Milano,et al.  Semi-Implicit Formulation of Differential-Algebraic Equations for Transient Stability Analysis , 2016, IEEE Transactions on Power Systems.

[8]  Jerzy Klamka,et al.  Controllability of dynamical systems. A survey , 2013 .

[9]  Yuanguo Zhu,et al.  Comparison theorems and distributions of solutions to uncertain fractional difference equations , 2020, J. Comput. Appl. Math..

[10]  Ioannis K. Dassios,et al.  A practical formula of solutions for a family of linear non-autonomous fractional nabla difference equations , 2017, J. Comput. Appl. Math..

[11]  Amar Debbouche,et al.  Fractional Evolution Integro-Differential Systems with Nonlocal Conditions , 2010 .

[12]  Ivan Matychyn,et al.  On time-optimal control of fractional-order systems , 2017, J. Comput. Appl. Math..

[13]  Yong Wang,et al.  The output feedback control synthesis for a class of singular fractional order systems. , 2017, ISA transactions.

[14]  Dumitru Baleanu,et al.  On non-homogeneous singular systems of fractional nabla difference equations , 2014, Appl. Math. Comput..

[15]  Andrzej Dzieliński,et al.  Point to point control of fractional differential linear control systems , 2011 .

[16]  Federico Milano,et al.  Modal Participation Factors of Algebraic Variables , 2020, IEEE Transactions on Power Systems.

[17]  Dumitru Baleanu,et al.  Caputo and related fractional derivatives in singular systems , 2018, Appl. Math. Comput..

[18]  Federico Milano,et al.  Participation Factors for Singular Systems of Differential Equations , 2020, Circuits Syst. Signal Process..

[19]  Federico Milano,et al.  The Möbius transform effect in singular systems of differential equations , 2019, Appl. Math. Comput..

[20]  Clara-Mihaela Ionescu,et al.  The role of fractional calculus in modeling biological phenomena: A review , 2017, Commun. Nonlinear Sci. Numer. Simul..

[21]  F. Lewis.  A survey of linear singular systems , 1986 .

[22]  Raja Muhammad Asif Zahoor,et al.  Design of normalized fractional SGD computing paradigm for recommender systems , 2019, Neural Computing and Applications.

[23]  Reyad El-Khazali,et al.  The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils , 2018, Entropy.

[24]  Ioannis K. Dassios,et al.  Stability and Robustness of Singular Systems of Fractional Nabla Difference Equations , 2016, Circuits, Systems, and Signal Processing.

[25]  Dumitru Baleanu,et al.  Duality of singular linear systems of fractional nabla difference equations , 2015 .

[26]  Federico Milano,et al.  Primal and Dual Generalized Eigenvalue Problems for Power Systems Small-Signal Stability Analysis , 2017, IEEE Transactions on Power Systems.

[27]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[28]  Amar Debbouche,et al.  Asymptotic Almost-Periodicity for a Class of Weyl-Like Fractional Difference Equations , 2019, Mathematics.

[29]  Ioannis K. Dassios,et al.  Optimal Solutions for Non-consistent Singular Linear Systems of Fractional Nabla Difference Equations , 2014, Circuits, Systems, and Signal Processing.

[30]  Zhen Wang,et al.  Synthesis of Sliding Mode Control for a Class of Uncertain Singular Fractional-Order Systems-Based Restricted Equivalent , 2019, IEEE Access.

[31]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[32]  Amar Debbouche,et al.  Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces , 2017 .

[33]  Weihua Deng,et al.  Remarks on fractional derivatives , 2007, Appl. Math. Comput..

[34]  José António Tenreiro Machado,et al.  Optimal variable-order fractional PID controllers for dynamical systems , 2018, J. Comput. Appl. Math..

[35]  Dumitru Baleanu,et al.  Optimal solutions for singular linear systems of Caputo fractional differential equations , 2018, Mathematical Methods in the Applied Sciences.

[36]  Federico Milano,et al.  Power System Modelling and Scripting , 2010 .

[37]  Carlos Lizama,et al.  Lebesgue regularity for differential difference equations with fractional damping , 2018 .