The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils
Abstract
:1. Introduction
- (1)
- Regular whenand det,
- (2)
- Singular whenorand det.
2. Basic Definitions and Preliminaries
3. Solution of FoLTI Systems Using ADM Method
4. The General Solution of FoLTI Singular Systems with Regular Pencils
5. Recursive form of FoLTI Systems with Regular Pencils
6. Illustrative Examples
7. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
- Das, S. Functional Fractional Calculus for System Identification and Controls; Springer: Berlin, Germany, 2008. [Google Scholar]
- Debnath, L. Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003, 54, 3413–3442. [Google Scholar] [CrossRef]
- Oustaloup, A.; le Lay, L.; Mathieu, B. Identification of non-integer order system in the time domain. In Proceedings of the IEEE CESA’96, SMC IMACS Multiconference, Computational Engineering in Systems Application, Symposium on Control, Optimization and Supervision, Lille, France, 9–12 July 1996. [Google Scholar]
- El-Khazali, R. Discretization of Fractional-Order Differentiators and Integrators. IFAC Proc. Vol. 2014, 47, 2016–2021. [Google Scholar] [CrossRef]
- El-Khazali, R. On the biquadratic approximation of fractional-order Laplacian operators. Analog Integr. Circuits Signal Process. 2015, 82, 503–517. [Google Scholar] [CrossRef]
- Ahmad, W.M.; El-Khazali, R.; Al-Assaf, Y. Stabilization of generalized fractional order chaotic systems using state feedback control. Chaos Solitons Fractals 2004, 22, 141–150. [Google Scholar] [CrossRef]
- EL-Khazali, R. On the state space modeling of fractional systems. IFAC Proc. Vol. 2006, 39, 499–504. [Google Scholar] [CrossRef]
- Trigeassou, J.-C.; Maamri, N. State space modeling of fractional differential equations and the initial condition problem. In Proceedings of the 6th International Multi-Conference on Systems, Signals and Devices, Djerba, Tunisia, 23–26 March 2009. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Oldham, K.B.; Spanier, J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Ahmad, W.M.; EL-Khazali, R. Fractional-order dynamical models of love. Chaos Solitons Fractals 2007, 33, 1367–1375. [Google Scholar] [CrossRef]
- Alsedá, L.L.; Cushing, J.M.; Elaydi, S.; Pinto, A.A. Difference Equations, Discrete Dynamical Systems and Applications. In Proceedings in Mathematics and Statistics; Springer: Berlin, Germany, 2016. [Google Scholar]
- Gandomi, A.H.; Yun, G.J.; Yang, X.S.; Talatahari, S. Chaos-enhanced accelerated particle swarm optimization. Commun. Nonlinear Sci. Numer. Simul. 2013, 18, 327–340. [Google Scholar] [CrossRef]
- Cánovas, J.S.; Muñoz-Guillermo, M. On the complexity of economic dynamics: An approach through topological entropy. Chaos Soliton Fractals 2017, 103, 163–176. [Google Scholar] [CrossRef]
- Torres, H.R.; Queirós, S.; Morais, P.; Oliveira, B.; Fonseca, J.C.; Vilaca, J.L. Kidney segmentation in ultrasound magnetic resonance and computed tomography images: A. systematic review. Comput. Methods Programs Biomed. 2018, 157, 49–67. [Google Scholar] [CrossRef] [PubMed]
- Jalab, H.A.; Ibrahim, R.W.; Ahmed, A. Image denoising algorithm based on the convolution of fractional tsallis entropy with the riesz fractional derivative. Neural Comput. Appl. 2017, 28, 217–223. [Google Scholar] [CrossRef]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Al-Zhour, Z. Efficient solutions of coupled matrix and matrix differential equations. Intell. Control Autom. 2012, 3, 176–187. [Google Scholar] [CrossRef]
- Campbell, S. Singular Systems of Differential Equations II; Pitman: London, UK, 1982. [Google Scholar]
- Kablar, N.; Lj, D. Debeljkovic, Finite-time stability of time varying linear singular systems. In Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, USA, 18 December 1998; pp. 3831–3836. [Google Scholar]
- Takaba, T.; Morihira, N.; Katayama, K. A generalized Lyapunov theorem for descriptor system. Syst. Control Lett. 1995, 24, 49–51. [Google Scholar] [CrossRef]
- Shahzad, A.; Jones, B.L.; Kerrigan, E.C.; Constantinides, G.A. An efficient algorithm for the solution of a coupled Sylvester equation appearing in descriptor systems. Automatica 2011, 47, 244–248. [Google Scholar] [CrossRef]
- Junsheng, D.; Jianye, A.; Mingyu, X. Solution of system of fractional differential equations by Adomian decomposition method. Appl. Math. J. Chin. Univ. Ser. B 2007, 22, 7–12. [Google Scholar]
- Gaxiola, G.; Bernal-Jaquez, R. Applying Adomian Decomposition Method to Solve Burgess Equation with a Non-linear Source. Int. J. Appl. Comput. Math 2017, 3, 213–224. [Google Scholar] [CrossRef]
- Dassios, I. Optimal Solutions for non-consistent Singular Linear Systems of Fractional Nabla Difference Equations. Circuits Syst. Signal Process. 2015, 34, 1769–1797. [Google Scholar] [CrossRef]
- Dassios, I.; Baleanu, D.I. Duality of singular linear systems of fractional nabla difference equations. Appl. Math. Model. 2015, 39, 4180–4195. [Google Scholar] [CrossRef]
- Dassios, I.; Baleanu, D.; Kalogeropoulos, G. On non-homogeneous singular systems of fractional nabla difference equations. Appl. Math. Comput. 2014, 227, 112–131. [Google Scholar] [CrossRef]
- Ibnbadis, A. Contribution to Analysis and Control of Linear Singular Fractional-Order Systems; Ministry of Higher Education and Scientific Research: Algiers, Algeria, 2017.
- Dai, L. Lecture Notes in Control and Information Sciences, The Theory of Matrices; Springer: Berlin, Germany; New York, NY, USA, 1989; Volume 2. [Google Scholar]
- Kailath, T. Linear Systems, Information and Systems Sciences Series; Prentice Hall: Englewood Cliffs, NJ, USA, 1980. [Google Scholar]
- Kagstrom, B. A perturbation analysis of the generalized Sylvester equation. SIAM J. Matrix Anal. Appl. 1994, 15, 1045–1060. [Google Scholar] [CrossRef]
- Caputo, M. Linear models of dissipation whose Q is almost frequency independent: Part II. Geophys. J. Int. 1967, 13, 529–539. [Google Scholar] [CrossRef]
- Kaczorek, T.; Rogowski, K. Fractional Linear Systems and Electrical Circuits; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar]
- Al-Zhour, Z. The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense. Alex. Eng. J. 2016, 55, 1675–1681. [Google Scholar] [CrossRef]
- Kaczorek, T. Singular fractional continuous-time and discrete-time linear systems. Acta Mech. Automat. 2013, 7, 26–33. [Google Scholar] [CrossRef]
- Jones, B.L.; Kerrigan, E.C.; Morrison, J.F. A modeling and filtering framework for the semi-discretized Navier-Stokes equations. In Proceedings of the European Control Conference, Budapest, Hungary, 23–26 August 2009; pp. 138–143. [Google Scholar]
- Gerdin, M. Computation of a Canonical form for Linear Differential-Algebraic Equations; Automatic Control Communication Systems; Linkopings Universitet: Linköping, Sweden, 2004. [Google Scholar]
- Guglielmi, N.; Overton, M.L.; Stewart, G.W. An efficient algorithm for computing the generalized null space decomposition. SIAM J. Matrix Anal. Appl. 2015, 36, 38–54. [Google Scholar] [CrossRef]
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Batiha, I.M.; El-Khazali, R.; AlSaedi, A.; Momani, S. The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils. Entropy 2018, 20, 400. https://doi.org/10.3390/e20060400
Batiha IM, El-Khazali R, AlSaedi A, Momani S. The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils. Entropy. 2018; 20(6):400. https://doi.org/10.3390/e20060400
Chicago/Turabian StyleBatiha, Iqbal M., Reyad El-Khazali, Ahmed AlSaedi, and Shaher Momani. 2018. "The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils" Entropy 20, no. 6: 400. https://doi.org/10.3390/e20060400
APA StyleBatiha, I. M., El-Khazali, R., AlSaedi, A., & Momani, S. (2018). The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils. Entropy, 20(6), 400. https://doi.org/10.3390/e20060400