Fractional proportional linear control systems: A geometric perspective on controllability and observability
Year 2024,
, 77 - 89, 15.06.2024
Khizra Bukhsh
,
Awais Younus
,
Aiman Mukheimer
,
Thabet Abdeljawad
Abstract
The paper presents a detailed analysis of control and observation of generalized Caputo proportional fractional time-invariant linear systems. The focus is on identifying controllable states and observable systems within the controllable subspace, null space, and unobservable subspace of the proposed system. The necessary conditions for the controllable subspace and the necessary and sufficient conditions for observability criteria are firmly established. The controllable subspace is treated geometrically as the set of controllable states, while the observable system is characterized by a zero unobservable subspace. The results are reinforced by examples and will immensely benefit future studies on fractional-order control systems.
Thanks
The authors A. Mukheimer and T. Abdeljawad would like to thank Prince sultan University for the support through TAS reseacrh lab.
References
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Year 2024,
, 77 - 89, 15.06.2024
Khizra Bukhsh
,
Awais Younus
,
Aiman Mukheimer
,
Thabet Abdeljawad
References
- D. R. Anderson, D. J. Ulness: Newly defined conformable derivatives, Adv. Dyn. Syst. Appl, 10 (2015), 109–137.
- Z. Al-Zhour: Controllability and observability behaviors of a non-homogeneous conformable fractional dynamical system compatible with some electrical applications, Alexandria Engineering Journal, 61 (2022), 1055–1067.
- K. Balachandran, M. Matar and J. J. Trujillo: Note on controllability of linear fractional dynamical systems, J. Control Decis., 3 (2016), 267–279.
- M. Bohner, K. S. Vidhyaa, E. Thandapani: Oscillation of noncanonical second- order advanced differential equations via canonical transform, Constr. Math. Anal., 5 (1) (2022), 7–13.
- K. Bukhsh, A. Younus: On the controllability and observability of fractional proportional linear systems, Internat. J. Systems Sci., 54 (2023), 1410–1422.
- A. Da Silva: Controllability of linear systems on solvable Lie groups, SIAM J. Control Optim., 54 (2016), 372–390.
- D. Ding, X. Zhang, J. Cao, N. Wang and D. Liang: Bifurcation control of complex networks model via PD controller, Neurocomputing, 175 (2016), 1–9.
- Z. Ge: Controllability and observability of stochastic singular systems in Banach spaces, J. Syst. Sci. Complex., 35 (2022), 194–204.
- K. A. Grasse, N. Ho: Simulation relations and controllability properties of linear and nonlinear control systems, SIAM J. Control Optim., 53 (2015), 1346–1374.
- T. L. Guo: Controllability and observability of impulsive fractional linear time-invariant system, Comput. Math. Appl., 64 (2012), 3171–3182.
- F. Jarad, M. A. Alqudah and T. Abdeljawad: On more general forms of proportional fractional operators, Open Math., 18 (2020), 167–176.
- F. Jarad, T. Abdeljawad and J. Alzabut: Generalized fractional derivatives generated by a class of local proportional derivatives, E. P. J. Special Topics, 226 (2017), 3457–3471.
- T. Kaczorek: Cayley–Hamilton theorem for fractional linear systems, In Theory and Applications of Non-integer Order Systems: 8th Conference on Non-integer Order Calculus and Its Applications, Zakopane, Poland, Springer International Publishing, (2017), 45–55.
- R. E. Kalman: Mathematical description of linear dynamical systems, Journal of the Society for Industrial and Applied Mathematics, Series A: Control, 1 (1963), 152–192.
- R. E. Kalman: Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana, 5 (1960), 102–119.
- A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006).
- A. A. Kilbas, O. I. Marichev and S. G. Samko: Fractional integrals and derivatives (theory and applications), CRC Press, India (1993).
- Y. Li, K. H. Ang and G. C. Chong: PID control system analysis and design, IEEE Control Systems Magazine, 26 (2006), 32–41.
- G. Mazanti,: Relative controllability of linear difference equations, SIAM J. Control Optim., 55 (2017), 3132–3153.
- M. Popolizio: On the matrix Mittag–Leffler function: theoretical properties and numerical computation, Mathematics, 7 (2019), 1140.
- I. Podlubny: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier (1998).
- J. D. J. Rubio, E. Orozco, D. A. Cordova, M. A. Islas, J. Pacheco, G. J. Gutierrez, ... and D. Mujica-Vargas: Modified
linear technique for the controllability and observability of robotic arms, IEEE Access, 10 (2022), 3366-3377.
- W. J. Rugh: Linear system theory, Prentice-Hall, Inc (1996).
- J. L. Šaji´c, S. Langthaler, J. Schröttner and C. Baumgartner: System identification and mathematical modeling of the pandemic spread COVID-19 in Serbia, IFAC-PapersOnLine, 55 (2022), 19–24.
- V. Singh, D. N. Pandey: Controllability of multi-term time-fractional differential systems, J. Control Decis., 7 (2020), 109–125.
- J.Wang, Y. Zheng, K. Li and Q. Xu: DeeP-LCC: Data-enabled predictive leading cruise control in mixed traffic flow, IEEE Transactions on Control Systems Technology, 31 (6) (2023), 2760–2776.
- L. Wang, Q. Yan and H. Yu: Constrained approximate null controllability of the coupled heat equation with impulse controls, SIAM J. Control Optim., 59 (2021), 3418–3446.
- J. Wei: The controllability of fractional control systems with control delay, Comput. Math. Appl., 64 (2012), 3153–3159.
- H. Zhang, I. Ahmad, G. Rahman and S. Ahmad: Investigation for Existence, Controllability & Observability of a
Fractional order Delay Dynamical System, Authorea Preprints, (2022).