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doi: 10.3934/dcdss.2020444

On the observability of conformable linear time-invariant control systems

Awais Younus 1, , Zoubia Dastgeer 1, , Nudrat Ishaq 1, , Abdul Ghaffar 2, , Kottakkaran Sooppy Nisar 3, and  Devendra Kumar 4,,

1. 

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
2. 

Informetrics Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3. 

Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia
4. 

Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India

* Corresponding author: Devendra Kumar

Received  November 2019 Revised  March 2020 Published  November 2020

In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of conformable Laplace transform.

Keywords: Conformable operator, time-invariant linear control systems, Gramian observability matrix, conformable Laplace transform.
Mathematics Subject Classification: Primary: 93B07, 34A38; Secondary: 34A08.
Citation: Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020444
References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.  Google Scholar

[2]

D. Baleanu, Z. B Güvenc and J. A. Tenneiro Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010. doi: 10.1007/978-90-481-3293-5.  Google Scholar

[3]

I. Benabbas and D. E. Teniou, Observability of wave equation with Ventcel dynamic condition, Evol. Equ. Control Theory, 7 (2018), 545-570.  doi: 10.3934/eect.2018026.  Google Scholar

[4]

R. Caponetto, G. Dongola, L. Fortuna and I. Petras, Fractional Order Systems: Modeling and Control Applications, World Scientific Series, vol. 72, 2010,200 pp. doi: 10.1142/7709.  Google Scholar

[5]

A. Escalante and Al dair-Pantoja, The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term, Eur. Phys. J. Plus, 134 (2019), 1-10.   Google Scholar

[6]

R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E, 51 (1995), 1-5.  doi: 10.1103/PhysRevE.51.R848.  Google Scholar

[7]

R. Khalil, M. Al Haroni, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

[8]

R. E. Kálmán, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana, 2 (1960), 102-119.   Google Scholar

[9]

R. E. Kálmán, Mathematical description of linear dynamical systems, J. SIAM Control Ser. A, 1 (1963), 152-192.   Google Scholar

[10]

N. A. Khan, O. A. Razzaq and M. Ayyaz, Some properties and applications of Conformable Fractional Laplace Transform (CFLT), J. Fract. Calc. Appl., 9 (2018), 72-81.   Google Scholar

[11]

D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Int. J. Heat Mass Transf., 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.  Google Scholar

[12]

V. Mohammadnezhad, M. Eslami and H. Rezazadeh, Stability analysis of linear conformable fractional differential equations system with time delays, Bol. Soc. Parana. Mat., 38 (2020), 159-171.  doi: 10.5269/bspm.v38i6.37010.  Google Scholar

[13]

K. S. Nisar, G. Rehman and K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), Paper No. 245, 9 pp. doi: 10.1186/s13660-019-2197-1.  Google Scholar

[14]

H. Rezazadeh, H. Aminikhah and A. H. Refahi Sheikhani, Stability analysis of conformable fractional systems, Iranian J. Numer. Anal. Opti., 7 (2017), 13-32.   Google Scholar

[15]

W. J. Rugh, Linear System Theory 2E, Prentice Hall, Upper Saddle River, NJ, 07458, 1996. Google Scholar

[16]

C. Ruiyang, G. Fudong, C. Yangquan and K. Chunhai, Regional observability for Hadamard-Caputo time fractional distributed parameter systems, Appl. Math. Comput., 360 (2019), 190-202.  doi: 10.1016/j.amc.2019.04.081.  Google Scholar

[17]

F. S. Silva, D. M. Moreira and M. A. Moret, Conformable Laplace Transform of fractional differential equations, Axioms, 55 (2018), 1-11.  doi: 10.3390/axioms7030055.  Google Scholar

[18]

E. Unal, A. Gokdogan and E. Celik, Solutions around a regular a singular point of a sequential conformable fractional differential equation, Kuwait Journal of Science, 44 (2017), 9-16.   Google Scholar

[19]

T. A. Yıldız, A. Jajarmi, B. Yıldız and D. Baleanu, New aspects of time fractional optimal control problems within operators with nonsingular kernel, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 407-428.  doi: 10.3934/dcdss.2020023.  Google Scholar

[20]

D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, (2017), 903–917. doi: 10.1007/s10092-017-0213-8.  Google Scholar

show all references

References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.  Google Scholar

[2]

D. Baleanu, Z. B Güvenc and J. A. Tenneiro Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010. doi: 10.1007/978-90-481-3293-5.  Google Scholar

[3]

I. Benabbas and D. E. Teniou, Observability of wave equation with Ventcel dynamic condition, Evol. Equ. Control Theory, 7 (2018), 545-570.  doi: 10.3934/eect.2018026.  Google Scholar

[4]

R. Caponetto, G. Dongola, L. Fortuna and I. Petras, Fractional Order Systems: Modeling and Control Applications, World Scientific Series, vol. 72, 2010,200 pp. doi: 10.1142/7709.  Google Scholar

[5]

A. Escalante and Al dair-Pantoja, The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term, Eur. Phys. J. Plus, 134 (2019), 1-10.   Google Scholar

[6]

R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E, 51 (1995), 1-5.  doi: 10.1103/PhysRevE.51.R848.  Google Scholar

[7]

R. Khalil, M. Al Haroni, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

[8]

R. E. Kálmán, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana, 2 (1960), 102-119.   Google Scholar

[9]

R. E. Kálmán, Mathematical description of linear dynamical systems, J. SIAM Control Ser. A, 1 (1963), 152-192.   Google Scholar

[10]

N. A. Khan, O. A. Razzaq and M. Ayyaz, Some properties and applications of Conformable Fractional Laplace Transform (CFLT), J. Fract. Calc. Appl., 9 (2018), 72-81.   Google Scholar

[11]

D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Int. J. Heat Mass Transf., 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.  Google Scholar

[12]

V. Mohammadnezhad, M. Eslami and H. Rezazadeh, Stability analysis of linear conformable fractional differential equations system with time delays, Bol. Soc. Parana. Mat., 38 (2020), 159-171.  doi: 10.5269/bspm.v38i6.37010.  Google Scholar

[13]

K. S. Nisar, G. Rehman and K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), Paper No. 245, 9 pp. doi: 10.1186/s13660-019-2197-1.  Google Scholar

[14]

H. Rezazadeh, H. Aminikhah and A. H. Refahi Sheikhani, Stability analysis of conformable fractional systems, Iranian J. Numer. Anal. Opti., 7 (2017), 13-32.   Google Scholar

[15]

W. J. Rugh, Linear System Theory 2E, Prentice Hall, Upper Saddle River, NJ, 07458, 1996. Google Scholar

[16]

C. Ruiyang, G. Fudong, C. Yangquan and K. Chunhai, Regional observability for Hadamard-Caputo time fractional distributed parameter systems, Appl. Math. Comput., 360 (2019), 190-202.  doi: 10.1016/j.amc.2019.04.081.  Google Scholar

[17]

F. S. Silva, D. M. Moreira and M. A. Moret, Conformable Laplace Transform of fractional differential equations, Axioms, 55 (2018), 1-11.  doi: 10.3390/axioms7030055.  Google Scholar

[18]

E. Unal, A. Gokdogan and E. Celik, Solutions around a regular a singular point of a sequential conformable fractional differential equation, Kuwait Journal of Science, 44 (2017), 9-16.   Google Scholar

[19]

T. A. Yıldız, A. Jajarmi, B. Yıldız and D. Baleanu, New aspects of time fractional optimal control problems within operators with nonsingular kernel, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 407-428.  doi: 10.3934/dcdss.2020023.  Google Scholar

[20]

D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, (2017), 903–917. doi: 10.1007/s10092-017-0213-8.  Google Scholar

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