# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2022020
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## Dynamical estimation of a noisy input in a system with a Caputo fractional derivative. The case of continuous measurements of a part of phase coordinates

 Ural Federal University, 12, Mira street, Krasovskii Institute of Mathematics and Mechanics UB RAS, 16, S. Kovalevskaya street, Ekaterinburg, 620108, Russia

Received  December 2021 Revised  March 2022 Early access April 2022

The problem of estimating (reconstructing) an unknown input for a system of nonlinear differential equations with the Caputo fractional derivative is considered. Information on the position of the system is available for observations and only a part of system's parameters can be measured. The case of measuring all phase coordinates is also presented. The measurements are continuous and the data obtained in them are noisy. The considered problem is ill-posed and, to solve it, we use the method of dynamic inversion. It is based on regularization methods and constructions of positional control theory. In particular, we use the Tikhonov regularization method also known as the smoothing functional method and the Krasovskii extremal aiming method. The approach to estimating an unknown input implies introducing an auxiliary system (a model) with an appropriate rule of forming a control. The proposed estimation algorithm gives approximations of an unknown input and is stable under informational noises and computational errors. As an example illustrating the elaborated technique, a biological model of human immunodeficiency virus disease is used for simulation. The simulation results demonstrate the importance of the approach to on-line estimating unobservable parameters in real processes.

Citation: Platon Surkov. Dynamical estimation of a noisy input in a system with a Caputo fractional derivative. The case of continuous measurements of a part of phase coordinates. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022020
##### References:
 [1] A. A. M. Arafa, S. Z. Rida and M. Khalil, Fractional modeling dynamics of HIV and CD4+T-cells during primary infection, Nonlinear Biomedical Physics, 6 (2012), 1-7. [2] Y. Bar-Shalom, X. R. Li and T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software, John Wiley & Sons, New York, 2004. [3] M. S. Blizorukova and V. I. Maksimov, Dynamic discrepancy method in the problem of reconstructing the input of a system with time delay control, Comput. Math. Math. Phys., 61 (2021), 359-367.  doi: 10.1134/S0965542521030040. [4] A. Boumenir, V. K. Tuan and W. Al-Khulaifi, Reconstructing a fractional integro-differential equation, Math. Methods Appl. Sci., 44 (2021), 3159-3166.  doi: 10.1002/mma.6648. [5] L. Bourdin, Cauchy–Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems, Differential and Integral Equations, 31 (2018), 559-594. [6] N. Botkin, V. Turova, B. Hosseini, J. Diepolder and F. Holzapfel, Tracking aircraft trajectories in the presence of wind disturbances, Math. Control Relat. Fields, 11 (2021), 499-520.  doi: 10.3934/mcrf.2021010. [7] S. Butera and M. Di Paola, A physically based connection between fractional calculus and fractal geometry, Annals of Physics, 350 (2014), 146-158.  doi: 10.1016/j.aop.2014.07.008. [8] F. Fagnani, V. Maksimov and L. Pandolfi, A recursive deconvolution approach to disturbance reduction, IEEE Trans. Automat. Control, 49 (2004), 907-921.  doi: 10.1109/TAC.2004.829596. [9] M. I. Gomoyunov, Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems, Fract. Calc. Appl. Anal., 21 (2018), 1238-1261.  doi: 10.1515/fca-2018-0066. [10] M. I. Gomoyunov, Differential games for fractional-order systems: Hamilton-Jacobi-Bellman-Isaacs equation and optimal feedback strategies, Mathematics, 6 (2021), 1667. [11] S. I. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Application, De Gruyter, Berlin, 2012. [12] K. J. Keesman, System Identification. An Introduction, Springer-Verlag, London, 2011. [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006. [14] E. K. Kostousova, External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms, Math. Control Relat. Fields, 11 (2021), 625-641.  doi: 10.3934/mcrf.2021015. [15] N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. [16] A. V. Kryazhimskii and Yu. S. Osipov, On positional calculation of $\Omega$-normal control in dynamical system, Problems Control Inform. Theory, 13 (1984), 425-436. [17] A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-10277-1. [18] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, R. I., 1986. [19] L. Ljung and T. Söderström, Theory and Practice of Recursive Identification, MIT Press, Cambridge, 1983. [20] V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems, VSP, Utrecht, 2002. doi: 10.1515/9783110944839. [21] V. I. Maksimov, On a modification of the dynamic regularization method, Differential Equations, 57 (2021), 1119-1123.  doi: 10.1134/S0012266121080152. [22] V. I. Maksimov, The methods of dynamical reconstruction of an input in a system of ordinary differential equations, J. Inverse and Ill-Posed Problems, 29 (2021), 125-156.  doi: 10.1515/jiip-2020-0040. [23] I. Matychyn and V. Onyshchenko, Time-optimal control of linear fractional systems with variable coefficients, Int. J. Appl. Math. Comput. Sci., 31 (2021), 375-386.  doi: 10.34768/amcs-2021-0025. [24] G. Nazir, K. Shah, A. Debbouche and R. A. Khan, Study of HIV mathematical model under nonsingular kernel type derivative of fractional order, Chaos Solitons Fractals, 139 (2020), 110095, 8 pp. doi: 10.1016/j.chaos.2020.110095. [25] J. P. Norton, An Introduction to Identification, Dover Publications Inc., New York, 2009. [26] Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach Science Publishers, Basel, 1995. [27] Yu. S. Osipov and V. I. Maksimov, On dynamical input reconstruction in a distributed second order equation, J. Inverse Ill-Posed Probl., 29 (2021), 707-719.  doi: 10.1515/jiip-2021-0004. [28] L. Pandolfi, On-line input identification and application to active noise cancellation, Annual Reviews in Control, 34 (2010), 245-261. [29] L. Pandolfi, Systems with Persistent Memory: Controllability, Stability, Identification, Springer, Cham, 2021. [30] I. Podlubny, Geometrical and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386. [31] V. Rozenberg, On a problem of dynamical input reconstruction for a system of special type under conditions of uncertainty, AIMS Math., 5 (2020), 4108-4120.  doi: 10.3934/math.2020263. [32] J. Shao and F. Meng, Gronwall–{B}ellman type inequalities and their applications to fractional differential equations, Abstr. Appl. Anal., (2013), Art. ID 217641, 7 pp. doi: 10.1155/2013/217641. [33] A. A. Stanislavsky, Probability interpretation of the integral of fractional order, Theoret. and Math. Phys., 138 (2004), 418-431.  doi: 10.1023/B:TAMP.0000018457.70786.36. [34] P. G. Surkov, Dynamic right-hand side reconstruction problem for a system of fractional differential equations, Differential Equation, 55 (2019), 849-858.  doi: 10.1134/S0012266119060120. [35] P. G. Surkov, Real-time reconstruction of external impact on fractional order system under measuring a part of coordinates, J. Comput. Appl. Math., 381 (2021), Paper No. 113039, 12 pp. doi: 10.1016/j.cam.2020.113039. [36] P. G. Surkov, Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach, Fract. Calc. Appl. Anal., 24 (2021), 895-922.  doi: 10.1515/fca-2021-0038. [37] P. G. Surkov, Real-time calculation of a Caputo fractional derivative from noisy data. The case of continuous measurements, Proceedings of the Steklov Institute of Mathematics, 315 (2021), S225-S235.  doi: 10.21538/0134-4889-2021-27-2-238-248. [38] V. E. Tarasov, Geometric interpretation of fractional-order derivative, Fract. Calc. Appl. Anal., 19 (2016), 1200-1221.  doi: 10.1515/fca-2016-0062. [39] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems, John Wiley & Sons, New York, 1977. [40] V. M. Veliov, On the relationship between continuous- and discrete-time control systems, CEJOR Cent. Eur. J. Oper. Res., 18 (2010), 511-523.  doi: 10.1007/s10100-010-0167-2. [41] H. Unbehauen and G. P. Rao, Identification of Continuous Systems, North-Holland Publishing Co., Amsterdam, 1987. [42] H. Unbehauen and G. P. Rao, A review of identification in continuous-time systems, Annual Reviews in Control, 2 (1998), 145-171.

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##### References:
 [1] A. A. M. Arafa, S. Z. Rida and M. Khalil, Fractional modeling dynamics of HIV and CD4+T-cells during primary infection, Nonlinear Biomedical Physics, 6 (2012), 1-7. [2] Y. Bar-Shalom, X. R. Li and T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software, John Wiley & Sons, New York, 2004. [3] M. S. Blizorukova and V. I. Maksimov, Dynamic discrepancy method in the problem of reconstructing the input of a system with time delay control, Comput. Math. Math. Phys., 61 (2021), 359-367.  doi: 10.1134/S0965542521030040. [4] A. Boumenir, V. K. Tuan and W. Al-Khulaifi, Reconstructing a fractional integro-differential equation, Math. Methods Appl. Sci., 44 (2021), 3159-3166.  doi: 10.1002/mma.6648. [5] L. Bourdin, Cauchy–Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems, Differential and Integral Equations, 31 (2018), 559-594. [6] N. Botkin, V. Turova, B. Hosseini, J. Diepolder and F. Holzapfel, Tracking aircraft trajectories in the presence of wind disturbances, Math. Control Relat. Fields, 11 (2021), 499-520.  doi: 10.3934/mcrf.2021010. [7] S. Butera and M. Di Paola, A physically based connection between fractional calculus and fractal geometry, Annals of Physics, 350 (2014), 146-158.  doi: 10.1016/j.aop.2014.07.008. [8] F. Fagnani, V. Maksimov and L. Pandolfi, A recursive deconvolution approach to disturbance reduction, IEEE Trans. Automat. Control, 49 (2004), 907-921.  doi: 10.1109/TAC.2004.829596. [9] M. I. Gomoyunov, Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems, Fract. Calc. Appl. Anal., 21 (2018), 1238-1261.  doi: 10.1515/fca-2018-0066. [10] M. I. Gomoyunov, Differential games for fractional-order systems: Hamilton-Jacobi-Bellman-Isaacs equation and optimal feedback strategies, Mathematics, 6 (2021), 1667. [11] S. I. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Application, De Gruyter, Berlin, 2012. [12] K. J. Keesman, System Identification. An Introduction, Springer-Verlag, London, 2011. [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006. [14] E. K. Kostousova, External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms, Math. Control Relat. Fields, 11 (2021), 625-641.  doi: 10.3934/mcrf.2021015. [15] N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. [16] A. V. Kryazhimskii and Yu. S. Osipov, On positional calculation of $\Omega$-normal control in dynamical system, Problems Control Inform. Theory, 13 (1984), 425-436. [17] A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-10277-1. [18] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, R. I., 1986. [19] L. Ljung and T. Söderström, Theory and Practice of Recursive Identification, MIT Press, Cambridge, 1983. [20] V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems, VSP, Utrecht, 2002. doi: 10.1515/9783110944839. [21] V. I. Maksimov, On a modification of the dynamic regularization method, Differential Equations, 57 (2021), 1119-1123.  doi: 10.1134/S0012266121080152. [22] V. I. Maksimov, The methods of dynamical reconstruction of an input in a system of ordinary differential equations, J. Inverse and Ill-Posed Problems, 29 (2021), 125-156.  doi: 10.1515/jiip-2020-0040. [23] I. Matychyn and V. Onyshchenko, Time-optimal control of linear fractional systems with variable coefficients, Int. J. Appl. Math. Comput. Sci., 31 (2021), 375-386.  doi: 10.34768/amcs-2021-0025. [24] G. Nazir, K. Shah, A. Debbouche and R. A. Khan, Study of HIV mathematical model under nonsingular kernel type derivative of fractional order, Chaos Solitons Fractals, 139 (2020), 110095, 8 pp. doi: 10.1016/j.chaos.2020.110095. [25] J. P. Norton, An Introduction to Identification, Dover Publications Inc., New York, 2009. [26] Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach Science Publishers, Basel, 1995. [27] Yu. S. Osipov and V. I. Maksimov, On dynamical input reconstruction in a distributed second order equation, J. Inverse Ill-Posed Probl., 29 (2021), 707-719.  doi: 10.1515/jiip-2021-0004. [28] L. Pandolfi, On-line input identification and application to active noise cancellation, Annual Reviews in Control, 34 (2010), 245-261. [29] L. Pandolfi, Systems with Persistent Memory: Controllability, Stability, Identification, Springer, Cham, 2021. [30] I. Podlubny, Geometrical and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386. [31] V. Rozenberg, On a problem of dynamical input reconstruction for a system of special type under conditions of uncertainty, AIMS Math., 5 (2020), 4108-4120.  doi: 10.3934/math.2020263. [32] J. Shao and F. Meng, Gronwall–{B}ellman type inequalities and their applications to fractional differential equations, Abstr. Appl. Anal., (2013), Art. ID 217641, 7 pp. doi: 10.1155/2013/217641. [33] A. A. Stanislavsky, Probability interpretation of the integral of fractional order, Theoret. and Math. Phys., 138 (2004), 418-431.  doi: 10.1023/B:TAMP.0000018457.70786.36. [34] P. G. Surkov, Dynamic right-hand side reconstruction problem for a system of fractional differential equations, Differential Equation, 55 (2019), 849-858.  doi: 10.1134/S0012266119060120. [35] P. G. Surkov, Real-time reconstruction of external impact on fractional order system under measuring a part of coordinates, J. Comput. Appl. Math., 381 (2021), Paper No. 113039, 12 pp. doi: 10.1016/j.cam.2020.113039. [36] P. G. Surkov, Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach, Fract. Calc. Appl. Anal., 24 (2021), 895-922.  doi: 10.1515/fca-2021-0038. [37] P. G. Surkov, Real-time calculation of a Caputo fractional derivative from noisy data. The case of continuous measurements, Proceedings of the Steklov Institute of Mathematics, 315 (2021), S225-S235.  doi: 10.21538/0134-4889-2021-27-2-238-248. [38] V. E. Tarasov, Geometric interpretation of fractional-order derivative, Fract. Calc. Appl. Anal., 19 (2016), 1200-1221.  doi: 10.1515/fca-2016-0062. [39] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems, John Wiley & Sons, New York, 1977. [40] V. M. Veliov, On the relationship between continuous- and discrete-time control systems, CEJOR Cent. Eur. J. Oper. Res., 18 (2010), 511-523.  doi: 10.1007/s10100-010-0167-2. [41] H. Unbehauen and G. P. Rao, Identification of Continuous Systems, North-Holland Publishing Co., Amsterdam, 1987. [42] H. Unbehauen and G. P. Rao, A review of identification in continuous-time systems, Annual Reviews in Control, 2 (1998), 145-171.
The component $z$ of system (76) and the component $w_1$ of model (79)
The unovservable component $x$ of system (76) and the component $w_2$ of model (79)
The input action $u$ and the reconstructed action $v^h$
The components u1, u2 and the components v1h, v2h
The input action Bu and its estimation Bvh