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doi: 10.3934/dcdss.2022027
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## Observer-based SMC for stochastic systems with disturbance driven by fractional Brownian motion

 1 School of Mathematical Sciences, Ocean University of China, Qingdao, China 2 School of Electronic and Information Engineering, Suzhou University of Science and Technology, Suzhou, China 3 Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

* Corresponding author: Baoping Jiang and Hamid Reza Karimi

Received  August 2021 Revised  December 2021 Early access February 2022

This paper investigates the problem of disturbance-observer-based sliding mode control for stabilization of stochastic systems driven by fractional Brownian motion (fBm). By proposing a novel disturbance observer, an integral-type sliding surface is put forward with the estimated disturbance error confined within a certain value. Meanwhile, by virtue of fractional infinitesimal operator and linear matrix inequality, a sufficient criterion is derived to guarantee the asymptotic stability of obtained sliding mode dynamics. Further, an observer-based sliding mode controller is designed to ensure finite-time reachability of state trajectories onto the predefined sliding surface. Lastly, an illustrative example is utilized to verify the reliability and applicability of the proposed control strategy.

Citation: Xin Meng, Cunchen Gao, Baoping Jiang, Hamid Reza Karimi. Observer-based SMC for stochastic systems with disturbance driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022027
##### References:
 [1] R. Calif and F. Schmitt, Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation, Journal of Wind Engineering and Industrial Aerodynamics, 109 (2012), 1-8. [2] W. Chen, J. Yang, L. Guo and S. Li, Disturbance-observer-based control and related methods–An overview, IEEE Transactions on Industrial Electronics, 63 (2015), 1083-1095. [3] Y. Chen, C. Tang and M. Roohi, Design of a model-free adaptive sliding mode control to synchronize chaotic fractional-order systems with input saturation: An application in secure communications, J. Franklin Inst., 358 (2021), 8109-8137.  doi: 10.1016/j.jfranklin.2021.08.007. [4] Q. Gao, G. Feng, L. Liu, J. Qiu and Y. Wang, An ISMC approach to robust stabilization of uncertain stochastic time-delay systems, IEEE Transactions on Industrial Electronics, 61 (2014), 6986-6994. [5] J. Huang, S. Ri, T. Fukuda and Y. Wang, A disturbance observer based sliding mode control for a class of underactuated robotic system with mismatched uncertainties, IEEE Trans. Automat. Control, 64 (2019), 2480-2487.  doi: 10.1109/tac.2018.2868026. [6] B. Jiang and C.-C. Gao, Decentralized adaptive sliding mode control of large-scale semi-Markovian jump interconnected systems with dead-zone input, IEEE Transactions on Automatic Control. doi: 10.1109/TAC.2021.3065658. [7] K. Khandani, A sliding mode observer design for uncertain fractional It$\hat{o}$ stochastic systems with state delay, Int. J. Gen. Syst., 48 (2019), 48-65.  doi: 10.1080/03081079.2018.1534846. [8] K. Khandani, V. J. Majd and M. Tahmasebi, Integral sliding mode control for robust stabilisation of uncertain stochastic time-delay systems driven by fractional Brownian motion, Internat. J. Systems Sci., 48 (2017), 828-837.  doi: 10.1080/00207721.2016.1216201. [9] K. Khandani, V. J. Majd and M. Tahmasebi, Comments on "Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach"[Nonlinear Dyn. 67, 2719–2726 (2012)], Nonlinear Dynamics, 82 (2015), 1605-1607.  doi: 10.1007/s11071-015-2249-0. [10] K. Khandani, V. J. Majd and M. Tahmasebi, Robust stabilization of uncertain time-delay systems with fractional stochastic noise using the novel fractional stochastic sliding approach and its application to stream water quality regulation, IEEE Trans. Automat. Control, 62 (2017), 1742-1751.  doi: 10.1109/TAC.2016.2594261. [11] R. Khasminskii, Stochastic Stability of Differential Equations, Completely revised and enlarged second edition. Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [12] E. Kim, A fuzzy disturbance observer and its application to control, IEEE Transactions on Fuzzy Systems, 10 (2002), 77-84. [13] C. Liu, S. Sun, C. Tao, Y. Shou and B. Xu, Sliding mode control of multi-agent system with application to UAV air combat, Computers and Electrical Engineering, 96 (2021), 107491. [14] Z. Liu, H. R. Karimi and J. Yu, Passivity-based robust sliding mode synthesis for uncertain delayed stochastic systems via state observer, Automatica J. IFAC, 111 (2020), 108596, 10 pp. doi: 10.1016/j.automatica.2019.108596. [15] Z. Liu and J. Yu, Non-fragile observer-based adaptive control of uncertain nonlinear stochastic Markovian jump systems via sliding mode technique, Nonlinear Analysis: Hybrid Systems, 38 (2020), 100931, 17 pp. doi: 10.1016/j.nahs.2020.100931. [16] S. Lu and W. Zhang, Robust $H_{\infty}$ filtering and control for a class of linear systems with fractional stochastic noise, Phys. A, 526 (2019), 120958, 11 pp. doi: 10.1016/j.physa.2019.04.194. [17] X. Meng, C. Gao, Z. Liu and B. Jiang, Robust $H_{\infty}$ control for a class of uncertain neutral-type systems with time-varying delays, Asian J. Control, 23 (2021), 1454-1465.  doi: 10.1002/asjc.2298. [18] X. Meng, Z. Wu, C. Gao, B. Jiang and H. Karimi, Finite-time projective synchronization control of variable-order fractional chaotic systems via sliding mode approach, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2021), 2503-2507. [19] M. Parvizian and K. Khandani, Mean square exponential stabilization of uncertain time-delay stochastic systems with fractional Brownian motion, Internat. J. Robust Nonlinear Control, 31 (2021), 9253-9266.  doi: 10.1002/rnc.5764. [20] Y. Ren, X. Cheng and R. Sakthivel, Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm, Appl. Math. Comput., 247 (2014), 205-212.  doi: 10.1016/j.amc.2014.08.095. [21] J. Song, Y. Niu, H.-K. Lam and Y. Zou, Asynchronous sliding mode control of singularly perturbed semi-Markovian jump systems: Application to an operational amplifier circuit, Automatica J. IFAC, 118 (2020), 109026, 8 pp. doi: 10.1016/j.automatica.2020.109026. [22] H. Sun, S. Li and X. Wang, Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1447-1464.  doi: 10.3934/dcdss.2020375. [23] J. Wang, C. Shao and Y. Chen, Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance, Mechatronics, 53 (2018), 8-19. [24] X.-J. Wei, Z.-J. Wu and H. R. Karimi, Disturbance observer-based disturbance attenuation control for a class of stochastic systems, Automatica J. IFAC, 63 (2016), 21-25.  doi: 10.1016/j.automatica.2015.10.019. [25] H. Yan, H. Zhang, X. Zhang, Y. Wang, S. Chen and F. Yang, Event-triggered sliding mode control of switched neural networks with mode-dependent average dwell time, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51 (2021), 1233-1243. [26] C. Zeng, Q. Yang and Y. Chen, Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach, Nonlinear Dynamics, 67 (2012), 2719-2726.  doi: 10.1007/s11071-011-0183-3. [27] X. Zhang, G. Zhang, Y. Yin and S. Je, Asynchronous sliding mode dissipative control for discrete-timeMarkov jump systems with application to automotive electronicthrottle body control system, Computers and Electrical Engineering, 96 (2021), 107496.

show all references

##### References:
 [1] R. Calif and F. Schmitt, Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation, Journal of Wind Engineering and Industrial Aerodynamics, 109 (2012), 1-8. [2] W. Chen, J. Yang, L. Guo and S. Li, Disturbance-observer-based control and related methods–An overview, IEEE Transactions on Industrial Electronics, 63 (2015), 1083-1095. [3] Y. Chen, C. Tang and M. Roohi, Design of a model-free adaptive sliding mode control to synchronize chaotic fractional-order systems with input saturation: An application in secure communications, J. Franklin Inst., 358 (2021), 8109-8137.  doi: 10.1016/j.jfranklin.2021.08.007. [4] Q. Gao, G. Feng, L. Liu, J. Qiu and Y. Wang, An ISMC approach to robust stabilization of uncertain stochastic time-delay systems, IEEE Transactions on Industrial Electronics, 61 (2014), 6986-6994. [5] J. Huang, S. Ri, T. Fukuda and Y. Wang, A disturbance observer based sliding mode control for a class of underactuated robotic system with mismatched uncertainties, IEEE Trans. Automat. Control, 64 (2019), 2480-2487.  doi: 10.1109/tac.2018.2868026. [6] B. Jiang and C.-C. Gao, Decentralized adaptive sliding mode control of large-scale semi-Markovian jump interconnected systems with dead-zone input, IEEE Transactions on Automatic Control. doi: 10.1109/TAC.2021.3065658. [7] K. Khandani, A sliding mode observer design for uncertain fractional It$\hat{o}$ stochastic systems with state delay, Int. J. Gen. Syst., 48 (2019), 48-65.  doi: 10.1080/03081079.2018.1534846. [8] K. Khandani, V. J. Majd and M. Tahmasebi, Integral sliding mode control for robust stabilisation of uncertain stochastic time-delay systems driven by fractional Brownian motion, Internat. J. Systems Sci., 48 (2017), 828-837.  doi: 10.1080/00207721.2016.1216201. [9] K. Khandani, V. J. Majd and M. Tahmasebi, Comments on "Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach"[Nonlinear Dyn. 67, 2719–2726 (2012)], Nonlinear Dynamics, 82 (2015), 1605-1607.  doi: 10.1007/s11071-015-2249-0. [10] K. Khandani, V. J. Majd and M. Tahmasebi, Robust stabilization of uncertain time-delay systems with fractional stochastic noise using the novel fractional stochastic sliding approach and its application to stream water quality regulation, IEEE Trans. Automat. Control, 62 (2017), 1742-1751.  doi: 10.1109/TAC.2016.2594261. [11] R. Khasminskii, Stochastic Stability of Differential Equations, Completely revised and enlarged second edition. Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [12] E. Kim, A fuzzy disturbance observer and its application to control, IEEE Transactions on Fuzzy Systems, 10 (2002), 77-84. [13] C. Liu, S. Sun, C. Tao, Y. Shou and B. Xu, Sliding mode control of multi-agent system with application to UAV air combat, Computers and Electrical Engineering, 96 (2021), 107491. [14] Z. Liu, H. R. Karimi and J. Yu, Passivity-based robust sliding mode synthesis for uncertain delayed stochastic systems via state observer, Automatica J. IFAC, 111 (2020), 108596, 10 pp. doi: 10.1016/j.automatica.2019.108596. [15] Z. Liu and J. Yu, Non-fragile observer-based adaptive control of uncertain nonlinear stochastic Markovian jump systems via sliding mode technique, Nonlinear Analysis: Hybrid Systems, 38 (2020), 100931, 17 pp. doi: 10.1016/j.nahs.2020.100931. [16] S. Lu and W. Zhang, Robust $H_{\infty}$ filtering and control for a class of linear systems with fractional stochastic noise, Phys. A, 526 (2019), 120958, 11 pp. doi: 10.1016/j.physa.2019.04.194. [17] X. Meng, C. Gao, Z. Liu and B. Jiang, Robust $H_{\infty}$ control for a class of uncertain neutral-type systems with time-varying delays, Asian J. Control, 23 (2021), 1454-1465.  doi: 10.1002/asjc.2298. [18] X. Meng, Z. Wu, C. Gao, B. Jiang and H. Karimi, Finite-time projective synchronization control of variable-order fractional chaotic systems via sliding mode approach, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2021), 2503-2507. [19] M. Parvizian and K. Khandani, Mean square exponential stabilization of uncertain time-delay stochastic systems with fractional Brownian motion, Internat. J. Robust Nonlinear Control, 31 (2021), 9253-9266.  doi: 10.1002/rnc.5764. [20] Y. Ren, X. Cheng and R. Sakthivel, Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm, Appl. Math. Comput., 247 (2014), 205-212.  doi: 10.1016/j.amc.2014.08.095. [21] J. Song, Y. Niu, H.-K. Lam and Y. Zou, Asynchronous sliding mode control of singularly perturbed semi-Markovian jump systems: Application to an operational amplifier circuit, Automatica J. IFAC, 118 (2020), 109026, 8 pp. doi: 10.1016/j.automatica.2020.109026. [22] H. Sun, S. Li and X. Wang, Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1447-1464.  doi: 10.3934/dcdss.2020375. [23] J. Wang, C. Shao and Y. Chen, Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance, Mechatronics, 53 (2018), 8-19. [24] X.-J. Wei, Z.-J. Wu and H. R. Karimi, Disturbance observer-based disturbance attenuation control for a class of stochastic systems, Automatica J. IFAC, 63 (2016), 21-25.  doi: 10.1016/j.automatica.2015.10.019. [25] H. Yan, H. Zhang, X. Zhang, Y. Wang, S. Chen and F. Yang, Event-triggered sliding mode control of switched neural networks with mode-dependent average dwell time, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51 (2021), 1233-1243. [26] C. Zeng, Q. Yang and Y. Chen, Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach, Nonlinear Dynamics, 67 (2012), 2719-2726.  doi: 10.1007/s11071-011-0183-3. [27] X. Zhang, G. Zhang, Y. Yin and S. Je, Asynchronous sliding mode dissipative control for discrete-timeMarkov jump systems with application to automotive electronicthrottle body control system, Computers and Electrical Engineering, 96 (2021), 107496.
A sample path of fBm with $H = 0.6$
State trajectories of open-loop systems
The curves of disturbance estimation
State trajectories of closed-loop systems
The curves of sliding mode surface function and control signal
State trajectories $x_{1}(t)$ with $\tau = 0.2,0.4,\cdot\cdot\cdot,1.0$, respectively
State trajectories $x_{2}(t)$ with $\tau = 0.2,0.4,\cdot\cdot\cdot,1.0$, respectively
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