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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. ChenJ. YangL. Guo and S. Li, Disturbance-observer-based control and related methods–An overview, IEEE Transactions on Industrial Electronics, 63 (2015), 1083-1095. 

[3]

Y. ChenC. 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. GaoG. FengL. LiuJ. 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. HuangS. RiT. 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. KhandaniV. 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. KhandaniV. 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. KhandaniV. 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. LiuS. SunC. TaoY. 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. MengC. GaoZ. 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. MengZ. WuC. GaoB. 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. RenX. 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. SunS. 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. WangC. 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. WeiZ.-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. YanH. ZhangX. ZhangY. WangS. 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. ZengQ. 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. ZhangG. ZhangY. 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. ChenJ. YangL. Guo and S. Li, Disturbance-observer-based control and related methods–An overview, IEEE Transactions on Industrial Electronics, 63 (2015), 1083-1095. 

[3]

Y. ChenC. 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. GaoG. FengL. LiuJ. 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. HuangS. RiT. 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. KhandaniV. 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. KhandaniV. 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. KhandaniV. 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. LiuS. SunC. TaoY. 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. MengC. GaoZ. 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. MengZ. WuC. GaoB. 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. RenX. 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. SunS. 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. WangC. 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. WeiZ.-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. YanH. ZhangX. ZhangY. WangS. 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. ZengQ. 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. ZhangG. ZhangY. 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. 

Figure 1.  A sample path of fBm with $ H = 0.6 $
Figure 2.  State trajectories of open-loop systems
Figure 3.  The curves of disturbance estimation
Figure 4.  State trajectories of closed-loop systems
Figure 5.  The curves of sliding mode surface function and control signal
Figure 6.  State trajectories $ x_{1}(t) $ with $ \tau = 0.2,0.4,\cdot\cdot\cdot,1.0 $, respectively
Figure 7.  State trajectories $ x_{2}(t) $ with $ \tau = 0.2,0.4,\cdot\cdot\cdot,1.0 $, respectively
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