May  2020, 3(2): 81-99. doi: 10.3934/mfc.2020007

A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty

College of Engineering, Department of Electrical Engineering, TAIF University KSA, TAIF, KSA

* Corresponding author: NasimUllah

Received  December 2019 Revised  January 2020 Published  May 2020

This research work proposes a novel triple mode sliding mode controller for a nonlinear system with measurement noise and uncertainty. The proposed control has the following goals (1) it ensures the transient and steady state robustness of the system in closed loop (2) it reduces chattering in the control signal with measurement noise. Fuzzy system is used to tune the appropriate order of the fractional operators for the proposed control system. Depending on the tuned range of the fractional operators, the proposed controller can operate effectively in the following three modes (1) classical sliding mode (SMC) (2) fractional order sliding mode (FSMC) (3) Integral sliding mode control (ISMC). With the noisy feedback, the performance of the classical SMC and SMC with boundary layer degrades significantly while ISMC shows better performance. However ISMC exhibits large transient overshoots.The proposed control method optimally selects the appropriate mode of the controller to ensure performance(transient and steady state) and suppresses the effect of noisy feedback. The proposed scheme is derived for the permanent magnet synchronous motor, s (PMSM) speed regulation problem which is subject to uncertainties, measurement noise and un-modeled dynamics as a case study. The effectiveness of proposed scheme is verified using numerical simulations.

Citation: Nasim Ullah, Ahmad Aziz Al-Ahmadi. A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty. Mathematical Foundations of Computing, 2020, 3 (2) : 81-99. doi: 10.3934/mfc.2020007
References:
[1]

M. P. Aghababa, A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems, Nonlinear Dynam., 78 (2014), 2129-2140.  doi: 10.1007/s11071-014-1594-8.

[2]

M. Asghar and N. Ullah, Performance comparison of wind turbine based doubly fed induction generator system using fault tolerant fractional and integer order controllers, Renewable Energy, 116 (2018), 244-264.  doi: 10.1016/j.renene.2017.01.008.

[3]

J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[4]

G. BartoliniA. Ferrara and E. Usai, Chattering avoidance by second-order sliding mode control, IEEE Trans. Automat. Control, 43 (1998), 241-246.  doi: 10.1109/9.661074.

[5]

A. BoulkrouneA. BouzeribaS. Hamel and T. Bouden, Adaptive fuzzy control-based projective synchronization of uncertain nonaffine chaotic systems, Complexity, 21 (2015), 180-192.  doi: 10.1002/cplx.21596.

[6]

B. ChenX. LiuK. Liu and C. Lin, Direct adaptive fuzzy control of nonlinear strict-feedback systems, Automatica J. IFAC, 45 (2009), 1530-1535.  doi: 10.1016/j.automatica.2009.02.025.

[7]

X. ChenS. TsuruokaT. Fukuda and T. Hori, Disturbance identification and its application for MIMO systems, IFAC Proceedings Volumes, 30 (1997), 1305-1310.  doi: 10.1016/S1474-6670(17)43022-9.

[8]

M.-S. Chen and M.-L. Tseng, A new design for noise-induced chattering reduction in sliding mode control, in Sliding Mode Control, 24, IntechOpen, Rijeka, (2011), 461–472. doi: 10.5772/15507.

[9]

W. Chen and Z. Zhang, Globally stable adaptive backstepping fuzzy control for output-feedback systems with unknown high-frequency gain sign, Fuzzy Sets and Systems, 161 (2010), 821-836.  doi: 10.1016/j.fss.2009.10.026.

[10]

W. ChenL. JiaoR. Li and J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Transactions on Fuzzy Systems, 18 (2010), 674-685.  doi: 10.1109/TFUZZ.2010.2046329.

[11]

S. Dadras and H. R. Momeni, Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 367-377.  doi: 10.1016/j.cnsns.2011.04.032.

[12]

H. Delavari, R. Ghaderi, A. N. Ranjbar, S. H. Hosseinnia and S. Momani, Adaptive fractional PID controller for robot manipulator, in Proceedings of FDA'10. The 4th IFAC Workshop Fractional Differentiation and its Applications, Badajoz, Spain, (2010), 1–7.

[13]

H. F. HoY. K. Wong and A. B. Rad, Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems, Simulation Modeling Practice and Theory, 17 (2009), 1199-1210.  doi: 10.1016/j.simpat.2009.04.004.

[14]

A. KawamuraH. Itoh and K. Sakamoto, Chattering reduction of disturbance observer based sliding mode control, IEEE Transactions on Industry Applications, 30 (1994), 456-461.  doi: 10.1109/28.287509.

[15]

G. LaiZ. LiuY. ZhangC. L. P. ChenS. Xie and Y. Liu, Fuzzy adaptive inverse compensation method to tracking control of uncertain nonlinear systems with generalized actuator dead zone, IEEE Transactions on Fuzzy Systems, 25 (2017), 191-204.  doi: 10.1109/TFUZZ.2016.2554152.

[16]

Y. Luo and Y. Q. Chen, Fractional Order Motion Controls, John Wiley & Sons, 2012. doi: 10.1002/9781118387726.

[17]

D. Matignon, Stability properties for generalized fractional differential systems, ESAIM: Proc., 5 (1998), 145-158.  doi: 10.1051/proc:1998004.

[18]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1993.

[19]

A. OustaloupX. Mreau and M Nouillant, The CRONE suspension, Control Engineering Practice, 4 (1996), 1101-1108.  doi: 10.1016/0967-0661(96)00109-8.

[20]

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.

[21]

A. Rhif, Stabilizing sliding mode control design and application for a DC motor: Speed control, CoRR, 2 (2012), 39-48.  doi: 10.5121/ijics.2012.2104.

[22]

S. Seshagiri and H. K. Khalil, Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators, Automatica J. IFAC, 41 (2005), 43-54.  doi: 10.1016/j.automatica.2004.08.013.

[23]

N. UllahM. A. AliR. Ahmad and A. Khattak, Fractional order control of static series synchronous compensator with parametric uncertainty, IET Generation, Transmission & Distribution, 11 (2017), 289-302. 

[24]

V. I. Utkin and H.-C. Chang, Sliding mode control on electro-mechanical systems, Math. Probl. Eng., 8 (2002), 451-471.  doi: 10.1080/10241230306724.

[25]

F. WangZ. Liu and G. Lai, Fuzzy adaptive control of nonlinear uncertain plants with unknown dead zone output, Fuzzy Sets and Systems, 263 (2015), 27-48.  doi: 10.1016/j.fss.2014.04.024.

[26]

J. WuW. Chen and J. Li, Fuzzy-approximation-based global adaptive control for uncertain strict-feedback systems with a priori known tracking accuracy, Fuzzy Sets and Systems, 273 (2015), 1-25.  doi: 10.1016/j.fss.2014.10.009.

[27]

J. YaoZ. Jiao and S. Han, Friction compensation for low velocity control of hydraulic flight motion simulator: A simple adaptive robust approach, Chinese Journal of Aeronautics, 26 (2013), 814-822. 

[28]

J. Yao and Z. Jiao, Friction compensation for hydraulic load simulator based on improved LuGre friction model, Journal of Beijing University of Aeronautics And Astronautics, 36 (2010), 812-815. 

[29]

K. D. YoungV. I. Utkin and U. Ozguner, A control engineer's guide to sliding mode control, IEEE Transactions on Control Systems Technology, 7 (1999), 328-342.  doi: 10.1109/87.761053.

[30]

B. ZhangY. Pi and Y. Luo, Fractional order sliding-mode control based on parameters auto-tuning for velocity control of permanent magnet synchronous motor, ISA Transactions, 51 (2012), 649-656.  doi: 10.1016/j.isatra.2012.04.006.

show all references

References:
[1]

M. P. Aghababa, A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems, Nonlinear Dynam., 78 (2014), 2129-2140.  doi: 10.1007/s11071-014-1594-8.

[2]

M. Asghar and N. Ullah, Performance comparison of wind turbine based doubly fed induction generator system using fault tolerant fractional and integer order controllers, Renewable Energy, 116 (2018), 244-264.  doi: 10.1016/j.renene.2017.01.008.

[3]

J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[4]

G. BartoliniA. Ferrara and E. Usai, Chattering avoidance by second-order sliding mode control, IEEE Trans. Automat. Control, 43 (1998), 241-246.  doi: 10.1109/9.661074.

[5]

A. BoulkrouneA. BouzeribaS. Hamel and T. Bouden, Adaptive fuzzy control-based projective synchronization of uncertain nonaffine chaotic systems, Complexity, 21 (2015), 180-192.  doi: 10.1002/cplx.21596.

[6]

B. ChenX. LiuK. Liu and C. Lin, Direct adaptive fuzzy control of nonlinear strict-feedback systems, Automatica J. IFAC, 45 (2009), 1530-1535.  doi: 10.1016/j.automatica.2009.02.025.

[7]

X. ChenS. TsuruokaT. Fukuda and T. Hori, Disturbance identification and its application for MIMO systems, IFAC Proceedings Volumes, 30 (1997), 1305-1310.  doi: 10.1016/S1474-6670(17)43022-9.

[8]

M.-S. Chen and M.-L. Tseng, A new design for noise-induced chattering reduction in sliding mode control, in Sliding Mode Control, 24, IntechOpen, Rijeka, (2011), 461–472. doi: 10.5772/15507.

[9]

W. Chen and Z. Zhang, Globally stable adaptive backstepping fuzzy control for output-feedback systems with unknown high-frequency gain sign, Fuzzy Sets and Systems, 161 (2010), 821-836.  doi: 10.1016/j.fss.2009.10.026.

[10]

W. ChenL. JiaoR. Li and J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Transactions on Fuzzy Systems, 18 (2010), 674-685.  doi: 10.1109/TFUZZ.2010.2046329.

[11]

S. Dadras and H. R. Momeni, Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 367-377.  doi: 10.1016/j.cnsns.2011.04.032.

[12]

H. Delavari, R. Ghaderi, A. N. Ranjbar, S. H. Hosseinnia and S. Momani, Adaptive fractional PID controller for robot manipulator, in Proceedings of FDA'10. The 4th IFAC Workshop Fractional Differentiation and its Applications, Badajoz, Spain, (2010), 1–7.

[13]

H. F. HoY. K. Wong and A. B. Rad, Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems, Simulation Modeling Practice and Theory, 17 (2009), 1199-1210.  doi: 10.1016/j.simpat.2009.04.004.

[14]

A. KawamuraH. Itoh and K. Sakamoto, Chattering reduction of disturbance observer based sliding mode control, IEEE Transactions on Industry Applications, 30 (1994), 456-461.  doi: 10.1109/28.287509.

[15]

G. LaiZ. LiuY. ZhangC. L. P. ChenS. Xie and Y. Liu, Fuzzy adaptive inverse compensation method to tracking control of uncertain nonlinear systems with generalized actuator dead zone, IEEE Transactions on Fuzzy Systems, 25 (2017), 191-204.  doi: 10.1109/TFUZZ.2016.2554152.

[16]

Y. Luo and Y. Q. Chen, Fractional Order Motion Controls, John Wiley & Sons, 2012. doi: 10.1002/9781118387726.

[17]

D. Matignon, Stability properties for generalized fractional differential systems, ESAIM: Proc., 5 (1998), 145-158.  doi: 10.1051/proc:1998004.

[18]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1993.

[19]

A. OustaloupX. Mreau and M Nouillant, The CRONE suspension, Control Engineering Practice, 4 (1996), 1101-1108.  doi: 10.1016/0967-0661(96)00109-8.

[20]

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.

[21]

A. Rhif, Stabilizing sliding mode control design and application for a DC motor: Speed control, CoRR, 2 (2012), 39-48.  doi: 10.5121/ijics.2012.2104.

[22]

S. Seshagiri and H. K. Khalil, Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators, Automatica J. IFAC, 41 (2005), 43-54.  doi: 10.1016/j.automatica.2004.08.013.

[23]

N. UllahM. A. AliR. Ahmad and A. Khattak, Fractional order control of static series synchronous compensator with parametric uncertainty, IET Generation, Transmission & Distribution, 11 (2017), 289-302. 

[24]

V. I. Utkin and H.-C. Chang, Sliding mode control on electro-mechanical systems, Math. Probl. Eng., 8 (2002), 451-471.  doi: 10.1080/10241230306724.

[25]

F. WangZ. Liu and G. Lai, Fuzzy adaptive control of nonlinear uncertain plants with unknown dead zone output, Fuzzy Sets and Systems, 263 (2015), 27-48.  doi: 10.1016/j.fss.2014.04.024.

[26]

J. WuW. Chen and J. Li, Fuzzy-approximation-based global adaptive control for uncertain strict-feedback systems with a priori known tracking accuracy, Fuzzy Sets and Systems, 273 (2015), 1-25.  doi: 10.1016/j.fss.2014.10.009.

[27]

J. YaoZ. Jiao and S. Han, Friction compensation for low velocity control of hydraulic flight motion simulator: A simple adaptive robust approach, Chinese Journal of Aeronautics, 26 (2013), 814-822. 

[28]

J. Yao and Z. Jiao, Friction compensation for hydraulic load simulator based on improved LuGre friction model, Journal of Beijing University of Aeronautics And Astronautics, 36 (2010), 812-815. 

[29]

K. D. YoungV. I. Utkin and U. Ozguner, A control engineer's guide to sliding mode control, IEEE Transactions on Control Systems Technology, 7 (1999), 328-342.  doi: 10.1109/87.761053.

[30]

B. ZhangY. Pi and Y. Luo, Fractional order sliding-mode control based on parameters auto-tuning for velocity control of permanent magnet synchronous motor, ISA Transactions, 51 (2012), 649-656.  doi: 10.1016/j.isatra.2012.04.006.

Figure 1.  (a): variation of $ \alpha $ with $ |S_{3} | $ (b): Variation of $ \beta $ with $ |S_{3} | $
Figure 2.  (a): Fuzzy sets of input variable $ |S_{3} | $(b): Fuzzy sets of out variable $ \alpha $(c): Fuzzy sets of out variable $ \beta $(d): Variation of $ \beta $ with $ \alpha $
Figure 3.  (a) speed regulation, (b) control signal, (c) sliding surface
Figure 4.  Enlarged view of (a) speed regulation, (b) control signal, (c) sliding surface
Figure 5.  (a) speed error, (b) control signal, (c) sliding surface with $ D(X, u, t) $
Figure 6.  Enlarged view of (a) speed error, (b) control signal, (c) sliding surface with $ D(X, u, t) $
Figure 7.  (a) speed error, (b) control signal, (c) sliding surface with $ D(X, u, t) $ and measurement noise
Figure 8.  Enlarged view of (a) speed error, (b) control signal, (c) sliding surface with $ D(X, u, t) $ and measurement noise
Figure 9.  Speed regulation (b) Speed error (c) Control signal (d) Sliding surface with with $ D(X, u, t) $ and measurement noise
Figure 10.  (a) Speed error (b) Control signal (c) Sliding surface with with $ D(X, u, t) $ and measurement noise
Figure 11.  (a) Adaptation of $ \alpha $ (b) Adaptation of $ \beta $
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