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February  2022, 5(1): 17-32. doi: 10.3934/mfc.2021017

Sliding mode observer based control for T-S fuzzy descriptor systems

1. 

Qinhuangdao Vocational and Technical College, Qinhuangdao, Hebei, 066100, China

2. 

School of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei, 066100, China

*Corresponding author: Dongyun Wang

Received  June 2021 Revised  August 2021 Published  February 2022 Early access  September 2021

In this paper, the problem of sliding mode observer (SMO) based sliding mode control (SMC) for nonlinear descriptor delay systems is studied. First, based on the T-S fuzzy dynamic modeling technique, the nonlinear descriptor system is transformed into a combination of local linear models. Then, a integral-type sliding surface (ITSS) based SMO is constructed for the error system. In the sequel, sufficient linear matrix inequality (LMI) conditions are established to ensure the admissibility of the sliding motions and obtain the observer gain matrix. Furthermore, two novel SMC laws are developed to ensure the reachability conditions and stabilize the descriptor systems. Finally, simulations are provided to show the effectiveness of the method.

Citation: Dongyun Wang. Sliding mode observer based control for T-S fuzzy descriptor systems. Mathematical Foundations of Computing, 2022, 5 (1) : 17-32. doi: 10.3934/mfc.2021017
References:
[1]

U. M. Al-SaggafM. Bettayeb and S. Djennoune, Super-twisting algorithm-based sliding mode observer for synchronization of nonlinear incommensurate fractional-order chaotic systems subject to unknown inputs, Arab. J. Sci. Eng., 42 (2017), 3065-3075.  doi: 10.1007/s13369-017-2548-5.

[2]

B. ChenC. LinX. Liu and K. Liu, Observer-based adaptive fuzzy control for a class of nonlinear delayed systems, IEEE Trans. Syst. Man, and Cybernetics: Systems, 46 (2016), 27-39.  doi: 10.1109/TSMC.2015.2420543.

[3]

M. Darouach, Observers and observer-based control for descriptor systems revisited, IEEE Trans. Automat. Control, 59 (2014), 1367-1373.  doi: 10.1109/TAC.2013.2292720.

[4] L. Dai, Singular Control Systems, Springer-Verlag, Berlin, 1989.  doi: 10.1007/BFb0002475.
[5] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor-Francis, London, UK, 1998.  doi: 10.1201/9781498701822.
[6]

E. Fridman and U. Shaked, A descriptor system approach to $H\infty$ control of linear time-delay systems, IEEE Trans. Automat. Control, 47 (2002), 253-270.  doi: 10.1109/9.983353.

[7]

Q. GaoG. FengZ. Y. Xi and Y. Wang, Robust $H\infty$ control of T-S fuzzy time-delay systems via a new sliding-mode control scheme, IEEE Transactions on Fuzzy Systems, 22 (2014), 459-465.  doi: 10.1109/TFUZZ.2013.2256914.

[8]

T. M. GuerraV. Estrada-Manzo and Z. Lendek, Observer design for Takagi-Sugeno descriptor models: An LMI approach, Automatica Automatica J. IFAC, 52 (2015), 154-159.  doi: 10.1016/j.automatica.2014.11.008.

[9]

K. Gu, V. Kharitonov and J. Chen, Stability of Time-Delay Systems, Control Engineering. Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0.

[10]

C. S. HanG. J. ZhangL. G. Wu and Q. S. Zeng, Sliding mode control of T-S fuzzy descriptor systems with time-delay, J. Franklin Inst., 349 (2012), 1430-1444.  doi: 10.1016/j.jfranklin.2011.07.001.

[11]

K. HfaiedhK. Dahech and T. Damak, A sliding mode observer for uncertain nonlinear systems based on multiple modes approach, International Journal of Automation and Computing, 14 (2017), 202-212.  doi: 10.1007/s11633-016-0970-x.

[12]

M. KlugE. B. CastelanV. J. S. Leite and L. F. P. Silva, Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models, Fuzzy Sets and Systems, 263 (2015), 92-111.  doi: 10.1016/j.fss.2014.05.019.

[13]

M. KchaouH. Gassara and A. El-Hajjaji, Robust observer-based control design for uncertain singular systems with time-delay, International J. Adapt. Control Signal Process., 28 (2014), 169-183.  doi: 10.1002/acs.2409.

[14]

J. H. LiQ. L. ZhangX. G. Yan and S. K. Spurgeon, Integral sliding mode control for Markovian jump T-S fuzzy descriptor systems based on the super-twisting algorithm, IET Control Theory Appl., 11 (2017), 1134-1143.  doi: 10.1049/iet-cta.2016.0862.

[15]

M. Liu and P. Shi, Sensor fault estimation and tolerant control for Ito stochastic systems with a descriptor sliding mode approach, Automatica J. IFAC, 49 (2013), 1242-1250.  doi: 10.1016/j.automatica.2013.01.030.

[16]

F. R. Lopez-EstradaC. M. Astorga-ZaragozaD. TheilliolJ. C. PonsartG. Valencia-Palomo and L. Torres, Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis, Internat. J. Systems Sci., 48 (2017), 3419-3430.  doi: 10.1080/00207721.2017.1384517.

[17]

Q. LiQ. L. ZhangY. J. Zhang and Y. C. An, Observer-based passive control for descriptor systems with time-delay, J. Syst. Engineering and Electronics, 20 (2009), 120-128. 

[18]

H. Y. LiJ. H. WangH. K. LamQ. Zhou and H. P. Du, Adaptive sliding mode control for interval type-2 fuzzy systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46 (2016), 1654-1663.  doi: 10.1109/TSMC.2016.2531676.

[19]

R. C. Li and Y. Yang, Fault tolerant control for T-S fuzzy stochastic singular systems, IEEE Transactions on Fuzzy Systems, (2020), 1–1. doi: 10.1109/TFUZZ.2020.3029303.

[20]

Y. C. MaX. R. Jia and Q. L. Zhang, Robust observer-based finite-time $H\infty$ control for discrete-time singular Markovian jumping system with time delay and actuator saturation, Nonlinear Anal. Hybrid Syst., 28 (2018), 1-22.  doi: 10.1016/j.nahs.2017.10.008.

[21]

S. Q. MaZ. Cheng and C. Zhang, Delay-dependent robust stability and stabilization for uncertain discrete singular systems with time-varying delays, IET Control Theory and Appl., 1 (2007), 1086-1095.  doi: 10.1049/iet-cta:20060131.

[22]

J. H. Li and Q. L. Zhang, An integral sliding mode control approach to observer-based stabilization of stochastic Ito descriptor systems, Neurocomputing, 173 (2016), 1330-1340.  doi: 10.1016/j.neucom.2015.09.006.

[23]

R. C. LiY. Yang and Q. L. Zhang, Neural network based adaptive SMO design for T-S fuzzy descriptor systems, IEEE Transactions on Fuzzy Systems, 28 (2020), 2605-2618.  doi: 10.1109/TFUZZ.2019.2945238.

[24]

I. R. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems Control Lett., 8 (1987), 351-357.  doi: 10.1016/0167-6911(87)90102-2.

[25]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its application to modeling and control, Readings in Fuzzy Sets for Intelligent Systems, (1993), 387–403. doi: 10.1016/B978-1-4832-1450-4.50045-6.

[26] V. Utkin, Sliding Mode in Control and Optimization, Springer-Verlag, Berlin, Germany, 1992.  doi: 10.1007/978-3-642-84379-2.
[27]

C. P. Tan and C. Edwards, An LMI approach for designing sliding mode observers, Internat. J. Control, 74 (2001), 1559-1568.  doi: 10.1080/00207170110081723.

[28]

C. P. Tan and C. Edwards, Sliding mode observers for robust detection and reconstruction of actuator and sensor faults, Internat. J. Robust Nonlinear Control, 13 (2003), 443-463.  doi: 10.1002/rnc.723.

[29]

X. H. WangC. P. Tan and D. H. Zhou, A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions, Automatica J. IFAC, 79 (2017), 290-295.  doi: 10.1016/j.automatica.2017.01.027.

[30]

S. Y. XuP. V. DoorenR. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Trans. Automat. Control, 47 (2002), 1122-1128.  doi: 10.1109/TAC.2002.800651.

[31]

S. YinH. J. GaoJ. B. Qiu and O Kaynak, Descriptor reduced-order sliding mode observers design for switched systems with sensor and actuator faults, Automatica J. IFAC, 76 (2017), 282-292.  doi: 10.1016/j.automatica.2016.10.025.

[32]

Y. ZhangQ. L. Zhang and G. F. Zhang, $H\infty$ control of T-S fuzzy fish population logistic model with the invasion of alien species, Neurocomputing, 173 (2016), 724-733. 

[33]

H. B. ZhangY. Y. Shen and G. Feng, Delay-dependent stability and $H\infty$ control for a class of fuzzy descriptor systems with time-delay, Fuzzy Sets and Systems, 160 (2009), 1689-1707.  doi: 10.1016/j.fss.2008.09.014.

show all references

References:
[1]

U. M. Al-SaggafM. Bettayeb and S. Djennoune, Super-twisting algorithm-based sliding mode observer for synchronization of nonlinear incommensurate fractional-order chaotic systems subject to unknown inputs, Arab. J. Sci. Eng., 42 (2017), 3065-3075.  doi: 10.1007/s13369-017-2548-5.

[2]

B. ChenC. LinX. Liu and K. Liu, Observer-based adaptive fuzzy control for a class of nonlinear delayed systems, IEEE Trans. Syst. Man, and Cybernetics: Systems, 46 (2016), 27-39.  doi: 10.1109/TSMC.2015.2420543.

[3]

M. Darouach, Observers and observer-based control for descriptor systems revisited, IEEE Trans. Automat. Control, 59 (2014), 1367-1373.  doi: 10.1109/TAC.2013.2292720.

[4] L. Dai, Singular Control Systems, Springer-Verlag, Berlin, 1989.  doi: 10.1007/BFb0002475.
[5] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor-Francis, London, UK, 1998.  doi: 10.1201/9781498701822.
[6]

E. Fridman and U. Shaked, A descriptor system approach to $H\infty$ control of linear time-delay systems, IEEE Trans. Automat. Control, 47 (2002), 253-270.  doi: 10.1109/9.983353.

[7]

Q. GaoG. FengZ. Y. Xi and Y. Wang, Robust $H\infty$ control of T-S fuzzy time-delay systems via a new sliding-mode control scheme, IEEE Transactions on Fuzzy Systems, 22 (2014), 459-465.  doi: 10.1109/TFUZZ.2013.2256914.

[8]

T. M. GuerraV. Estrada-Manzo and Z. Lendek, Observer design for Takagi-Sugeno descriptor models: An LMI approach, Automatica Automatica J. IFAC, 52 (2015), 154-159.  doi: 10.1016/j.automatica.2014.11.008.

[9]

K. Gu, V. Kharitonov and J. Chen, Stability of Time-Delay Systems, Control Engineering. Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0.

[10]

C. S. HanG. J. ZhangL. G. Wu and Q. S. Zeng, Sliding mode control of T-S fuzzy descriptor systems with time-delay, J. Franklin Inst., 349 (2012), 1430-1444.  doi: 10.1016/j.jfranklin.2011.07.001.

[11]

K. HfaiedhK. Dahech and T. Damak, A sliding mode observer for uncertain nonlinear systems based on multiple modes approach, International Journal of Automation and Computing, 14 (2017), 202-212.  doi: 10.1007/s11633-016-0970-x.

[12]

M. KlugE. B. CastelanV. J. S. Leite and L. F. P. Silva, Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models, Fuzzy Sets and Systems, 263 (2015), 92-111.  doi: 10.1016/j.fss.2014.05.019.

[13]

M. KchaouH. Gassara and A. El-Hajjaji, Robust observer-based control design for uncertain singular systems with time-delay, International J. Adapt. Control Signal Process., 28 (2014), 169-183.  doi: 10.1002/acs.2409.

[14]

J. H. LiQ. L. ZhangX. G. Yan and S. K. Spurgeon, Integral sliding mode control for Markovian jump T-S fuzzy descriptor systems based on the super-twisting algorithm, IET Control Theory Appl., 11 (2017), 1134-1143.  doi: 10.1049/iet-cta.2016.0862.

[15]

M. Liu and P. Shi, Sensor fault estimation and tolerant control for Ito stochastic systems with a descriptor sliding mode approach, Automatica J. IFAC, 49 (2013), 1242-1250.  doi: 10.1016/j.automatica.2013.01.030.

[16]

F. R. Lopez-EstradaC. M. Astorga-ZaragozaD. TheilliolJ. C. PonsartG. Valencia-Palomo and L. Torres, Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis, Internat. J. Systems Sci., 48 (2017), 3419-3430.  doi: 10.1080/00207721.2017.1384517.

[17]

Q. LiQ. L. ZhangY. J. Zhang and Y. C. An, Observer-based passive control for descriptor systems with time-delay, J. Syst. Engineering and Electronics, 20 (2009), 120-128. 

[18]

H. Y. LiJ. H. WangH. K. LamQ. Zhou and H. P. Du, Adaptive sliding mode control for interval type-2 fuzzy systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46 (2016), 1654-1663.  doi: 10.1109/TSMC.2016.2531676.

[19]

R. C. Li and Y. Yang, Fault tolerant control for T-S fuzzy stochastic singular systems, IEEE Transactions on Fuzzy Systems, (2020), 1–1. doi: 10.1109/TFUZZ.2020.3029303.

[20]

Y. C. MaX. R. Jia and Q. L. Zhang, Robust observer-based finite-time $H\infty$ control for discrete-time singular Markovian jumping system with time delay and actuator saturation, Nonlinear Anal. Hybrid Syst., 28 (2018), 1-22.  doi: 10.1016/j.nahs.2017.10.008.

[21]

S. Q. MaZ. Cheng and C. Zhang, Delay-dependent robust stability and stabilization for uncertain discrete singular systems with time-varying delays, IET Control Theory and Appl., 1 (2007), 1086-1095.  doi: 10.1049/iet-cta:20060131.

[22]

J. H. Li and Q. L. Zhang, An integral sliding mode control approach to observer-based stabilization of stochastic Ito descriptor systems, Neurocomputing, 173 (2016), 1330-1340.  doi: 10.1016/j.neucom.2015.09.006.

[23]

R. C. LiY. Yang and Q. L. Zhang, Neural network based adaptive SMO design for T-S fuzzy descriptor systems, IEEE Transactions on Fuzzy Systems, 28 (2020), 2605-2618.  doi: 10.1109/TFUZZ.2019.2945238.

[24]

I. R. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems Control Lett., 8 (1987), 351-357.  doi: 10.1016/0167-6911(87)90102-2.

[25]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its application to modeling and control, Readings in Fuzzy Sets for Intelligent Systems, (1993), 387–403. doi: 10.1016/B978-1-4832-1450-4.50045-6.

[26] V. Utkin, Sliding Mode in Control and Optimization, Springer-Verlag, Berlin, Germany, 1992.  doi: 10.1007/978-3-642-84379-2.
[27]

C. P. Tan and C. Edwards, An LMI approach for designing sliding mode observers, Internat. J. Control, 74 (2001), 1559-1568.  doi: 10.1080/00207170110081723.

[28]

C. P. Tan and C. Edwards, Sliding mode observers for robust detection and reconstruction of actuator and sensor faults, Internat. J. Robust Nonlinear Control, 13 (2003), 443-463.  doi: 10.1002/rnc.723.

[29]

X. H. WangC. P. Tan and D. H. Zhou, A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions, Automatica J. IFAC, 79 (2017), 290-295.  doi: 10.1016/j.automatica.2017.01.027.

[30]

S. Y. XuP. V. DoorenR. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Trans. Automat. Control, 47 (2002), 1122-1128.  doi: 10.1109/TAC.2002.800651.

[31]

S. YinH. J. GaoJ. B. Qiu and O Kaynak, Descriptor reduced-order sliding mode observers design for switched systems with sensor and actuator faults, Automatica J. IFAC, 76 (2017), 282-292.  doi: 10.1016/j.automatica.2016.10.025.

[32]

Y. ZhangQ. L. Zhang and G. F. Zhang, $H\infty$ control of T-S fuzzy fish population logistic model with the invasion of alien species, Neurocomputing, 173 (2016), 724-733. 

[33]

H. B. ZhangY. Y. Shen and G. Feng, Delay-dependent stability and $H\infty$ control for a class of fuzzy descriptor systems with time-delay, Fuzzy Sets and Systems, 160 (2009), 1689-1707.  doi: 10.1016/j.fss.2008.09.014.

Figure 1.  Trajectories of the open-loop system
Figure 2.  Trajectories of the open-loop system
Figure 3.  Trajectories of the error system
Figure 4.  Trajectories of the closed-loop system
Figure 5.  Sliding surfaces
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