# American Institute of Mathematical Sciences

April  2021, 14(4): 1301-1328. doi: 10.3934/dcdss.2020412

## Event-based fault detection for interval type-2 fuzzy systems with measurement outliers

 1 College of Engineering, Bohai University, Jinzhou 121013, Liaoning, China 2 School of Mathematics and Physics, Bohai University, Jinzhou 121013, Liaoning, China

* Corresponding author: Hong Xue

Received  January 2020 Revised  April 2020 Published  July 2020

This paper investigates the event-based fault detection (FD) problem for a category of discrete-time interval type-2 fuzzy systems with measurement outliers. For the sake of decreasing the utilization of limited communication bandwidth, an event-based mechanism is introduced. Based on the saturation function technique, a novel event-based FD observer is first designed to reduce the influence of outliers in the dynamic systems. Then, on the basis of Lyapunov stability theory, sufficient conditions are provided to ensure that the error system satisfies the $\mathcal{H}_{\infty}$ performance and the $\mathcal{H}_{\infty}$ fault performance in different cases, respectively. In contrast to the existing event-based FD results, the false alarm, which is induced by measurement outliers, can be effectively avoided by the designed FD observer with saturation function. Lastly, some simulation results are given to verify the effectiveness of the method presented in this paper.

Citation: Qi Li, Hong Xue, Changxin Lu. Event-based fault detection for interval type-2 fuzzy systems with measurement outliers. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1301-1328. doi: 10.3934/dcdss.2020412
##### References:
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Yang, Observer-based fault detection for T-S fuzzy systems subject to measurement outliers, Neurocomputing, 335 (2019), 21-36.  doi: 10.1016/j.neucom.2019.01.047.  Google Scholar [34] Z. Wang and D. Liu, Data-based controllability and observability analysis of linear discrete-time systems, IEEE Transactions on Neural Networks, 22 (2011), 2388-2392.   Google Scholar [35] Z. Wang, Y. Xu, R. Lu and H. Peng, Finite-time state estimation for coupled Markovian neural networks with sensor nonlinearities, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 630-638.  doi: 10.1109/TNNLS.2015.2490168.  Google Scholar [36] C. Wu, L. Wu, J. Liu and Z.-P. Jiang, Active defense-based resilient sliding mode control under denial-of-service attacks, IEEE Transactions on Information Forensics and Security, 15 (2019), 237-249.  doi: 10.1109/TIFS.2019.2917373.  Google Scholar [37] D. Yao, H. Li, R. Lu and Y. Shi, Distributed sliding mode tracking control of second-order nonlinear multi-agent systems: An event-triggered approach, IEEE Transactions on Cybernetics, 2020, 1–11. doi: 10.1109/TCYB.2019.2963087.  Google Scholar [38] D. Yang, X. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar [39] T. Zhang, C. P. Chen, L. Chen, X. Xu and B. Hu, Design of highly nonlinear substitution boxes based on I-Ching operators, IEEE Transactions on Cybernetics, 48 (2018), 3349-3358.  doi: 10.1109/TCYB.2018.2846186.  Google Scholar [40] L. Zhang, H.-K. Lam, Y. Sun and H. Liang, Fault detection for fuzzy semi-markov jump systems based on interval type-2 fuzzy approach, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1016/j.fss.2018.05.003.  Google Scholar [41] Z. Zhang, H. Liang, C. Wu and C. K. Ahn, Adaptive event-triggered output feedback fuzzy control for nonlinear networked systems with packet dropouts and actuator failure, IEEE Transactions on Fuzzy Systems, 27 (2019), 1793-1806.  doi: 10.1109/TFUZZ.2019.2891236.  Google Scholar [42] C. Zhang, J. Hu, J. Qiu and Q. Chen, Event-triggered nonsynchronized $\mathcal{H}_{\infty}$ filtering for discrete-time T–S fuzzy systems based on piecewise lyapunov functions, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2330-2341.   Google Scholar [43] T. Zhang, X. Wang, X. Xu and C. P. Chen, GCB-net: Graph convolutional broad network and its application in emotion recognition, IEEE Transactions on Affective Computing, 2019, 1–1. doi: 10.1109/TAFFC.2019.2937768.  Google Scholar [44] X. Zhao, H. Mo, K. Yan and L. Li, Type-2 fuzzy control for driving state and behavioral decisions of unmanned vehicle, IEEE/CAA Journal of Automatica Sinica, 7 (2020), 178-186.  doi: 10.1109/JAS.2019.1911810.  Google Scholar [45] Q. Zhou, W. Wang, H. Ma and H. Li, Event-triggered fuzzy adaptive containment control for nonlinear multi-agent systems with unknown bouc-wen hysteresis input, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2961642.  Google Scholar [46] S. Zhu, Y. Liu, Y. Lou and J. Cao, Stabilization of logical control networks: An event-triggered control approach, Science China Information Sciences, 63 (2020), 112203, 11 pp. doi: 10.1007/s11432-019-9898-3.  Google Scholar [47] Z. Zhu, Y. Pan, Q. Zhou and C. Lu, Event-triggered adaptive fuzzy control for stochastic nonlinear systems with unmeasured states and unknown backlash-like hysteresis, IEEE Transactions on Fuzzy Systems, 2020, 1–1. doi: 10.1109/TFUZZ.2020.2973950.  Google Scholar [48] Q. Zhou, W. Wang, H. Liang, M. Basin and B. Wang, Observer-based event-triggered fuzzy adaptive bipartite containment control of multi-agent systems with input quantization, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2953573.  Google Scholar

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##### References:
 [1] A. Alessandri and L. Zaccarian, Stubborn state observers for linear time-invariant systems, Automatica, 88 (2018), 1-9.  doi: 10.1016/j.automatica.2017.10.022.  Google Scholar [2] A. Alessandri and M. Awawdeh, Moving-horizon estimation with guaranteed robustness for discrete-time linear systems and measurements subject to outliers, Automatica, 67 (2016), 85-93.  doi: 10.1016/j.automatica.2016.01.015.  Google Scholar [3] Z. Chen, Q.-L. Han, Y. Yan and Z.-G. Wu, How often should one update control and estimation: Review of networked triggering techniques, Science China Information Sciences, 63 (2020), Paper No. 150201. doi: 10.1007/s11432-019-2637-9.  Google Scholar [4] S. Court, K. Kunisch and L. Pfeiffer, Hybrid optimal control problems for a class of semilinear parabolic equations, Discrete & Continuous Dynamical Systems-S, 11 (2018), 1031-1060.  doi: 10.3934/dcdss.2018060.  Google Scholar [5] H. Gao, Y. Zhao, J. Lam and K. Chen, $\mathcal{H}_ {\infty}$ fuzzy filtering of nonlinear systems with intermittent measurements, IEEE Transactions on Fuzzy Systems, 17 (2008), 291-300.   Google Scholar [6] Y. Gao, F. Xiao, J. Liu and and R. Wang, Distributed soft fault detection for interval type-2 fuzzy-model-based stochastic systems with wireless sensor networks, IEEE Transactions on Industrial Informatics, 15 (2018), 334-347.  doi: 10.1109/TII.2018.2812771.  Google Scholar [7] H. Gao and C. Wang, Comments and further results on "A descriptor system approach to $\mathcal{H}_{\infty}$ control of linear time-delay systems", IEEE Transactions on Automatic Control, 48 (2003), 520-525.  doi: 10.1109/TAC.2003.809154.  Google Scholar [8] M. A. Gandhi and L. Mili, Robust kalman filter based on a generalized maximum-likelihood-type estimator, IEEE Transactions on Signal Processing, 58 (2010), 2509-2520.  doi: 10.1109/TSP.2009.2039731.  Google Scholar [9] E. Grigorieva and E. Khailov, Determination of the optimal controls for an ebola epidemic model, Discrete & Continuous Dynamical Systems-S, 11 (2018), 1071-1101.  doi: 10.3934/dcdss.2018062.  Google Scholar [10] N. Hayek, Infinite-horizon multiobjective optimal control problems for bounded processes, Discrete & Continuous Dynamical Systems-S, 11 (2018), 1121-1141.  doi: 10.3934/dcdss.2018064.  Google Scholar [11] H. Hassani, J. Zarei, M. Chadli and J. Qiu, Unknown input observer design for interval type-2 T–S fuzzy systems with immeasurable premise variables, IEEE Transactions on Cybernetics, 47 (2017), 2639-2650.  doi: 10.1109/TCYB.2016.2602300.  Google Scholar [12] H. K. Lam and L. D. Seneviratne, Stability analysis of interval type-2 fuzzy-model-based control systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 617-628.  doi: 10.1109/TSMCB.2008.915530.  Google Scholar [13] X. Li, X. Zhang and S. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica, 76 (2017), 378-382.  doi: 10.1016/j.automatica.2016.08.009.  Google Scholar [14] Z. Li, L. Gao, W. Chen and Y. Xu, Distributed adaptive cooperative tracking of uncertain nonlinear fractional-order multi-agent systems, IEEE/CAA Journal of Automatica Sinica, 7 (2020), 292-300.  doi: 10.1109/JAS.2019.1911858.  Google Scholar [15] H. Li, J. Yu, C. Hilton and H. Liu, Adaptive sliding-mode control for nonlinear active suspension vehicle systems using T–S fuzzy approach, IEEE Transactions on Industrial Electronics, 60 (2013), 3328-3338.  doi: 10.1109/TIE.2012.2202354.  Google Scholar [16] X. Li, X. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.  Google Scholar [17] X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.  Google Scholar [18] H. Liang, X. Guo, Y. Pan and T. Huang, Event-triggered fuzzy bipartite tracking control for network systems based on distributed reduced-order observers, IEEE Transactions on Fuzzy Systems, 2020, 1–1. doi: 10.1109/TFUZZ.2020.2982618.  Google Scholar [19] H. Liang, L. Zhang, Y. Sun and T. Huang, Containment control of semi-markovian multiagent systems with switching topologies, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 1–11. doi: 10.1109/TSMC.2019.2946248.  Google Scholar [20] R. Lu, W. Yu, J. Lü and A. Xue, Synchronization on complex networks of networks, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014), 2110-2118.  doi: 10.1109/TNNLS.2014.2305443.  Google Scholar [21] J. M. Mendel, R. I. John and F. Liu, Interval type-2 fuzzy logic systems made simple, IEEE Transactions on Fuzzy Systems, 14 (2006), 808-821.  doi: 10.1109/TFUZZ.2006.879986.  Google Scholar [22] J. Na, Y. Huang, X. Wu, S.-F. Su and G. Li, Adaptive finite-time fuzzy control of nonlinear active suspension systems with input delay, IEEE Transactions on Cybernetics, 50 (2020), 2639-2650.  doi: 10.1109/TCYB.2019.2894724.  Google Scholar [23] J. Na, B. Jing, Y. Huang, G. Gao and C. Zhang, Unknown system dynamics estimator for motion control of nonlinear robotic systems, IEEE Transactions on Industrial Electronics, 67 (2020), 3850-3859.  doi: 10.1109/TIE.2019.2920604.  Google Scholar [24] Z. Ning, J. Yu, Y. Pan and H. Li, Adaptive event-triggered fault detection for fuzzy stochastic systems with missing measurements, IEEE Transactions on Fuzzy Systems, 26 (2018), 2201-2212.  doi: 10.1109/TFUZZ.2017.2780799.  Google Scholar [25] Y. Pan, P. Du, H. Xue and H. K. Lam, Singularity-free fixed-time fuzzy control for robotic systems with user-defined performance, IEEE Transactions on Fuzzy Systems, 2020, 1–1. doi: 10.1109/TFUZZ.2020.2999746.  Google Scholar [26] Y. Pan and G. H. Yang, Event-triggered fault detection filter design for nonlinear networked systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2017), 1851-1862.   Google Scholar [27] Y. Pan and G. H. Yong, Event-driven fault detection for discrete-time interval type-2 fuzzy systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 1–2. doi: 10.1109/TSMC.2019.2945063.  Google Scholar [28] C. Peng, Q. L. Han and D. Yue, To transmit or not to transmit: A discrete event-triggered communication scheme for networked Takagi–Sugeno fuzzy systems, IEEE Transactions on Fuzzy Systems, 21 (2013), 164-170.  doi: 10.1109/TFUZZ.2012.2199994.  Google Scholar [29] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, 52 (2007), 1680-1685.  doi: 10.1109/TAC.2007.904277.  Google Scholar [30] M. Tong, W. Lin, X. Huo, Z. Jin and C. Miao, A model-free fuzzy adaptive trajectory tracking control algorithm based on dynamic surface control, International Journal of Advanced Robotic Systems, 17 (2020), 1729881419894417. Google Scholar [31] C. S. Tseng, B. S. Chen and H. J. Uang, Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model, IEEE Transactions on Fuzzy Systems, 9 (2001), 381-392.   Google Scholar [32] W. Wang, H. Liang, Y. Pan and T. Li, Prescribed performance adaptive fuzzy containment control for nonlinear multi-agent systems using disturbance observer, IEEE Transactions on Cybernetics, 2020, 1–13. doi: 10.1109/TCYB.2020.2969499.  Google Scholar [33] X. L. Wang and G. H. Yang, Observer-based fault detection for T-S fuzzy systems subject to measurement outliers, Neurocomputing, 335 (2019), 21-36.  doi: 10.1016/j.neucom.2019.01.047.  Google Scholar [34] Z. Wang and D. Liu, Data-based controllability and observability analysis of linear discrete-time systems, IEEE Transactions on Neural Networks, 22 (2011), 2388-2392.   Google Scholar [35] Z. Wang, Y. Xu, R. Lu and H. Peng, Finite-time state estimation for coupled Markovian neural networks with sensor nonlinearities, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 630-638.  doi: 10.1109/TNNLS.2015.2490168.  Google Scholar [36] C. Wu, L. Wu, J. Liu and Z.-P. Jiang, Active defense-based resilient sliding mode control under denial-of-service attacks, IEEE Transactions on Information Forensics and Security, 15 (2019), 237-249.  doi: 10.1109/TIFS.2019.2917373.  Google Scholar [37] D. Yao, H. Li, R. Lu and Y. Shi, Distributed sliding mode tracking control of second-order nonlinear multi-agent systems: An event-triggered approach, IEEE Transactions on Cybernetics, 2020, 1–11. doi: 10.1109/TCYB.2019.2963087.  Google Scholar [38] D. Yang, X. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar [39] T. Zhang, C. P. Chen, L. Chen, X. Xu and B. Hu, Design of highly nonlinear substitution boxes based on I-Ching operators, IEEE Transactions on Cybernetics, 48 (2018), 3349-3358.  doi: 10.1109/TCYB.2018.2846186.  Google Scholar [40] L. Zhang, H.-K. Lam, Y. Sun and H. Liang, Fault detection for fuzzy semi-markov jump systems based on interval type-2 fuzzy approach, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1016/j.fss.2018.05.003.  Google Scholar [41] Z. Zhang, H. Liang, C. Wu and C. K. Ahn, Adaptive event-triggered output feedback fuzzy control for nonlinear networked systems with packet dropouts and actuator failure, IEEE Transactions on Fuzzy Systems, 27 (2019), 1793-1806.  doi: 10.1109/TFUZZ.2019.2891236.  Google Scholar [42] C. Zhang, J. Hu, J. Qiu and Q. Chen, Event-triggered nonsynchronized $\mathcal{H}_{\infty}$ filtering for discrete-time T–S fuzzy systems based on piecewise lyapunov functions, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2330-2341.   Google Scholar [43] T. Zhang, X. Wang, X. Xu and C. P. Chen, GCB-net: Graph convolutional broad network and its application in emotion recognition, IEEE Transactions on Affective Computing, 2019, 1–1. doi: 10.1109/TAFFC.2019.2937768.  Google Scholar [44] X. Zhao, H. Mo, K. Yan and L. Li, Type-2 fuzzy control for driving state and behavioral decisions of unmanned vehicle, IEEE/CAA Journal of Automatica Sinica, 7 (2020), 178-186.  doi: 10.1109/JAS.2019.1911810.  Google Scholar [45] Q. Zhou, W. Wang, H. Ma and H. Li, Event-triggered fuzzy adaptive containment control for nonlinear multi-agent systems with unknown bouc-wen hysteresis input, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2961642.  Google Scholar [46] S. Zhu, Y. Liu, Y. Lou and J. Cao, Stabilization of logical control networks: An event-triggered control approach, Science China Information Sciences, 63 (2020), 112203, 11 pp. doi: 10.1007/s11432-019-9898-3.  Google Scholar [47] Z. Zhu, Y. Pan, Q. Zhou and C. Lu, Event-triggered adaptive fuzzy control for stochastic nonlinear systems with unmeasured states and unknown backlash-like hysteresis, IEEE Transactions on Fuzzy Systems, 2020, 1–1. doi: 10.1109/TFUZZ.2020.2973950.  Google Scholar [48] Q. Zhou, W. Wang, H. Liang, M. Basin and B. Wang, Observer-based event-triggered fuzzy adaptive bipartite containment control of multi-agent systems with input quantization, IEEE Transactions on Fuzzy Systems, 2019, 1–1. doi: 10.1109/TFUZZ.2019.2953573.  Google Scholar
Estimation errors of the system
Trajectory of $\bar{\sigma}(k)$
Event-based release instants and release interval
Evaluation function $J(k)$ and threshold $\mathcal{J}_{th}$
Evaluation function $J(k)$ and threshold $\mathcal{J}_{th}$
Evaluation function $J(k)$ and threshold $\mathcal{J}_{th}$
Tunnel diode circuit
Estimation errors of the system
Trajectory of $\bar{\sigma}(k)$
Event-based release instants and release interval
Evaluation function $J(k)$ and threshold $\mathcal{J}_{th}$
Evaluation function $J(k)$ and threshold $\mathcal{J}_{th}$
Evaluation function $J(k)$ and threshold $\mathcal{J}_{th}$
Lower and upper membership functions of the system
 ${\underline{u}}_{\mathcal{L}11}(x_{1})=1-e^{-\frac{x_{1}^{2}}{1.5}}$ ${ \bar{u}}_{\mathcal{L}11}(x_{1})=1-e^{-\frac{x_{1}^{2}}{1.2}}$ ${\underline{u}}_{\mathcal{L}21}(x_{1})=1-{\bar{u}}_{\mathcal{L}11}(x_{1})$ ${\bar{u}}_{\mathcal{L}21}(x_{1})=1-{\underline{u}}_{\mathcal{L}11}(x_{1})$
 ${\underline{u}}_{\mathcal{L}11}(x_{1})=1-e^{-\frac{x_{1}^{2}}{1.5}}$ ${ \bar{u}}_{\mathcal{L}11}(x_{1})=1-e^{-\frac{x_{1}^{2}}{1.2}}$ ${\underline{u}}_{\mathcal{L}21}(x_{1})=1-{\bar{u}}_{\mathcal{L}11}(x_{1})$ ${\bar{u}}_{\mathcal{L}21}(x_{1})=1-{\underline{u}}_{\mathcal{L}11}(x_{1})$
Lower and upper membership functions of the observer
 ${\underline{u}}_{\mathcal{V}11}(x_{1})=e^{-\frac{x_{1}^{2}}{0.4}}$ ${\bar{ u}}_{\mathcal{V}11}(x_{1})=e^{-\frac{x_{1}^{2}}{0.4}}$ ${\underline{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V} 11}(x_{1})$ ${\bar{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V }11}(x_{1})$
 ${\underline{u}}_{\mathcal{V}11}(x_{1})=e^{-\frac{x_{1}^{2}}{0.4}}$ ${\bar{ u}}_{\mathcal{V}11}(x_{1})=e^{-\frac{x_{1}^{2}}{0.4}}$ ${\underline{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V} 11}(x_{1})$ ${\bar{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V }11}(x_{1})$
Lower and upper membership functions of the system
 ${\underline{u}}_{\mathcal{L}11}(x_{1})=\frac{\bar{e}_{\max }-\bar{e}}{\bar{e }_{\max }-\bar{e}_{\min }},\partial=0.03$ ${\bar{u}}_{\mathcal{L} 11}(x_{1})=\frac{\bar{e}_{\max }-\bar{e}}{\bar{e}_{\max }-\bar{e}_{\min }} ,\partial=0.01$ ${\underline{u}}_{\mathcal{L}21}(x_{1})=\frac{\bar{e}-\bar{e}_{\min}}{\bar{e} _{\max }-\bar{e}_{\min }},\partial=0.01$ ${\bar{u}}_{\mathcal{L}21}(x_{1})= \frac{\bar{e}-\bar{e}_{\min}}{\bar{e}_{\max }-\bar{e}_{\min }},\partial=0.03$
 ${\underline{u}}_{\mathcal{L}11}(x_{1})=\frac{\bar{e}_{\max }-\bar{e}}{\bar{e }_{\max }-\bar{e}_{\min }},\partial=0.03$ ${\bar{u}}_{\mathcal{L} 11}(x_{1})=\frac{\bar{e}_{\max }-\bar{e}}{\bar{e}_{\max }-\bar{e}_{\min }} ,\partial=0.01$ ${\underline{u}}_{\mathcal{L}21}(x_{1})=\frac{\bar{e}-\bar{e}_{\min}}{\bar{e} _{\max }-\bar{e}_{\min }},\partial=0.01$ ${\bar{u}}_{\mathcal{L}21}(x_{1})= \frac{\bar{e}-\bar{e}_{\min}}{\bar{e}_{\max }-\bar{e}_{\min }},\partial=0.03$
Lower and upper membership functions of the observer
 ${\underline{u}}_{\mathcal{V}11}(x_{1})=0.4e^{-\frac{x_{1}^{2}}{1.2}}$ ${ \bar{u}}_{\mathcal{V}11}(x_{1})=0.4e^{-\frac{x_{1}^{2}}{1.2}}$ ${\underline{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V} 11}(x_{1})$ ${\bar{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V }11}(x_{1})$
 ${\underline{u}}_{\mathcal{V}11}(x_{1})=0.4e^{-\frac{x_{1}^{2}}{1.2}}$ ${ \bar{u}}_{\mathcal{V}11}(x_{1})=0.4e^{-\frac{x_{1}^{2}}{1.2}}$ ${\underline{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V} 11}(x_{1})$ ${\bar{u}}_{\mathcal{V}21}(x_{1})=1-{\underline{u}}_{\mathcal{V }11}(x_{1})$
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