July  2022, 15(7): 1839-1858. doi: 10.3934/dcdss.2022014

Robust dynamic sliding mode control design for interval type-2 fuzzy systems

1. 

School of Electrical Engineering, Chungbuk National University, Cheongju, 28644, South Korea

2. 

Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India

3. 

School of Information and Communication Engineering, Chungbuk National University, Cheongju 28644, South Korea

*Corresponding authors: Oh-Min Kwon; Seong-Gon Choi

Received  November 2021 Revised  December 2021 Published  July 2022 Early access  January 2022

This paper discusses the problem of stabilization of interval type-2 fuzzy systems with uncertainties, time delay and external disturbance using a dynamic sliding mode controller. The sliding surface function, which is based on both the system's state and control input vectors, is used during the control design process. The sliding mode dynamics are presented by defining a new vector that augments the system state and control vectors. First, the reachability of the addressed sliding mode surface is demonstrated. Second, the required sufficient conditions for the system's stability and the proposed control design are derived by using extended dissipative theory and an asymmetric Lyapunov-Krasovskii functional approach. Unlike some existing sliding mode control designs, the one proposed in this paper does not require the control coefficient matrices of all linear subsystems to be the same, reducing the method's conservatism. Finally, numerical examples are provided to demonstrate the viability and superiority of the proposed design method.

Citation: Ramasamy Kavikumar, Boomipalagan Kaviarasan, Yong-Gwon Lee, Oh-Min Kwon, Rathinasamy Sakthivel, Seong-Gon Choi. Robust dynamic sliding mode control design for interval type-2 fuzzy systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1839-1858. doi: 10.3934/dcdss.2022014
References:
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S. DongM. FangP. ShiZ. G. Wu and D. Zhang, Dissipativity-based control for fuzzy systems with asynchronous modes and intermittent measurements, IEEE Trans. Cybern., 50 (2020), 2389-2399.  doi: 10.1109/TCYB.2018.2887060.

[2]

P. DuY. PanH. Li and H. K. Lam, Nonsingular finite-time event-triggered fuzzy control for large-scale nonlinear systems, IEEE Trans. Fuzzy Syst., 29 (2021), 2088-2099.  doi: 10.1109/TFUZZ.2020.2992632.

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Z. DuY. Kao and J. H. Park, Interval type-2 fuzzy sampled-data control of time-delay systems, Inf. Sci., 487 (2019), 193-207.  doi: 10.1016/j.ins.2019.03.009.

[4]

Q. GaoG. FengZ. XiY. Wang and J. Qiu, Robust $\mathcal{H}_{\infty}$ control of T-S fuzzy time-delay systems via a new sliding-mode control scheme, IEEE Trans. Fuzzy Syst., 22 (2014), 459-465. 

[5]

K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003. doi: 10.1007/978-1-4612-0039-0.

[6]

S. HanS. K. Kommuri and S. M. Lee, Affine transformed IT2 fuzzy event-triggered control under deception attacks, IEEE Trans. Fuzzy Syst., 29 (2021), 322-335.  doi: 10.1109/TFUZZ.2020.2999779.

[7]

W. JiJ. QiuL. Wu and H. K. Lam, Fuzzy-affine-model-based output feedback dynamic sliding mode controller design of nonlinear systems, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 1652-1661.  doi: 10.1109/TSMC.2019.2900050.

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N. N. KarnikJ. M. Mendel and Q. Liang, Type-2 fuzzy logic systems, IEEE Trans. Fuzzy Syst., 7 (1999), 643-658.  doi: 10.1109/91.811231.

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B. LiJ. XiaH. ZhangH. Shen and Z. Wang, Event-triggered adaptive fuzzy tracking control for stochastic nonlinear systems, J. Frank. Inst., 357 (2020), 9505-9522.  doi: 10.1016/j.jfranklin.2020.07.023.

[10]

J. LiQ. ZhangX. G. Yan and S. K. Spurgeon, Observer-based fuzzy integral sliding mode control for nonlinear descriptor systems, IEEE Trans. Fuzzy Syst., 26 (2018), 2818-2832. 

[11]

Z. LiH. YanH. ZhangH. K. Lam and M. Wang, Aperiodic sampled-data-based control for interval type-2 fuzzy systems via refined adaptive event-triggered communication scheme, IEEE Trans. Fuzzy Syst., 29 (2021), 310-321. 

[12]

Z. LianY. HeC. K. ZhangP. Shi and M. Wu, Robust $H_{\infty}$ control for T-S fuzzy systems with state and input time-varying delays via delay-product-type functional method, IEEE Trans. Fuzzy Syst., 27 (2019), 1917-1930.  doi: 10.1109/TFUZZ.2019.2892356.

[13]

Z. LianP. Shi and C. C. Lim, Hybrid-triggered interval type-2 fuzzy control for networked systems under attacks, Inf. Sci., 567 (2021), 332-347.  doi: 10.1016/j.ins.2021.03.050.

[14]

Q. Liang and J. M. Mendel, Interval type-2 fuzzy logic systems: Theory and design, IEEE Trans. Fuzzy Syst., 8 (2000), 535-550. 

[15]

X. LiuJ. XiaJ. Wang and H. Shen, Interval type-2 fuzzy passive filtering for nonlinear singularly perturbed PDT-switched systems and its application, J. Syst. Sci. Complex., 34 (2020), 2195-2218.  doi: 10.1007/s11424-020-0106-9.

[16]

Y. Pan and G. H. Yang, Event-driven fault detection for discrete-time interval type-2 fuzzy systems, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 4959-4968.  doi: 10.1109/TSMC.2019.2945063.

[17]

P. G. ParkW. I. Lee and S. Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Frank. Inst., 352 (2015), 1378-1396.  doi: 10.1016/j.jfranklin.2015.01.004.

[18]

J. QiuW. Ji and M. Chadli, A novel fuzzy output feedback dynamic sliding mode controller design for two-dimensional nonlinear systems, IEEE Trans. Fuzzy Syst., 29 (2021), 2869-2877.  doi: 10.1109/TFUZZ.2020.3008271.

[19]

R. SakthivelR. KavikumarA. MohammadzadehO. M. Kwon and B. Kaviarasan, Fault estimation for mode-dependent IT2 fuzzy systems with quantized output signals, IEEE Trans. Fuzzy Syst., 29 (2021), 298-309.  doi: 10.1109/TFUZZ.2020.3018509.

[20]

S. Saravanan and K. S. Hong, An event-triggered extended dissipative control for Takagi-Sugeno fuzzy systems with time-varying delay via free-matrix-based integral inequality, J. Frank. Inst., 357 (2020), 7696-7717.  doi: 10.1016/j.jfranklin.2020.05.035.

[21]

A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013), 2860-2866.  doi: 10.1016/j.automatica.2013.05.030.

[22]

H. ShenM. XingZ. G. Wu and J. H. Park, Fault-tolerant control for fuzzy switched singular systems with persistent dwell-time subject to actuator fault, Fuzzy Sets Syst., 392 (2020), 60-76.  doi: 10.1016/j.fss.2019.08.011.

[23]

Z. ShengC. LinB. Chen and Q. G. Wang, Asymmetric Lyapunov-Krasovskii functional method on stability of time-delay systems, Int. J. Robust Nonlinear Control, 31 (2021), 2847-2854.  doi: 10.1002/rnc.5417.

[24]

Z. Sheng, C. Lin, B. Chen and Q. G. Wang, An asymmetric Lyapunov-Krasovskii functional method on stability and stabilization for T-S fuzzy systems with time delay, IEEE Trans. Fuzzy Syst., 2021.

[25]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications 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]

J. WangJ. XiaH. ShenM. Xing and J. H. Park, $H_\infty$ synchronization for fuzzy Markov jump chaotic systems with piecewise-constant transition probabilities subject to PDT switching rule, IEEE Trans. Fuzzy Syst., 29 (2021), 3082-3092. 

[27]

J. Wang, C. Yang, J. Xia, Z. G. Wu and H. Shen, Observer-based sliding mode control for networked fuzzy singularly perturbed systems under weighted try-once-discard protocol, IEEE Trans. Fuzzy Syst., (2021), 1–1. doi: 10.1109/TFUZZ.2021.3070125.

[28]

L. Wang and H. K. Lam, $H_{\infty}$ control for continuous-time Takagi-Sugeno fuzzy model by applying generalized Lyapunov function and introducing outer variables, Automatica, 125 (2021), Paper No. 109409, 5 pp. doi: 10.1016/j.automatica.2020.109409.

[29]

S. WenM. Z. Q. ChenZ. ZengX. Yu and T. Huang, Fuzzy control for uncertain vehicle active suspension systems via dynamic sliding-mode approach, IEEE Trans. Syst. Man Cybern. Syst., 47 (2017), 24-32.  doi: 10.1109/TSMC.2016.2564930.

[30]

G. WuG. H. Yang and H. Wang, ISS control synthesis of T-S fuzzy systems with multiple transmission channels under denial of service, J. Frank. Inst., 358 (2021), 3010-3032.  doi: 10.1016/j.jfranklin.2021.02.014.

[31]

L. XieM. Fu and C. E. De Souza, $H_{\infty}$ control and quadratic stabilization of systems with parameter uncertainty via output feedback, IEEE Trans. Automat. Control, 37 (1992), 1253-1256.  doi: 10.1109/9.151120.

[32]

T. Xu, J. Xia, S. Wang, Y. Lian and H. Zhang, Extended dissipativity-based non-fragile sampled-data control of fuzzy Markovian jump systems with incomplete transition rates, Appl. Math. Comput., 380 (2020), 125258, 20 pp. doi: 10.1016/j.amc.2020.125258.

[33]

Y. XueB. C. Zheng and X. Yu, Robust sliding mode control for T-S fuzzy systems via quantized state feedback, IEEE Trans. Fuzzy Syst., 26 (2018), 2261-2272.  doi: 10.1109/TFUZZ.2017.2771467.

[34]

Y. YangJ. XiaJ. ZhaoX. Li and Z. Wang, Multiobjective nonfragile fuzzy control for nonlinear stochastic financial systems with mixed time delays, Nonlinear Anal. Model. Control, 24 (2019), 696-717.  doi: 10.15388/na.2019.5.2.

[35]

B. ZhangW. X. Zheng and S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I Reg. Papers, 60 (2013), 1250-1263.  doi: 10.1109/TCSI.2013.2246213.

[36]

Z. ZhangY. NiuZ. Cao and J. Song, Security sliding mode control of interval type-2 fuzzy systems subject to cyber attacks: The stochastic communication protocol case, IEEE Trans. Fuzzy Syst., 29 (2021), 240-251.  doi: 10.1109/TFUZZ.2020.2972785.

[37]

Z. ZhangY. Niu and H. Zhao, Secure sliding mode control of interval type-2 fuzzy systems against intermittent denial-of-service attacks, Int. J. Robust Nonlinear Control, 31 (2021), 1866-1884.  doi: 10.1002/rnc.5219.

[38]

J. ZhaoS. Xu and J. H. Park, Improved criteria for the stabilization of T-S fuzzy systems with actuator failures via a sampled-data fuzzy controller, Fuzzy Sets Syst., 392 (2020), 154-169.  doi: 10.1016/j.fss.2019.09.004.

[39]

Y. ZhaoJ. WangF. Yan and Y. Shen, Adaptive sliding mode fault-tolerant control for type-2 fuzzy systems with distributed delays, Inf. Sci., 473 (2019), 227-238.  doi: 10.1016/j.ins.2018.09.002.

[40]

S. Zhou and Y. Han, Extended dissipativity and control synthesis of interval type-2 fuzzy systems via line-integral Lyapunov function, IEEE Trans. Fuzzy Syst., 28 (2020), 2631-2644.  doi: 10.1109/TFUZZ.2019.2945258.

show all references

References:
[1]

S. DongM. FangP. ShiZ. G. Wu and D. Zhang, Dissipativity-based control for fuzzy systems with asynchronous modes and intermittent measurements, IEEE Trans. Cybern., 50 (2020), 2389-2399.  doi: 10.1109/TCYB.2018.2887060.

[2]

P. DuY. PanH. Li and H. K. Lam, Nonsingular finite-time event-triggered fuzzy control for large-scale nonlinear systems, IEEE Trans. Fuzzy Syst., 29 (2021), 2088-2099.  doi: 10.1109/TFUZZ.2020.2992632.

[3]

Z. DuY. Kao and J. H. Park, Interval type-2 fuzzy sampled-data control of time-delay systems, Inf. Sci., 487 (2019), 193-207.  doi: 10.1016/j.ins.2019.03.009.

[4]

Q. GaoG. FengZ. XiY. Wang and J. Qiu, Robust $\mathcal{H}_{\infty}$ control of T-S fuzzy time-delay systems via a new sliding-mode control scheme, IEEE Trans. Fuzzy Syst., 22 (2014), 459-465. 

[5]

K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003. doi: 10.1007/978-1-4612-0039-0.

[6]

S. HanS. K. Kommuri and S. M. Lee, Affine transformed IT2 fuzzy event-triggered control under deception attacks, IEEE Trans. Fuzzy Syst., 29 (2021), 322-335.  doi: 10.1109/TFUZZ.2020.2999779.

[7]

W. JiJ. QiuL. Wu and H. K. Lam, Fuzzy-affine-model-based output feedback dynamic sliding mode controller design of nonlinear systems, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 1652-1661.  doi: 10.1109/TSMC.2019.2900050.

[8]

N. N. KarnikJ. M. Mendel and Q. Liang, Type-2 fuzzy logic systems, IEEE Trans. Fuzzy Syst., 7 (1999), 643-658.  doi: 10.1109/91.811231.

[9]

B. LiJ. XiaH. ZhangH. Shen and Z. Wang, Event-triggered adaptive fuzzy tracking control for stochastic nonlinear systems, J. Frank. Inst., 357 (2020), 9505-9522.  doi: 10.1016/j.jfranklin.2020.07.023.

[10]

J. LiQ. ZhangX. G. Yan and S. K. Spurgeon, Observer-based fuzzy integral sliding mode control for nonlinear descriptor systems, IEEE Trans. Fuzzy Syst., 26 (2018), 2818-2832. 

[11]

Z. LiH. YanH. ZhangH. K. Lam and M. Wang, Aperiodic sampled-data-based control for interval type-2 fuzzy systems via refined adaptive event-triggered communication scheme, IEEE Trans. Fuzzy Syst., 29 (2021), 310-321. 

[12]

Z. LianY. HeC. K. ZhangP. Shi and M. Wu, Robust $H_{\infty}$ control for T-S fuzzy systems with state and input time-varying delays via delay-product-type functional method, IEEE Trans. Fuzzy Syst., 27 (2019), 1917-1930.  doi: 10.1109/TFUZZ.2019.2892356.

[13]

Z. LianP. Shi and C. C. Lim, Hybrid-triggered interval type-2 fuzzy control for networked systems under attacks, Inf. Sci., 567 (2021), 332-347.  doi: 10.1016/j.ins.2021.03.050.

[14]

Q. Liang and J. M. Mendel, Interval type-2 fuzzy logic systems: Theory and design, IEEE Trans. Fuzzy Syst., 8 (2000), 535-550. 

[15]

X. LiuJ. XiaJ. Wang and H. Shen, Interval type-2 fuzzy passive filtering for nonlinear singularly perturbed PDT-switched systems and its application, J. Syst. Sci. Complex., 34 (2020), 2195-2218.  doi: 10.1007/s11424-020-0106-9.

[16]

Y. Pan and G. H. Yang, Event-driven fault detection for discrete-time interval type-2 fuzzy systems, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 4959-4968.  doi: 10.1109/TSMC.2019.2945063.

[17]

P. G. ParkW. I. Lee and S. Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Frank. Inst., 352 (2015), 1378-1396.  doi: 10.1016/j.jfranklin.2015.01.004.

[18]

J. QiuW. Ji and M. Chadli, A novel fuzzy output feedback dynamic sliding mode controller design for two-dimensional nonlinear systems, IEEE Trans. Fuzzy Syst., 29 (2021), 2869-2877.  doi: 10.1109/TFUZZ.2020.3008271.

[19]

R. SakthivelR. KavikumarA. MohammadzadehO. M. Kwon and B. Kaviarasan, Fault estimation for mode-dependent IT2 fuzzy systems with quantized output signals, IEEE Trans. Fuzzy Syst., 29 (2021), 298-309.  doi: 10.1109/TFUZZ.2020.3018509.

[20]

S. Saravanan and K. S. Hong, An event-triggered extended dissipative control for Takagi-Sugeno fuzzy systems with time-varying delay via free-matrix-based integral inequality, J. Frank. Inst., 357 (2020), 7696-7717.  doi: 10.1016/j.jfranklin.2020.05.035.

[21]

A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013), 2860-2866.  doi: 10.1016/j.automatica.2013.05.030.

[22]

H. ShenM. XingZ. G. Wu and J. H. Park, Fault-tolerant control for fuzzy switched singular systems with persistent dwell-time subject to actuator fault, Fuzzy Sets Syst., 392 (2020), 60-76.  doi: 10.1016/j.fss.2019.08.011.

[23]

Z. ShengC. LinB. Chen and Q. G. Wang, Asymmetric Lyapunov-Krasovskii functional method on stability of time-delay systems, Int. J. Robust Nonlinear Control, 31 (2021), 2847-2854.  doi: 10.1002/rnc.5417.

[24]

Z. Sheng, C. Lin, B. Chen and Q. G. Wang, An asymmetric Lyapunov-Krasovskii functional method on stability and stabilization for T-S fuzzy systems with time delay, IEEE Trans. Fuzzy Syst., 2021.

[25]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications 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]

J. WangJ. XiaH. ShenM. Xing and J. H. Park, $H_\infty$ synchronization for fuzzy Markov jump chaotic systems with piecewise-constant transition probabilities subject to PDT switching rule, IEEE Trans. Fuzzy Syst., 29 (2021), 3082-3092. 

[27]

J. Wang, C. Yang, J. Xia, Z. G. Wu and H. Shen, Observer-based sliding mode control for networked fuzzy singularly perturbed systems under weighted try-once-discard protocol, IEEE Trans. Fuzzy Syst., (2021), 1–1. doi: 10.1109/TFUZZ.2021.3070125.

[28]

L. Wang and H. K. Lam, $H_{\infty}$ control for continuous-time Takagi-Sugeno fuzzy model by applying generalized Lyapunov function and introducing outer variables, Automatica, 125 (2021), Paper No. 109409, 5 pp. doi: 10.1016/j.automatica.2020.109409.

[29]

S. WenM. Z. Q. ChenZ. ZengX. Yu and T. Huang, Fuzzy control for uncertain vehicle active suspension systems via dynamic sliding-mode approach, IEEE Trans. Syst. Man Cybern. Syst., 47 (2017), 24-32.  doi: 10.1109/TSMC.2016.2564930.

[30]

G. WuG. H. Yang and H. Wang, ISS control synthesis of T-S fuzzy systems with multiple transmission channels under denial of service, J. Frank. Inst., 358 (2021), 3010-3032.  doi: 10.1016/j.jfranklin.2021.02.014.

[31]

L. XieM. Fu and C. E. De Souza, $H_{\infty}$ control and quadratic stabilization of systems with parameter uncertainty via output feedback, IEEE Trans. Automat. Control, 37 (1992), 1253-1256.  doi: 10.1109/9.151120.

[32]

T. Xu, J. Xia, S. Wang, Y. Lian and H. Zhang, Extended dissipativity-based non-fragile sampled-data control of fuzzy Markovian jump systems with incomplete transition rates, Appl. Math. Comput., 380 (2020), 125258, 20 pp. doi: 10.1016/j.amc.2020.125258.

[33]

Y. XueB. C. Zheng and X. Yu, Robust sliding mode control for T-S fuzzy systems via quantized state feedback, IEEE Trans. Fuzzy Syst., 26 (2018), 2261-2272.  doi: 10.1109/TFUZZ.2017.2771467.

[34]

Y. YangJ. XiaJ. ZhaoX. Li and Z. Wang, Multiobjective nonfragile fuzzy control for nonlinear stochastic financial systems with mixed time delays, Nonlinear Anal. Model. Control, 24 (2019), 696-717.  doi: 10.15388/na.2019.5.2.

[35]

B. ZhangW. X. Zheng and S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I Reg. Papers, 60 (2013), 1250-1263.  doi: 10.1109/TCSI.2013.2246213.

[36]

Z. ZhangY. NiuZ. Cao and J. Song, Security sliding mode control of interval type-2 fuzzy systems subject to cyber attacks: The stochastic communication protocol case, IEEE Trans. Fuzzy Syst., 29 (2021), 240-251.  doi: 10.1109/TFUZZ.2020.2972785.

[37]

Z. ZhangY. Niu and H. Zhao, Secure sliding mode control of interval type-2 fuzzy systems against intermittent denial-of-service attacks, Int. J. Robust Nonlinear Control, 31 (2021), 1866-1884.  doi: 10.1002/rnc.5219.

[38]

J. ZhaoS. Xu and J. H. Park, Improved criteria for the stabilization of T-S fuzzy systems with actuator failures via a sampled-data fuzzy controller, Fuzzy Sets Syst., 392 (2020), 154-169.  doi: 10.1016/j.fss.2019.09.004.

[39]

Y. ZhaoJ. WangF. Yan and Y. Shen, Adaptive sliding mode fault-tolerant control for type-2 fuzzy systems with distributed delays, Inf. Sci., 473 (2019), 227-238.  doi: 10.1016/j.ins.2018.09.002.

[40]

S. Zhou and Y. Han, Extended dissipativity and control synthesis of interval type-2 fuzzy systems via line-integral Lyapunov function, IEEE Trans. Fuzzy Syst., 28 (2020), 2631-2644.  doi: 10.1109/TFUZZ.2019.2945258.

Figure 1.  IT2 fuzzy membership functions
Figure 2.  State responses of the closed-loop system and control input in $ L_{2}-L_{\infty} $ case
Figure 3.  State responses of the closed-loop system and control input in $ H_{\infty} $ case
Figure 4.  State responses of the closed-loop system and control input in passivity case
Figure 5.  State responses of the closed-loop system and control input in dissipativity case
Figure 6.  State responses of the open-loop system
Table 1.  Four cases of extended dissipative performance
Performance $ \vartheta_{1} $ $ \vartheta_{2} $ $ \vartheta_{3} $ $ \vartheta_{4} $
$ L_2-L_{\infty} $ 0 0 $ \lambda^{2}I $ $ I $
$ H_{\infty} $ $ -I $ 0 $ \lambda^{2}I $ 0
Passivity 0 $ I $ $ \lambda I $ 0
Dissipativity $ Q=-I $ $ S=I $ $ R=(2-\lambda)I $ 0
Performance $ \vartheta_{1} $ $ \vartheta_{2} $ $ \vartheta_{3} $ $ \vartheta_{4} $
$ L_2-L_{\infty} $ 0 0 $ \lambda^{2}I $ $ I $
$ H_{\infty} $ $ -I $ 0 $ \lambda^{2}I $ 0
Passivity 0 $ I $ $ \lambda I $ 0
Dissipativity $ Q=-I $ $ S=I $ $ R=(2-\lambda)I $ 0
Table 2.  Minimum disturbance attenuation index for different $ d $
Performance $ d=0.4 $ $ d=0.5 $ $ d=0.6 $ $ d=0.7 $ $ d=0.8 $ $ d=0.9 $ $ d=1.0 $
$ L_{2}-L_{\infty} $ 0.8142 0.8200 0.8204 0.8371 0.8643 0.9008 0.9339
$ H_{\infty} $ 0.8021 1.1488 1.1654 1.1895 1.2163 1.2440 1.2828
Passivity 1.7813 1.7984 1.8144 1.8454 1.8931 1.9716 2.0705
Dissipativity 0.9158 0.9166 0.9231 0.9271 0.9281 0.9282 0.9356
Performance $ d=0.4 $ $ d=0.5 $ $ d=0.6 $ $ d=0.7 $ $ d=0.8 $ $ d=0.9 $ $ d=1.0 $
$ L_{2}-L_{\infty} $ 0.8142 0.8200 0.8204 0.8371 0.8643 0.9008 0.9339
$ H_{\infty} $ 0.8021 1.1488 1.1654 1.1895 1.2163 1.2440 1.2828
Passivity 1.7813 1.7984 1.8144 1.8454 1.8931 1.9716 2.0705
Dissipativity 0.9158 0.9166 0.9231 0.9271 0.9281 0.9282 0.9356
Table 3.  Sliding surface gain matrices $ \mathcal{H}_{x} $ and $ \mathcal{H}_{u} $ for $ d = 0.4 $
Performance $ \mathcal{H}_{x} $ $ \mathcal{H}_{u} $
$ L_2-L_{\infty} $ [0.1660 -2.1859 -3.1432] 9.5060
$ H_{\infty} $ [-1.3400 -5.3997 -7.5747] 19.4401
Passivity [-0.2907 -1.2659 -1.8723] 5.0976
Dissipativity [-0.2416 -1.5850 -2.4005] 6.8844
Performance $ \mathcal{H}_{x} $ $ \mathcal{H}_{u} $
$ L_2-L_{\infty} $ [0.1660 -2.1859 -3.1432] 9.5060
$ H_{\infty} $ [-1.3400 -5.3997 -7.5747] 19.4401
Passivity [-0.2907 -1.2659 -1.8723] 5.0976
Dissipativity [-0.2416 -1.5850 -2.4005] 6.8844
Table 4.  Minimum $ H_{\infty} $ disturbance attenuation index for various $ d $
$ d $ $ 0.5 $ $ 1.0 $ $ 1.5 $ $ 2.0 $ $ 2.5 $
$ \lambda_{\min} $ 0.0044 0.0102 0.0171 0.0286 0.0322
$ d $ $ 0.5 $ $ 1.0 $ $ 1.5 $ $ 2.0 $ $ 2.5 $
$ \lambda_{\min} $ 0.0044 0.0102 0.0171 0.0286 0.0322
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