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H∞ Asynchronous Deconvolution Fuzzy Filter Design for Nonlinear Singular Markov Jump Systems with Time-Varying Delays

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Abstract

This thesis focuses on the asynchronous deconvolution fuzzy filtering for time-varying delays Lur’e singular Markov jump fuzzy systems via the Takagi–Sugeno fuzzy control technique. The stochastic admissibility and \(H_\infty\) performance index of Lur’e singular Markov jump fuzzy systems are obtained by establishing mode-dependent Lyapunov-Krasovskill functional. In addition, the asynchrony phenomenon between the original system modes and filter modes is described by a hidden Markov model. The parallel distribute compensation approach is used to design an asynchronous deconvolution fuzzy filter, and the desired filter parameters are achieved via solving linear matrix inequalities. Lastly, the validity and practicality of the proposed method are shown by a numerical example and a single-link robot arm.

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Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request.

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Acknowledgements

The authors would like to thank the Editors and the Referees for the important comments and valuable suggestions for improving the paper. This work was supported by National Natural Science Foundation of China under Grants 62173174, 61773191, 62173183, 61973148; Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant 2019KJI010; Natural Science Foundation of Shandong Province for Key Projects under Grant ZR2020KA010; Graduate education high-quality curriculum construction project for Shandong Province under Grant SDYKC20185.

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Authors and Affiliations

  1. School of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, People’s Republic of China

    Yuexia Yin, Guangming Zhuang, Jianwei Xia & Wei Sun

  2. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Qian Ma

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  1. Yuexia Yin
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  2. Guangming Zhuang
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Correspondence to Guangming Zhuang.

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Appendix

Appendix

Conditions of Theorem 2

$$\begin{aligned}&\hat{\varXi }_{iir\eta 11}=\left[ \begin{array}{cccc} \varXi _{iir\eta 11}&\,\varXi _{iir\eta 12}&\,\varXi _{iir\eta 13}&\,\varXi _{iir\eta 14}\\ *&\,\varXi _{iir\eta 22}&\,\varXi _{iir\eta 23}&\,\varXi _{iir\eta 24}\\ *&\,*&\,\varXi _{iir\eta 33}&\,\varXi _{iir\eta 34}\\ *&\,*&\,*&\,\varXi _{iir\eta 44} \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 12}=\left[ \begin{array}{cccc} \varXi _{iir\eta 15}&\,\varXi _{iir\eta 16}&\,\varXi _{iir\eta 17}\\ \varXi _{iir\eta 25}&\,\varXi _{iir\eta 26}&\,\varXi _{iir\eta 27}\\ \varXi _{iir\eta 35}&\,0&\,\varXi _{iir\eta 37}\\ \varXi _{iir\eta 45}&\,0&\,\varXi _{iir\eta 47} \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 13}=\left[ \begin{array}{cccc} \varXi _{iir\eta 18}&\,\varXi _{iir\eta 19}&\,\varXi _{iir\eta 110}\\ \varXi _{iir\eta 28}&\,\varXi _{iir\eta 29}&\,\varXi _{iir\eta 210}\\ \varXi _{iir\eta 38}&\,\varXi _{iir\eta 39}&\,0\\ \varXi _{iir\eta 48}&\,\varXi _{iir\eta 49}&\,0 \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 22}=\left[ \begin{array}{cccc} \varXi _{iir\eta 55}&\,\varXi _{iir\eta 56}&\,0\\ *&\,\varXi _{iir\eta 66}&\,0\\ *&\,*&\,-2{\breve{\varLambda }}\\ \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 33}=\left[ \begin{array}{cccc} -\chi I&\,0&\,\varXi _{iir\eta 810}\\ *&\,-\chi I&\,\varXi _{iir\eta 910}\\ *&\,*&\,-I\\ \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 33}=\left[ \begin{array}{cccc} -2\chi I&\,0&\,\varXi _{ijr\eta 810}\\ *&\,-2\chi I&\,\varXi _{ijr\eta 910}\\ *&\,*&\,-2I\\ \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 12}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 15}&\,\tilde{\varXi }_{ijr\eta 16}&\,\tilde{\varXi }_{ijr\eta 17}\\ \tilde{\varXi }_{ijr\eta 25}&\,\tilde{\varXi }_{ijr\eta 26}&\,\tilde{\varXi }_{ijr\eta 27}\\ \tilde{\varXi }_{ijr\eta 35}&\,0&\,\tilde{\varXi }_{ijr\eta 37}\\ \tilde{\varXi }_{ijr\eta 45}&\,0&\,\tilde{\varXi }_{ijr\eta 47} \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 13}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 18}&\,\tilde{\varXi }_{ijr\eta 19}&\,\tilde{\varXi }_{ijr\eta 110}\\ \tilde{\varXi }_{ijr\eta 28}&\,\tilde{\varXi }_{ijr\eta 29}&\,\tilde{\varXi }_{ijr\eta 210}\\ \tilde{\varXi }_{ijr\eta 38}&\,\tilde{\varXi }_{ijr\eta 39}&\,0\\ \tilde{\varXi }_{ijr\eta 48}&\,\tilde{\varXi }_{ijr\eta 49}&\,0 \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 22}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 55}&\,\tilde{\varXi }_{ijr\eta 56}&\,0\\ *&\,\tilde{\varXi }_{ijr\eta 66}&\,0\\ *&\,*&\,-4{\breve{\varLambda }}\\ \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 11}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 11}&\,\tilde{\varXi }_{ijr\eta 12}&\,\tilde{\varXi }_{ijr\eta 13}&\,\tilde{\varXi }_{ijr\eta 14}\\ *&\,\tilde{\varXi }_{ijr\eta 22}&\,\tilde{\varXi }_{ijr\eta 23}&\,\tilde{\varXi }_{ijr\eta 24}\\ *&\,*&\,\tilde{\varXi }_{ijr\eta 33}&\,\tilde{\varXi }_{ijr\eta 34}\\ *&\,*&\,*&\,\varXi _{ijr\eta 44} \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 23}=\left[ \begin{array}{cccc} 0&\,0&\,\varXi _{iir\eta 510}\\ *&\,0&\,0\\ *&\,*&\,0\\ \end{array}\right] , \check{\varXi }_{iir\eta 23}=\left[ \begin{array}{cccc} 0&\,0&\,\tilde{\varXi }_{ijr\eta 510}\\ *&\,0&\,0\\ *&\,*&\,0\\ \end{array}\right] , \\&\hat{\varTheta }_{iir\eta 11}=sym(Y_{r1}A_{ir}+{\mathcal {B}}_{fi\eta }C_{ir})+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E\\ \,&-E^{T}Z_{11}E, \varXi _{iir\eta 26}=E^{T}_{f}Z_{22}E_{f},\\&\hat{\varTheta }_{iir\eta 12}={\mathcal {A}}_{fi\eta }+A^{T}_{ir}Y^{T}_{r2}+C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta }+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}\\ \,&-E^{T}Z_{12}E_{f}, \hat{\varTheta }_{iir\eta 24}=E^{T}_{f}P^{T}_{r3}+U_{r4}R^{T}_{2}-Y+{\mathcal {A}}^{T}_{fi\eta },\\&\hat{\varTheta }_{iir\eta 13}=E^{T}P^{T}_{r1}+U_{r1}R^{T}_{1}-Y_{r1}+A^{T}_{ir}Y^{T}_{r3}+C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta }, \\&\hat{\varTheta }_{iir\eta 14}=E^{T}P_{r2}+U_{r2}R^{T}_{2}-Y+A^{T}_{ir}Y^{T}_{r4}+C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta },\\&\hat{\varTheta }_{iir\eta 22}=sym({\mathcal {A}}_{fi\eta })+\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f}-E_{f}^{T}Z_{22}E_{f},\\&\hat{\varTheta }_{iir\eta 23}=E^{T}_{f}P^{T}_{r2}+U_{r3}R^{T}_{1}-Y_{r2}+{\mathcal {A}}^{T}_{fi\eta }, \\&\check{\varTheta }_{ijr\eta 11}=sym(Y_{r1}A_{ir}+Y_{r1}A_{jr}+{\mathcal {B}}_{fj\eta }C_{ir}+{\mathcal {B}}_{fi\eta }C_{jr})+\\ \,&2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E-2E^{T}Z_{11}E,\\&\check{\varTheta }_{ijr\eta 12}={\mathcal {A}}_{fj\eta }+{\mathcal {A}}_{fi\eta }+A^{T}_{ir}Y^{T}_{r2}+C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }\\ \,&+A^{T}_{jr}Y^{T}_{r2}+C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta }+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}-2E^{T}Z_{12}E_{f},\\&\check{\varTheta }_{ijr\eta 13}=2E^{T}P^{T}_{r1}+2U_{r1}R^{T}_{1}-2Y_{r1}+A^{T}_{ir}Y^{T}_{r3}\\ \,&+A^{T}_{jr}Y^{T}_{r3}+C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta },\\&\check{\varTheta }_{ijr\eta 14}=2E^{T}P_{r2}+2U_{r2}R^{T}_{2}-2Y+A^{T}_{ir}Y^{T}_{r4}+A^{T}_{jr}Y^{T}_{r4}\\ \,&+C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta },\\&\check{\varTheta }_{ijr\eta 22}=sym({\mathcal {A}}_{fj\eta }+{\mathcal {A}}_{fi\eta })+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f}\\ \,&-2E^{T}_{f}Z_{22}E_{f},\\&\check{\varTheta }_{ijr\eta 23}=2E^{T}_{f}P^{T}_{r2}+2U_{r3}R^{T}_{1}-2Y_{r2}+{\mathcal {A}}^{T}_{fj\eta }+{\mathcal {A}}^{T}_{fi\eta },\\&\check{\varTheta }_{ijr\eta 24}=2E^{T}_{f}P^{T}_{r3}+2U_{r4}R^{T}_{2}-2Y+{\mathcal {A}}^{T}_{fj\eta }+{\mathcal {A}}^{T}_{fi\eta },\\&\varXi _{iir\eta 11}=sym(Y_{r1}A_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{ir})\\ \,&+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E+Q_{11}-E^{T}Z_{11}E,\\&\varXi _{iir\eta 12}=A^{T}_{ir}Y^{T}_{r2}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta }+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}\\ \,&+Q_{12}-E^{T}Z_{12}E_{f}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta },\\&\varXi _{iir\eta 13}=U_{r1}R^{T}_{1}+E^{T}P_{r1}-Y_{r1}+A^{T}_{ir}Y^{T}_{r3}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta },\\&\varXi _{iir\eta 14}=U_{r2}R^{T}_{2}+E^{T}P_{r2}+A^{T}_{ir}Y^{T}_{r4}-Y+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta },\\&\varXi _{iir\eta 15}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}+E^{T}Z_{11}E+Y_{r1}A_{dir},\\&\varXi _{iir\eta 66}=-(1-\mu )Q_{22}-E^{T}_{f}Z_{22}E_{f},\\&\varXi _{iir\eta 16}=E^{T}Z_{12}E_{f}, \varXi _{iir\eta 18}=Y_{r1}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir},\\&\varXi _{iir\eta 17}=Y_{r1}D_{ir}+L^{T}O,\varXi _{iir\eta 27}=Y_{r2}D_{ir}, \\&\varXi _{iir\eta 19}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir}+Y_{r1}H_{ir},\\&\varXi _{iir\eta 22}=sym(\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta })+\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f}+Q_{22}\\ \,&-E^{T}_{f}Z_{22}E_{f}, \varXi _{iir\eta 44}=-Y^{T}-Y+\tau ^{2}Z_{22},\\&\varXi _{iir\eta 23}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+E^{T}_{f}P^{T}_{r2}+U_{r3}R^{T}_{1}-Y_{r2},\\&\varXi _{iir\eta 24}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+E^{T}_{f}P^{T}_{r3}+U_{r4}R^{T}_{2}-Y, \\&\varXi _{iir\eta 25}=Y_{r2}A_{dir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}+E^{T}_{f}Z^{T}_{12}E,\\ \\&\varXi _{iir\eta 28}=Y_{r2}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir}, \\&\varXi _{iir\eta 29}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir}+Y_{r2}H_{ir},\\&\varXi _{iir\eta 33}=-sym(Y_{r3})+\tau ^{2}Z_{11},\\&{\varXi _{iir\eta 34}=-Y^{T}_{r4}-Y+\tau ^{2}Z_{12},}\\&\varXi _{iir\eta 35}=Y_{r3}A_{dir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}, \\&\varXi _{iir\eta 38}=Y_{r3}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir},\\&\varXi _{iir\eta 37}=Y_{r3}D_{ir}, \varXi _{iir\eta 39}=Y_{r3}H_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir}, \\&\varXi _{iir\eta 45}=Y_{r4}A_{dir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}, \varXi _{iir\eta 47}=Y_{r4}D_{ir},\\&\varXi _{iir\eta 48}=Y_{r4}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir}, \\&\varXi _{iir\eta 49}=Y_{r4}H_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir},\\&\varXi _{iir\eta 55}=-(1-\mu )Q_{11}-E^{T}Z_{11}E,\\&\varXi _{iir\eta 56}=-(1-\mu )Q_{12}-E^{T}Z_{12}E_{f},\\&\varXi _{iir\eta 110}=[\sqrt{\pi _{r1}}C^{T}_{ir}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{ir}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{ir}D^{T}_{fi\kappa }],\\&\varXi _{iir\eta 210}=[\sqrt{\pi _{r1}}C^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{fi\kappa }~],\\&\varXi _{iir\eta 510}=[\sqrt{\pi _{r1}}C^{T}_{dir}D^{T}_{fi1}~\sqrt{\pi _{r2}}C^{T}_{dir}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{dir}D^{T}_{fi\kappa }],\\&\varXi _{iir\eta 810}=[\sqrt{\pi _{r1}}F^{T}_{ir}D^{T}_{fi1}-I\quad \sqrt{\pi _{r2}}F^{T}_{ir}D^{T}_{fi2}-I \\&\cdots \sqrt{\pi _{r\kappa }}F^{T}_{ir}D^{T}_{fi\kappa }-I],\\&\varXi _{iir\eta 910}=[\sqrt{\pi _{r1}}G^{T}_{ir}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}G^{T}_{ir}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}G^{T}_{ir}D^{T}_{fi\kappa }],\\&\tilde{\varXi }_{ijr\eta 11}=2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E+2Q_{11}-2E^{T}Z_{11}E+sym\\ \,&{(Y_{r1}A_{ir}+Y_{r1}A_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{ir}),}\\&{\tilde{\varXi }_{ijr\eta 12}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta }+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fj\eta }}+A^{T}_{ir}Y^{T}_{r2}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+A^{T}_{jr}Y^{T}_{r2} +\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta }\\ \,&+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}+2Q_{12}-2E^{T}Z_{12}E_{f},\\&\tilde{\varXi }_{ijr\eta 13}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+2E^{T}P_{r1}+2U_{r1}R^{T}_{1}\\ \,&-2Y_{r1}+A^{T}_{ir}Y^{T}_{r3}+A^{T}_{jr}Y^{T}_{r3}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta },\\&\tilde{\varXi }_{ijr\eta 14}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+2E^{T}P_{r2}-2Y\\ \,&+A^{T}_{ir}Y^{T}_{r4}+A^{T}_{jr}Y^{T}_{r4}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta }+2U_{r2}R^{T}_{2},\\ \\&\tilde{\varXi }_{ijr\eta 15}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}+Y_{r1}A_{dir}+Y_{r1}A_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}+2E^{T}Z_{11}E,\\&\tilde{\varXi }_{ijr\eta 16}=2E^{T}Z_{12}E_{f}, \tilde{\varXi }_{ijr\eta 17}=2Y_{r1}D_{ir}+2L^{T}O,\\&\tilde{\varXi }_{ijr\eta 34}=-2Y^{T}_{r4}-2Y+2\tau ^{2}Z_{12},\\&\tilde{\varXi }_{ijr\eta 18}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}+Y_{r1}B_{ir}+Y_{r1}B_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir}, {\tilde{\varXi }_{ijr\eta 19}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}+Y_{r1}H_{ir}}\\ \,&{+Y_{r1}H_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir},}\\&\tilde{\varXi }_{ijr\eta 22}=2Q_{22}-2E^{T}_{f}Z_{22}E_{f}+sym(\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta }\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fj\eta })+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f},\\&\tilde{\varXi }_{ijr\eta 23}=2E^{T}_{f}P^{T}_{r2}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+2U_{r3}R^{T}_{1}-2Y_{r2}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fj\eta },\\&\tilde{\varXi }_{ijr\eta 24}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+2E^{T}_{f}P^{T}_{r3}-2Y+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fj\eta }\\ \,&+2U_{r4}R^{T}_{2}, \tilde{\varXi }_{ijr\eta 26}=2E^{T}_{f}Z_{12}E_{f},\\&\tilde{\varXi }_{ijr\eta 25}=Y_{r2}A_{dir}+Y_{r2}A_{djr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}+2E^{T}_{f}Z^{T}_{12}E, \\&\tilde{\varXi }_{ijr\eta 27}=2Y_{r2}D_{ir}, \tilde{\varXi }_{ijr\eta 28}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}+Y_{r2}B_{ir}\\ \,&+Y_{r2}B_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir},\\&\tilde{\varXi }_{ijr\eta 29}=Y_{r2}H_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}+Y_{r2}H_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir},\\&\tilde{\varXi }_{ijr\eta 33}=-2Y^{T}_{r3}-2Y_{r3}+2\tau ^{2}Z_{11},\\&\tilde{\varXi }_{ijr\eta 44}=-2Y^{T}-2Y+2\tau ^{2}Z_{22},\\&\tilde{\varXi }_{ijr\eta 35}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}+Y_{r3}A_{dir}+Y_{r3}A_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}, \\&\tilde{\varXi }_{ijr\eta 45}=Y_{r4}A_{dir}+Y_{r4}A_{djr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}, \tilde{\varXi }_{ijr\eta 37}=2Y_{r3}D_{ir},\\&\tilde{\varXi }_{ijr\eta 39}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}+Y_{r3}H_{ir}+Y_{r3}H_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir}, \tilde{\varXi }_{ijr\eta 48}=Y_{r4}B_{ir}+Y_{r4}B_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}{+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir},}\\&\tilde{\varXi }_{ijr\eta 49}=Y_{r4}H_{ir}+Y_{r4}H_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir}, \tilde{\varXi }_{ijr\eta 55}=-2(1-\mu )Q_{11}-2E^{T}Z_{11}E,\\&\tilde{\varXi }_{ijr\eta 38}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}+Y_{r3}B_{ir}+Y_{r3}B_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir}, \tilde{\varXi }_{ijr\eta 47}=2Y_{r4}D_{ir},\\&\tilde{\varXi }_{ijr\eta 56}=-2(1-\mu )Q_{12}-2E^{T}Z_{12}E_{f},\\&\tilde{\varXi }_{ijr\eta 66}=-2(1-\mu )Q_{22}-2E^{T}_{f}Z_{22}E_{f},\\&\tilde{\varXi }_{ijr\eta 110}=[\sqrt{\pi _{r1}}C^{T}_{ir}D^{T}_{fj1}\quad \sqrt{\pi _{r2}}C^{T}_{ir}D^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{ir}D^{T}_{fj\kappa }~]\\ \,&+[~\sqrt{\pi _{r1}}C^{T}_{jr}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{jr}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{jr}D^{T}_{fi\kappa }],\\&\tilde{\varXi }_{ijr\eta 210}=[\sqrt{\pi _{r1}}C^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{fi2}~\cdots ~\sqrt{\pi _{r\kappa }}C^{T}_{fi\kappa }~]\\ \,&+[~\sqrt{\pi _{r1}}C^{T}_{fj1}\quad \sqrt{\pi _{r2}}C^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{fj\kappa }],\\&\tilde{\varXi }_{ijr\eta 510}=[\sqrt{\pi _{r1}}C^{T}_{dir}D^{T}_{fj1}~\sqrt{\pi _{r2}}C^{T}_{dir}D^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{dir}D^{T}_{fj\kappa }]\\ \,&+[\sqrt{\pi _{r1}}C^{T}_{djr}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{djr}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{djr}D^{T}_{fi\kappa }],\\&\tilde{\varXi }_{ijr\eta 810}=[\sqrt{\pi _{r1}}F^{T}_{ir}D^{T}_{fj1}-I~\sqrt{\pi _{r2}}F^{T}_{ir}D^{T}_{fj2}-I\cdots \\ \,&\sqrt{\pi _{r\kappa }}F^{T}_{ir}D^{T}_{fj\kappa }-I] +[\sqrt{\pi _{r1}}F^{T}_{jr}D^{T}_{fi1}-I~\sqrt{\pi _{r2}}F^{T}_{jr}D^{T}_{fi2}-I\\ \,&\cdots \sqrt{\pi _{r\kappa }}F^{T}_{jr}D^{T}_{fi\kappa }-I],\\&\tilde{\varXi }_{ijr\eta 910}=[\sqrt{\pi _{r1}}G^{T}_{ir}D^{T}_{fj1}\quad \sqrt{\pi _{r2}}G^{T}_{ir}D^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}G^{T}_{ir}D^{T}_{fj\kappa }]\\ \,&+[\sqrt{\pi _{r1}}G^{T}_{jr}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}G^{T}_{jr}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}G^{T}_{jr}D^{T}_{fi\kappa }]. \end{aligned}$$

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Yin, Y., Zhuang, G., Xia, J. et al. H∞ Asynchronous Deconvolution Fuzzy Filter Design for Nonlinear Singular Markov Jump Systems with Time-Varying Delays. Int. J. Fuzzy Syst. 25, 763–779 (2023). https://doi.org/10.1007/s40815-022-01400-1

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