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Event-Triggered \(H_\infty \) State Estimation for Coupled and Switched Genetic Regulatory Networks

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Abstract

This paper investigates the problem of event-triggered \(H_\infty \) state estimation for switched genetic regulatory networks with static coupling via a sojourn-probability-dependent approach. The measurements of the network are evaluated by the event-triggers which are only undertaken at the switching times. By employing a time-delay approach, the estimation can be achieved by determining the exponential mean-square stability of the switched system with time-varying delay and known sojourn probability, while the system prescribes an \(H_\infty \) performance level. A co-design approach for the event-triggered mechanism and the estimators is presented by means of a novel Lyapunov–Krasovskii functional combining with refined Jensen-based inequalities. Finally, a numerical example is given to demonstrate the effectiveness of the designed estimators.

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Acknowledgements

This work was jointly supported by the National Natural Science Foundation of China under Grant No. 61573201, Project of Flagship-Major Construction of Jiangsu Higher Education Institutions of China under Grant No. PPZY2015B135, Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant Nos. KYCX17\(_{-}\)1915 and KYCX18\(_{-}\)2423, and Applied Basic Research-Industrial Innovation Project of Nantong City under Grant No. GY12017025.

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  1. School of Information Science and Technology, Nantong University, Nantong, 226019, China

    Suying Sheng, Xiaomei Zhang & Qingqing Lu

  2. School of Electrical Engineering, Nantong University, Nantong, 226019, China

    Guoping Lu

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Appendix A

Appendix A

The event-triggered parameters and the estimator gains are given as

$$\begin{aligned} U_1= & {} \begin{bmatrix} 0.0023&\quad -\,0.0067\\ -\,0.0067&\quad 0.0200 \end{bmatrix}, U_2=\begin{bmatrix} 0.0002&\quad 0.0008\\ 0.0008&\quad 0.0065 \end{bmatrix},\\ U_3= & {} \begin{bmatrix} 0.0010&\quad -0.0012\\ -\,0.0012&\quad 0.0017 \end{bmatrix}, A_{1f1}=\begin{bmatrix} 0.0680&\quad 0.4452&\quad 0.7029&\quad -\,0.7867\\ -\,0.4647&\quad 1.1069&\quad 0.5471&\quad -\,0.6486\\ -\,0.3780&\quad 0.6515&\quad 1.0854&\quad -\,0.7946\\ -\,0.4017&\quad 0.7698&\quad 0.3564&\quad 0.0467 \end{bmatrix},\\ A_{2f1}= & {} \begin{bmatrix} 0.6122&\quad -\,0.4157&\quad -\,0.1012&\quad 0.3131\\ -\,0.0165&\quad 0.3306&\quad -\,0.0032&\quad 0.1549\\ 0.2878&\quad -\,0.2409&\quad 0.3251&\quad 0.0574\\ 0.1053&\quad -\,0.1432&\quad -\,0.3478&\quad 0.9406 \end{bmatrix},\\ A_{3f1}= & {} \begin{bmatrix} 0.7058&\quad -\,0.4571&\quad -\,0.3184&\quad 0.6256\\ -\,0.0786&\quad 0.5091&\quad -\,0.1825&\quad 0.2790\\ 0.3069&\quad -\,0.3220&\quad 0.1344&\quad 0.4256\\ 0.0243&\quad -\,0.0098&\quad -\,0.2552&\quad 0.8769 \end{bmatrix},\\ G_{1f1}= & {} \begin{bmatrix} 0.0074&\quad 1.1864&\quad 1.1279&\quad -\,1.3413\\ -\,0.4896&\quad 1.4504&\quad 0.8266&\quad -\,0.7351\\ -\,0.7367&\quad 1.0441&\quad 1.8424&\quad -\,0.9246\\ -\,0.5387&\quad 0.8027&\quad 0.8533&\quad 0.1982 \end{bmatrix},\\ G_{2f1}= & {} \begin{bmatrix} 1.0053&\quad -\,0.8937&\quad -\,0.1585&\quad 0.4102\\ 0.1064&\quad 0.2852&\quad -\,0.0501&\quad 0.1027\\ 0.2100&\quad -\,0.6078&\quad 0.8754&\quad -\,0.1136\\ 0.3789&\quad -\,0.7475&\quad -\,0.6250&\quad 1.6746 \end{bmatrix},\\ G_{3f1}= & {} \begin{bmatrix} 0.7963&\quad -\,0.1944&\quad -\,0.4551&\quad -\,0.2871\\ -\,0.3050&\quad 1.1184&\quad -\,0.6247&\quad -\,0.1386\\ -\,0.1745&\quad -\,0.0415&\quad 0.3783&\quad -\,0.3539\\ -\,0.2277&\quad 0.1923&\quad -\,0.3842&\quad 0.6582 \end{bmatrix},\\ B_{1f1}= & {} \begin{bmatrix} 0.2240&\quad 0.1478\\ 0.1701&\quad 0.1231\\ 0.2425&\quad 0.1566\\ 0.1374&\quad 0.1146 \end{bmatrix}, B_{2f1}=\begin{bmatrix} 0.1385&\quad -\,0.0647\\ 0.1239&\quad -\,0.0856\\ 0.1459&\quad -\,0.0734\\ 0.1066&\quad -\,0.0970 \end{bmatrix}, \\ B_{3f1}= & {} \begin{bmatrix} 0.0801&\quad 0.0862\\ 0.0683&\quad 0.0536\\ 0.0838&\quad 0.0830\\ 0.0564&\quad 0.0481 \end{bmatrix}, \ A_{1f2}=\begin{bmatrix} 0.0765&\quad 0.5008&\quad 0.7907&\quad -\,0.8850\\ -\,0.5227&\quad 1.2453&\quad 0.6155&\quad -\,0.7296\\ -\,0.4252&\quad 0.7330&\quad 1.2210&\quad -\,0.8939\\ -\,0.4519&\quad 0.8660&\quad 0.4010&\quad 0.0525 \end{bmatrix},\\ A_{2f2}= & {} \begin{bmatrix} 0.6887&\quad -\,0.4677&\quad -\,0.1139&\quad 0.3522\\ -\,0.0186&\quad 0.3719&\quad -\,0.0036&\quad 0.1742\\ 0.3238&\quad -\,0.2710&\quad 0.3658&\quad 0.0645\\ 0.1185&\quad -\,0.1611&\quad -\,0.3912&\quad 1.0581 \end{bmatrix},\\ A_{3f2}= & {} \begin{bmatrix} 0.7940&\quad -\,0.5143&\quad -\,0.3582&\quad 0.7038\\ -\,0.0884&\quad 0.5727&\quad -\,0.2053&\quad 0.3139\\ 0.3452&\quad -\,0.3622&\quad 0.1512&\quad 0.4788\\ 0.0274&\quad -\,0.0110&\quad -\,0.2871&\quad 0.9866 \end{bmatrix},\\ G_{1f2}= & {} \begin{bmatrix} 0.0084&\quad 1.3347&\quad 1.2689&\quad -\,1.5090\\ -\,0.5508&\quad 1.6317&\quad 0.9299&\quad -\,0.8270\\ -\,0.8287&\quad 1.1747&\quad 2.0727&\quad -\,1.0402\\ -\,0.6060&\quad 0.9031&\quad 0.9599&\quad 0.2230 \end{bmatrix},\\ G_{2f2}= & {} \begin{bmatrix} 1.1310&\quad -\,1.0054&\quad -\,0.1783&\quad 0.4615\\ 0.1197&\quad 0.3209&\quad -\,0.0564&\quad 0.1156\\ 0.2363&\quad -\,0.6838&\quad 0.9848&\quad -\,0.1278\\ 0.4263&\quad -\,0.8410&\quad -\,0.7031&\quad 1.8839 \end{bmatrix},\\ G_{3f2}= & {} \begin{bmatrix} 0.8959&\quad -\,0.2187&\quad -\,0.5120&\quad -\,0.3229\\ -\,0.3431&\quad 1.2582&\quad -\,0.7028&\quad -\,0.1559\\ -\,0.1963&\quad -\,0.0467&\quad 0.4256&\quad -\,0.3981\\ -\,0.2562&\quad 0.2164&\quad -\,0.4322&\quad 0.7405 \end{bmatrix},\ B_{1f2}=\begin{bmatrix} -\,0.1986&\quad -\,0.1312\\ -\,0.1509&\quad -\,0.1091\\ -\,0.2150&\quad -\,0.1391\\ -\,0.1219&\quad -\,0.1014 \end{bmatrix}, \\ B_{2f2}= & {} \begin{bmatrix} -\,0.1228&\quad 0.0574\\ -\,0.1098&\quad 0.0760\\ -\,0.1294&\quad 0.0651\\ -\,0.0945&\quad 0.0860 \end{bmatrix},\ B_{3f2}=\begin{bmatrix} -\,0.0710&\quad -\,0.0764\\ -\,0.0605&\quad -\,0.0476\\ -\,0.0744&\quad -\,0.0736\\ -\,0.0500&\quad -\,0.0427 \end{bmatrix}. \end{aligned}$$

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Sheng, S., Zhang, X., Lu, Q. et al. Event-Triggered \(H_\infty \) State Estimation for Coupled and Switched Genetic Regulatory Networks. Circuits Syst Signal Process 38, 4420–4445 (2019). https://doi.org/10.1007/s00034-019-01073-6

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