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Robust Non-fragile \(H_{\infty }\) Fuzzy Control for Uncertain Nonlinear-Delayed Hyperbolic PDE Systems

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Abstract

In this paper, we consider the robust non-fragile \(H_{\infty }\) fuzzy control for uncertain nonlinear delayed hyperbolic partial differential equation (PDE) systems, which can represent the dynamics of most industrial processes, such as plug-flow reactors, fixed-bed reactors, tubular heat exchangers, traffic flows, and wave equations. Initially, a Takagi–Sugeno (T–S) fuzzy-delayed hyperbolic PDE model is presented to describe the uncertain nonlinear delayed hyperbolic PDE system. Subsequently, based on the T–S fuzzy delayed hyperbolic PDE model, spatial linear matrix inequality (SLMI) based sufficient conditions to ensure exponential stability with an \(H_{\infty }\) performance are obtained via utilizing the Lyapunov direct method. Then, to solve the SLMIs, the robust non-fragile \(H_{\infty }\) fuzzy control problem for uncertain nonlinear delayed hyperbolic PDE systems is formulated as an LMI feasibility problem. Finally, two examples on a Lotka–Volterra PDE system and a non-isothermal plug-flow reactor (PFR) are presented to illustrate the effectiveness of the presented control method.

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Acknowledgements

This work was supported in part by the National Key Research and Development Program of China under Grants 2021ZD0112301 and 2021ZD0112302, in part by the National Natural Science Foundations of China under Grants 62073011, 62203326, and 61973135 and in part by the Shandong Provincial Natural Science Foundation under Grant ZR2021MF 004.

Author information

Authors and Affiliations

  1. School of Electrical Engineering, University of Jinan, Jinan, 250022, China

    Xu Zhang

  2. Faculty of Information Technology, Beijing Laboratory of Smart Environmental Protection, Beijing Key Laboratory of Computational Intelligence and Intelligent System, and Engineering Research Center of Intelligence Perception and Autonomous Control Ministry of Education, and Beijing Institute of Articial Intelligence, Beijing University of Technology, Beijing, 100124, China

    Zi-Peng Wang & Jun-Fei Qiao

  3. Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, China

    Huai-Ning Wu

  4. School of Control Science and Engineering, Tiangong University, Tianjin, 300387, China

    Xiao-Wei Zhang

  5. Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon Tong, Hong Kong

    Han-Xiong Li

Authors
  1. Xu Zhang
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  2. Zi-Peng Wang
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  3. Huai-Ning Wu
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Corresponding author

Correspondence to Zi-Peng Wang.

Appendices

Appendix 1

Differentiating V(t) along the solution of the system (9) yields

$$\begin{aligned}&{\dot{V}}(t)+2\delta V(t)\nonumber \\&\quad \le \int _{l_1}^{l_2} [z^{T}(x,t)P(x){\varTheta }(x) z_x(x,t)\nonumber \\&\qquad +z_x^{T}(x,t){\varTheta }^{T}(x)P(x) z(x,t) ]dx\nonumber \\&\qquad +2\int _{l_1}^{l_2}\sum \limits _{i=1}^r\sum \limits _{j=1}^rh_i(\varpi (x,t))h_j(\varpi (x,t))z^{T}(x,t)P(x)\nonumber \\&\qquad \times \{A_i(x)z(x,t)+A_{\tau i}(x)z(x,t-\tau (t))\nonumber \\&\qquad +G_i(x)K_j(x)z(x,t)\}dx\nonumber \\&\qquad +2\delta \int _{l_1}^{l_2} z^{T}(x,t)P(x) z(x,t)dx\nonumber \\&\qquad +\int _{l_1}^{l_2}z^{T}(x,t)Q_1(x)z(x,t)dx\nonumber \\&\qquad -(1-\nu )e^{-2\delta {{\bar{\tau }}}}\int _{l_1}^{l_2}z^{T}(x,t-\tau (t))\nonumber \\&\qquad \times Q_1(x)z(x,t-\tau (t))dx\nonumber \\&\qquad +\int _{l_1}^{l_2}z^{T}(x,t)Q_2(x)z(x,t)d\nonumber \\&\qquad -e^{-2\delta {{\bar{\tau }}}}\int _{l_1}^{l_2}z^{T}(x,t-{\bar{\tau }})Q_2(x)z(x,t-{\bar{\tau }})dx. \end{aligned}$$
(35)

Integrating by parts in (35), we obtain

$$\begin{aligned}&\int _{l_1}^{l_2} z^{T}(x,t)P(x){\varTheta }(x) z_x(x,t)dx\nonumber \\&\quad =z^{T}(x,t)P(x){\varTheta }(x) z(x,t)|_{x=l_1}^{x=l_2}\nonumber \\&\qquad -\int _{l_1}^{l_2}(z^{T}(x,t)P(x){\varTheta }(x))_x z(x,t)dx\nonumber \\&\quad =\int _{l_1}^{l_2}(\vartheta (x-l_2)-\vartheta (x-l_1))z^{T}(x,t)P(x){\varTheta }(x) z(x,t)dx\nonumber \\&\qquad -\int _{l_1}^{l_2} z_x^{T}(x,t)P(x){\varTheta }(x) z(x,t)dx\nonumber \\&\qquad -\int _{l_1}^{l_2} z^{T}(x,t)(P(x) {\varTheta }(x))_x z(x,t)dx \end{aligned}$$
(36)

where \(\vartheta (x)\) is the Dirac delta function defined as follows:

$$\begin{aligned} \vartheta (x)=\left\{ \begin{aligned}&\infty ,&x=0\\&0,&x\ne 0 \end{aligned} \right. \end{aligned}$$

and has the following property:

$$\begin{aligned} \int _{-\infty }^{+\infty }f(x)\vartheta (x-x_0)ds=f(x_0). \end{aligned}$$

From (36), one has the following equality considering \(P(x){\varTheta }(x)={\varTheta }^T(x)P(x)\):

$$\begin{aligned}&\int _{l_1}^{l_2} [z^{T}(x,t)P(x){\varTheta }(x) z_x(x,t)\nonumber \\&\qquad +z_x^{T}(x,t){\varTheta }^{T}(x)P(x) z(x,t) ]dx\nonumber \\&\quad =x\int _{l_1}^{l_2}(\vartheta (x-l_2)-\vartheta (x-l_1))z^{T}(x,t)P(x){\varTheta }(x) z(x,t)dx\nonumber \\&\qquad -\int _{l_1}^{l_2} z^{T}(x,t)(P(x) {\varTheta }(x))_x z(x,t)dx. \end{aligned}$$
(37)

Based on (35)–(37), we have

$$\begin{aligned}&{\dot{V}}(t)+2\delta V(t)\nonumber \\&\quad \le \int _{l_1}^{l_2}\sum \limits _{i=1}^r\sum \limits _{j=1}^rh_i(\varpi (x,t))h_j(\varpi (x,t))\nonumber \\&\qquad \times \xi (x,t){\varPsi }_{ij}(x)\xi (x,t)dx \end{aligned}$$
(38)

with \(\xi (x,t)=[ \ z^T(x,t) \ z^T(x,t-\tau (t)) \ z^T(z,t-{\bar{\tau }}) \ ]^T\).

It is shown from (38) that if the following matrix inequalities hold:

$$\begin{aligned} {\varPsi }_{ij}(x)=\left[ \begin{array}{ccc}{\varPsi }_{11}(x)&{} {\varPsi }_{12}(x)&{} {\varPsi }_{13}(x)\\ *&{} {\varPsi }_{22}(x) &{} {\varPsi }_{23}(x)\\ * &{} * &{} {\varPsi }_{33}(x) \end{array} \right] <0 \end{aligned}$$

with

$$\begin{aligned} {\varPsi }_{11}(x)= & {} \psi (x)+P(x)A_i(x)+A_i^T(x)P(x)\\&+P(x)G_i(x)K_j(x)+K_j^T(x)G_i^T(x)P(x)\\ {\varPsi }_{12}(x)= & {} P(x)A_{\tau i}(x), \ {\varPsi }_{13}(x)=0\\ {\varPsi }_{22}(x)= & {} -(1-\nu )e^{-2\delta {\bar{\tau }}}Q_1(x), \ {\varPsi }_{23}(x)=0\\ {\varPsi }_{33}(x)= & {} -e^{-2\delta {\bar{\tau }}}Q_2(x)\\ \psi (x)= & {} \varphi (x)-P_x(x){\varTheta }(x)-P(x){\varTheta }_x(x)\\&+2\delta P(x)+Q_1(x)+Q_2(x)\\ \varphi (x)= & {} (\vartheta (x-l_2)-\vartheta (x-l_1))P(x){\varTheta }(x). \end{aligned}$$

Then the closed-loop fuzzy delayed hyperbolic PDE system (13) is exponentially stable with a decay rate \(\delta\).

Define \(Q(x)\triangleq P^{-1}(x)\), \({\tilde{K}}_j(x)\triangleq K_j(x) Q(x)\), \({\bar{Q}}_1(x)\) \(\triangleq Q(x)Q_1(x)Q(x)\), and \({\bar{Q}}_2(x)\triangleq Q(x)Q_2(x)Q(x)\). Pre- and postmultiplying both sides of \({\varPsi }_{ij}(x)\) by \(\text {diag}\{Q(x)\)\(Q(x), Q(x)\}\) and using the property \(Q_x(x)=-Q(x)\)\(Q^{-1}_x(x)Q(x)\). Subsequently, by Lemma 2, we can obtain Theorem 1. \(\square\)

Appendix 2

From (9), (10), and (35), we have

$$\begin{aligned}&\ \ \ {\dot{V}}(t)+2\delta V(t)+\Vert y_c(\cdot ,t)\Vert _2^2-\gamma ^2\Vert w(\cdot ,t)\Vert _2^2\nonumber \\&\quad \le \int _{l_1}^{l_2}(\vartheta (x-l_2)-\vartheta (x-l_1))z^{T}(x,t)P(x){\varTheta }(x) z(x,t)dx\nonumber \\&\qquad -\int _{l_1}^{l_2} z^{T}(x,t)\{P_x(x){\varTheta }(x)+P(x){\varTheta }_x(x)\}z(x,t)dx\nonumber \\&\qquad +2\int _{l_1}^{l_2}\sum \limits _{i=1}^r\sum \limits _{j=1}^rh_i(\varpi (x,t))h_j(\varpi (x,t))z^{T}(x,t)\nonumber \\&\qquad \times \{P(x){\bar{A}}_i(x,t)z(x,t)+P(x){\bar{A}}_{\tau i}(x,t)z(x,t-\tau (t))\nonumber \\&\qquad +P(x){\bar{G}}_i(x,t){\bar{K}}_j(x,t)z(x,t)+P(x){\bar{G}}_{wi}(x,t)w(x,t)\}\nonumber \\&\qquad +\int _{l_1}^{l_2}\sum \limits _{j=1}^rh_j(\varpi (x,t))z^T(x,t)(C+D{\bar{K}}_j(x,t))^T \nonumber \\&\qquad \times (C+D{\bar{K}}_j(x,t))z(x,t)dx\nonumber \\&\qquad -\gamma ^2\int _{l_1}^{l_2}w^T(x,t)w(x,t)dx\nonumber \\&\qquad +2\delta \int _{l_1}^{l_2} z^{T}(x,t)P(x) z(x,t)dx\nonumber \\&\qquad +\int _{l_1}^{l_2}z^{T}(x,t)Q_1(x)z(x,t)dx\nonumber \\&\qquad -(1-\nu )e^{-2\delta {{\bar{\tau }}}}\int _{l_1}^{l_2}z^{T}(x,t-\tau (t))Q_1(x)z(x,t-\tau (t))dx\nonumber \\&\qquad +\int _{l_1}^{l_2}z^{T}(x,t)Q_2(x)z(x,t)dx\nonumber \\&\qquad -e^{-2\delta {{\bar{\tau }}}}\int _{l_1}^{l_2}z^{T}(x,t-{\bar{\tau }})Q_2(x)z(x,t-{\bar{\tau }})dx. \end{aligned}$$
(39)

Defining \({\bar{\xi }}(x,t)=[ \ z^T(x,t) \ z^T(x,t-\tau (t)) \ z^T(z,t-{\bar{\tau }}) \ w^T(x,t) \ ]^T\), one has

$$\begin{aligned}&{\dot{V}}(t)+2\delta V(t)+\Vert y_c(\cdot ,t)\Vert _2^2-\gamma ^2\Vert w(\cdot ,t)\Vert _2^2\nonumber \\&\quad \le \int _{l_1}^{l_2}\sum \limits _{i=1}^r\sum \limits _{j=1}^rh_i(\varpi (x,t))h_j(\varpi (x,t)\nonumber \\&\qquad \times {\bar{\xi }}^T(x,t){\varPhi }_{ij}(x){\bar{\xi }}(x,t)dx \end{aligned}$$
(40)

where

$$\begin{aligned} {\varPhi }_{ij}(x)=\left[ \begin{array}{cccc} {\varPhi }_{11}(x) &{} {\varPhi }_{12}(x) &{} {\varPhi }_{13} &{} {\varPhi }_{14}(x)\\ * &{} {\varPhi }_{22}(x) &{} {\varPhi }_{23} &{} {\varPhi }_{24}\\ * &{} * &{} {\varPhi }_{33}(x) &{} {\varPhi }_{34}\\ * &{} * &{} * &{} {\varPhi }_{44} \end{array}\right] \end{aligned}$$

with

$$\begin{aligned} {\varPhi }_{11}(x)= &\,\, {} \psi (x)+P(x){\bar{G}}_i(x,t){\bar{K}}_j(x,t)+P(x){\bar{A}}_i(x,t)\\&\,\,+{\bar{A}}^T_i(x,t)P(x)+{\bar{K}}^T_j(x,t){\bar{G}}^T_i(x,t)P(x)\\&\,\,+(C+D{\bar{K}}_j(x,t))^T(C+D{\bar{K}}_j(x,t))\\ {\varPhi }_{12}(x)= &\,\, {} P(x){\bar{A}}_{\tau i}(x,t), {\varPhi }_{13}=0 \\ {\varPhi }_{14}(x)= &\,\, {} P(x){\bar{G}}_{wi}(x,t), {\varPhi }_{22}(x)=-(1-\nu )e^{-2\delta {\bar{\tau }}}Q_1(x)\\ {\varPhi }_{23}= &\,\, {} 0, {\varPhi }_{24}=0, {\varPhi }_{33}(x)=-e^{-2\delta {\bar{\tau }}} Q_2(x)\\ {\varPhi }_{34}= &\,\, {} 0, {\varPhi }_{44}=-\gamma ^2 I\\ \psi (x)= &\,\, {} \varphi (x)-P_x(x){\varTheta }(x)-P(x){\varTheta }_x(x)+2\delta P(x)\\&+Q_1(x)+Q_2(x)\\ \varphi (x)= & {} (\vartheta (x-l_2)-\vartheta (x-l_1))P(x){\varTheta }(x). \end{aligned}$$

Using Lemma 1, one has

$$\begin{aligned}&P(x)G_i(x){\varDelta } K_j(x,t)+{\varDelta } K^T_j(x,t)G^T_i(x)P(x)\nonumber \\&\quad \le \varsigma _{ij}P(x)G_i(x)M_{j}(x)[P(x)G_i(x)M_{j}(x)]^T\nonumber \\&\qquad +\varsigma _{ij}^{-1}N^T_{j}(x)N_{j}(x),\nonumber \\&P(x){\varDelta } A_i(x,t)+{\varDelta } A^T_i(x,t)P(x)\nonumber \\&\quad \le \sigma _{ij}P(x)H_i(x)(P(x)H_i(x))^T+\sigma _{ij}^{-1}E_{1i}^T(x)E_{1i}(x),\nonumber \\&P(x){\varDelta } A_{\tau i}(x,t)+{\varDelta } A^T_{\tau i}(x,t)P(x)\nonumber \\&\quad \le \varrho _{ij}P(x)H_i(x)(P(x)H_i(x))^T+\varrho _{ij}^{-1}E_{2i}^T(x)E_{2i}(x),\nonumber \\&P(x){\varDelta } G_i(x,t){\bar{K}}_j(x,t)+{\bar{K}}^T_j(x,t){\varDelta } G^T_i(x,t)P(x)\nonumber \\&\quad \le \rho _{ij}P(x)H_i(x)(P(x)H_i(x))^T\nonumber \\&\qquad +\rho _{ij}^{-1}(E_{3i}(x){\bar{K}}_j(x,t))^T E_{3i}(x){\bar{K}}_j(x,t),\nonumber \\&P(x){\varDelta } G_{wi}(x,t)+{\varDelta } G^T_{wi}(x,t)P(x)\nonumber \\&\quad \le \theta _{ij}P(x)H_i(x)(P(x)H_i(x))^T+\theta _{ij}^{-1}E_{4i}^T(x)E_{4i}(x). \end{aligned}$$
(41)

With the help of Lemma 1, the terms \(D{\varDelta } K_j(x,t)\) and \(E_{3i}{\varDelta } K_j(x,t)\) in (41) can be solved by parameters \(\iota _{ij}\) and \(\kappa _{ij}\), respectively. Then, from (39)–(41) and using Schur complement, the matrices \({\varPhi }_{ij}(x)\) in (40) can be transformed as follows:

$$\begin{aligned} {\varLambda }_{ij}(x)=\left[ \begin{array}{ccc} {\varLambda }_{1,1}(x) &{} \cdots &{} {\varLambda }_{1,19}\\ \vdots &{} \ddots &{} \vdots \\ * &{} \cdots &{} {\varLambda }_{19,19} \end{array}\right] \end{aligned}$$

with

$$\begin{aligned} {\varLambda }_{1,1}(x)= & {} \psi (x)+P(x)A_i(x)+A^T_i(x)P(x)\\&+P(x)G_i(x)K_j(x)+K^T_j(x)G^T_i(x)P(x)\\ {\varLambda }_{1,2}(x)= & {} P(x)A_{\tau i}(x), \ {\varLambda }_{1,3}=0\\ {\varLambda }_{1,4}(x)= & {} P(x)G_{wi}(x), \ {\varLambda }_{1,5}(x)=(C+D K_j(x))^T\\ {\varLambda }_{1,6}(x)= & {} P(x)G_i(x)M_{j}(x), \ {\varLambda }_{1,7}(x)=N_{j}^T(x)\\ {\varLambda }_{1,8}(x)= & {} P(x)H_i(x), \ {\varLambda }_{1,9}(x)=E_{1i}^T(x)\\ {\varLambda }_{1,10}(x)= & {} P(x)H_i(x), \ {\varLambda }_{1,11}(x)=E_{2i}^T(x)\\ {\varLambda }_{1,12}(x)= & {} P(x)H_i(x), \ {\varLambda }_{1,13}(x)=K^T_j(x)E_{3i}^T(x)\\ {\varLambda }_{1,14}(x)= & {} P(x)H_i(x), \ {\varLambda }_{1,15}(x)=E_{4i}^T(x)\\ {\varLambda }_{1,16}(x)= & {} N_{j}^T(x), \ {\varLambda }_{1,17}(x)=N_{j}^T(x)\\ {\varLambda }_{1,18}= & {} 0, \ {\varLambda }_{1,19}=0\\ {\varLambda }_{2,2}(x)= & {} (\nu -1)e^{-2\delta {\bar{\tau }}}Q_1(x)\\ {\varLambda }_{3,3}(x)= & {} -e^{-2\delta {\bar{\tau }}} Q_2(x)\\ {\varLambda }_{4,4}= & {} -\gamma ^2 I, \ {\varLambda }_{5,5}=-I\\ {\varLambda }_{6,6}= & {} -\varsigma _{ij}^{-1}I, \ {\varLambda }_{7,7}=-\varsigma _{ij}I\\ {\varLambda }_{8,8}= & {} -\sigma _{ij}^{-1}I, \ {\varLambda }_{9,9}=-\sigma _{ij}I\\ {\varLambda }_{10,10}= & {} -\varrho _{ij}^{-1}I, \ {\varLambda }_{11,11}=-\varrho _{ij}I\\ {\varLambda }_{12,12}= & {} -\rho _{ij}^{-1}I, \ {\varLambda }_{13,13}=-\rho _{ij}I\\ {\varLambda }_{14,14}= & {} -\theta _{ij}^{-1}I, \ {\varLambda }_{15,15}=-\theta _{ij}I\\ {\varLambda }_{16,16}= & {} -\iota _{ij}I, \ {\varLambda }_{17,17}=-\kappa _{ij}I\\ {\varLambda }_{17,18}(x)= & {} \iota _{ij}E_{3i}(x)M_{j}(x), \ {\varLambda }_{18,18}=-\iota _{ij}I\\ {\varLambda }_{18,19}(x)= & {} \kappa _{ij}DM_{j}(x), \ {\varLambda }_{19,19}=-\kappa _{ij}I\\ \psi (x)= & {} \varphi (x)-P_x(x){\varTheta }(x)-P(x){\varTheta }_x(x)+2\delta P(x)\\&+Q_1(x)+Q_2(x)\\ \varphi (x)= & {} (\vartheta (x-l_2)-\vartheta (x-l_1))P(x){\varTheta }(x). \end{aligned}$$

It is obvious that the inequality is ensured as follows:

$$\begin{aligned} {\dot{V}}(t)+2\delta V(t)+\Vert y_c(\cdot ,t)\Vert _2^2-\gamma ^2\Vert w(\cdot ,t)\Vert _2^2<0 \end{aligned}$$
(42)

if

$$\begin{aligned} {\varLambda }_{ij}(x)<0. \end{aligned}$$
(43)

Pre- and post–multiplying both sides of \({\varLambda }_{ij}(x)\) by diag \(\{Q(x), Q(x), Q(x), I, I, \varsigma _{ij}I, I, \sigma _{ij}I, I, \varrho _{ij}I, I, \rho _{ij}I, I\), \(\theta _{ij}I, I, I, I, I, I\}\) and using the property \(Q_x(x)=-Q(x)\)\(Q^{-1}_x(x)\) Q(x). Then, by Lemma 2, we can obtain Theorem 2. \(\square\)

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Zhang, X., Wang, ZP., Wu, HN. et al. Robust Non-fragile \(H_{\infty }\) Fuzzy Control for Uncertain Nonlinear-Delayed Hyperbolic PDE Systems. Int. J. Fuzzy Syst. 25, 851–867 (2023). https://doi.org/10.1007/s40815-022-01409-6

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