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

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.

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. $\square$