Fuzzy Intermittent Control for Nonlinear PDE-ODE Coupled Systems

Abstract

This paper introduces a fuzzy intermittent control issue for nonlinear PDE-ODE coupled system under spatially point measurements (SPMs), which can be represented by an ordinary differential equation (ODE) and a partial differential equation (PDE). Firstly, the nonlinear coupled system is aptly characterized by the Takagi–Sugeno (T–S) fuzzy PDE-ODE coupled model. Subsequently, based on T–S fuzzy model, a novel Lyapunov function (LF) is provided to design a fuzzy intermittent controller ensuring exponential stability of the closed-loop coupled system. The stabilization conditions are presented by means of a group of space-dependent linear matrix inequalities (SDLMIs). Finally, simulation results are given to illustrate the effectiveness of the proposed design method in the control of a hypersonic rocket car (HRC).

Appendices

Appendix 1

Proof of Theorem 1

It follows from (18) and (19) that

$$\begin{aligned} \Xi _{1vm}^{wn}(z,t)< & {} 0\end{aligned}$$

(40)

$$\begin{aligned} \Xi _{2}^{vw}(z,t)< & {} 0 \end{aligned},$$

(41)

in which

$$\begin{aligned} \Xi _{1vm}^{wn}(z,t)= \,& {} \left[ \begin{array}{cccc}\Phi _{1}^{vm}(t)&{} \Phi _{1}^{v}(z,t)&{} 0 &{}0\\ * &{}\Phi _{1}^{wn}(z,t)&{}0 &{}\Phi _{1}^{n}(z,t)\\ * &{} * &{}\Phi _{1}(z,t) &{} 0\\ *&{}*&{}*&{}-\frac{\pi ^{2}}{\Delta _s^{2}}\Gamma (t)\\ \end{array} \right] \\ \Xi _{2}^{vw}(z,t)= \,& {} \left[ \begin{array}{cccc}\Phi _{2}^{v}(t) &{}\Phi _{2}^{v}(z,t) &{}0 \\ * &{}\Phi _{2}^{w}(z,t)&{}0 \\ * &{} * &{}\Phi _{2}(z,t) \\ \end{array} \right] \end{aligned},$$

where

$$\begin{aligned} \Phi _{1}^{vm}(t)= \,& {} \frac{1}{l_2-l_1}[P_1(t)(\theta _1(t)+2\xi )+\rho _{10}(t)(P_{11}-P_{12}) \\{} & {} +P_1(t)A_{1v}+A_{1v}^{T}P_1(t)+P_1(t)B_{1v}K_m\\{} & {} +(B_{1v}K_m)^{T}P_1(t)]\\ \Phi _{2}^{v}(t)= \,& {} \frac{1}{l_2-l_1}[P_2(t)(\theta _2(t)+2\xi )+\rho _{20}(t)(P_{21}-P_{22})\\{} & {} +P_2(t)A_{1v}+A_{1v}^{T}P_2(t)]\\ \Phi _{1}^{v}(z,t)= \,& {} P_1(t)G_{1v}+G_{2v}^{T}(z)Q_1(t)\\ \Phi _{2}^{v}(z,t)= \,& {} P_2(t)G_{1v}+G_{2v}^{T}(z)Q_2(t)\\ \Phi _{1}^{wn}(z,t)= \,& {} Q_1(t)(\theta _1(t)+2\xi )+\rho _{10}(t)(Q_{11}-Q_{12})\\{} & {} +Q_1(t)A_{2w}(z)+A_{2w}^{T}(z)Q_1(t)\\{} & {} +Q_1(t)B_{u_2}(z)W_{ns}+(B_{u_2}(z)W_{ns})^{T}Q_1(t)\\{} & {} -\frac{\pi ^{2}}{(l_2-l_1)^{2}}U_1(t)\\ \Phi _{2}^{w}(z,t)= \,& {} Q_2(t)(\theta _2(t)+2\xi )+\rho _{20}(t)(Q_{21}-Q_{22})\\{} & {} +Q_2(t)A_{2w}(z)+A_{2w}^{T}(z)Q_2(t)\\{} & {} -\frac{\pi ^{2}}{(l_2-l_1)^{2}}U_2(t)\\ \Phi _{1}^{n}(z,t)= \,& {} Q_1(t)B_{u_2}(z)W_{ns}\\ \Phi _{1}(z,t)= \,& {} -Q_1(t)\Upsilon (z)-\Upsilon ^{T}(z)Q_1(t)+U_1(t) \\ \Phi _{2}(z,t)= \,& {} -Q_2(t)\Upsilon (z)-\Upsilon ^{T}(z)Q_2(t)+U_2(t). \end{aligned}$$

Next, the stability analysis is made for the fuzzy coupled system (13) via a novel LF. To this end, a novel LF candidate is introduced as follows:

$$\begin{aligned} V(t)=V_1(t)+V_2(t) \end{aligned},$$

(42)

in which

$$\begin{aligned} V_1(t)= \,& {} \eta _{\sigma (t)}(t)x^{T}(t)P_{\sigma (t)}(t)x(t) \\ V_2(t)= \,& {} \eta _{\sigma (t)}(t)\int _\Omega T^{T}(z,t)Q_{\sigma (t)}(t)T(z,t)dz \end{aligned}$$

where $\eta _h(t)=\psi _h(t)e^{2\xi (t-t_0)}$, $\psi _h(t)=\mu _h^{\rho _{h1}(t)}$, $h\in \overline{1,2}$. $P_h(t)=\sum \limits _{i=1}^{2}\rho _{hi}(t)P_{hi}$, $Q_h(t)=\sum \limits _{i=1}^{2}\rho _{hi}(t)Q_{hi}$, $\theta _1(t)=\rho _{10}(t)\ln \mu _1$, and $\theta _2(t)=\rho _{20}(t)\ln \mu _2$.

For $t\in J _{1,k}$ with any given $k\in \mathbb {N}_0$, taking the derivative of V(t) with respect to time along the trajectory of (13a) yields

$$\begin{aligned} \dot{V}_1(t)= \,& {} \eta _1(t)x^{T}(t)\{P_1(t)(\theta _1(t)+2\xi )+\rho _{10}(t)(P_{11}\nonumber \\{} & {} -P_{12})\}x(t)+2\eta _1(t)\sum \limits _{v=1}^{r_1}\sum \limits _{m=1}^{r_1}\varrho _v(\theta (t))\varrho _m(\theta (t))\nonumber \\{} & {} \times x^{T}(t)P_1(t)\{A_{1v}x(t)+B_{1v}K_mx(t)\nonumber \\{} & {} +G_{1v}\int _\Omega {T(z,t)}dz\}. \end{aligned}$$

(43)

Denote $\kappa (z,t)\triangleq T(\overline{z}_s, t)-T(z, t)$. Then, we have $y_s(t)=T(z, t)+\kappa (z,t)$. Using (13b) gives the following relationship:

$$\begin{aligned} \dot{V}_2(t)= \,& {} \eta _1(t)\int _\Omega T^{T}(z,t)\{Q_1(t)(\theta _1(t)+2\xi )\nonumber \\{} \,& {} +\rho _{10}(t)(Q_{11}-Q_{12})\}T(z,t)dz\nonumber \\{} & {} +2\eta _1(t)\int _\Omega T^{T}(z,t)Q_1(t)T_{t}(z,t)dz\nonumber \\= \,& {} \eta _1(t)\int _\Omega T^{T}(z,t)\{Q_1(t)(\theta _1(t)+2\xi )\nonumber \\{} \,& {} +\rho _{10}(t)(Q_{11}-Q_{12})\}T(z,t)dz\nonumber \\{} & {} +2\eta _1(t)\int _\Omega \sum \limits _{v=1}^{r_1}\sum \limits _{w=1}^{r_2} \sum \limits _{n=1}^{r_2}\varrho _v(\theta (t))\varpi _w(\vartheta (z,t))\nonumber \\{} & {} \times \varpi _n(\vartheta (\bar{z}_s,t))T^{T}(z,t)Q_1(t)\{(A_{2w}(z)\nonumber \\{} & {} +B_{u_2}(z)W_{ns})T(z,t)+G_{2v}(z)x(t)\nonumber \\{} & {} +B_{u_2}(z)W_{ns}\kappa (z, t)\}dz\nonumber \\{} & {} +2\eta _1(t)\int _\Omega T^{T}(z,t)Q_1(t)\{\Upsilon (z)T_{z}(z,t)\}_zdz. \end{aligned}$$

(44)

Then, utilizing the boundary conditions (2), for any matrix $U_1(t)=\sum \limits _{i=1}^{2}\sum \limits _{l=1}^{2}\rho _{1i}(t)\zeta _{1l}(t)U_{1il}>0$, one has from Lemma 1 that

$$\begin{aligned}{} \,& {} 2\int _\Omega T^{T}(z,t)Q_1(t)\{\Upsilon (z)T_{z}(z,t)\}_zdz\nonumber \\= \,& {} -2\int _\Omega T_{z}^{T}(z,t)Q_1(t)\Upsilon (z)T_{z}(z,t)dz\nonumber \\\le & {} -\int _\Omega T_{z}^{T}(z,t)\{Q_1(t)\Upsilon (z)+\Upsilon ^{T}(z)Q_1(t)\nonumber \\{} & {} -U_1(t)\}T_{z}(z,t)dz\nonumber \\{} & {} -\frac{\pi ^{2}}{(l_2-l_1)^{2}}\int _\Omega T^{T}(z,t)U_1(t)T(z,t)dz. \end{aligned}$$

(45)

Considering $\kappa ^{2}_z(z,t)=T^{2}_z(z,t)$, for any matrix $\Gamma (t)=\sum \limits _{i=1}^{2}\sum \limits _{l=1}^{2} \rho _{1i}(t)\zeta _{1l}(t)\Gamma _{1il}>0$, one can get from Lemma 2 that

$$\begin{aligned}{} & {} -\frac{\pi ^{2}}{\Delta _s^{2}}\int _{z_s}^{z_{s+1}}\kappa ^{T}(z,t)\Gamma (t)\kappa (z, t)dz\nonumber \\= \,& {} -\frac{\pi ^{2}}{\Delta _s^{2}}\int _{z_s}^{\overline{z}_{s}}\kappa ^{T}(z, t)\Gamma (t)\kappa (z,t)dz\nonumber \\{} & {} -\frac{\pi ^{2}}{\Delta _s^{2}}\int _{\overline{z}_s}^{z_{s+1}}\kappa ^{T}(z, t)\Gamma (t)\kappa (z,t)dz\nonumber \\\le & {} -\int _{z_s}^{z_{s+1}}\kappa ^{T}_{z}(z, t)\Gamma (t)\kappa _{z}(z,t)dz\nonumber \\= \,& {} -\int _{z_s}^{z_{s+1}}T_z(z,t)\Gamma (t) T_z(z,t)dz. \end{aligned}$$

(46)

Using (40) and (43)–(46), we obtain

$$\begin{aligned} \dot{V}(t)\le & {} \eta _1(t)\sum \limits _{s=0}^{c-1} \int _{z_{s}}^{z_{s+1}}\sum \limits _{v=1}^{r_1}\sum \limits _{m=1}^{r_1}\sum \limits _{w=1}^{r_2} \sum \limits _{n=1}^{r_2}\varrho _v(\theta (t))\\{} & {} \times \varrho _m(\theta (t))\varpi _w(\vartheta (z,t))\varpi _n(\vartheta (\bar{z}_s,t))\varphi _1^{T}(z,t)\\{} & {} \times \Xi _{1vm}^{wn}(z,t)\varphi _1(z,t)dz<0, t\in J _{1,k} \end{aligned}$$

where $\varphi _1(z,t)=[x^{T}(t) \ T^{T}(z,t) \ T_z^{T}(z,t) \ \kappa ^{T}(z, t)]^{T}$. It follows that

$$\begin{aligned} V(t)\le V(t_k),t\in {J}_{1,k}, k\in \mathbb {N}_0. \end{aligned}$$

(47)

For $t\in J _{2,k}, k\in \mathbb {N}_0$, taking the derivative of V(t) with respect to time along the trajectory of (13b) yields

$$\begin{aligned} \dot{V}_1(t)= \,& {} \eta _2(t)x^{T}(t)\{P_2(t)(\theta _2(t)+2\xi )+\rho _{20}(t)\nonumber \\{} & {} \times (P_{21}-P_{22})\}x(t)+2\eta _2(t)\sum \limits _{v=1}^{r_1}\varrho _v(\theta (t))\nonumber \\{} & {} \times x^{T}(t)P_2(t)\{A_{1v}x(t)+G_{1v}\int _\Omega {T(z,t)}dz\}\end{aligned}$$

(48)

$$\begin{aligned} \dot{V}_2(t)= \,& {} \eta _2(t)\int _\Omega T^{T}(z,t)\{Q_2(t)(\theta _2(t)+2\xi )\nonumber \\{} & {} +\rho _{20}(t)(Q_{21}-Q_{22})\}T(z,t)dz\nonumber \\{} & {} +2\eta _2(t)\int _\Omega T^{T}(z,t)Q_2T_{t}(z,t)dz\nonumber \\= \,& {} \eta _2(t)\int _\Omega T^{T}(z,t)\{Q_2(t)(\theta _2(t)+2\xi )+\rho _{20}(t)\nonumber \\{} & {} \times (Q_{21}-Q_{22})\}T(z,t)dz+2\eta _2(t)\int _\Omega \sum \limits _{v=1}^{r_1} \sum \limits _{w=1}^{r_2}\nonumber \\{} & {} \times \varrho _v(\theta (t))\varpi _w(\vartheta (z,t))T^{T}(z,t)Q_2(t)\nonumber \\{} & {} \times \{A_{2w}(z)T(z,t)+G_{2v}(z)x(t)\}dz\nonumber \\{} & {} +2\eta _2(t)\int _\Omega T^{T}(z,t)Q_2(t)\nonumber \\{} & {} \times \{\Upsilon (z)T_{z}(z,t)\}_zdz. \end{aligned}$$

(49)

Then, utilizing the boundary conditions (2), for any matrix $U_2(t)=\sum \limits _{i=1}^{2}\sum \limits _{l=1}^{2}\rho _{2i}(t)\zeta _{2l}(t)U_{2il}>0$, one has from Lemma 1 that

$$\begin{aligned}{} & {} 2\int _\Omega T^{T}(z,t)Q_2(t)\{\Upsilon (z)T_{z}(z,t)\}_zdz\nonumber \\= \,& {} -2\int _\Omega T_{z}^{T}(z,t)Q_2(t)\Upsilon (z)T_{z}(z,t)dz\nonumber \\\le & {} -\int _\Omega T_{z}^{T}(z,t)\{Q_2(t)\Upsilon (z)+\Upsilon ^{T}(z)Q_2(t)\nonumber \\{} & {} -U_2(t)\}T_{z}(z,t)dz\nonumber \\{} & {} -\frac{\pi ^{2}}{(l_2-l_1)^{2}}\int _\Omega T^{T}(z,t)U_2(t)T(z,t)dz. \end{aligned}$$

(50)

Using (41) and (48)–(50), we obtain

$$\begin{aligned} \dot{V}(t)\le & {} \eta _2(t)\int _\Omega \sum \limits _{v=1}^{r_1}\sum \limits _{w=1}^{r_2}\varrho _v(\theta (t))\varpi _w(\vartheta (z,t))\\{} & {} \times \varphi _2^{T}(z,t)\Xi _{2}^{vw}(z,t)\varphi _2(z,t)dz<0,t\in J _{2,k} \end{aligned}$$

where $\varphi _2(z,t)=[x(t) \ T(z,t) \ T_z(z,t)]^{T}$. It follows that

$$\begin{aligned} V(t)\le V(s_k), t\in J _{2,k}, k\in \mathbb {N}_0. \end{aligned}.$$

(51)

Estimate V(t) at the switching instants $t_k$ and $s_k$, $k\in \mathbb {N}_0$. From the definitions of $P_h(t)$ and $\psi _h(t)$, $h\in \overline{1,2}$, one has

$$\begin{aligned} P_1(t_k)= \,& {} P_{12}, P_2(t_k^{-})=P_{21}, Q_1(t_k)=Q_{12}\\ Q_2(t_k^{-})= \,& {} Q_{21}, \psi _1(t_k)=1,\psi _2(t_k^{-})=\mu _2\\ P_1(s_k^{-})\,&=&P_{11}, P_2(s_k)=P_{22}, Q_1(s_k^{-})=Q_{11}\\ Q_2(s_k)= \,& {} Q_{22}, \psi _1(s_k^{-})=\mu _1,\psi _2(s_k)=1. \end{aligned}$$

Then, according to [32] and [34], for any $k\in \mathbb {N}_0$, one obtains from (16) and (17):

$$\begin{aligned} V(t_k)= \,& {} e^{2\xi (t_k-t_0)}\{x^{T}(t_k)P_{12}x(t_k) \nonumber \\{} & {} +\int _\Omega T^{T}(z,t_k)Q_{12}T(z, t_k)dz\}\nonumber \\\le & {} e^{2\xi (t_k-t_0)}\{\mu _2 x^{T}(t_k)P_{21}x(t_k)\nonumber \\{} & {} +\mu _2\int _\Omega T^{T}(z,t_k)Q_{21}T(z,t_k)dz\}\nonumber \\= \,& {} V(t_k^{-})\end{aligned}$$

(52)

$$\begin{aligned} V(s_k)= \,& {} e^{2\xi (s_k-t_0)} \{x^{T}(s_k)P_{22}x(s_k)\nonumber \\{} & {} +\int _\Omega T^{T}(z,s_k)Q_{22}T(z,s_k)dz\}\nonumber \\\le & {} e^{2\xi (s_k-t_0)}\{\mu _1 x^{T}(s_k)P_{11}x(s_k)\nonumber \\{} & {} +\mu _1\int _\Omega T^{T}(z,s_k)Q_{11}T(z,s_k)\}dz\nonumber \\= \,& {} V(s_k^{-}). \end{aligned}$$

(53)

By jointly applying equations (47), (51), (52), and (53), one can obtain $V(t)\le V(t_0), t\ge t_0$. Thus, we conclude that $\Vert x(t)\Vert _2+\Vert T(\cdot ,t)\Vert _2\le \sqrt{\frac{M\lambda _1}{\lambda _0}}(\Vert x(t_0)\Vert _2+\Vert T(\cdot , t_0)\Vert _2)e^{-\xi (t-t_0)}$, $\text{where}\,M=\max \{1, \mu _1, \mu _2\}$, $\lambda _1=\max \{\lambda _{\max }(P_i), i\in \overline{1,2}\}$, and $\lambda _0=\min \{\lambda _{\min }(P_i), i\in \overline{1,2}\}$. Therefore, the nonlinear closed-loop coupled system (13) is exponential stable over $S_\sigma (\delta _{11}, \delta _{12}; \delta _{21}, \delta _{22})$.

Appendix 2

Proof of Theorem 2

Define $P_{hi}=X_{hi}^{-1}, Q_{hi}=Y_{hi}^{-1}, K_m=\bar{K}_mX_{0}^{-1},$ $W_{ns}=\bar{W}_{ns}Y_{0}^{-1},$ $U_{1il}=Y_{1i}^{-1}\bar{U}_{1il}Y_{1i}^{-1}, U_{2il}=Y_{2i}^{-1}\bar{U}_{2il}Y_{2i}^{-1},$ $\Gamma _{1il}=Y_{1i}^{-1}\bar{\Gamma }_{1il}Y_{1i}^{-1}, h,i,l\in \overline{1,2}$. Applying Schur complement and the matrix inequalities: $-X_{11}X_{12}^{-1}X_{11}\le -2\varepsilon _1 X_{11}+\varepsilon _1^{2}X_{12}$, $-Y_{11}Y_{12}^{-1}Y_{11}\le -2\varepsilon _2 Y_{11}+\varepsilon _2^{2}Y_{12}$. By applying Lemma 2, we can derive from (21) that

$$\begin{aligned} \left[ \begin{array}{cccc}\Theta _{1il}^{vmwn}(z) &{}\hat{X}_{1i}&{}\hat{Y}_{1i} \\ * &{}-\theta _{i}(X_0+X_0^{T})&{}0\\ * &{}* &{}-\epsilon _{i}(Y_0+Y_0^{T}) \\ \end{array} \right] <0 \end{aligned},$$

(54)

where $\hat{X}_{1i}=\mathcal {A}_1^{T}(X_{1i}-X_0)^{T}+\mathcal {B}_1(\theta _{i}X_0)$, $\hat{Y}_{1i}=\mathcal {A}_2^{T}(Y_{1i}-Y_0)^{T}+\mathcal {B}_2(\epsilon _{i}Y_0)$, $\Theta _{1il}^{vmwn}(z)=\bar{\Xi }_{1il}^{vmwn}(z)+\mathcal {B}_1X_0\mathcal {A}_1+ (\mathcal {B}_1X_0\mathcal {A}_1)^{T}+\mathcal {B}_2Y_0\mathcal {A}_2+(\mathcal {B}_2 Y_0\mathcal {A}_2)^{T}$, $\mathcal {A}_1=[I \ 0 \ 0 \ 0]$, $\mathcal {B}_1=\text {col}(B_{1v}K_m, 0, 0, 0)$, $\mathcal {A}_2=[0 \ I \ 0 \ I]$, $\mathcal {B}_2=\text {col}(0, B_{u_2}(z)W_{ns}, 0, 0)$, and

$$\begin{aligned} \bar{\Xi }_{1il}^{vmwn}(z)= \,& {} \left[ \begin{array}{cccc}\bar{\Phi }_{1il}^{vm} &{}\bar{\Phi }_{1i}^{v}(z)&{} 0 &{}0\\ * &{}\bar{\Phi }_{1il}^{wn}(z)&{}0 &{}0\\ * &{} * &{}\bar{\Phi }_{1il}(z) &{}0\\ *&{}*&{}*&{}-\frac{\pi ^{2}}{\Delta _s^{2}}\bar{\Gamma }_{1il}\\ \end{array} \right] \end{aligned},$$

(55)

in which $\bar{\Phi }_{1il}^{vm}=\frac{1}{l_2-l_1}\{(\frac{\ln \mu _1}{\delta _{1l}} +2\xi )X_{1i}+\frac{1}{\delta _{1l}}X_{1i}(P_{11}-P_{12})X_{1i} +A_{1v}X_{1i}+X_{1i}A_{1v}^{T}\}$, $\bar{\Phi }_{1i}^{v}(z)=G_{1v}Y_{1i}+X_{1i}G_{2v}^{T}(z)$, and $\bar{\Phi }_{1il}^{wn}(z)=(\frac{\ln \mu _1}{\delta _{1l}}+2\xi )Y_{1i} +\frac{1}{\delta _{1l}}Y_{1i}(Q_{11}-Q_{12})Y_{1i}+A_{2w}(z)Y_{1i} +Y_{1i}A_{2w}^{T}(z)-\frac{\pi ^{2}}{(l_2-l_1)^{2}}\bar{U}_{1il}, \bar{\Phi }_{1il}(z)=-\Upsilon (z)Y_{1i}-Y_{1i}\Upsilon ^{T}(z)+\bar{U}_{1il}$.

Based on Lemma 3, it can be concluded that the matrix inequalities (54) imply that

$$\begin{aligned} \hat{\Xi }_{1il}^{vmwn}(z)<0,i,l=1,2 \end{aligned},$$

(56)

$$\begin{aligned} \hat{\Xi }_{1il}^{vmwn}(z)= \,& {} \bar{\Xi }_{1il}^{vmwn}(z)+\mathcal {B}_1X_{1i}\mathcal {A}_1+ (\mathcal {B}_1X_{1i}\mathcal {A}_1)^{T}\\{} & {} +\mathcal {B}_2Y_{1i}\mathcal {A}_2+(\mathcal {B}_2 Y_{1i}\mathcal {A}_2)^{T}. \end{aligned} $$

Pre- and post-multiplying (56) by $\text {diag}\{P_{1i},Q_{1i},Q_{1i},Q_{1i}\}$ yields that the matrix inequalities (56) are equivalent to (18) with $h=1$. Pre- and post-multiplying (19) by $\text {diag}\{P_{1i},Q_{1i},Q_{1i}\}$ yields with $h=2$. Therefore, it can be concluded that the conclusion of this theorem directly follows from Theorem 1, which completes the proof.