Outlier-resistant observer-based fuzzy sampled-data boundary control for the hot strip mill cooling process

Abstract

This paper proposes an outlier-resistant non-fragile (ORNF) observer-based boundary control strategy for the hot strip mill (HSM) cooling process. First, measurement outliers and observer parameter perturbations are unavoidable due to the disturbances or faults of the HSM cooling process, so the ORNF observer is developed using the saturation function and the non-fragile control theory to improve the observation accuracy. Then, a sampled-data (SD) measurement is utilized, which avoids frequent updating of temperature sensors. Furthermore, a SD boundary control strategy is proposed by using the above observed temperature information, which only requires an actuator to be installed on one side of the system to regulate the temperature, reducing the control cost of the HSM cooling process effectively. Finally, sufficient conditions to ensure the stability of the HSM cooling process are derived through rigorous mathematical proof, and simulation results are presented to verify the effectiveness and superiority of the proposed method.

Appendix

Proof of Theorem 1

The following Lyapunov function is constructed:

$$\begin{aligned} V(t)=&\int _{0}^{l}p(e^2(x,t)+s^2(x,t))dx. \end{aligned}$$

(33)

Then, along the (16) and (18), the derivative is calculated for (33) with respect to time as

$$\begin{aligned} \dot{V}(t) =&\int _{0}^{l} 2p(e(x,t)e_t(x,t)+s(x,t)s_t(x,t))dx. \end{aligned}$$

(34)

Next, substituting (16) and (18) into (34) yields

$$\begin{aligned}&\int _{0}^{l} 2pe(x,t)e_t(x,t)dx \nonumber \\&= \int _{0}^{l} 2pe(x,t)\{ \alpha e_{xx}(x,t){+} \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t))\sigma _i e(x,t)\nonumber \\&\quad - {\tilde{L}}_i\phi (y(t_k) - {\hat{y}}(t_k))\} dx, \end{aligned}$$

(35)

and

$$\begin{aligned}&\int _{0}^{l} 2ps(x,t)s_t(x,t)dx\nonumber \\&\quad = \int _{0}^{l} 2ps(x,t)\{ \alpha s_{xx}(x,t) \nonumber \\&\qquad + \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t)) \sigma _is(x,t)\} dx, \end{aligned}$$

(36)

where using integral by parts and boundary condition (19), then $\int _{0}^{l} 2p\alpha e(x,t)e_{xx}(x,t)dx$ in (35) is further deduced as

$$\begin{aligned} \int _{0}^{l} 2p\alpha e(x,t)e_{xx}(x,t)dx =&\int _{0}^{l} {l^{-1}}2p\alpha de^2(l,t)dx\nonumber \\&-2\int _{0}^{l} p\alpha e_x^2(x,t) dx. \end{aligned}$$

(37)

Similarly, combining integral by parts and boundary condition (17), $\int _{0}^{l} 2p\alpha s(x,t)s_{xx}(x,t)dx$ in (36) is reexpressed as

$$\begin{aligned}&\int _{0}^{l} 2p\alpha s(x,t)s_{xx}(x,t)dx\nonumber \\&= 2pg\alpha s(l,t)\{ \sum \limits _{j = 1}^r \varpi _j(\varphi (x,t))k_j \{ s(l,t_k)- e(l,t_k)\} \}\nonumber \\&\quad + \int _{0}^{l}{l^{-1}}2p\alpha ds^2(l,t) dx- 2\int _{0}^{l} p\alpha s_x^2(x,t)dx. \end{aligned}$$

(38)

Employing Lemma 1 in [33], $- 2\int _{0}^{l} p\alpha e_x^2(x,t) dx$ in (37) and $- 2\int _{0}^{l} p\alpha s_x^2(x,t) dx$ in (38) are estimated as:

$$\begin{aligned}&- 2\int _{0}^{l} p\alpha e_x^2(x,t) dx\nonumber \\&\quad \le - 0.5l^{-2}\pi ^2\int _{0}^{l} p\alpha \{ e(x,t) - e(l,t)\}^2dx, \end{aligned}$$

(39)

and

$$\begin{aligned}&- 2\int _{0}^{l} p\alpha s_x^2(x,t) dx\nonumber \\&\quad \le -0.5l^{-2}\pi ^2\int _{0}^{l} p\alpha \{ s(x,t) - s(l,t)\}^2dx. \end{aligned}$$

(40)

Define $\omega (l,t)=s(l,t) - s(l,t_k)$ and $m(l,t) = e(l,t) - e(l,t_k)$, which means that

$$\begin{aligned} \begin{aligned} s(l,t) - s(l,t_k) - \omega (l,t) = 0, \end{aligned} \end{aligned}$$

(41)

and

$$\begin{aligned} \begin{aligned} e(l,t) - e(l,t_k)- m(l,t) = 0, \end{aligned} \end{aligned}$$

(42)

from which it is straightforward to obtain that the following equation holds:

$$\begin{aligned} 0 =&2\int _{0}^{l} [\omega (l,t)q_1 + s(l,t_k)q_2] [ s(l,t)\nonumber \\&- s(l,t_k) - \omega (l,t) ] dx, \end{aligned}$$

(43)

and

$$\begin{aligned} 0 =&2\int _{0}^{l} [ m(l,t)q_3 + e(l,t_k)q_4 ][ e(l,t)\nonumber \\&- e(l,t_k) - m(l,t) ]dx, \end{aligned}$$

(44)

where $q_{\varrho }\ (\varrho =1,2,3,4)$ are arbitrary positive constants.

Then, according to Lemma 1, it is direct to obtain

$$\begin{aligned} \begin{aligned} \phi (e(l,t_k)) = {\tilde{\upsilon }} e(l,t_k) + \vartheta (e(l,t_k)), \end{aligned} \end{aligned}$$

(45)

and

$$\begin{aligned} \begin{aligned} \vartheta (e(l,t_k))\{ \vartheta (e(l,t_k)) - \upsilon e(l,t_k)\} \le 0. \end{aligned} \end{aligned}$$

(46)

Therefore, based on (34)–(46), the following formula can be derived:

$$\begin{aligned} \dot{V}(t) \le&\int _{0}^{l} 2pe(x,t)\left\{ \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t))\sigma _i e(x,t) \right. \\&-\left. {\tilde{L}}_i\{ {\tilde{\upsilon }} e(l,t_k) + \vartheta (e(l,t_k))\right\} dx\\&+ \int _{0}^{l} {l^{-1}}2p\alpha de^2(l,t)dx\\&\mathrm{+} \int _{0}^{l} 2p s(x,t)\left\{ \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t))\sigma _i s(x,t)\} dx\right. \\&\mathrm{+} 2p g\alpha s(l,t)\{ \sum \limits _{j = 1}^r \varpi _j(\varphi (x,t))k_j \{ s(l,t_k) \\&- e(l,t_k)\} \} + \int _{0}^{l}{l^{-1}}2p\alpha ds^2(l,t) dx\\&- 0.5l^{-2}\pi ^2\int _{0}^{l} p\alpha \{ e(x,t) - e(l,t)\}^2dx\\&-0.5l^{-2}\pi ^2\int _{0}^{l} p\alpha \{ s(x,t) - s(l,t)\}^2 dx\\&+ 2\int _{0}^{l} [ \omega (l,t)q_1+ s(l,t_k)q_2 ][ s(l,t) \\&- s(l,t_k) - \omega (l,t) ]dx + 2\int _{0}^{l} [ m(l,t)q_3\\&+ e(l,t_k)q_4 ][ e(l,t) - e(l,t_k)- m(l,t) ]dx\\&-\gamma _1\omega ^2(l,t) + \gamma _1c_1-\gamma _2 m^2(l,t)+\gamma _2c_2\\&-\left. 2\rho \vartheta (e(l,t_k))\{ \vartheta (e(l,t_k)) - \upsilon e(l,t_k)\right\} ,\\ \end{aligned}$$

where $\omega ^2(l,t) \le c_1,m^2(l,t) \le c_2$ with given positive parameters $c_1$ and $c_2$ and $\rho >0$ is a given constant.

Defining $\zeta {=}col[e(x,t),e(l,t),\vartheta (e(l,t_k)),s(x,t),s(l,t), s(l,t_k),\omega (l,t),e(l,t_k),m(l,t)]$, one gets

$$\begin{aligned} \dot{V}(t)+\beta V(t) \le&\int _{0}^{l} \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t))\sum \limits _{i = 1}^r \varpi _j(\varphi (x,t))\nonumber \\&\times \zeta ^T(x,t)\varPsi \zeta (x,t)dx+\gamma c_1{+}\gamma c_2, \end{aligned}$$

(47)

where $\varPsi =\varOmega -\varXi <0$,

$$\begin{aligned} \begin{aligned} \varOmega = \left[ {\begin{array}{*{20}{c}} \hat{\varOmega }_{11}&{}0&{}{\hat{\varOmega }}_{13}\\ {*}&{}\varOmega _{22}&{}\varOmega _{23}\\ {*}&{}{*}&{}{\hat{\varOmega }}_{33} \end{array}} \right] \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \varXi = \left[ {\begin{array}{*{20}{c}} \varGamma _{11}&{}0&{}\varGamma _{13}\\ *&{}0&{}0\\ *&{}*&{}0 \end{array}} \right] \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \hat{\varOmega }_{11}&= \left[ {\begin{array}{*{20}{c}} \aleph _1&{}0.25l^{-2}\pi ^2p\alpha &{}- pL_i\\ *&{}\aleph _2&{}0\\ *&{}*&{}-2\rho \end{array}} \right] ,\nonumber \\ {\hat{\varOmega }}_{13}&= \left[ {\begin{array}{*{20}{c}} 0&{}- p{\tilde{\upsilon }} L_i &{}0\\ 0&{}q_4+\rho \upsilon &{}q_3\\ 0&{} 0&{}0 \end{array}}\right] ,\nonumber \\ {\hat{\varOmega }}_{33}&= \left[ {\begin{array}{*{20}{c}} - 2q_1 - \gamma _1&{}0&{}0\\ *&{}- 2q_4&{}-q_3 - q_4\\ *&{}*&{}- 2q_3-\gamma _2 \end{array}} \right] ,\nonumber \\ \varGamma _{11}&= \left[ {\begin{array}{*{20}{c}} 0&{}0&{}p(D_i\varLambda _i(t)F_i)\\ {*}&{}0&{}0\\ {*}&{}{*}&{}0 \end{array}} \right] ,\nonumber \\ \varGamma _{13}&= \left[ {\begin{array}{*{20}{c}} 0&{}p{\tilde{\upsilon }} (D_i\varLambda _i(t)F_i)&{}0\\ {*}&{}0&{}0\\ {*}&{}{*}&{}0 \end{array}} \right] . \end{aligned}$$

Next, defining $\varPhi _i=col[pD_i,0,0,0,0,0,0,0,0]$ and $\varTheta _i = [0,0,F_i,0,0,0,0,{\tilde{\upsilon }} F_i,0]$, then we can obtain $\varXi =\varPhi \varLambda (t)\varTheta +*$, where $\varLambda _i^T(t)\varLambda _i(t) \le I$.

Therefore, one has

$$\begin{aligned} \begin{aligned} \varPsi = \varOmega -\varXi = \varOmega -\{\varPhi _i\varLambda _i(t)\varTheta _i+* \}<0. \end{aligned} \end{aligned}$$

Inspired by [8], the following equation is valid:

$$\begin{aligned} \begin{aligned} - \{ \varPhi _i\varLambda _i(t)\varTheta _i +* \} \le \chi ^{-1}\varPhi _i\varPhi _i^T + \chi \varTheta _i^T\varTheta _i. \end{aligned} \end{aligned}$$

(48)

Then, $\varPsi = \varOmega - \varXi$ can be further derived as

$$\begin{aligned} \begin{aligned} \varOmega + \chi ^{-1}\varPhi _i\varPhi _i^T + \chi \varTheta _i^T\varTheta _i<0, \end{aligned} \end{aligned}$$

(49)

which, by utilizing the Schur Complement Lemma, is reformulated as (21), i.e., $\varUpsilon <0$.

Since $\varUpsilon <0$, based on (47), we can get

$$\begin{aligned} \begin{aligned} \dot{V}(t)+\beta V(t) \le \gamma _1c_1+\gamma _2c_2. \end{aligned} \end{aligned}$$

(50)

Defining $c=\gamma _1c_1+\gamma _2c_2$, it can be deduced that

$$\begin{aligned} V(t) \le&e^{ - \beta t}V(0) + \int _0^t e^{ - \beta (t - s)}cds\nonumber \\ =&e^{-\beta t}V(0) + c\beta ^{-1}. \end{aligned}$$

(51)

Therefore, according to Definition 1 in [37, 38], we complete the proof. $\blacksquare$

Proof of Theorem 3

Similar to the process of (33)–(40), one can directly get

$$\begin{aligned} \dot{V}(t) \le&\int _{0}^{l} 2pe(x,t)\{ \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t))\sigma _i e(x,t)\\&\mathrm{-} {\tilde{L}}_i\{ {\tilde{\upsilon }} e(l,t){+} \vartheta (e(l,t))\} dx\\&+ \int _{0}^{l} {l^{-1}}2p\alpha de^2(l,t)dx\\&+ \int _{0}^{l}{l^{-1}} 2p\alpha ds^2(l,t) dx\\&+ \int _{0}^{l} 2ps(x,t)\left\{ \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t)) \sigma _is(x,t)\} dx\right. \\&+ 2pg\alpha s(l,t)\{ \sum \limits _{j = 1}^r \varpi _j(\varphi (x,t))k_j \{ s(l,t)\\&- e(l,t)\} \}{-} 0.5l^{-2}\pi ^2\!\int _{0}^{l}\!\! p\alpha \{ e(x,t) {-} e(l,t)\}^2dx\\&-0.5l^{-2}\pi ^2\int _{0}^{l} p\alpha \{ s(x,t) - s(l,t)\}^2 dx\\&-\left. 2\rho \vartheta (e(l,t))\{ \vartheta (e(l,t)) - \upsilon e(l,t)\right\} . \end{aligned}$$

Defining $\zeta = col[e(x,t),e(l,t),\vartheta (e(l,t)),s(x,t),s(l,t)]$, one has

$$\begin{aligned} \dot{V}(t)+\beta V(t) \le&\int _{0}^{l} \sum \limits _{i = 1}^r \varpi _i(\varphi (x,t))\sum \limits _{i = 1}^r \varpi _j(\varphi (x,t))\nonumber \\&\times \zeta ^T(x,t)\varPsi '\zeta (x,t)dx, \end{aligned}$$

(52)

where $\varPsi '=\varOmega '-\varXi '<0$,

$$\begin{aligned} \varOmega '&= \left[ {\begin{array}{*{20}{c}} {\tilde{\varOmega }}_{11}&{}{\tilde{\varOmega }}_{12}\\ {*}&{}{\tilde{\varOmega }}_{22} \end{array}} \right] ,\nonumber \\ {\tilde{\varOmega }}_{11}&= \left[ {\begin{array}{*{20}{c}} \aleph _1&{} 0.5\pi ^2l^{-2}p\alpha -p{\tilde{\upsilon }} L_i&{}- pL_i\\ *&{}\aleph _2&{}\rho \upsilon \\ *&{}*&{}-2\rho \end{array}} \right] ,\nonumber \\ {\tilde{\varOmega }}_{12}&=\left[ {\begin{array}{*{20}{c}} 0&{}0\\ 0&{}- {l }^{-1}pg\alpha k_j\\ 0&{}0 \end{array}} \right] ,\nonumber \\ {\tilde{\varOmega }}_{22}&=\left[ {\begin{array}{*{20}{c}} \aleph _3&{}{0.25l^{-2}\pi ^2p\alpha }\\ *&{}\ell _1 \end{array}} \right] , \end{aligned}$$

$\varXi '=\varPhi '_i\varLambda _i(t)\varTheta ' _i+*$, $\varPhi '_i = col[pD_i,0,0,0,0]$, and $\varTheta ' _i = [0,{\tilde{\upsilon }} F_i,F_i,0,0]$.

Next, similar to the steps of (48)–(51), it can be deduced that (29) holds, which completes the proof. $\blacksquare$