Robust Exponential $H_{\infty }$ Filtering for Discrete-Time Switched Fuzzy Systems with Time-Varying Delay

Abstract

This paper considers $H_{\infty }$ filtering of discrete-time uncertain switched nonlinear systems with time-varying delay via Takagi–Sugeno fuzzy model which is used to approximate each nonlinear subsystem. Fuzzy piecewise Lyapunov–Krasovskii functionals and average dwell time approaches are utilized for the exponential stability analysis and $H_{\infty }$ filter design. A new sufficient condition is obtained to guarantee the exponential stability with a prescribed $H_{\infty }$ performance index for the filtering error system. Filter parameter matrices can be obtained by solving sets of linear matrix inequalities. An example of an uncertain single-link robot arm is given to demonstrate the effectiveness of the proposed method.

Appendix

$$\begin{aligned}&{V_{i1}}(k + 1) - \alpha {V_{i1}}(k) \nonumber \\&\quad = \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{r_i}}\sum \limits _{j' = 1}^{{r_i}}\sum \limits _{\eta = 1}^{{r_i}}\sum \limits _{\eta ' = 1}^{{r_i}} {\psi _{i}(\sigma (k)){h_{ij}}(k){h_{ij'}}(k){h_{i\eta }}(k){h_{i\eta '}}(k)} \nonumber \\&\qquad \times \, \left[ {{\xi ^T}(k)\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ \end{array}} \right] {\bar{P}_i}(k + 1){{\left[ {\begin{array}{cc} {\tilde{A}^T_{ij'\eta '}} \\ {\tilde{A}^T_{dij'\eta '}} \\ {\tilde{B}^T_{ij'\eta '}} \\ \end{array}} \right] }^T}\xi (k)} -{\tilde{x}^T}(k)\alpha {\bar{P}_i}(k)\tilde{x}(k)\right] \nonumber \\&\quad \le \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{r_i}}\sum \limits _{\eta = 1}^{{r_i}} \psi _{i}(\sigma (k)){h_{ij}}(k){h_{i\eta }(k)}\nonumber \\&\qquad \times \left[ {{\xi ^T}(k)\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ \end{array}} \right] {\bar{P}_i}(k + 1){{\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ \end{array}} \right] }^T}\xi (k)}-\,{\tilde{x}^T}(k)\alpha {\bar{P}_i}(k)\tilde{x}(k)\right] .\quad \end{aligned}$$

(48)

$$\begin{aligned}&{V_{i2}}(k+1) - \alpha {V_{i2}}(k)\nonumber \\&\quad = {\alpha ^k}\left[ {\sum \limits _{l=k + 1 - {d_i}(k + 1)}^k {{\tilde{x}^T}(l){\alpha ^{ - l}}{\bar{Q}_i}(l)\tilde{x}(l)} - \sum \limits _{l=k - {d_i}(k)}^{k - 1} {{\tilde{x}^T}(l){\alpha ^{ - l}}{\bar{Q}_i}(l)\tilde{x}(l)} } \right] \nonumber \\&\quad = {\alpha ^k}\left[ {{\tilde{x}^T}(k){\alpha ^{ - k}}{\bar{Q}_i}(k)\tilde{x}(k) \,{-}\, {\tilde{x}^T}(k \,{-}\, {d_i}(k)){\alpha ^{ - \left( {k \,{-}\, {d_i}(k)} \right) }}{\bar{Q}_i}(k \,{-}\, {d_i}(k))\tilde{x}(k \,{-}\, {d_i}(k))} \right. \nonumber \\&\qquad + \sum \limits _{k + 1 - {d_i}(k + 1)}^{k - 1} {{\tilde{x}^T}(l){\alpha ^{ - l}}{\bar{Q}_i}(l)\tilde{x}(l)} - \sum \limits _{k + 1 - {d_i}(k)}^{k - 1} {\left. {{\tilde{x}^T}(l){\alpha ^{ - l}}{\bar{Q}_i}(l)\tilde{x}(l)} \right] } \nonumber \\&\quad \le {\tilde{x}^T}(k){\bar{Q}_i}(k)\tilde{x}(k) - {\tilde{x}^T}(k - {d_i}(k))\left( {{\alpha ^{{d_i}(k)}}{\bar{Q}_i}(k - {d_i}(k))} \right) \tilde{x}(k - {d_i}(k)) \nonumber \\&\qquad + \sum \limits _{l = k + 1 - {d_{{M_i}}}}^{k - {d_{{m_i}}}} {{\tilde{x}^T}(l){\alpha ^{k - l}}{\bar{Q}_i}(l)\tilde{x}(l)} \nonumber \\&\quad \le {\tilde{x}^T}(k){\bar{Q}_i}(k)\tilde{x}(k) - {\tilde{x}^T}(k - {d_i}(k))\left( {{\alpha ^{{d_{{M_i}}}}}{\bar{Q}_i}(k - {d_i}(k))} \right) \tilde{x}(k - {d_i}(k)) \nonumber \\&\qquad + \sum \limits _{l = k + 1 - {d_{{M_i}}}}^{k - {d_{{m_i}}}} {{\tilde{x}^T}(l){\alpha ^{k - l}}{\bar{Q}_i}(l)\tilde{x}(l)}. \end{aligned}$$

(49)

$$\begin{aligned}&{V_{i3}}(k + 1) - \alpha {V_{i3}}(k) \nonumber \\&\quad = \sum \limits _{j = - {d_{{M_i}}} + 2}^{ - {d_{{m_i}}} + 1} {\left[ {{\tilde{x}^T}(k) {{\bar{Q}_i}(k)}\tilde{x}(k)} \right. } - {\tilde{x}^T}(k - 1 + j)\nonumber \\&\qquad \qquad \times \left. {\left( {{\alpha ^{ - \left( {- 1 + j} \right) }}{\bar{Q}_i}(k - 1 + j)} \right) \tilde{x}(k - 1 + j)} \right] \nonumber \\&\quad = ({d_{{M_i}}} - {d_{{m_i}}}){\tilde{x}^T}(k){\bar{Q}_i}(k)\tilde{x}(k) - \sum \limits _{l = k+ 1 - {d_{{M_i}}}}^{k - {d_{{m_i}}}} {{\tilde{x}^T}(l)\left( {{\alpha ^{k - l}}{\bar{Q}_i}(l)} \right) \tilde{x}(l)}. \end{aligned}$$

(50)

$$\begin{aligned}&{V_{i4}}(k + 1) - \alpha {V_{i4}}(k)\nonumber \\&\quad = {\alpha ^k}\sum \limits _{p = - {d_{{M_i}}}}^{ - 1} {\left[ {\sum \limits _{l = k + p + 1}^k {{\triangle \tilde{x}^T}(l){\alpha ^{ - l}}} {\bar{Z}_i}(l)\triangle \tilde{x}(l) - \sum \limits _{l = k + p}^{k - 1} {{\triangle \tilde{x}^T}(l)} {\alpha ^{ - l}}{\bar{Z}_i}(l)\triangle \tilde{x}(l)} \right] } \nonumber \\&\quad = \sum \limits _{p = - {d_{{M_i}}}}^{ - 1} {\left( {{\triangle \tilde{x}^T}(k){\bar{Z}_i}(k)\triangle \tilde{x}(k) - {\triangle \tilde{x}^T}(k + p){\alpha ^{ - p}}{\bar{Z}_i}(k + p)\triangle \tilde{x}(k + p)} \right) } \nonumber \\&\quad \le {d_{{M_i}}}{\triangle \tilde{x}^T}(k){\bar{Z}_i}(k)\triangle \tilde{x}(k) - \sum \limits _{l = k - {d_i}(k)}^{k - 1} {{\triangle \tilde{x}^T}(l){\alpha ^{d_{{M_i}}}}{\bar{Z}_i}(l)\triangle \tilde{x}(l)} . \end{aligned}$$

(51)

$$\begin{aligned}&\quad {V_{i4}}(k + 1) - \alpha {V_{i4}}(k) \nonumber \\&\quad \le \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{r_i}}\sum \limits _{j' = 1}^{{r_i}}\sum \limits _{\eta = 1}^{{r_i}}\sum \limits _{\eta ' = 1}^{{r_i}} {\psi _{i}(\sigma (k)){h_{ij}}(k){h_{ij'}}(k){h_{i\eta }}(k){h_{i\eta '}}(k)} \nonumber \\&\qquad \times \,\left[ {{\xi ^T}(k)\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }-I} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ \end{array}} \right] {{d_{{M_i}}}\bar{Z}_i}(k){{\left[ {\begin{array}{cc} {\tilde{A}^T_{ij'\eta '}-I} \\ {\tilde{A}^T_{dij'\eta '}} \\ {\tilde{B}^T_{ij'\eta '}} \\ \end{array}} \right] }^T}\xi (k)}\right. \nonumber \\&\qquad \left. -{\sum \limits _{l = k - {d_i}(k)}^{k - 1} {{\triangle {\tilde{x}}^T}(l)} \alpha ^{d_{M_{i}}}{\bar{Z}_i}(l)\triangle {\tilde{x}}(l)}\right] \nonumber \\&\quad \le \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{r_i}}\sum \limits _{\eta = 1}^{{r_i}}\psi _{i}(\sigma (k)){h_{ij}}(k){h_{i\eta }}(k)\nonumber \\&\qquad \times \left[ {{\xi ^T}(k)\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }-I} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ \end{array}} \right] {d_{{M_i}}}{\bar{Z}_i}(k){{\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }-I} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ \end{array}} \right] }^T}\xi (k)}\right. \nonumber \\&\qquad \left. -{\sum \limits _{l = k - {d_i}(k)}^{k - 1} {{\triangle {\tilde{x}}^T}(l)} \alpha ^{d_{M_{i}}}{\bar{Z}_i}(l)\triangle {\tilde{x}}(l)}\right] . \end{aligned}$$

(52)

$$\begin{aligned}&{V_i}(k + 1) - \alpha {V_i}(k)+\tilde{z}^T(k)\tilde{z}(k)-\gamma ^2{w}^T(k){w}(k) \le \frac{1}{{{d_i}(k)}}\nonumber \\&\quad \sum \limits _{l = k - {d_i}(k)}^{k - 1}\sum \limits _{i =1}^{n} \sum \limits _{l = 1}^{{r_i}} {\sum \limits _{n = 1}^{{r_i}} {\sum \limits _{m = 1}^{{r_i}}\sum \limits _{j = 1}^{{r_i}}\sum \limits _{\eta = 1}^{{r_i}}\psi _{i}(\sigma (k)) {{h_{il}}(k){h_{in}}(k\!-\!d_i(k))} } } {h_{im}}(k\!+\!1){h_{ij}}(k){h_{i\eta }}(k)\nonumber \\&\qquad \left\{ \zeta ^{\mathrm{T}}(k,l)\left( {\left[ {\begin{array}{cccc} -{\alpha }\bar{P}_{i\eta }+(d_{M_i}-d_{m_i}+1)\bar{Q}_{i\eta } &{} 0 &{}0&{}0 \\ * &{} -\alpha ^{d_{M_i}}\bar{Q}_{in}&{} 0&{}0 \\ * &{} * &{} -\gamma ^2I&{}0 \\ *&{}*&{}*&{}-\frac{1}{{{d_{{M_i}}}}}{\alpha ^{{d_{{M_i}}}}}{\bar{Z}_{il}}\\ \end{array}} \right] }\right. \right. \nonumber \\&\qquad +\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ 0\\ \end{array}} \right] {\bar{P}_{im}}{{\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ 0\\ \end{array}} \right] }^T} \left. \left. +\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }-I} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ 0\\ \end{array}} \right] {d_{{M_i}}}{\bar{Z}_{i\eta }}{{\left[ {\begin{array}{cc} {\tilde{A}^T_{ij\eta }-I} \\ {\tilde{A}^T_{dij\eta }} \\ {\tilde{B}^T_{ij\eta }} \\ 0\\ \end{array}} \right] }^T}\right. \right. \nonumber \\&\qquad +\left[ \! {\begin{array}{cc} {\tilde{H}^T_{ij\eta }} \\ {\tilde{C}^T_{dij\eta }} \\ {\tilde{G}^T_{ij\eta }} \\ 0\\ \end{array}} \!\right] {{\left[ \! {\begin{array}{cc} {\tilde{H}^T_{ij\eta }} \\ {\tilde{C}^T_{dij\eta }} \\ {\tilde{G}^T_{ij\eta }} \\ 0\\ \end{array}} \!\right] }^T}\left. \left. +\left[ {\begin{array}{cccc} \bar{N}_{ij}&{}-\bar{N}_{ij}&{}0&{}-\bar{N}_{ij}\\ \end{array}} \right] +\left[ {\begin{array}{cccc} \bar{N}_{ij}&{}-\bar{N}_{ij}&{}0&{}-\bar{N}_{ij}\\ \end{array}} \right] ^T\right) \zeta (k,l)\right\} .\nonumber \\ \end{aligned}$$

(53)