Decentralized PI Controller Design for Robust Perfect Adaptation in Noisy Time-Delayed Genetic Regulatory Networks

Abstract

In cellular scales, precise regulation of protein expression is challenging due to time delays and stochastic noises impairing the practical implementation of the existing controllers. To cope with such regulation issues in gene regulatory networks (GRNs), this paper designs a novel decentralized proportional–integral controller with an implementable structure that achieves robust perfect adaptation (RPA) in the presence of stochastic noises and time delays. In this regard, we first define a new augmented state space that incorporates the integral dynamics of the controller. Next, we exploit an appropriate Lyapunov–Krasovskii functional that enables us to quantify sufficient conditions for noise-to-state stability of the time-delayed GRNs in the form of linear matrix inequalities (LMIs). Then, we provide the proportional and integral gains of the decentralized controller using a modification of the variables along with a special structure of the derived LMIs. Finally, the proposed decentralized controller’s applicability and effectiveness are investigated using a synthetic network (known as Repressilator), and a GRN derived from a practical-based simulator. Due to its simple configuration, this feedback control strategy does not only provide RPA in cellular homeostasis but it can be universally implemented in any cellular environment.

Appendices

Appendix

Proof of Relationship (17)

Based on Ito’s formula [64], the infinitesimal generator of the proposed Lyapunov function equals to:

$$\begin{aligned} \mathcal {L}V(t)={} & {} \sum _{i=1}^{3} \mathcal {L}V_i(t)\nonumber \\ \mathcal {L}V_1(t)={} & {} {{{{\bar{x}}}}^{T}}\left( t \right) P {\bar{r}}_x \left( t \right) + {\bar{r}}^T_x\left( t \right) P {\bar{x}}\left( t \right) +{{{{\bar{y}}}}^{T}}\left( t \right) Q {\bar{r}}_y \left( t \right) \nonumber \\{} & {} {\bar{r}}^T_y\left( t \right) Q {\bar{y}}\left( t \right) +{{{{\bar{x}}}}^{T}}\left( t \right) {{R}_{1}}{\bar{x}}\left( t \right) +{{{{\bar{y}}}}^{T}}\left( t \right) {{R}_{2}}{\bar{y}}\left( t \right) \nonumber \\{} & {} + \frac{1}{2}Tr\left\{ \bar{\Sigma }_{x}^{T}\left( t \right) \left( P+{{P}^{T}} \right) {{{\bar{\Sigma }}}_{x}}\left( t \right) \right\} \nonumber \\{} & {} +\frac{1}{2}Tr\left\{ \bar{\Sigma }_{x}^{T}\left( t \right) \left( P+{{P}^{T}} \right) {{{\bar{\Sigma }}}_{x}}\left( t \right) \right\} \nonumber \\ \mathcal {L}V_2(t)={} & {} -{{e}^{-\alpha \sigma \left( t \right) }}\left( 1-{\dot{\sigma }}\left( t \right) \right) {{{{\bar{x}}}}^{T}}\left( t-\sigma \left( t \right) \right) {{R}_{1}}{\bar{x}}\left( t-\sigma \left( t \right) \right) \nonumber \\{} & {} -{{e}^{-\alpha \tau \left( t \right) }}\left( 1-{\dot{\tau }}\left( t \right) \right) {{{{\bar{y}}}}^{T}}\left( t-\tau \left( t \right) \right) {{R}_{2}}{\bar{y}}\left( t-\tau \left( t \right) \right) \nonumber \\{} & {} -\alpha {{e}^{-\alpha t}}\int _{t-\sigma \left( t \right) }^{t}{{{e}^{\alpha s}}{{{{\bar{x}}}}^{T}}\left( s \right) {{R}_{1}}{\bar{x}}\left( s \right) ds} \nonumber \\{} & {} -\alpha {{e}^{-\alpha t}}\int _{t-\tau \left( t \right) }^{t}{{{e}^{\alpha s}}{{{{\bar{y}}}}^{T}}\left( s \right) {{R}_{2}}{\bar{y}}\left( s \right) ds} \nonumber \\ \mathcal {L}V_3(t)={} & {} +\bar{\sigma }{{{{\bar{r}}}}_{x}}^{T}\left( t \right) {{Z}_{1}}{{{{\bar{r}}}}_{x}}\left( t \right) +\bar{\tau }{{{{\bar{r}}}}_{y}}^{T}\left( t \right) {{Z}_{1}}{{{{\bar{r}}}}_{y}}\left( t \right) \nonumber \\{} & {} -{{e}^{-\alpha t}}\int _{t-\sigma \left( t \right) }^{t}{{{e}^{\alpha \theta }}}{{{{\bar{r}}}}_{x}}^{T}\left( \theta \right) {{Z}_{1}}{{{{\bar{r}}}}_{x}}\left( \theta \right) d\theta \nonumber \\{} & {} -{{e}^{-\alpha t}}\int _{t-\tau \left( t \right) }^{t}{{{e}^{\alpha \theta }}}{{{{\bar{r}}}}_{y}}^{T}\left( \theta \right) {{Z}_{2}}{{{{\bar{r}}}}_{y}}\left( \theta \right) d\theta \nonumber \\{} & {} -\alpha {{e}^{-\alpha t}}\int _{-\bar{\sigma }}^{0}{\int _{t+\theta }^{t}{{{e}^{\alpha s}}{{r}_{{{\bar{x}}}}}^{T}\left( s \right) {{Z}_{1}}{{r}_{{{\bar{x}}}}}\left( s \right) ds}}d\theta \nonumber \\{} & {} -\alpha {{e}^{-\alpha t}}\int _{-\bar{\tau }}^{0}{\int _{t+\theta }^{t}{{{e}^{\alpha s}}{{r}_{{{\bar{y}}}}}^{T}\left( s \right) {{Z}_{2}}{{r}_{{{\bar{y}}}}}\left( s \right) ds}}d\theta \nonumber \\{} & {} -{{e}^{-\alpha t}}\int _{t-\sigma \left( t \right) }^{t}{{{e}^{\alpha \theta }}Tr\left\{ {{{\bar{\Sigma }}}^{T}}_{x}\left( \theta \right) {{Z}_{1}}{{{\bar{\Sigma }}}_{x}}\left( \theta \right) \right\} d\theta } \nonumber \\{} & {} -{{e}^{-\alpha t}}\int _{t-\tau \left( t \right) }^{t}{{{e}^{\alpha \theta }}Tr\left\{ {{{\bar{\Sigma }}}^{T}}_{y}\left( \theta \right) {{Z}_{2}}{{{\bar{\Sigma }}}_{y}}\left( \theta \right) \right\} d\theta } \nonumber \\{} & {} -\alpha {{e}^{-\alpha t}}\int _{-\bar{\sigma }}^{0}{\int _{t+\theta }^{t}{{{e}^{\alpha s}}Tr\left\{ \bar{\Sigma }_{x}^{T}\left( s \right) {{Z}_{1}}{{{\bar{\Sigma }}}_{x}}\left( s \right) \right\} ds}}d\theta \nonumber \\{} & {} -\alpha {{e}^{-\alpha t}}\int _{-\bar{\tau }}^{0}{\int _{t+\theta }^{t}{{{e}^{\alpha s}}Tr\left\{ \bar{\Sigma }_{y}^{T}\left( s \right) {{Z}_{2}}{{{\bar{\Sigma }}}_{y}}\left( s \right) \right\} ds}}d\theta \nonumber \\{} & {} +\bar{\sigma }Tr\left\{ \bar{\Sigma }_{x}^{T}\left( t \right) {{\varepsilon }_{{{Z}_{1}}}}P{{{\bar{\Sigma }}}_{x}}\left( t \right) \right\} \nonumber \\{} & {} +\bar{\tau }Tr\left\{ \bar{\Sigma }_{y}^{T}\left( t \right) {{\varepsilon }_{{{Z}_{2}}}}Q{{{\bar{\Sigma }}}_{x}}\left( t \right) \right\} \end{aligned}$$

(36)

By defining ${{\xi }^{T}}\left( t \right) =\left[ \begin{matrix} {{{\bar{x}}}^{T}}\left( t \right) &{} \quad {{{\bar{x}}}^{T}}\left( t-\sigma \left( t \right) \right) \\ \end{matrix} \right]$ and ${{\eta }^{T}}\left( t \right) =\left[ \begin{matrix} {{{\bar{y}}}^{T}}\left( t \right) &{} \quad {{{\bar{y}}}^{T}}\left( t-\tau \left( t \right) \right) \\ \end{matrix} \right]$, and using Ito’s isometry [64] and the Jensen inequality [65], the following inequalities can be obtained:

$$\begin{aligned}{} & {} 2{{\xi }^{T}}\left( t \right) {\tilde{N}}\left( {\bar{x}}\left( t \right) -{\bar{x}}\left( t-\sigma \left( t \right) \right) \right) \le \nonumber \\{} & {} {{\xi }^{T}}\left( t \right) {\tilde{N}}{{\left( \bar{\sigma }{{\left( t \right) }^{-1}}{{Z}_{1}} \right) }^{-1}}{\tilde{N}}{}^{T}\xi \left( t \right) + {{\xi }^{T}}\left( t \right) {\tilde{N}}{{Z}_{1}}^{-1}{\tilde{N}}{}^{T}\xi \left( t \right) \nonumber \\{} & {} +{{\left( \int _{t-\sigma \left( t \right) }^{t}{{{{{\bar{r}}}}_{x}}\left( s \right) ds} \right) }^{T}}{{\left( \bar{\sigma }\left( t \right) \right) }^{-1}}{{Z}_{1}}\left( \int _{t-\sigma \left( t \right) }^{t}{{{{{\bar{r}}}}_{x}}\left( s \right) ds} \right) \nonumber \\{} & {} +{{\left( \int _{t-\sigma \left( t \right) }^{t}{{{{\bar{\Sigma }}}_{x}}\left( s \right) d{{\omega }_{x}}} \right) }^{T}}{{Z}_{1}}\left( \int _{t-\sigma \left( t \right) }^{t}{{{{\bar{\Sigma }}}_{x}}\left( s \right) d{{\omega }_{x}}} \right) \end{aligned}$$

(37)

$$\begin{aligned}{} & {} 2{{\xi }^{T}}\left( t \right) {\tilde{N}}\left( {\bar{x}}\left( t \right) -{\bar{x}}\left( t-\sigma \left( t \right) \right) \right) \le \nonumber \\{} & {} \bar{\sigma }\left( t \right) {{\xi }^{T}}\left( t \right) {\tilde{N}}{{Z}_{1}}^{-1}{\tilde{N}}{}^{T}\xi \left( t \right) + {{\xi }^{T}}\left( t \right) {\tilde{N}}{{Z}_{1}}^{-1}{\tilde{N}}{}^{T}\xi \left( t \right) \nonumber \\{} & {} \quad +\int _{t-\sigma \left( t \right) }^{t}{{{{{\bar{r}}}}_{x}}^{T}\left( s \right) {{Z}_{1}}{{{{\bar{r}}}}_{x}}\left( s \right) ds} \nonumber \\{} & {} \quad +\int _{t-\sigma \left( t \right) }^{t}{Tr\left\{ \bar{\Sigma }_{x}^{T}\left( s \right) {{Z}_{1}}{{{\bar{\Sigma }}}_{x}}\left( s \right) \right\} ds} \end{aligned}$$

(38)

and

$$\begin{aligned}{} & {} 2{{\eta }^{T}}\left( t \right) {\tilde{M}}\left( {\bar{y}}\left( t \right) -{\bar{y}}\left( t-\tau \left( t \right) \right) \right) \le \nonumber \\{} & {} {{\eta }^{T}}\left( t \right) {\tilde{M}}{{\left( \bar{\tau }{{\left( t \right) }^{-1}}{{Z}_{2}} \right) }^{-1}}{\tilde{M}}{}^{T}\eta \left( t \right) + {{\eta }^{T}}\left( t \right) {\tilde{M}}{{Z}_{2}}^{-1}{\tilde{M}}{}^{T}\eta \left( t \right) \nonumber \\{} & {} +{{\left( \int _{t-\tau \left( t \right) }^{t}{{{{{\bar{r}}}}_{y}}\left( s \right) ds} \right) }^{T}}{{\left( \bar{\tau }\left( t \right) \right) }^{-1}}{{Z}_{2}}\left( \int _{t-\tau \left( t \right) }^{t}{{{{{\bar{r}}}}_{y}}\left( s \right) ds} \right) \nonumber \\{} & {} +{{\left( \int _{t-\tau \left( t \right) }^{t}{{{{\bar{\Sigma }}}_{y}}\left( s \right) d{{\omega }_{y}}} \right) }^{T}}{{Z}_{2}}\left( \int _{t-\tau \left( t \right) }^{t}{{{{\bar{\Sigma }}}_{y}}\left( s \right) d{{\omega }_{y}}} \right) \end{aligned}$$

(39)

$$\begin{aligned} \begin{aligned}&2{{\eta }^{T}}\left( t \right) {\tilde{M}}\left( {\bar{y}}\left( t \right) -{\bar{y}}\left( t-\tau \left( t \right) \right) \right) \le \\&\bar{\sigma }\left( t \right) {{\eta }^{T}}\left( t \right) {\tilde{M}}{{Z}_{2}}^{-1}{\tilde{M}}{}^{T}\eta \left( t \right) + {{\eta }^{T}}\left( t \right) {\tilde{M}}{{Z}_{2}}^{-1}{\tilde{M}}{}^{T}\eta \left( t \right) \\&+\int _{t-\tau \left( t \right) }^{t}{{{{{\bar{r}}}}_{y}}^{T}\left( s \right) {{Z}_{2}}{{{{\bar{r}}}}_{y}}\left( s \right) ds} \\&+\int _{t-\tau \left( t \right) }^{t}{Tr\left\{ \bar{\Sigma }_{y}^{T}\left( s \right) {{Z}_{2}}{{{\bar{\Sigma }}}_{y}}\left( s \right) \right\} ds} \\ \end{aligned} \end{aligned}$$

(40)

Now, we can write:

$$\begin{aligned} \begin{aligned}&\mathcal {L}V\left( t \right) +\alpha V\left( t \right) \le \\&{{{{\bar{x}}}}^{T}}\left( t \right) P {\bar{r}}_x \left( t \right) + {\bar{r}}^T_x\left( t \right) P {\bar{x}}\left( t \right) +{{{{\bar{y}}}}^{T}}\left( t \right) Q {\bar{r}}_y \left( t \right) \\&+ {\bar{r}}^T_y\left( t \right) Q {\bar{y}}\left( t \right) +{{{{\bar{x}}}}^{T}}\left( t \right) {{R}_{1}}{\bar{x}}\left( t \right) +{{{{\bar{y}}}}^{T}}\left( t \right) {{R}_{2}}{\bar{y}}\left( t \right) \\&-2{{{{\bar{f}}}}^{T}}\left( {\bar{y}}\left( t-\tau \left( t \right) \right) \right) \bar{\Lambda }\\&\times \left( {\bar{f}}\left( {\bar{y}}\left( t-\tau \left( t \right) \right) \right) -{{{{\bar{K}}}}_{f}}{\bar{y}}\left( t-\tau \left( t \right) \right) \right) \\&+{{{{\bar{x}}}}^{T}}\left( t \right) {{R}_{1}}{\bar{x}}\left( t \right) +{{{{\bar{y}}}}^{T}}\left( t \right) {{R}_{2}}{\bar{y}}\left( t \right) \\&-{{e}^{-\alpha \bar{\sigma }}}\left( 1-{{{\bar{\sigma }}}_{d}} \right) {{{{\bar{x}}}}^{T}}\left( t-\sigma \left( t \right) \right) {{R}_{1}}{\bar{x}}\left( t-\sigma \left( t \right) \right) \\&-{{e}^{-\alpha \bar{\tau }}}\left( 1-{{{\bar{\tau }}}_{d}} \right) {{{{\bar{y}}}}^{T}}\left( t-\tau \left( t \right) \right) {{R}_{2}}{\bar{y}}\left( t-\tau \left( t \right) \right) \\&+\bar{\sigma }{{{{\bar{r}}}}_{x}}^{T}\left( t \right) {{Z}_{1}}{{{{\bar{r}}}}_{x}}\left( t \right) +\bar{\tau }{{{{\bar{r}}}}^{T}}_{y}\left( t \right) {{Z}_{2}}{{{{\bar{r}}}}_{y}}\left( t \right) \\ {}&+{{e}^{-\alpha \bar{\sigma }}}\left\{ \bar{\sigma }\left( t \right) {{\xi }^{T}}\left( t \right) {\tilde{N}}{{Z}_{1}}^{-1}{\tilde{N}}{}^{T}\xi \left( t \right) \right\} \\&+{{e}^{-\alpha \bar{\sigma }}}\left\{ +{{\xi }^{T}}\left( t \right) {\tilde{N}}{{Z}_{1}}^{-1}{\tilde{N}}{}^{T}\xi \left( t \right) \right\} \\&+{{e}^{-\alpha \bar{\sigma }}}\left\{ -2{{\xi }^{T}}\left( t \right) {\tilde{N}}\left( {\bar{x}}\left( t \right) -{\bar{x}}\left( t-\sigma \left( t \right) \right) \right) \right\} \\&+{{e}^{-\alpha \bar{\tau }}}\left\{ \bar{\tau }\left( t \right) {{\eta }^{T}}\left( t \right) {\tilde{M}}{{Z}_{2}}^{-1}{\tilde{M}}{}^{T}\eta \left( t \right) \right\} \\&+{{e}^{-\alpha \bar{\tau }}}\left\{ {{\eta }^{T}}\left( t \right) {\tilde{M}}{{Z}_{2}}^{-1}{\tilde{M}}{}^{T}\eta \left( t \right) \right\} \\ {}&-{{e}^{-\alpha \bar{\tau }}}\left\{ 2{{\eta }^{T}}\left( t \right) {\tilde{M}}\left( {\bar{y}}\left( t \right) -{\bar{y}}\left( t-\tau \left( t \right) \right) \right) \right\} \\&+\frac{1}{2}Tr\left\{ \bar{\Sigma }_{x}^{T}\left( t \right) \left( P+{{P}^{T}} \right) {{{\bar{\Sigma }}}_{x}}\left( t \right) \right\} \\&+\frac{1}{2}Tr\left\{ \bar{\Sigma }_{y}^{T}\left( t \right) \left( Q+{{Q}^{T}} \right) {{{\bar{\Sigma }}}_{y}}\left( t \right) \right\} \\&+\bar{\sigma }Tr\left\{ \bar{\Sigma }_{x}^{T}\left( t \right) {{Z}_{1}}{{{\bar{\Sigma }}}_{x}}\left( t \right) \right\} \\&+\bar{\tau }Tr\left\{ \bar{\Sigma }_{y}^{T}\left( t \right) {{Z}_{2}}{{{\bar{\Sigma }}}_{y}}\left( t \right) \right\} \\ \end{aligned} \end{aligned}$$

(41)

By considering $\begin{matrix} {{{{\bar{y}}}}^{T}}\left( t-\tau \left( t \right) \right) &{} \quad {{{{\bar{f}}}}^{T}}\left( {{{{\bar{y}}}}^{T}}\left( t-\tau \left( t \right) \right) \right) \\ \end{matrix} ]^{T},$ we have:

$$\begin{aligned} \begin{aligned}&\mathcal {L}V\left( t \right) +\alpha V\left( t \right) \le \\&{{\chi }^{T}}\left( t \right) \left[ \begin{matrix} {{{\tilde{\Omega }}}_{11}} &{} \quad 0 &{} \quad {{{\tilde{\Omega }}}_{13}} &{} \quad 0 &{} \quad {{{\tilde{\Omega }}}_{15}} \\ * &{} \quad {{{\tilde{\Omega }}}_{22}} &{} \quad {{{\tilde{\Omega }}}_{23}} &{} \quad {{{\tilde{\Omega }}}_{24}} &{} \quad 0 \\ * &{} \quad * &{} \quad {{{\tilde{\Omega }}}_{33}} &{} \quad 0 &{} \quad 0 \\ * &{} \quad * &{} \quad * &{} \quad {{{\tilde{\Omega }}}_{44}} &{} \quad {{{\tilde{\Omega }}}_{45}} \\ * &{} \quad * &{} \quad * &{} \quad * &{} \quad {{{\tilde{\Omega }}}_{55}} \\ \end{matrix} \right] \chi \left( t \right) + \\&+\bar{\sigma }{{\chi }^{T}}\left( t \right) {{{{\bar{R}}}}_{x}}{{Z}_{1}}{\bar{R}}_{x}^{T}\chi \left( t \right) +\bar{\tau }{{\chi }^{T}}\left( t \right) {{{{\bar{R}}}}_{y}}{{Z}_{2}}{{{{\bar{R}}}}^{T}}_{y}\chi \left( t \right) \\&+{{e}^{-\alpha \bar{\sigma }}}\left\{ \left( \bar{\sigma }+1 \right) {{\chi }^{T}}\left( t \right) {{{{\tilde{N}}}}_{e}}{{Z}_{1}}^{-1}{\tilde{N}}_{e}^{T}\chi \left( t \right) \right\} \\&+{{e}^{-\alpha \bar{\tau }}}\left\{ \left( \bar{\tau }+1 \right) {{\chi }^{T}}\left( t \right) {{{{\tilde{M}}}}_{e}}{{Z}_{2}}^{-1}{\tilde{M}}_{e}^{T}\chi \left( t \right) \right\} \\ \end{aligned} \end{aligned}$$

(42)

where

$$\begin{aligned} \begin{aligned}&{{{\tilde{\Omega }}}_{11}}=-P{\bar{A}}-{{{{\bar{A}}}}^{T}}P+\alpha P+P{{{{\bar{B}}}}_{x}}{{{{\bar{K}}}}_{x}}+{{{{\bar{K}}}}_{x}}{{{{\bar{B}}}}^{T}}_{x}P+{{R}_{1}}\\&-{{e}^{-\alpha \bar{\sigma }}}\left( {{{{\tilde{N}}}}_{1}}+{{{{\tilde{N}}}}^{T}}_{1} \right) ,{{{\tilde{\Omega }}}_{13}}={{e}^{-\alpha \bar{\sigma }}}\left( {{{{\tilde{N}}}}_{1}}-{{{{\tilde{N}}}}^{T}}_{2} \right) ,{{{\tilde{\Omega }}}_{15}}=P{\bar{B}},\\ {}&{{{\tilde{\Omega }}}_{22}}=-Q{\bar{C}}-{{{{\bar{C}}}}^{T}}Q+Q{{{{\bar{B}}}}_{y}}{{{{\bar{K}}}}_{y}}+{\bar{K}}_{y}^{T}{{{{\bar{B}}}}^{T}}{{Q}_{y}}+\alpha Q+{{R}_{2}}\\&-{{e}^{-\alpha \bar{\tau }}}\left( {{{{\tilde{M}}}}_{1}}+{{{{\tilde{M}}}}^{T}}_{1} \right) , {{{\tilde{\Omega }}}_{23}}=Q{\bar{D}},{{{\tilde{\Omega }}}_{24}}={{e}^{-\alpha \bar{\tau }}}\left( {{{{\tilde{M}}}}_{1}}-M_{2}^{T} \right) ,\\&{{{\tilde{\Omega }}}_{33}}=-{{e}^{-\alpha \bar{\sigma }}}\left( \left( 1-{{\sigma }_{d}} \right) {{R}_{1}}+{{{{\tilde{N}}}}_{2}}+{{{{\tilde{N}}}}^{T}}_{2} \right) ,\\&{{{\tilde{\Omega }}}_{44}}=-{{e}^{-\alpha \bar{\tau }}}\left( \left( 1-{{\tau }_{d}} \right) {{R}_{2}}+{{{{\tilde{M}}}}_{2}}+{{{{\tilde{M}}}}^{T}}_{2} \right) , {{{\tilde{\Omega }}}_{45}}={\bar{K}}_{f}^{T}\bar{\Lambda },\\&{{{\tilde{\Omega }}}_{55}}=-2\bar{\Lambda }, {\bar{R}}_{x}^{T}=\left[ \begin{matrix} -{\bar{A}}+{{{{\bar{B}}}}_{x}}{{{{\bar{K}}}}_{x}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad {{\bar{B}}} \\ \end{matrix} \right] ,\\&\quad {\bar{R}}_{y}^{T}=\left[ \begin{matrix} 0 &{} \quad -{\bar{C}}+{{{{\bar{B}}}}_{y}}{{{{\bar{K}}}}_{y}} &{} \quad {{\bar{D}}} &{} \quad 0 &{} \quad 0 \\ \end{matrix} \right] ,\\&{{{{\tilde{N}}}}^{T}}_{e}=\left[ \begin{matrix} {\tilde{N}}_{1}^{T} &{} 0 &{} {\tilde{N}}_{2}^{T} &{} \quad 0 &{} \quad 0 \\ \end{matrix} \right] , \\&{{{{\tilde{M}}}}^{T}}_{e}=\left[ \begin{matrix} {\tilde{M}}_{1}^{T} &{} 0 &{} {\tilde{M}}_{2}^{T} &{} \quad 0 &{} \quad 0 \\ \end{matrix} \right] \\ \end{aligned} \end{aligned}$$

By using Schur’s lemma, pre and post multiplying with

$$\begin{aligned} \begin{aligned} diag\left\{ {{P}^{-1}},{{Q}^{-1}},{{P}^{-1}},{{Q}^{-1}},{{\Lambda }^{-1}},Z_{1}^{-1},Z_{1}^{-1},Z_{2}^{-1},Z_{2}^{-1} \right\} \end{aligned} \end{aligned}$$

and defining $U={{P}^{-1}}$, $V={{Q}^{-1}}$, $X={{{\bar{K}}}_{x}}{{P}^{-1}}$, $Y={{{\bar{K}}}_{y}}{{Q}^{-1}}$, $\Gamma ={{\Lambda }^{-1}}$, ${{R}_{1}}={{\varepsilon }_{{{R}_{1}}}}P$, ${{R}_{2}}={{\varepsilon }_{{{R}_{2}}}}Q$, ${{N}_{1}}={{P}^{-1}}{{{\tilde{N}}}_{1}}{{P}^{-1}}$, ${{N}_{2}}={{P}^{-1}}{{{\tilde{N}}}_{2}}{{P}^{-1}}$, ${{M}_{1}}={{Q}^{-1}}{{{\tilde{M}}}_{1}}{{Q}^{-1}}$, ${{M}_{2}}={{Q}^{-1}}{{{\tilde{M}}}_{2}}{{Q}^{-1}}$, ${{Z}_{1}}={{\varepsilon }_{{{Z}_{1}}}}P$ and ${{Z}_{2}}={{\varepsilon }_{{{Z}_{2}}}}Q$, the LMIs presented in (13) can be deduced.