A novel robust $H_{\infty }$ fuzzy state feedback plus state-derivative feedback controller design for nonlinear time-varying delay systems

Abstract

This paper investigates the problem of designing a robust $H_{\infty }$ state feedback plus state-derivative feedback control mechanism for a class of uncertain nonlinear time-varying delay systems described by a Takagi–Sugeno fuzzy model. A linear matrix inequality approach is applied to derive a robust controller for such a system. The proposed controller satisfies design requirements that ensure that the closed-loop system is asymptotically stable and meets pre-prescribed $H_{\infty }$ performance index values. Finally, the illustrative examples are simulated to illustrate the effectiveness of the proposed methodology.

Appendices

Appendix 1: Proof of Theorem 1

Proof

Write the quadratic Lyapunov–Krasovskii functional V(x(t)) as

$$\begin{aligned} V\left( {x(t)} \right) = {x^{\mathrm{T}}}(t)Qx(t) + \beta \int ^{t}_{t-\tau (t)}{x^{\mathrm{T}}(V)Sx(V)}\mathrm{d}V, \end{aligned}$$

(35)

where $Q = {P^{- 1}}>0, S=W^{-1}>0$ and $\beta =\frac{1}{1-\tau _d}$. Differentiate $V\left( {x(t)} \right)$ along the closed-loop system (14); using the fact that for any vector x(t) and $x(t-\tau (t))$ and matrix G,

$$\begin{aligned}&x^{\mathrm{T}}(t)Gx(t-\tau (t))+{x^{\mathrm{T}}(t-\tau (t))G^Tx(t)}\nonumber \\ &\quad \le \,x^{\mathrm{T}}(t)GR^{-1}G^Tx(t)+x^{\mathrm{T}}(t-\tau (t))Rx(t-\tau (t)), \end{aligned}$$

(36)

where R is a positive definite matrix, we have

$$\begin{aligned} \dot{V}\left( {x(t)} \right)& = {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \Big [ {{{x}^{\mathrm{T}}}(t)} \Big (({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q\nonumber \\&\quad +\,Q{E_{ij}} ({A_i}+B_iK_{s_j})+ {\beta }S\Big )x(t)+{x}^{\mathrm{T}}\big (t-{\tau }(t)\big ){A^{\mathrm{T}}_{d_{i}}}{E^{\mathrm{T}}_{ij}}Qx(t)\nonumber \\&\quad +\,x^{\mathrm{T}}(t)QE_{ij}A_{{{d_{i}}}}x\big (t-{\tau }(t)\big )-\beta \big (1-{\dot{\tau }{(t)}}\big )x^{\mathrm{T}}\big (t-\tau (t)\big )Sx\big (t-{\tau }(t)\big )\nonumber \\&\quad +\,w^{\mathrm{T}}(t)B^{\mathrm{T}}_wE^{\mathrm{T}}_{ij}Qx(t)+x^{\mathrm{T}}(t)QE_{ij}B_ww(t)\Big ]\nonumber \\& \le {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \Big [{{{x}^{\mathrm{T}}}(t)} \Big (({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q\nonumber \\&\quad +\,Q{E_{ij}} ({A_i}+B_iK_{s_j})+ {\beta }S\Big )x(t)+ x^{\mathrm{T}}(t)QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Qx(t)\nonumber \\&\quad +\,x^{\mathrm{T}}\big (t-\tau (t)\big )Sx\big (t-\tau (t)\big )- x^{\mathrm{T}}\big (t-\tau (t)\big )Sx\big (t-\tau (t)\big )\nonumber \\&\quad +\,w^{\mathrm{T}}(t)B^{\mathrm{T}}_wE^{\mathrm{T}}_{ij}Qx(t)+x^{\mathrm{T}}(t)QE_{ij}B_ww(t)\Big ]\nonumber \\& = {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \Big [{{{x}^{\mathrm{T}}}(t)} \Big (({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q+Q{E_{ij}} ({A_i}+B_iK_{s_j})\nonumber \\&\quad +\,QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q+ {\beta }S\Big )x(t)+w^{\mathrm{T}}(t)B^{\mathrm{T}}_wE^{\mathrm{T}}_{ij}Qx(t) \nonumber \\&\quad +\,x^{\mathrm{T}}(t)QE_{ij}B_ww(t)\Big ]. \end{aligned}$$

(37)

By adding and subtracting the equation

$$\begin{aligned} - {z^{\mathrm{T}}}(t)z(t) + {\gamma ^2}\sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}}\big [{w^{\mathrm{T}}}(t)w(t)\big ] \end{aligned}$$

(38)

to and from (37), we acquire

$$\begin{aligned} \dot{V} \left( {x(t)} \right)& = {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \left[ {\begin{array}{cc} {{{x}^{\mathrm{T}}}(t)}&{{{w}^{\mathrm{T}}}(t)} \end{array}} \right] \nonumber \\&\quad \times \,\left( {\begin{array}{cc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q +Q{E_{ij}} ({A_i}+B_iK_{s_j})\\ +QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q + {\beta }S+C^{\mathrm{T}}_iC_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ B^{\mathrm{T}}_wE^{\mathrm{T}}_{ij}Q&{}{- {\gamma ^2}I} \end{array}} \right) \left[ {\begin{array}{c} {x(t)}\\ {w(t)} \end{array}} \right] \nonumber \\&\quad -\,{z^{\mathrm{T}}}(t)z(t) + {\gamma ^2}\sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}}[{w^{\mathrm{T}}}(t)w(t)], \end{aligned}$$

(39)

where

$$\begin{aligned} z(t) = \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} C_ix(t). \end{aligned}$$

(40)

Next, let us consider Theorem 1. By applying the Schur complement, we have

$$\begin{aligned}&\left( {\begin{array}{cccc} \Phi _{ij} &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {P+Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i}&{}0&{}{-\, \frac{1}{\beta }W}&{}{{{(*)}^{\mathrm{T}}}}\\ {C_iP}+C_iY^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i&{}0&{}0&{}{-\, I} \end{array}} \right) \nonumber \\&\quad +\,\left( {\begin{array}{cccc} \left( {\begin{array}{c} B_i(Y_{s_j}+Y_{d_j})P^{-1}\\ (Y_{s_j}+Y_{d_j})^TB^{\mathrm{T}}_i \end{array}} \right) &{}0 &{}0 &{}0\\ 0&{}0 &{}0 &{}0\\ 0&{}0 &{}0 &{}0\\ 0&{}0 &{}0 &{}0 \end{array}} \right) < 0. \end{aligned}$$

(41)

By applying the algebraic inequality

$$\begin{aligned} aX{b} + b^TX{a^{\mathrm{T}}} \le (a + b) X (a + b)^{\mathrm{T}}, \end{aligned}$$

(42)

(41) then yields

$$\begin{aligned}&\left( {\begin{array}{cccc} \Phi _{ij} &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {P+Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i}&{}0&{}{-\, \frac{1}{\beta }W}&{}{{{(*)}^{\mathrm{T}}}}\\ {C_iP}+C_iY^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i&{}0&{}0&{}{-\, I} \end{array}} \right) \nonumber \\&\quad +\,\left( {\begin{array}{cccc} \left( {\begin{array}{c} B_iY_{s_j}P^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i\\ +B_iY_{d_j}P^{-1}Y_{s_j}^TB^{\mathrm{T}}_i \end{array}} \right) &{}0 &{}0 &{}0\\ 0&{}0 &{}0 &{}0\\ 0&{}0 &{}0 &{}0\\ 0&{}0 &{}0 &{}0 \end{array}} \right) < \,0, \end{aligned}$$

(43)

and by substituting $\Phi _{ij}$ shown in Theorem 1 of (43), we have

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} {P}{A^{\mathrm{T}}_i} + {A_i}{P} + Y_{s_j}^TB_i^{\mathrm{T}}+ B_iY_{s_j} + B_i{Y_{d_j}}A^{\mathrm{T}}_i \\ + A_i{Y_{d_j}^{\mathrm{T}}}{B_i^{\mathrm{T}}} + {A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}\\ +B_iY_{s_j}P^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i +B_iY_{d_j}P^{-1}Y_{s_j}^TB^{\mathrm{T}}_i \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {P+Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i}&{}0&{}{-\, \frac{1}{\beta }W}&{}{{{(*)}^{\mathrm{T}}}}\\ {C_iP}+C_iY^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i&{}0&{}0&{}{-\, I} \end{array}} \right) < 0 \end{aligned}$$

(44)

or

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} {P} (A_i+B_iY_{s_j}P^{-1})^{\mathrm{T}} + (A_i + B_iY_{s_j}P^{-1})P\\ +B_iY_{d_j}P^{-1}P(A_i+B_iY_{s_j}P^{-1})^{\mathrm{T}}\\ + (A_i+B_iY_{s_j}P^{-1})PP^{-1}Y^{\mathrm{T}}_{d_j}B_i^{\mathrm{T}}+ A_{d_i}WA^{\mathrm{T}}_{d_i} \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {P+PP^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i}&{}0&{}{-\, \frac{1}{\beta }W}&{}{{{(*)}^{\mathrm{T}}}}\\ {C_iP}+C_iPP^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i&{}0&{}0&{}{-\, I} \end{array}} \right) <0, \end{aligned}$$

(45)

with $K_{d_j}=Y_{d_j}P^{-1}$ and $K_{s_j}=Y_{s_j}P^{-1}.$ Then, (45) yields

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} {(I+B_iK_{d_j})}P{(A_i+B_iK{s_j})^{\mathrm{T}}} \\ + {(A_i+B_iK{s_j})}{P}{(I+B_iK_{d_j})^{\mathrm{T}}}\\ + {A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}} \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}\\ {P{(I+B_iK_{d_j})}^{\mathrm{T}}}&{}0&{}{-\, \frac{1}{\beta }W}&{}{{{(*)}^{\mathrm{T}}}}\\ {C_iP{(I+B_iK_{d_j})}^{\mathrm{T}}}&{}0&{}0&{}{-\, I} \end{array}} \right) < 0. \end{aligned}$$

(46)

Pre- and post-multiply both sides of (46) by $\left( {\begin{array}{ccc} {{(I+B_iK_{d_j})}^{-1}}&{}0&{}0\\ 0&{}I&{}0\\ 0&{}0&{}I \end{array}} \right)$ and $\left( {\begin{array}{ccc} {{(I+B_iK_{d_j})}^{-T}}&{}0&{}0\\ 0&{}I&{}0\\ 0&{}0&{}I \end{array}} \right),$ respectively, and we have

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} P{{({A_i}+B_iK_{s_j})^{\mathrm{T}}}}{{({I}+B_iK_{d_j})^{-T}}} \\ + {{({I}+B_iK_{d_j})^{-1}}}{{({A_i}+B_iK_{s_j})}}P\\ +{{({I}+B_iK_{d_j})^{-1}}}{A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}{{({I}+B_iK_{d_j})^{-T}}} \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}{{(I+B_iK_{d_j})}^{-T}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {P}&{}0&{}{-\, \frac{1}{\beta }W}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {C_iP}&{}0&{}0&{}{-\, I} \end{array}} \right) < 0 \end{aligned}$$

(47)

or, in a more compact form,

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} P{{({A_i}+B_iK_{s_j})^{\mathrm{T}}}}{E^{T}_{ij}}+ {E_{ij}}{{({A_i}+B_iK_{s_j})}}{P}\\ +{E_{ij}}{A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}{E^{T}_{ij}} \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}{E^{T}_{ij}}&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {P}&{}0&{}{-\, \frac{1}{\beta }W}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {C_iP}&{}0&{}0&{}{-\, I} \end{array}} \right) < 0. \end{aligned}$$

(48)

By multiplying both sides of (48) by $\left( {\begin{array}{cccc} Q&{} 0&{} 0&{} 0\\ 0&{} I&{} 0&{} 0\\ 0&{} 0&{} I&{} 0\\ 0&{} 0&{} 0&{} I \end{array}} \right),$ we obtain

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} {{({A_i}+B_iK_{s_i})^{\mathrm{T}}}}{E^{T}_{ii}}Q+ Q{E_{ii}}{{({A_i}+B_iK_{s_i})}}\\ + Q{E_{ii}}{A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}{E^{T}_{ii}}Q \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}{E^{T}_{ii}}Q&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {I}&{}0&{}{-\, \frac{1}{\beta }W}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {C_i}&{}0&{}0&{}{-\, I} \end{array}} \right) < 0, \end{aligned}$$

(49)

$i=1, 2, 3\ldots , r\,$, and

$$\begin{aligned}&\left( {\begin{array}{cccc} \left( {\begin{array}{c} {{({A_i}+B_iK_{s_j})^{\mathrm{T}}}}{E^{T}_{ij}}Q+ Q{E_{ij}}{{({A_i}+B_iK_{s_j})}}\\ + Q{E_{ij}}{A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}{E^{T}_{ij}}Q \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}{E^{T}_{ii}}Q&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {I}&{}0&{}{-\, \frac{1}{\beta }W}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {C_i}&{}0&{}0&{}{-\, I} \end{array}} \right) \nonumber \\&\quad +\,\left( {\begin{array}{cccc} \left( {\begin{array}{c} {{({A_j}+B_jK_{s_i})^{\mathrm{T}}}}{E^{T}_{ji}}Q+ Q{E_{ji}}{{({A_j}+B_jK_{s_i})}}\\ + Q{E_{ji}}{A_{{d}_j}}W{A^{\mathrm{T}}_{{d}_j}}{E^{T}_{ji}}Q \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {{B^{\mathrm{T}}_w}}{E^{T}_{ji}}Q&{}{-\, \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {I}&{}0&{}{-\, \frac{1}{\beta }W}&{} \,{{{(*)}^{\mathrm{T}}}}\\ {C_i}&{}0&{}0&{}{-\, I} \end{array}} \right) < 0, \end{aligned}$$

(50)

$i<j \le r$. Apply the Schur complement to (49)–(50), and rearrange them; we then have

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_i})^{\mathrm{T}}{E^{\mathrm{T}}_{ii}}Q +Q{E_{ii}} ({A_i}+B_iK_{s_i})\\ + QE_{ii}A_{d_i}WA^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ii}Q+ {\beta }W^{-1}+C^{\mathrm{T}}_iC_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ B^{\mathrm{T}}_wE^{\mathrm{T}}_{ii}Q&{}{-\, {\gamma ^2}I} \end{array}} \right) < 0, \end{aligned}$$

(51)

$i=1, 2, 3\ldots , r$, and

$$\begin{aligned}&\left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q +Q{E_{ij}} ({A_i}+B_iK_{s_j})\\ + QE_{ij}A_{d_i}WA^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q + {\beta }W^{-1}+C^{\mathrm{T}}_iC_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ B^{\mathrm{T}}_wE^{\mathrm{T}}_{ij}Q&{}{- {\gamma ^2}I} \end{array}} \right) \nonumber \\&\quad +\,\left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{j}}+B_jK_{s_i})^{\mathrm{T}}{E^{\mathrm{T}}_{ji}}Q +Q{E_{ji}} ({A_j}+B_jK_{s_i})\\ + QE_{ji}A_{d_j}WA^{\mathrm{T}}_{d_j}E^{\mathrm{T}}_{ji}Q + {\beta }W^{-1}+C^{\mathrm{T}}_jC_j \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ B^{\mathrm{T}}_wE^{\mathrm{T}}_{ji}Q&{}{- {\gamma ^2}I} \end{array}} \right) < 0, \end{aligned}$$

(52)

$i<j \le r$. From (51)–(52), it is clear that

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q +Q{E_{ij}} ({A_i}+B_iK_{s_j})\\ + QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q + {\beta }S+C^{\mathrm{T}}_iC_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ B^{\mathrm{T}}_wE^{\mathrm{T}}_{ij}Q&{}{- {\gamma ^2}I} \end{array}} \right) < 0, \end{aligned}$$

(53)

where $W={S^{-1}}$ and $i, j = 1, 2,\ldots , r$. As (53) is less than zero, $\mu _n\ge 0$ and $\sum \nolimits _{n = 1}^r {\mu _n} = 1$, (39) becomes

$$\begin{aligned} \dot{V} \left( {x(t)} \right) \le - {z^{\mathrm{T}}}(t)z(t) + {\gamma ^2}{w^{\mathrm{T}}}(t)w(t). \end{aligned}$$

(54)

Integrating both sides of (54) yields

$$\begin{aligned} \int _0^{{T_{\mathrm{f}}}} {\dot{V}\left( {x(t)} \right) } \,\mathrm{d}t \le \int _0^{{T_{\mathrm{f}}}} {\Big [ {- {{z}^{\mathrm{T}}}}(t)z(t)\, + {\gamma ^2}{{w}^{\mathrm{T}}}(t){w(t)} \Big ]\mathrm{d}t}, \end{aligned}$$

(55)

$$\begin{aligned} V\left( {x({T_{\mathrm{f}}})} \right) - V\left( {x(0)} \right) \, \le \int _0^{{T_{\mathrm{f}}}} {\Big [ {- {{z}^{\mathrm{T}}}}(t)} z(t) + {\gamma ^2}{{w}^{\mathrm{T}}}(t) {w(t)} \Big ]\mathrm{d}t. \end{aligned}$$

(56)

As $V(x(0)) = 0$ and $V\left( {x({T_{\mathrm{f}}})} \right) \ge 0$ for all, ${T_{\mathrm{f}}} \ne 0,$ we have

$$\begin{aligned} \int _0^{{T_{\mathrm{f}}}} {{{z}^{\mathrm{T}}}(t)} z(t)\,\mathrm{d}t\, \le {\gamma ^2}\left[ {\int _0^{{T_{\mathrm{f}}}} {{{w}^{\mathrm{T}}}(t)w(t)}} \mathrm{d}t\right] . \end{aligned}$$

(57)

Hence, the inequality (8) holds. This is the case when ${{w}}(t) = 0$, and (54) becomes $\dot{V}(t) \le - {z^{\mathrm{T}}}(t)z(t) \le 0$. Therefore, the system (14) is asymptotically stable, and (b) is achieved. This completes the proof. $\square$

Appendix 2: Proof of Theorem 2

Proof

Choose the quadratic Lyapunov–Krasovskii functional V(x(t)) as

$$\begin{aligned} V\left( {x(t)} \right) = {x^{\mathrm{T}}}(t)Qx(t) + \beta \int _{t-\tau (t)}^{t}x^{\mathrm{T}}(V)Sx(V)\mathrm{d}V, \end{aligned}$$

(58)

where $Q = {P^{- 1}}>0, S=W^{-1}>0$ and $\beta =\frac{1}{1-\tau _d}$. Differentiate $V\left( {x(t)} \right)$ along the closed-loop system (24); from (36), we have

$$\begin{aligned} \dot{V}\left( {x(t)} \right)& = {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \Big [ {{{x}^{\mathrm{T}}}(t)} \Big (({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q\nonumber \\&\quad +\,Q{E_{ij}} ({A_i}+B_iK_{s_j})+ {\beta }S\Big )x(t)+{x}^{\mathrm{T}}\big (t-{\tau }(t)\big ){A^{\mathrm{T}}_{d_{i}}}{E^{\mathrm{T}}_{ij}}Qx(t)\nonumber \\&\quad +\,x^{\mathrm{T}}(t)QE_{ij}A_{{{d_{i}}}}x\big (t-{\tau }(t)\big )- \beta \big (1-{\dot{\tau }{(t)}}\big )x^{\mathrm{T}}\big (t-\tau (t)\big )Sx\big (t-{\tau }(t)\big )\nonumber \\&\quad +\,\tilde{w}^{\mathrm{T}}(t)\tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ij}Qx(t)+x^{\mathrm{T}}(t)QE_{ij}\tilde{B}_{w_i}\tilde{w}(t)\Big ]\nonumber \\& \le {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \Big [{{{x}^{\mathrm{T}}}(t)} \Big (({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q\nonumber \\&\quad +\,Q{E_{ij}} ({A_i}+B_iK_{s_j})+ {\beta }S\Big )x(t) + x^{\mathrm{T}}(t)QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Qx(t)\nonumber \\&\quad +\,x^{\mathrm{T}}\big (t-\tau (t)\big )Sx\big (t-\tau (t)\big )-x^{\mathrm{T}}\big (t-\tau (t)\big )Sx\big (t-\tau (t)\big )\nonumber \\&\quad +\,\tilde{w}^{\mathrm{T}}(t)\tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ij}Qx(t)+x^{\mathrm{T}}(t)QE_{ij}\tilde{B}_{w_i}\tilde{w}(t)\Big ]\nonumber \\& = {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \Big [{{{x}^{\mathrm{T}}}(t)} \Big (({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q+Q{E_{ij}} ({A_i}+B_iK_{s_j})\nonumber \\&\quad +\,QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q+ {\beta }S\Big )x(t)+ \tilde{w}^{\mathrm{T}}(t)\tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ij}Qx(t) \nonumber \\&\quad +\,x^{\mathrm{T}}(t)QE_{ij}\tilde{B}_{w_i}\tilde{w}(t)\Big ]. \end{aligned}$$

(59)

By adding and subtracting $- {\tilde{z}^{\mathrm{T}}}(t)\tilde{z}(t)\;+\;{\gamma ^2}\sum \nolimits _{i = 1}^r {\sum \nolimits _{j = 1}^r} {\sum \nolimits _{m = 1}^r} {\sum \nolimits _{n = 1}^r} {{\mu _i}{\mu _j}{\mu _m}{\mu _n}} \left[ {\tilde{w}^{\mathrm{T}}}(t)\tilde{w}(t) \right]$ to and from (59), we have

$$\begin{aligned} \dot{V} \left( {x(t)} \right)& = {} \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r{\sum \limits _{m = 1}^r{\sum \limits _{n = 1}^r {{\mu _i}{\mu _j}{\mu _m}{\mu _n}}}}} \left[ {\begin{array}{cc} {{{x}^{\mathrm{T}}}(t)}&{{{\tilde{w}}^{\mathrm{T}}}(t)} \end{array}} \right] \nonumber \\&\quad \times \,\left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q +Q{E_{ij}} ({A_i}+B_iK_{s_j})\\ + QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q+ {\beta }S+C^{\mathrm{T}}_iC_i \end{array}} \right) &{}\;{{{(*)}^{\mathrm{T}}}}\\ \tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ij}Q&{}{- {\gamma ^2}I} \end{array}} \right) \left[ {\begin{array}{c} {x(t)}\\ {\tilde{w}(t)} \end{array}} \right] \nonumber \\&\quad -\,{\tilde{z}^{\mathrm{T}}}(t)\tilde{z}(t) + {\gamma ^2}\sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r{\sum \limits _{m = 1}^r{\sum \limits _{n = 1}^r {{\mu _i}{\mu _j}{\mu _m}{\mu _n}}}}}[{\tilde{w}^{\mathrm{T}}}(t)\tilde{w}(t)], \end{aligned}$$

(60)

where

$$\begin{aligned} \tilde{z}(t) = \sum \limits _{i = 1}^r {\sum \limits _{j = 1}^r {{\mu _i}{\mu _j}}} \tilde{C}_ix(t). \end{aligned}$$

(61)

Next, let us consider Theorem 2. We apply the Schur complement and the algebraic inequality (42) and in turn have

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} {P}{A^{\mathrm{T}}_i} + {A_i}{P} + Y_{s_j}^TB_i^{\mathrm{T}} + B_iY_{s_j} + B_i{Y_{d_j}}A^{\mathrm{T}}_i \\ + A_i{Y_{d_j}^{\mathrm{T}}}{B_i^{\mathrm{T}}} + {A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}\\ + B_iY_{s_j}P^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i + B_iY_{d_j}P^{-1}Y_{s_j}^TB^{\mathrm{T}}_i \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} {{{(*)}^{\mathrm{T}}}} \\ {{\tilde{B}^{\mathrm{T}}_{w_i}}}&{}{- \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} {{{(*)}^{\mathrm{T}}}}\\ {P+Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i}&{}0&{}{- \frac{1}{\beta }W}&{} {{{(*)}^{\mathrm{T}}}}\\ {\tilde{C}_iP}+\tilde{C}_iY^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i&{}0&{}0&{}{- I} \end{array}} \right) < 0 \end{aligned}$$

(62)

or

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} {P} (A_i+B_iY_{s_j}P^{-1})^{\mathrm{T}} + (A_i + B_iY_{s_j}P^{-1})P\\ +B_iY_{d_j}P^{-1}P(A_i+B_iY_{s_j}P^{-1})^{\mathrm{T}}\\ +(A_i+B_iY_{s_j}P^{-1})PP^{-1}Y^{\mathrm{T}}_{d_j}B_i^{\mathrm{T}}+ A_{d_i}WA^{\mathrm{T}}_{d_i} \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} {{{(*)}^{\mathrm{T}}}} \\ {{\tilde{B}^{\mathrm{T}}_{w_i}}}&{}{- \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} {{{(*)}^{\mathrm{T}}}}\\ {P+PP^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i}&{}0&{}{- \frac{1}{\beta }W}&{} {{{(*)}^{\mathrm{T}}}}\\ {\tilde{C}_iP}+\tilde{C}_iPP^{-1}Y^{\mathrm{T}}_{d_j}B^{\mathrm{T}}_i&{}0&{}0&{}{- I} \end{array}} \right) <0, \end{aligned}$$

(63)

with $K_{d_j}=Y_{d_j}P^{-1}$ and $K_{s_j}=Y_{s_j}P^{-1}.$ Pre- and post-multiply both sides of (63) by $\left( {\begin{array}{ccc} {{(I+B_iK_{d_j})}^{-1}}&{}0&{}0\\ 0&{}I&{}0\\ 0&{}0&{}I \end{array}} \right)$ and $\left( {\begin{array}{ccc} {{(I+B_iK_{d_j})}^{-T}}&{}0&{}0\\ 0&{}I&{}0\\ 0&{}0&{}I \end{array}} \right),$ respectively, and we have

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} P{{({A_i}+B_iK_{s_j})^{\mathrm{T}}}}{E^{T}_{ij}}+ {E_{ij}}{{({A_i}+B_iK_{s_j})}}{P}\\ + {E_{ij}}{A_{{d}_i}}W{A^{\mathrm{T}}_{{d}_i}}{E^{T}_{ij}} \end{array}} \right) &{}{{{(*)}^{\mathrm{T}}}}&{}{{{(*)}^{\mathrm{T}}}}&{} {{{(*)}^{\mathrm{T}}}}\\ {{\tilde{B}^{\mathrm{T}}_{w_i}}}{E^{T}_{ij}}&{}{- \gamma ^2{I}}&{}{{{(*)}^{\mathrm{T}}}}&{} {{{(*)}^{\mathrm{T}}}}\\ {P}&{}0&{}{- \frac{1}{\beta }W}&{} {{{(*)}^{\mathrm{T}}}}\\ {\tilde{C}_iP}&{}0&{}0&{}{- I} \end{array}} \right) < 0. \end{aligned}$$

(64)

By multiplying both sides of (64) by $\left( {\begin{array}{cccc} Q &{}0 &{}0 &{}0\\ 0 &{}I &{}0 &{}0\\ 0 &{}0 &{}I &{}0\\ 0 &{}0 &{}0 &{}I \end{array}} \right),$ applying the Schur complement and rearranging them, we have

$$\begin{aligned} \left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_i})^{\mathrm{T}}{E^{\mathrm{T}}_{ii}}Q +Q{E_{ii}} ({A_i}+B_iK_{s_i})\\ + QE_{ii}A_{d_i}WA^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ii}Q+ {\beta }W^{-1}+\tilde{C}^{\mathrm{T}}_i\tilde{C}_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ \tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ii}Q&{}{- {\gamma ^2}I} \end{array}} \right) < 0,\quad i=1, 2, 3\ldots , r, \end{aligned}$$

(65)

and

$$\begin{aligned}&\left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q +Q{E_{ij}} ({A_i}+B_iK_{s_j})\\ + QE_{ij}A_{d_i}WA^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q+ {\beta }W^{-1}+\tilde{C}^{\mathrm{T}}_i\tilde{C}_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ \tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ij}Q&{}{- {\gamma ^2}I} \end{array}} \right) \nonumber \\&\quad +\,\left( {\begin{array}{cccc} \left( {\begin{array}{c} ({{A}_{j}}+B_jK_{s_i})^{\mathrm{T}}{E^{\mathrm{T}}_{ji}}Q +Q{E_{ji}} ({A_j}+B_jK_{s_i})\\ + QE_{ji}A_{d_j}WA^{\mathrm{T}}_{d_j}E^{\mathrm{T}}_{ji}Q+ {\beta }W^{-1}+\tilde{C}^{\mathrm{T}}_j\tilde{C}_j \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ \tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ji}Q&{}{- {\gamma ^2}I} \end{array}} \right) < 0, \end{aligned}$$

(66)

$i<j \le r$. Using (65)–(66) and the fact that

$$\begin{aligned} \sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{\sum \limits _{m = 1}^r}{\sum \limits _{n = 1}^r}{{\mu _i}{\mu _j}{\mu _m}{\mu _n}}M^{\mathrm{T}}_{ij}N_{mn} \le \frac{1}{2}\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{{\mu _i}{\mu _j}} \left[ M^{\mathrm{T}}_{ij}M_{ij}+N_{ij}N^{\mathrm{T}}_{ij}\right] , \end{aligned}$$

(67)

we have

$$\begin{aligned} \left( {\begin{array}{cc} \left( {\begin{array}{c} ({{A}_{i}}+B_iK_{s_j})^{\mathrm{T}}{E^{\mathrm{T}}_{ij}}Q +Q{E_{ij}} ({A_i}+B_iK_{s_j})\\ + QE_{ij}A_{d_i}S^{-1}A^{\mathrm{T}}_{d_i}E^{\mathrm{T}}_{ij}Q+ {\beta }S+\tilde{C}^{\mathrm{T}}_i\tilde{C}_i \end{array}} \right) &{} {{{(*)}^{\mathrm{T}}}}\\ \tilde{B}^{\mathrm{T}}_{w_i}E^{\mathrm{T}}_{ij}Q&{}{- {\gamma ^2}I} \end{array}} \right) < 0, \end{aligned}$$

(68)

where $W={S^{-1}}$ and $i, j = 1, 2,\ldots , r$. As (68) is less than zero, $\mu _n\ge 0$ and $\sum \nolimits _{n = 1}^r {\mu _n} = 1$, (60) becomes

$$\begin{aligned} \dot{V} \left( {x(t)} \right) \le - {\tilde{z}^{\mathrm{T}}}(t)\tilde{z}(t) + {\gamma ^2}\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{\sum \limits _{m = 1}^r}{\sum \limits _{n = 1}^r}{{\mu _i}{\mu _j}{\mu _m}{\mu _n}}[{\tilde{w}^{\mathrm{T}}}(t)\tilde{w}(t)]. \end{aligned}$$

(69)

Integrating both sides of (69) yields

$$\begin{aligned} \int _0^{{T_{\mathrm{f}}}} \dot{V}\left( {x(t)} \right) \mathrm{d}t& \le {} \int _0^{{T_{\mathrm{f}}}} {\left[ {- {{\tilde{z}}^{\mathrm{T}}}} \right. (t)}\tilde{z}(t)\nonumber \\&\left.\quad +\,{\gamma ^2}\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{\sum \limits _{m = 1}^r}{\sum \limits _{n = 1}^r}{{\mu _i}{\mu _j}{\mu _m}{\mu _n}}[{\tilde{w}^{\mathrm{T}}}(t)\tilde{w}(t)] \right ] \mathrm{d}t, \end{aligned}$$

(70)

$$\begin{aligned} V\left( {x({T_{\mathrm{f}}})} \right) - V\left( {x(0)} \right)& \le {} \int _0^{{T_{\mathrm{f}}}}{\left[ {- {{\tilde{z}}^{\mathrm{T}}}} \right. (t)} \tilde{z}(t)\nonumber \\&\left.\quad +\,{\gamma ^2}\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{\sum \limits _{m = 1}^r}{\sum \limits _{n = 1}^r}{{\mu _i}{\mu _j}{\mu _m}{\mu _n}}[{\tilde{w}^{\mathrm{T}}}(t)\tilde{w}(t)] \right ] \mathrm{d}t. \end{aligned}$$

(71)

As $V(x(0)) = 0$ and $V\left( {x({T_{\mathrm{f}}})} \right) \ge 0$ for all, ${T_{\mathrm{f}}} \ne 0,$ we have

$$\begin{aligned}&\int _0^{{T_{\mathrm{f}}}} {{{\tilde{z}}^{\mathrm{T}}}(t)} \tilde{z}(t)\mathrm{d}t\nonumber \\&\quad \le {\gamma ^2}\Big [\int _0^{{T_{\mathrm{f}}}}\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{\sum \limits _{m = 1}^r}{\sum \limits _{n = 1}^r}{{\mu _i}{\mu _j}{\mu _m}{\mu _n}}[{\tilde{w}^{\mathrm{T}}}(t)\tilde{w}(t)] \mathrm{d}t\Big ]. \end{aligned}$$

(72)

Putting $\tilde{z}(t)$ and $\tilde{w}(t)$, respectively, given in (61) and (24) into (72) and using the fact that $\Vert F(x(t),t)\Vert \le \rho , {\lambda }^2 = 1 + \rho ^2 \sum \nolimits _{i = 1}^r \sum \nolimits _{j = 1}^r \left[ \parallel {H^{\mathrm{T}}_{2_i}H_{2_j}}\parallel \right]$ and (67), we have

$$\begin{aligned}&\int _0^{{T_{\mathrm{f}}}} \sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{{\mu _i}{\mu _j}} \Big (2\lambda ^2x^{\mathrm{T}}(t)C^{\mathrm{T}}_iC_ix(t) +2\lambda ^2\rho ^2x^{\mathrm{T}}(t)H^{\mathrm{T}}_{4_i}H_{4_i}x(t)\Big )\mathrm{d}t\nonumber \\&\quad \le \, {\gamma ^2}{\lambda ^2}\Big [\int _0^{{T_{\mathrm{f}}}}[{{w}^{\mathrm{T}}}(t){w}(t)] \mathrm{d}t\Big ]. \end{aligned}$$

(73)

By adding and subtracting

$$\begin{aligned} {\lambda }^2{z}^{\mathrm{T}}(t) {z}(t)& = {} {\lambda }^2 \sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{{\mu _i}{\mu _j}} \Big (x^{\mathrm{T}}(t) {\big (C_i+F \big (x(t),t \big )H_{4_i} \big )}^{\mathrm{T}}\nonumber \\&\quad \times \,{\big (C_i+F \big (x(t),t \big )H_{4_i} \big )} x(t)\Big ) \end{aligned}$$

(74)

to and from (73), one obtains

$$\begin{aligned}&\int _0^{{T_{\mathrm{f}}}} \Big [{\lambda }^2{z}^{\mathrm{T}}(t){z}(t) + \sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{{\mu _i}{\mu _j}} \Big [ \Big (2\lambda ^2x^{\mathrm{T}}(t)C^{\mathrm{T}}_iC_ix(t)\nonumber \\&\quad +\,2\lambda ^2\rho ^2x^{\mathrm{T}}(t)H^{\mathrm{T}}_{4_i}H_{4_i}x(t)-\Big ({\lambda }^2 \Big (x^{\mathrm{T}}(t) {\big (C_i+F \big (x(t),t \big )H_{4_i} \big )}^{\mathrm{T}}\nonumber \\&\quad \times \,{\big (C_i+F \big (x(t),t \big )H_{4_i} \big )} x(t)\Big )\Big )\Big ] \Big ] \mathrm{d}t\le {\gamma ^2}{\lambda }^2\Big [ \int _0^{{T_{\mathrm{f}}}}[{{w}^{\mathrm{T}}}(t){w}(t)] \mathrm{d}t\Big ]. \end{aligned}$$

(75)

From the triangular inequality and the fact that $\Vert F(x(t),t)\Vert \le \rho$, we have

$$\begin{aligned}&{\lambda }^2\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{{\mu _i}{\mu _j}} \Big [ \Big (x^{\mathrm{T}}(t) {\big (C_i+F \big (x(t),t \big )H_{4_i} \big )}^{\mathrm{T}}\times {\big (C_i+F \big (x(t),t \big )H_{4_i}\big )} x(t)\Big )\Big ] \nonumber \\&\quad \le \,\sum \limits _{i = 1}^r{\sum \limits _{j = 1}^r}{{\mu _i}{\mu _j}} \Big [ 2\lambda ^2x^{\mathrm{T}}(t)C^{\mathrm{T}}_iC_ix(t)+2\lambda ^2\rho ^2x^{\mathrm{T}}(t)H^{\mathrm{T}}_{4_i}H_{4_i}x(t)\Big ]. \end{aligned}$$

(76)

Applying (76)–(75), we have

$$\begin{aligned} \int _0^{{T_{\mathrm{f}}}} {{{z}^{\mathrm{T}}}(t)} z(t)\,\mathrm{d}t\, \le {\gamma ^2}\left[ {\int _0^{{T_{\mathrm{f}}}} {{{w}^{\mathrm{T}}}(t)w(t)}} \mathrm{d}t\right] . \end{aligned}$$

(77)

Hence, the inequality (8) holds. This is the case when ${{w}}(t) = 0$, and (69) becomes $\dot{V}(t) \le - {z^{\mathrm{T}}}(t)z(t) \le 0$. Therefore, the closed-loop system (24) is asymptotically stable, and (b) is achieved. This completes the proof. $\square$