Fuzzy adaptive finite-time consensus tracking control of high-order nonlinear multi-agent networks with dead zone

Abstract

This paper studies distributed singularity-free finite-time consensus tracking control for quite a large class of high-order (powers are positive odd integers) nonlinear multi-agent networks in the presence of unknown asymmetric dead zone. Achieving finite-time consensus tracking for such dynamics is extremely challenging because feedback linearization and backstepping methods successfully developed for low-order systems fail to work, and some appropriate exponential terms typically arising from finite-time stability are difficult to design due to the existence of high powers and strong couplings among distinct agents. To this purpose, an adding-one-power-integrator methodology is skillfully incorporated into the finite-time stability theory so as to stabilize the closed-loop system. Over the course of design, a variable-separable lemma is utilized to extract the unknown asymmetric dead-zone input in a “linear-like” manner and fuzzy logic systems are utilized to estimate the unknown system continuous nonlinearities over some compact sets. The singularity issue typical of finite-time control is overcome by delicately introducing a switching function. It is rigorously proved that the consensus tracking error eventually converges to a residual set, whose size can be made as small as desired, in finite time, while guaranteeing the boundedness of all closed-loop signals. Comparative simulations are finally provided to verify the effectiveness of the presented scheme on the existing control methods.

Appendix: Proof of Theorem 1

Step  $i,1~\left( {i \in \left\{ {1, \ldots ,M} \right\} } \right)$: According to (1), differentiating both sides of (7) yields

$$\begin{aligned} {{\dot{z}}_{i,1}}&= \left( {{d_i} + {\mu _i}} \right) \left( h_{i,1}({x_{i,1}}){x_{i,2}^{{r_{i,1}}} + {\varphi _{i,1}}({x_i})} \right) - {\mu _i}\varphi \left( {{x_0},t} \right) \nonumber \\&\quad - \sum \limits _{j \in {{\mathcal {N}}_i}} {{a_{ij}}\left( {h_{j,1}({x_{j,1}})x_{j,2}^{{r_{j,1}}} + {\varphi _{j,1}}({x_j})} \right) }. \end{aligned}$$

(16)

Consider the following Lyapunov function as

$$\begin{aligned} {V_{i,1}} = \frac{{z_{i,1}^{{s_i} - {r_{i,1}} + 2}}}{{{s_i} - {r_{i,1}} + 2}} + \frac{1}{{2{k_{i,1}}}}{\tilde{\varXi }} _{i,1}^2. \end{aligned}$$

(17)

It follows from (16) that the time derivative of ${V_{i,1}}$ is

$$\begin{aligned} {{\dot{V}}_{i,1}}&=z_{i,1}^{{s_i} - {r_{i,1}} + 1}\left( {{d_i} + {\mu _i}} \right) h_{i,1}({x_{i,1}})\left( {x_{i,2}^{{r_{i,1}}} - \nu _{i,1}^{{r_{i,1}}}} \right) \nonumber \\&\quad -\frac{1}{{{k_{i,1}}}}{{{\tilde{\varXi }} }_{i,1}}{{\dot{{\hat{\varXi }}} }_{i,1}}+ z_{i,1}^{{s_i} - {r_{i,1}} + 1}{{\mathcal {F}}_{i,1}}({Z_{i,1}})\nonumber \\&\quad + z_{i,1}^{\left( {{s_i} - {r_{i,1}} + 1} \right) \beta }\left( {{d_i} + {\mu _i}} \right) h_{i,1}({x_{i,1}})\nu _{i,1}^{{r_{i,1}}} \end{aligned}$$

(18)

with ${{\mathcal {F}}_{i,1}}({Z_{i,1}})$ being a function defined by

$$\begin{aligned} {{\mathcal {F}}_{i,1}}({Z_{i,1}}) =&-\left( {{d_i} + {\mu _i}} \right) h_{i,1}({x_{i,1}})\nu _{i,1}^{{r_{i,1}}}\left( z_{i,1}^{\left( {{s_i} - {r_{i,1}} + 1} \right) \left( {\beta - 1} \right) }-1\right) \nonumber \\&\quad - \sum \limits _{j \in {{\mathcal {N}}_i}} {{a_{ij}}\left( {h_{j,1}({x_{j,1}})x_{j,2}^{{r_{j,1}}} + {\varphi _{j,1}}({x_j})} \right) } \nonumber \\&\quad + \left( {{d_i} + {\mu _i}} \right) {\varphi _{i,1}}({x_i})- {\mu _i}\varphi \left( {{x_0},t} \right) , \end{aligned}$$

(19)

where ${Z_{i,1}} = {\left[ x_i^T,x_{j,{j \in {{\mathcal {N}}_i}}}^T,{\mu _i}x_0\right] ^T}$. To facilitate the control design, we use FLS $\varTheta _{i,1}^{ * T}{\psi _{i,1}}({Z_{i,1}})$ to approximate the unknown continuous function ${{\mathcal {F}}_{i,1}}({Z_{i,1}})$, it holds that

$$\begin{aligned} \begin{aligned} {{\mathcal {F}}_{i,1}}({Z_{i,1}}) =\varTheta _{i,1}^{ * T}{\psi _{i,1}}({Z_{i,1}}) + {o_{i,1}}({Z_{i,1}}),{\mathrm{}}\left| {{o_{i,1}}({Z_{i,1}})} \right| {\le } {\varpi _{i,1}}, \end{aligned}\nonumber \\ \end{aligned}$$

(20)

where ${o_{i,1}}({Z_{i,1}})$ is an approximation error and ${\varpi _{i,1}} > 0$. Using Lemma 3 yields

$$\begin{aligned}&z_{i,1}^{{s_i} - {r_{i,1}} + 1}{{\mathcal {F}}_{i,1}} = z_{i,1}^{{s_i} - {r_{i,1}} + 1}\varTheta _{i,1}^{*T}{\psi _{i,1}} + z_{i,1}^{{s_i} - {r_{i,1}} + 1}{o_{i,1}}\nonumber \\&\le \frac{{{s_i} - {r_{i,1}} + 1}}{{{s_i} + 1}}\phi _{i,1}^{{{{\bar{r}}}_{i,1}}}z_{i,1}^{{s_i} + 1}{\left\| {\varTheta _{i,1}^*} \right\| ^{{{{\bar{r}}}_{i,1}}}}{\left( {\psi _{i,1}^T{\psi _{i,1}}} \right) ^\frac{{{{{\bar{r}}}_{i,1}}}}{{\mathrm{2}}}}\nonumber \\&\quad + \frac{{{r_{i,1}}}}{{{s_i} + 1}}\lambda _{i,1}^{ - {{}{\underline{r}}_{i,1}}}\varpi _{i,1}^{{{}{\underline{r}}_{i,1}}} + \frac{{{r_{i,1}}}}{{{s_i} + 1}}\phi _{i,1}^{ - {{}{\underline{r}}_{i,1}}}\nonumber \\&\quad + \frac{{{s_i} - {r_{i,1}} + 1}}{{{s_i} + 1}}\lambda _{i,1}^{{{{\bar{r}}}_{i,1}}}z_{i,1}^{{s_i} + 1}\nonumber \\&\le z_{i,1}^{{s_i} + 1}\Big ( {\phi _{i,1}^{{{{\bar{r}}}_{i,1}}}{\varXi _{i,1}}{{\left( {\psi _{i,1}^T{\psi _{i,1}}} \right) }^\frac{{{{{\bar{r}}}_{i,1}}}}{{\mathrm{2}}}}{\mathrm{+ }}\lambda _{i,1}^{{{{\bar{r}}}_{i,1}}}} \Big ){\mathrm{+ }}{b_{i,1}}, \end{aligned}$$

(21)

where ${\varXi _{i,1}}{\mathrm{= }}{\left\| {\varTheta _{i,1}^*} \right\| ^{{{{\bar{r}}}_{i,1}}}}$ and ${b_{i,1}} = \phi _{i,1}^{ - {{}{\underline{r}}_{i,1}}} + \lambda _{i,1}^{ - {{}{\underline{r}}_{i,1}}}\varpi _{i,1}^{{{}{\underline{r}}_{i,1}}}$. Recalling (21), we can rewrite (18) as

$$\begin{aligned} {{\dot{V}}_{i,1}}&\le z_{i,1}^{{s_i} + 1}\Big ( {\phi _{i,1}^{{{{\bar{r}}}_{i,1}}}{\varXi _{i,1}}{{\left( {\psi _{i,1}^T{\psi _{i,1}}} \right) }^\frac{{{{{\bar{r}}}_{i,1}}}}{{\mathrm{2}}}}{\mathrm{+ }}\lambda _{i,1}^{{{{\bar{r}}}_{i,1}}}} \Big )- \frac{{{{{\tilde{\varXi }} }_{i,1}}{{\dot{{\hat{\varXi }}} }_{i,1}}}}{{{k_{i,1}}}} \nonumber \\&\quad + z_{i,1}^{{s_i} - {r_{i,1}} + 1}\left( {{d_i} + {\mu _i}} \right) h_{i,1}({x_{i,1}})\left( {x_{i,2}^{{r_{i,1}}} - \nu _{i,1}^{{r_{i,1}}}} \right) \nonumber \\&\quad + z_{i,1}^{\left( {{s_i} - {r_{i,1}} + 1} \right) \beta }\left( {{d_i} + {\mu _i}} \right) h_{i,1}({x_{i,1}})\nu _{i,1}^{{r_{i,1}}}+ {b_{i,1}}. \end{aligned}$$

(22)

In light of (8), (11) and (12), when $\left| z_{i,1}\right| \ge {\tau _{i,1}}$, (22) can be rewritten as

(23)

Apparently, when $\left| z_{i,1}\right| < {\tau _{i,1}}$, we can obtain the derivative of $V_{i,1}$ as

(24)

Under the condition of $z_{i,1}<{\tau _{i,1}}<1$, one has

$$\begin{aligned}&- \frac{1}{2}\tau _{i,1}^{\beta - 2}\left( {\left( {4 - \beta } \right) z_{i,1}^2 + \left( {\beta - 2} \right) \tau _{i,1}^{ - 2}z_{i,1}^4} \right) \nonumber \\&\quad = - \tau _{i,1}^{\beta {-} 2}z_{i,1}^2 {-} \frac{1}{2}\left( {2 {-} \beta } \right) \tau _{i,1}^{\beta {-} 2}z_{i,1}^2 {+} \frac{1}{2}\left( {2 {-} \beta } \right) \tau _{i,1}^{\beta {-} 4}z_{i,1}^4\nonumber \\&\quad \le {-} \tau _{i,1}^{\beta {-} 2}z_{i,1}^2 {-} \frac{1}{2}\left( {2 {-} \beta } \right) \tau _{i,1}^{\beta {-} 2}z_{i,1}^2 {+} \frac{1}{2}\left( {2 {-} \beta } \right) \tau _{i,1}^{\beta {-} 2}z_{i,1}^2\nonumber \\&\quad = - \tau _{i,1}^{\beta - 2}z_{i,1}^2 - z_{i,1}^\beta + z_{i,1}^\beta \nonumber \\&\quad \le - z_{i,1}^\beta + \tau _{i,1}^\beta . \end{aligned}$$

(25)

Substituting (25) into (24) gives

(26)

In theory, to get the small local consensus tracking error, we can choose $\tau _{i,1}$ small enough. So, the subsequent stability analysis is done for the case of $\left| z_{i,1}\right| \ge {\tau _{i,1}}$. By using Lemmas 2 and 3, it follows that

$$\begin{aligned}&\left| {z_{i,1}^{{s_i} - {r_{i,1}} + 1}h_{i,1}({x_{i,1}})\left( {x_{i,2}^{{r_{i,1}}} - \nu _{i,1}^{{r_{i,1}}}} \right) } \right| \nonumber \\&\quad \le {r_{i,1}}{{\bar{h}}_{i,1}}\left| {z_{i,1}^{{s_i} {-} {r_{i,1}} {+} 1}} \right| \left| {{z_{i,2}}} \right| \Big ( {x_{i,2}^{{r_{i,1}} {-} 1} {+} {{\left( {{z_{i,1}^\beta }{\xi _{i,1}}} \right) }^{{r_{i,1}} {-} 1}}} \Big )\nonumber \\&\quad \le \frac{1}{{{s_i} + 1}}\hbar _{i,1}^{ - {s_i}}{\big ( {{{\mathrm{2}}^{{r_{i,1}} - {\mathrm{2}}}}{r_{i,1}}{{\bar{h}}_{i,1}}\xi _{i,1}^{{r_{i,1}} - 1}} \big )^{{s_i} + 1}}z_{i,2}^{{s_i} + 1} \nonumber \\&\qquad + \frac{{2{s_i}}}{{{s_i} + 1}}{\hbar _{i,1}}z_{i,1}^{{s_i} + 1} + \frac{{{s_i} - {r_{i,1}} + 1}}{{{s_i} + 1}}{\hbar _{i,1}}z_{i,1}^{{s_i} + 1}\nonumber \\&\qquad + {\underline{r}}_{i,1}^{ - 1}\hbar _{i,1}^{ - \frac{{{s_i} - {r_{i,1}} + 1}}{{{r_{i,1}}}}}{\left( {{{\mathrm{2}}^{{r_{i,1}} - {\mathrm{2}}}}{{\bar{h}}_{i,1}}{r_{i,1}}} \right) ^{{{}{\underline{r}}_{i,1}}}}z_{i,2}^{{s_i} + 1}\nonumber \\&\qquad + \frac{1}{{{s_i} + 1}}\hbar _{i,1}^{ - {s_i}}{\big ( {{r_{i,1}}{{\bar{h}}_{i,1}}\xi _{i,1}^{{r_{i,1}} - 1}} \big )^{{s_i} + 1}}z_{i,2}^{{s_i} + 1}\nonumber \\&\quad \le z_{i,1}^{\left( {{s_i} + 1} \right) \beta } + {\left( {{d_i} + {\mu _i}} \right) ^{ - 1}}z_{i,2}^{{s_i} + 1}{\varUpsilon _{i,1}}. \end{aligned}$$

(27)

Combining (27) with (23) results in

(28)

Noting the fact that

$$\begin{aligned} \begin{aligned} {\upsilon _{i,1}}{{\tilde{\varXi }} _{i,1}}{{{\hat{\varXi }}} _{i,1}}\le \frac{1}{2}{\upsilon _{i,1}}{\varXi _{i,1}^2} - \frac{1}{2}{\upsilon _{i,1}}{{\tilde{\varXi }} _{i,1}^2}, \end{aligned} \end{aligned}$$

(29)

then, it follows that

(30)

Step $i,m~\left( {i \in \left\{ {1, \ldots ,M} \right\} ,m \in \left\{ {2, \ldots ,{n_i} - 1} \right\} } \right)$: Consider the following Lyapunov function as

$$\begin{aligned} \begin{aligned} {V_{i,m}} = {V_{i,m - 1}} + \frac{{z_{i,m}^{{s_i} - {r_{i,m}} + 2}}}{{{s_i} - {r_{i,m}} + 2}} + \frac{1}{{2{k_{i,m}}}}{\tilde{\varXi }} _{i,m}^2. \end{aligned} \end{aligned}$$

(31)

The derivative of ${V_{i,m}}$ along (1), (30) and (31) can be written as

(32)

with ${{\mathcal {F}}_{i,m}}({Z_{i,m}})$ being a function defined by

$$\begin{aligned}&{{\mathcal {F}}_{i,m}}({Z_{i,m}}) =\nonumber \\&\quad - \sum \limits _{n = 1}^{m - 1} {\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {x_{i,n}}}}\left( {h_{i,n}({{\bar{x}}_{i,n}})x_{i,n + 1}^{{r_{i,n}}} + {\varphi _{i,n}}({x_i})} \right) }\nonumber \\&\quad - \sum \limits _{j \in {{\mathcal {N}}_i}} {\frac{{\partial {\nu _{i,{m - 1}}}}}{{\partial {x_{j,1}}}}\left( {h_{j,1}({ x_{j,1}})x_{j,2}^{{r_{j,1}}} + {\varphi _{j,1}}({x_j})} \right) } \nonumber \\&\quad - \sum \limits _{n = 1}^{m - 1} {\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {{{{\hat{\varXi }}} }_{i,n}}}}{{\dot{{\hat{\varXi }}} }_{i,n}}}-\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {x_0}}}\varphi \left( {{x_0},t} \right) \nonumber \\&\quad - z_{i,m}^{\left( {{s_i} - {r_{i,m}} + 1} \right) \left( {\beta - 1} \right) }h_{i,m}({{\bar{x}}_{i,m}})\nu _{i,m}^{{r_{i,m}}}\nonumber \\&\quad +h_{i,m}({{\bar{x}}_{i,m}})\nu _{i,m}^{{r_{i,m}}}+{\varphi _{i,m}}({x_i}), \end{aligned}$$

(33)

where ${Z_{i,m}}=\Big [ x_i^T, x_{j,{j \in {{\mathcal {N}}_i}}}^T, \frac{{\partial {\nu _{i,m - 1}}}}{{\partial {x_{j,1}}}},\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {x_{i,1}}}}, \ldots ,\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {x_{i,m - 1}}}},$ $\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {{{{\hat{\varXi }}} }_{i,1}}}},\ldots ,\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {{{{\hat{\varXi }}} }_{i,m - 1}}}},{{{{\hat{\varXi }}} }_{i,1}}, \ldots ,{{{{\hat{\varXi }}} }_{i,m - 1}},\frac{{\partial {\nu _{i,m - 1}}}}{{\partial {x_0}}},{\mu _i}{x_0} \Big ]^T$.

Following similar lines as (21), we can obtain

$$\begin{aligned} z_{i,m}^{{s_i} - {r_{i,m}} + 1}{{\mathcal {F}}_{i,m}}({Z_{i,m}})&\le z_{i,m}^{{s_i} + 1}\phi _{i,m}^{{{{\bar{r}}}_{i,m}}}{\varXi _{i,m}}{\left( {\psi _{i,m}^T{\psi _{i,m}}} \right) ^{\frac{{{{{\bar{r}}}_{i,m}}}}{2}}}\nonumber \\&\quad +z_{i,m}^{{s_i} + 1}\lambda _{i,m}^{{{{\bar{r}}}_{i,m}}}{\mathrm{+ }}{b_{i,m}}, \end{aligned}$$

(34)

where ${\varXi _{i,m}}{\mathrm{= }}{\left\| {\varTheta _{i,m}^*} \right\| ^{{{{\bar{r}}}_{i,m}}}}$ and ${b_{i,m}} = \phi _{i,m}^{ - {{}{\underline{r}}_{i,m}}} + \lambda _{i,m}^{ - {{}{\underline{r}}_{i,m}}}\varpi _{i,m}^{{{}{\underline{r}}_{i,m}}}$. Substituting (34) to (32) results in

(35)

Similar to Step i, 1, in light of (9), (12) and (35), ${{\dot{V}}_{i,m}}$ can be represented by

(36)

Again, using Lemmas 2 and 3, one has

$$\begin{aligned}&\left| {z_{i,m}^{{s_i} - {r_{i,m}} + 1}h_{i,m}({{\bar{x}}_{i,m}})\left( {x_{i,m + 1}^{{r_{i,m}}} - \nu _{i,m}^{{r_{i,m}}}} \right) } \right| \nonumber \\&\quad \le {r_{i,m}}{\bar{h}}_{i,m}\left| {z_{i,m}^{{s_i} - {r_{i,m}} + 1}} \right| \left| {{z_{i,m + 1}}} \right| \big ( {x_{i,m + 1}^{{r_{i,m}} - 1} + \nu _{i,m}^{{r_{i,m}} - 1}} \big )\nonumber \\&\quad \le z_{i,m}^{\left( {{s_i} + 1} \right) \beta } + z_{i,m + 1}^{{s_i} + 1}{\varUpsilon _{i,m}}. \end{aligned}$$

(37)

Recalling (37) and using Young’s inequality to ${\upsilon _{i,m}}{{\tilde{\varXi }} _{i,m}}{{{\hat{\varXi }}} _{i,m}}$, the time derivative of ${V_{i,{n_i}}}$ can be given by

(38)

Step $i,{n_i}~\left( {i \in \left\{ {1, \ldots ,M} \right\} } \right)$: Consider the following Lyapunov function as

$$\begin{aligned} \begin{aligned} {V_{i,{n_i}}} = {V_{i,{n_i} - 1}} + \frac{{z_{i,{n_i}}^{{s_i} - {r_{i,{n_i}}} + 2}}}{{{s_i} - {r_{i,{n_i}}} + 2}} + \frac{1}{{2{k_{i,{n_i}}}}}{\tilde{\varXi }} _{i,{n_i}}^2. \end{aligned} \end{aligned}$$

(39)

In view of (1), (38) and (39) and applying Lemma 5, the time derivative of ${V_{i,{n_i}}}$ satisfies

(40)

with ${{\mathcal {F}}_{i,{n_i}}}\left( {{Z_{i,{n_i}}}} \right)$ being a function defined by

$$\begin{aligned}&{{\mathcal {F}}_{i,{n_i}}}({Z_{i,{n_i}}}) =\nonumber \\&\quad - h_{i,{n_i}}({x_i})\ell _{1,i}\kappa _i^{{r_{i,{n_i}}}}{{{{{u_i}}}^{{r_{i,{n_i}}}}}}\Big ( {z_{i,{n_i}}^{( {{s_i} - {r_{i,{n_i}}} + 1} )\left( {\beta - 1}\right) }-1}\Big )\nonumber \\&\quad - \sum \limits _{n = 1}^{{n_i} - 1} {\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {x_{i,n}}}}\left( h_{i,{n}}({x_i}){x_{i,n + 1}^{{r_{i,n}}} + {\varphi _{i,n}}({x_i})} \right) }\nonumber \\&\quad - \sum \limits _{j \in {{\mathcal {N}}_i}} {\frac{{\partial {\nu _{i,{{n_i} - 1}}}}}{{\partial {x_{j,1}}}}\left( h_{j,{1}}({x_{j,1}}){x_{j,2}^{{r_{j,1}}} + {\varphi _{j,1}}({x_j})} \right) }\nonumber \\&\quad - \sum \limits _{n = 1}^{{n_i} - 1} {\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {{{{\hat{\varXi }}} }_{i,n}}}}{{\dot{{\hat{\varXi }}} }_{i,n}}}-\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {x_0}}}\varphi \left( {{x_0},t} \right) \nonumber \\&\quad +h_{i,{n_i}}({x_i})\ell _{2,i}{{{{{\varDelta _i}}}^{{r_{i,{n_i}}}}}}+{\varphi _{i,{n_i}}}({x_i}), \end{aligned}$$

(41)

where ${Z_{i,{n_i}}}=\Big [ x_i^T,x_{j,{j \in {{\mathcal {N}}_i}}}^T,\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {x_{j,1}}}},\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {x_{i,1}}}}, \ldots ,\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {x_{i,{n_i} - 1}}}},$ $\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {{{{\hat{\varXi }}} }_{i,1}}}},\ldots ,\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {{{{\hat{\varXi }}} }_{i,{n_i} - 1}}}},{{{{\hat{\varXi }}} }_{i,1}}, \ldots ,{{{{\hat{\varXi }}} }_{i,{n_i} - 1}},\frac{{\partial {\nu _{i,{n_i} - 1}}}}{{\partial {x_0}}},{\mu _i}{x_0} \Big ]^T$. Similar to (21) and (34), the following inequality holds that

$$\begin{aligned} z_{i,{n_i}}^{{s_i} - {r_{i,{n_i}}} + 1}{{\mathcal {F}}_{i,{n_i}}}({Z_{i,{n_i}}})\le&z_{i,{n_i}}^{{s_i} + 1}\phi _{i,{n_i}}^{{{{\bar{r}}}_{i,{n_i}}}}{\varXi _{i,{n_i}}}{\left( {\psi _{i,{n_i}}^T{\psi _{i,{n_i}}}} \right) ^\frac{{{{{\bar{r}}}_{i,{n_i}}}}}{2}}\nonumber \\&\quad +z_{i,{n_i}}^{{s_i} + 1}\lambda _{i,{n_i}}^{{{{\bar{r}}}_{i,{n_i}}}}{\mathrm{+ }}{b_{i,{n_i}}}, \end{aligned}$$

(42)

where ${\varXi _{i,{n_i}}}={\left\| {\varTheta _{i,{n_i}}^*} \right\| ^{{{{\bar{r}}}_{i,{n_i}}}}}$ and ${b_{i,{n_i}}} = \phi _{i,{n_i}}^{ - {{}{\underline{r}}_{i,{n_i}}}} + \lambda _{i,{n_i}}^{ - {{}{\underline{r}}_{i,{n_i}}}}\varpi _{i,{n_i}}^{{{}{\underline{r}}_{i,{n_i}}}}$. Using (42), (40) can be rewritten as

(43)

Substituting (10) and (12) into (43) yields

(44)

where above inequality holds owing to ${\zeta _{i,1}} = \left( {{d_i} + {\mu _i}} \right)$, ${\zeta _{i,n}} = 1,\left( {n = 2, \ldots ,{n_i}} \right)$ and the fact that

$$\begin{aligned} \vartheta _i^{\frac{{{r_{i,n}} - 1}}{{{s_i} + 1}}}z_{i,n}^{\left( {{s_i} - {r_{i,n}} + 2} \right) \beta }&\le \frac{{{r_{i,n}} - 1}}{{{s_i} + 1}}{\vartheta _i} + \frac{{{s_i} - {r_{i,n}} + 2}}{{{s_i} + 1}}z_{i,n}^{\left( {{s_i} + 1} \right) \beta }\nonumber \\&\quad \le {\vartheta _i} + z_{i,n}^{\left( {{s_i} + 1} \right) \beta },\left( {n = 1,2, \ldots ,{n_i}} \right) , \end{aligned}$$

(45)

with ${\vartheta _i} > 0$ being a constant.

Now, consider the total Lyapunov function

$$\begin{aligned} \begin{aligned} V = \sum \limits _{i = 1}^M {{V_{i,{n_i}}}} = \sum \limits _{i = 1}^M {\sum \limits _{m = 1}^{{n_i}} {\Bigg \{ {\frac{{z_{i,m}^{{s_i} - {r_{i,m}} + 2}}}{{{s_i} - {r_{i,m}} + 2}} + \frac{1}{{2{k_{i,m}}}}{\tilde{\varXi }} _{i,m}^2} \Bigg \}} }. \end{aligned}\nonumber \\ \end{aligned}$$

(46)

In view of Lemma 4, it holds that

$$\begin{aligned} {V^\beta }&= {\Bigg \{ {\sum \limits _{i = 1}^M {\sum \limits _{m = 1}^{{n_i}} {\bigg ( {\frac{{z_{i,m}^{{s_i} - {r_{i,m}} + 2}}}{{{s_i} - {r_{i,m}} + 2}} + \frac{1}{{2{k_{i,m}}}}{\tilde{\varXi }} _{i,m}^2} \bigg )} } } \Bigg \}^\beta }\nonumber \\&\quad \le \sum \limits _{i = 1}^M {\sum \limits _{m = 1}^{{n_i}} {\Bigg \{ {{{ \bigg ( {\frac{{z_{i,m}^{{s_i} - {r_{i,m}} + 2}}}{{{s_i} - {r_{i,m}} + 2}}} \bigg )}^{\beta } } + {{\bigg ( {\frac{1}{{2{k_{i,m}}}}{\tilde{\varXi }} _{i,m}^2} \bigg )}^\beta }} \Bigg \}} } \nonumber \\&\quad \le \sum \limits _{i = 1}^M {\sum \limits _{m= 1}^{{n_i}} {\Bigg \{ {\frac{1}{{{{\left( {{s_i} - {r_{i,m}} + 2} \right) }^\beta }}}z_{i,m}^{\left( {{s_i} - {r_{i,m}} + 2} \right) \beta } + {{\bigg ( {\frac{1}{{2{k_{i,m}}}}{\tilde{\varXi }} _{i,m}^2} \bigg )}^\beta }}\Bigg \}} }. \end{aligned}$$

(47)

Then, we can rewrite (44) as

(48)

where . After applying Lemma 3 to deal with the terms ${\Big [ {\frac{1}{2}{\upsilon _{i,m}}{\tilde{\varXi }} _{i,m}^2} \Big ]^\beta }$ and choosing design parameters $\upsilon = 1$, $\omega = \frac{1}{2}{\upsilon _{i,m}}{\tilde{\varXi }} _{i,m}^2$, ${q_1} = 1 - \beta$, ${q_2} = \beta$ and $\varsigma = {\beta ^{{\beta \big / {\left( {1 - \beta } \right) }}}}$, respectively, the following inequality holds that

$$\begin{aligned} \begin{aligned} {\bigg [ {\frac{1}{2}{\upsilon _{i,m}}{\tilde{\varXi }} _{i,m}^2} \bigg ]^\beta } \le \left( {1 - \beta } \right) \varsigma + \frac{1}{2}{\upsilon _{i,m}}{\tilde{\varXi }} _{i,m}^2. \end{aligned} \end{aligned}$$

(49)

Substituting (49) into (48) results in

(50)

Invoking (50), one gets

$$\begin{aligned} \begin{aligned} \dot{V}\le - \alpha {V^\beta } + \varrho , \end{aligned}\end{aligned}$$

(51)

where $\alpha =\min \left\{ {{{\left( {{s_i} - {r_{i,m}} + 2} \right) }^\beta }{\alpha _i},1 \le i \le M} \right\}$ and ${{+b_{i,m}} + \frac{1}{2}{{\left( {{s_i} - {r_{i,m}} + 2} \right) }^\beta }{\alpha _i}{\upsilon _{i,m}}\varXi _{i,m}^2} \Big \}$.

In accordance with Lemma 1, the final reaching time can be calculated by

$$\begin{aligned} {T_{{\mathrm{reach}}}} = \frac{1}{{( {1 - \beta } ){\theta _0}\alpha }}\Bigg [ {{V^{1 - \beta }}\left( {{z_i}( 0 ),{\varXi _i}( 0 )} \right) - {{\bigg ( {\frac{\varrho }{{( {1 - {\theta _0}})\alpha }}} \bigg )}^{\frac{{1 - \beta }}{\beta }}}} \Bigg ],\nonumber \\ \end{aligned}$$

(52)

where $0 {<} {\theta _0} {\le } 1$ is a constant, ${z_i}\left( 0 \right) {=} \left[ {{z_{i,1}}\left( 0 \right) ,{z_{i,2}}\left( 0 \right) , } \right.$ ${\left. { \ldots ,{z_{i,m}}\left( 0 \right) } \right] ^T}$ and ${\varXi _i}\left( 0 \right) = \left[ {\varXi _{i,1}}\left( 0 \right) ,{\varXi _{i,2}}\left( 0 \right) , \ldots , {\varXi _{i,m}}\left( 0 \right) \right] ^T$. Thus, when $\forall t \ge {T_{{\mathrm{reach}}}}$, the trajectories of the closed-loop networks are bounded by the following compact set

$$\begin{aligned} \begin{aligned} \varOmega = \left\{ {\left( {{z_i},{\varXi _i}} \right) \Big |V \le {{\bigg ( {\frac{\varrho }{{\left( {1 - {\theta _0}} \right) \alpha }}} \bigg )}^{\frac{1}{\beta }}}} \right\} . \end{aligned} \end{aligned}$$

(53)

Thus, from definition 1, it can be concluded that the closed-loop networks are SGPFS. The local tracking errors converge to a small neighborhood of the origin in finite time ${T_{{\mathrm{reach}}}}$. When $\forall t \ge {T_{{\mathrm{reach}}}}$, we can further obtain

$$\begin{aligned} \left\| {{z_1}} \right\| \le \sqrt{\sum \nolimits _{i = 1}^M {{{\left[ {\left( {{s_i} - {r_{i,1}} + 2} \right) {{\left( {\frac{\varrho }{{( {1 - {\theta _0}} )\alpha }}} \right) }^{\frac{1}{\beta }}}} \right] }^{\frac{2}{{{s_i} - {r_{i,1}} + 2}}}}} }.\nonumber \\ \end{aligned}$$

(54)

Consequently, from the inequalities below (7), one gets that $\left\| \delta \right\| \le \frac{{\left\| {{z_1}} \right\| }}{{ {\underline{\sigma }} ( {\bar{{\mathcal {L}}} + {\mathcal {B}}} )}}$. This completes the proof of Theorem 1. $\square$