Simplified optimized finite-time containment control for a class of multi-agent systems with actuator faults

Abstract

In this paper, for a class of multi-agent systems with unknown dynamic functions and unknown actuator faults, the optimized finite-time containment control problem based on optimized backstepping technology is investigated. The optimal control strategy is obtained through a simplified reinforcement learning algorithm with the structure of identifier-critic-actor. Based on such a structure, the identifier, the critic and the actor are applied to estimate unknown dynamic functions, evaluate system performance and implement control behavior, respectively. The updating laws of the actor and critic are derived based on the gradient descent method of a simple positive function rather than the square of Bellman residual, which makes the updating laws simpler and eliminates the harsh persistent excitation condition. In addition, in order to eliminate the effect of actuator faults on system stability, a network-based fault observer is constructed to observe online actuator faults. Furthermore, the designed finite-time optimal distributed containment controllers ensure that the followers can ultimately converge to the convex hull composed by leaders within a predetermined finite time. Finally, a numerical simulation result and a practical example are presented to verify the effectiveness of the proposed control method.

Appendix

To prove Theorem 1, the following steps are given.

Step 1: According to the system (1), the derivative of the containment error ${z}_{i,1}$ can be rewritten as follows

$$\begin{aligned}&{\dot{z}}_{i,1}= -\sum _{j=1}^{Q}a_{i,j}( x_{j,2}+f_{j,1})-\!\!\sum _{r=Q+1}^{Q+P}\!\!a_{i,r}{\dot{y}}_{r,d}\nonumber \\&\quad +d_{i}(z_{i,2}+{{\hat{\alpha }}}_{i,1}^{*}+f_{i,1}). \end{aligned}$$

(53)

Then, ${{\tilde{\varXi }}}_{f_{i,1}}$ $={{\hat{\varXi }}}_{f_{i,1}}-{\varXi }_{f_{i,1}}^{*}, {\tilde{\varXi }}_{c_{i,1}}={\hat{\varXi }}_{c_{i,1}}-\varXi _{J_{i,1}}^{*}$ and ${\tilde{\varXi }} _{a_{i,1}}={\hat{\varXi }}_{a_{i,1}}-\varXi _{J_{i,1}}^{*}$ are defined as the identifier, critic, and actor NN weight errors, and the Lyapunov function candidate for subsystem $z_{i,1}$ is selected as follows

$$\begin{aligned} V_{i,1} =&\frac{1}{2}z_{i,1}^{2}\!\!+\!\frac{1}{2}{\tilde{\varXi }}_{f_{i,1}}^{T}\varGamma _{i,1}^{-1}{{\tilde{\varXi }}}_{f_{i,1}} \\&\quad +\!\frac{1}{2}{\tilde{\varXi }}_{c_{i,1}}^{T}{\tilde{\varXi }}_{c_{i,1}}\!\!+\!\frac{1}{2}{\tilde{\varXi }} _{a_{i,1}}^{T}{\tilde{\varXi }}_{a_{i,1}}. \end{aligned}$$

Then, the time derivative of $V_{i,1}$ is given by

$$\begin{aligned}&{\dot{V}}_{i,1}=z_{i,1}{\dot{z}}_{i,1}-{\tilde{\varXi }}_{f_{i,1}}^{T}\varGamma _{i,1}^{-1}\dot{{\hat{\varXi }}}_{f_{i,1}}\\&\quad -{\tilde{\varXi }}_{c_{i,1}}^{T}\dot{{\hat{\varXi }}}_{c_{i,1}} -{\tilde{\varXi }}_{a_{i,1}}^{T}\dot{{\hat{\varXi }}}_{a_{i,1}}. \end{aligned}$$

Furthermore, ${\dot{V}}_{i,1}$ along (12), (14) and (16) is as follows

$$\begin{aligned}&{\dot{V}}_{i,1}= z_{i,1}(d_{i}{\hat{\alpha }}_{i,1}^{*}+F_{i,1}) -\ \gamma _{c_{i,1}}{\tilde{\varXi }}_{c_{i,1}}^{T}\ S_{J_{i,1}}\ S_{J_{i,1}}^{T}{{\hat{\varXi }} }_{c_{i,1}} \nonumber \\&\quad -{\tilde{\varXi }}_{a_{i,1}}^{T}S_{J_{i,1}}S_{J_{i,1}}^{T}\!(\gamma _{a_{i,1}}({{\hat{\varXi }}}_{a_{i,1}}\!\! -\!{{\hat{\varXi }}}_{c_{i,1}})\!+\!\gamma _{c_{i,1}}{{\hat{\varXi }}}_{c_{i,1}}) \nonumber \\&\quad +{\tilde{\varXi }}_{f_{i,1}}^{T}(S_{f_{i,1}}z_{i,1}-\sigma _{i,1}{{\hat{\varXi }}} _{f_{i,1}}) -\frac{3}{4}z_{i,1}^{2}-\frac{d_{i}}{2}z_{i,1}^{2}\nonumber \\&\quad +d_{i}z_{i,1}z_{i,2}. \end{aligned}$$

Based on (7) and (15), one has

$$\begin{aligned} {\dot{V}}_{i,1}=&-\beta _{i,1}z_{i,1}^{2l}-\frac{ 1}{2}z_{i,1}{\hat{\varXi }}_{a_{i,1}}^{T}S_{J_{i,1}}-\left( \frac{3}{4} +\frac{d_{i}}{2}\right) z_{i,1}^{2} \nonumber \\&-\sigma _{i,1}{\tilde{\varXi }}_{f_{i,1}}^{T}{\hat{\varXi }}_{f_{i,1}}\!\!-\!\gamma _{c_{i,1}} {\tilde{\varXi }}_{c_{i,1}}^{T}\!S_{J_{i,1}}\!S_{J_{i,1}}^{T}{\hat{\varXi }}_{c_{i,1}} \nonumber \\&-{\tilde{\varXi }}_{a_{i,1}}^{T}S_{J_{i,1}}S_{J_{i,1}}^{T}(\gamma _{a_{i,1}}( {\hat{\varXi }}_{a_{i,1}}-{\hat{\varXi }}_{c_{i,1}})\nonumber \\&+\gamma _{c_{i,1}}{\hat{\varXi }}_{c_{i,1}})+z_{i,1}\varepsilon _{f_{i,1}}+d_{i}z_{i,1}z_{i,2}. \end{aligned}$$

(54)

By adopting Young’s inequality, the following inequalities hold

$$\begin{aligned}&-\frac{1}{2}z_{i,1}{\hat{\varXi }}_{a_{i,1}}^{T}S_{J_{i,1}}\le \frac{1}{4}z_{i,1}^{2}+\frac{1}{4}({\hat{\varXi }}_{a_{i,1}}^{T} S_{J_{i,1}})^2,\nonumber \\&d_{i}z_{i,1}z_{i,2}\le \frac{d_{i}}{2}z_{i,1}^{2}+\frac{d_{i}}{2}z_{i,2}^{2},\nonumber \\&z_{i,1}\varepsilon _{f_{i,1}}\le \frac{1}{2}z_{i,1}^{2}+\frac{1}{2}\varepsilon _{f_{i,1}}^{2}. \end{aligned}$$

(55)

Substituting the inequalities (55) into (54), one has

$$\begin{aligned} {\dot{V}}_{i, 1} \le&-\sigma _{i, 1} {\tilde{\varXi }}_{f_{i, 1}}^{T} {\tilde{\varXi }}_{f_{i, 1}}\!-\!\gamma _{c_{i, 1}} {\tilde{\varXi }}_{c_{i, 1}}^{T} S_{J_{i, 1}} S_{J_{i, 1}}^{T} {\hat{\varXi }}_{c_{i, 1}} \nonumber \\&-\beta _{i,1} z_{i, 1}^{2 l}+\frac{1}{2} \varepsilon _{f_{i, 1}}^{2}+\frac{1}{4} ({\hat{\varXi }}_{a_{i, 1}}^{T} S_{J_{i, 1}})^2\nonumber \\&-\gamma _{a_{i, 1}} {\tilde{\varXi }}_{a_{i, 1}}^{T} S_{J_{i, 1}} S_{J_{i, 1}}^{T} {\hat{\varXi }}_{a_{i, 1}}+\frac{1}{2} d_{i} z_{i, 2}^{2} \nonumber \\&+(\gamma _{a_{i, 1}}-\gamma _{c_{i, 1}}) {\tilde{\varXi }}_{a_{i, 1}}^{T} S_{J_{i, 1}} S_{J_{i, 1}}^{T} {\hat{\varXi }}_{c_{i, 1}}. \end{aligned}$$

(56)

On the basis of ${\tilde{\varXi }}_{f_{i,1}}={\hat{\varXi }}_{f_{i,1}}-\varXi _{f_{i,1}}^{*}$, we can obtain

$$\begin{aligned} {\tilde{\varXi }}_{f_{i,1}}^{T}{\hat{\varXi }}_{f_{i,1}}=\ \frac{1}{2}(\Vert {\tilde{\varXi }} _{f_{i,1}}\Vert ^2+\Vert {\hat{\varXi }}_{f_{i,1}}\Vert ^2-\Vert \varXi _{f_{i,1}}^{*}\Vert ^2). \end{aligned}$$

(57)

Similarly, owing to ${\tilde{\varXi }}_{{{\bar{o}}}_{i,1}}={\hat{\varXi }}_{{{\bar{o}}}_{i,1}}-\varXi ^*_{{{\bar{o}}}_{i,1}}$ with ${{\bar{o}}}=a,c$, the following equation can be derived

$$\begin{aligned}&{\tilde{\varXi }}_{{{\bar{o}}}_{i,1}}^{T}S_{J_{i,1}}S_{J_{i,1}}^{T}{\hat{\varXi }}_{{{\bar{o}}}_{i,1}}\nonumber \\&= \frac{1}{2}({\tilde{\varXi }}_{{{\bar{o}}}_{i,1}}^{T}S_{J_{i,1}})^2 \!+\!\frac{1}{2}({\hat{\varXi }}_{{{\bar{o}}}_{i,1}}^{T}S_{J_{i,1}})^2 \!-\!\frac{1}{2}(\varXi _{J_{i,1}}^{*T}S_{J_{i,1}}) ^{2}. \end{aligned}$$

(58)

Substituting (57) and (58) into (56), one gets

$$\begin{aligned}&{\dot{V}}_{i,1} \le -\beta _{i,1}z_{i,1}^{2l} - \frac{\sigma _{i,1}}{2}(\Vert {{\tilde{\varXi }}}_{f,1}\Vert ^2-\Vert \varXi _{{f_{i,1}}}^{*}\Vert ^2)+ \frac{1}{2}{d_i}z_{i,2}^2\\&\quad - \frac{{{\gamma _{{c_{i,1}}}}}}{2}({{{{\tilde{\varXi }}}}_{{c_{i,1}}}}{S_{{J_{i,1}}}})^2\!-\!\frac{{{\gamma _{{a_{i,1}}}}}}{2}({{{{\tilde{\varXi }}}}_{{a_{i,1}}}}{S_{{J_{i,1}}}})^2 +\frac{1}{2}\varepsilon _{{f_{i,1}}}^2\\&\quad -\left( {\frac{{{\gamma _{{a_{i,1}}}}}}{2} - \frac{1}{4}}\right) \left( {{\hat{\varXi }}}_{{a_{i,1}}}^T{S_{{J_{i,1}}}}\right) ^2- \frac{{{\gamma _{{c_{i,1}}}}}}{2}({{\hat{\varXi }}}_{{c_{i,1}}}^T{S_{{J_{i,1}}}})^2\\&\quad + ( {{\gamma _{{a_{i,1}}}} - {\gamma _{{c_{i,1}}}}} ){{\tilde{\varXi }}}_{{a_{i,1}}}^T{S_{{J_{i,1}}}}S_{{J_{i,1}}}^T{{{{\hat{\varXi }}}}_{{c_{i,1}}}}\\&\quad + ( {\frac{{{\gamma _{{c_{i,1}}}}}}{2} + \frac{{{\gamma _{{a_{i,1}}}}}}{2}} ){( {\varXi _{{J_{i,1}}}^{*T}{S_{{J_{i,1}}}}} )^2}. \end{aligned}$$

By using Young’s inequality, one has

$$\begin{aligned}&({{\gamma _{{a_{i,1}}}}-{\gamma _{{c_{i,1}}}}}) {\tilde{\varXi }}_{{ a_{i,1}}}^{T}{S_{{J_{i,1}}}}S_{{J_{i,1}}}^{T}{{{\hat{\varXi }}}_{{c_{i,1} }}}\nonumber \\&\quad \le \frac{{{\gamma _{{a_{i,1}}}}-{\gamma _{{c_{i,1}}}}}}{2}({\tilde{\varXi }}_{{ a_{i,1}}}^{T}{S_{{J_{i,1}}}})^2 +\frac{{{\gamma _{{a_{i,1}}}}-{\gamma _{{c_{i,1}}}}}}{2}({\hat{\varXi }}_{{c_{i,1}} }^{T}{S_{{J_{i,1}}}})^2. \end{aligned}$$

Further, it yields

$$\begin{aligned} {\dot{V}}_{i, 1} \le&-\beta _{i,1} z_{i, 1}^{2 l}-\frac{\sigma _{i, 1}}{2} {\tilde{\varXi }}_{f_{i, 1}}^{T} \ {\tilde{\varXi }}_{f_{i, 1}}\!-\!\frac{\gamma _{c_{i, 1}}}{2} ({\tilde{\varXi }}_{c_{i, 1}}^{T}\ S_{J_{i, 1}})^2\nonumber \\&-\frac{\gamma _{c_{i, 1}}}{2} ({\tilde{\varXi }}_{a_{i, 1}}^{T} S_{J_{i, 1}})^2\!-\!(\gamma _{c_{i, 1}}\!-\!\frac{\gamma _{a_{i, 1}}}{2})({\hat{\varXi }}_{c_{i, 1}}^{T} S_{J_{i, 1}})^{2}\nonumber \\&+ ( {\frac{{{\gamma _{{c_{i,1}}}}}}{2} + \frac{{{\gamma _{{a_{i,1}}}}}}{2}} ){( {\varXi _{{J_{i,1}}}^{*T}{S_{{J_{i,1}}}}} )^2}\!+\!\frac{1}{2} \varepsilon _{f_{i, 1}}^{2}\!+\!\frac{1}{2} d_{i} z_{i, 2}^{2}\nonumber \\&-(\frac{\gamma _{a_{i, 1}}}{2}-\frac{1}{4})({\hat{\varXi }}_{a_{i, 1}}^{T} S_{J_{i, 1}})^{2}+\frac{\sigma _{i, 1}}{2} \varXi _{f_{i, 1}}^{* T} \varXi _{f_{i, 1}}^{*}. \end{aligned}$$

(59)

Based on (17), (59) can be rewritten as

$$\begin{aligned} {\dot{V}}_{i, 1} \le&-\beta _{i,1} z_{i, 1}^{2 l}-\frac{\sigma _{i, 1}}{2} {\tilde{\varXi }}_{f_{i, 1}}^{T} {\tilde{\varXi }}_{f_{i,1}}+\frac{1}{2} d_{i} z_{i, 2}^{2}+c_{i, 1}\nonumber \\&-\frac{\gamma _{c_{i, 1}}}{2}( {\tilde{\varXi }}_{c_{i, 1}}^{T} S_{J_{i, 1}})^2 -\frac{\gamma _{c_{i, 1}}}{2} ({\tilde{\varXi }}_{a_{i, 1}}^{T} S_{J_{i, 1}})^2, \end{aligned}$$

(60)

where $c_{i,1}=(\gamma _{c_{i,1}}/2+\gamma _{a_{i,1}}/2)( \varXi _{J_{i,1}}^{*T}S_{J_{i,1}}) ^{2}+\varepsilon _{f_{i,1}}^{2}/2 + (\sigma _{i,1}/2)\varXi _{f_{i,1}}^{*T}\varXi _{f_{i,1}}^{*}$. Since all terms in $c_{i,1}$ are bounded, there exists a positive constant $C_{i,1}$ such that $|c_{i,1}(t)| \le C_{i,1}$.

Denote $\zeta _{\varGamma _{i,1}^{-1}}^{\max }$ and $\zeta _{S_{J_{i,1}}}^{\min }$ as the maximum eigenvalue and the minimum eigenvalue of $\varGamma _{i,1}^{-1}$ and $S_{J_{i,1}}S_{J_{i,1}}^T$, respectively. Then, one has

$$\begin{aligned} -{\tilde{\varXi }}_{{f_{i,1}}}^{T}{{{\tilde{\varXi }}}_{{f_{i,1}}}}&\le -\frac{1}{{{ \zeta _{\varGamma _{i,1}^{-1}}^{\max }}}}{\tilde{\varXi }}_{{f_{i,1}}}^{T}\varGamma _{i,1}^{-1}{{{\tilde{\varXi }}}_{{f_{i,1}}}}, \nonumber \\ -({\tilde{\varXi }}_{{c_{i,1}}}^{T}{S_{{J_{i,1}}}})^2&\le -\zeta _{{S_{{J_{i,1}}}}}^{\min }{\tilde{\varXi }}_{{c_{i,1}} }^{T}{{{\tilde{\varXi }}}_{{c_{i,1}}}}, \nonumber \\ -({\tilde{\varXi }}_{{a_{i,1}}}^{T}{S_{{J_{i,1}}}})^2&\le -\zeta _{{S_{{J_{i,1}}}}}^{\min }{\tilde{\varXi }}_{{a_{i,1}} }^{T}{{{\tilde{\varXi }}}_{{a_{i,1}}}}. \end{aligned}$$

(61)

Substituting (61) into (60), we obtain

$$\begin{aligned} {\dot{V}}_{i, 1} \le&-\beta _{i,1} z_{i, 1}^{2 l}\!-\!\frac{\sigma _{i, 1}}{2 \zeta _{\varGamma _{i, 1}^{-1}}^{\max }} {\tilde{\varXi }}_{f_{i, 1}}^{T} \varGamma _{i, 1}^{-1} {\tilde{\varXi }}_{f_{i, 1}}\!+\!\frac{1}{2} d_{i} z_{i, 2}^{2}+C_{i, 1}\\&-\frac{\gamma _{c_{i, 1}}}{2} \zeta _{J_{i, 1}}^{\min } {\tilde{\varXi }}_{c_{i, 1}}^{T} {\tilde{\varXi }}_{c_{i, 1}} -\frac{\gamma _{c_{i, 1}}}{2} \zeta _{J_{i, 1}}^{\min } {\tilde{\varXi }}_{a_{i, 1}}^{T} {\tilde{\varXi }}_{a_{i, 1}}. \end{aligned}$$

Step m ($2\le m\le n-1$): The Lyapunov function of the mth subsystem is defined as follows

$$\begin{aligned} V_{i,m}=&V_{i,m-1}+\frac{1}{2}z^2_{i,m}-\frac{1}{2}{\tilde{\varXi }} _{f_{i,m}}^{T}\varGamma _{i,m}^{-1}{\tilde{\varXi }} _{f_{i,m}}-\!\!\frac{1}{2}{\tilde{\varXi }}_{c_{i,m}}^{T}{\tilde{\varXi }}_{c_{i,m}}\\&-\frac{1}{2}{\tilde{\varXi }}_{a_{i,m}}^{T}{\tilde{\varXi }}_{a_{i,m}}. \end{aligned}$$

Then, the derivative of Lyapunov candidate function $V_{i,m}$ is as follows

$$\begin{aligned} \dot{V}_{i,m}=&\dot{V}_{i,m-1}+z_{i,m}\dot{z}_{i,m}-{\tilde{\varXi }} _{f_{i,m}}^{T}\varGamma _{i,m}^{-1} \dot{{\hat{\varXi }}}_{f_{i,m}} \!\!-{\tilde{\varXi }}_{c_{i,m}}^{T}\dot{{\hat{\varXi }}}_{c_{i,m}}\\&-{\tilde{\varXi }}_{a_{i,m}}^{T}\dot{{\hat{\varXi }}}_{a_{i,m}}. \end{aligned}$$

Based on (27)–(29), one has

$$\begin{aligned}&{\dot{V}}_{i,m}= z_{i,m}(-\beta _{i,m}z_{i,m}^{2l-1}-\frac{1}{2} {\hat{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}}-{\hat{\varXi }}_{f_{i,m}}^{T}S_{f_{i,m}} \nonumber \\&\quad +\varXi _{f_{i,m}}^{*T}S_{f_{i,m}}+\varepsilon _{f_{i,m}})+{\dot{V}}_{i,m-1}-{\tilde{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}}\nonumber \\&\quad \times S_{J_{i,m}}^{T} (\gamma _{a_{i,m}}({\hat{\varXi }}_{a_{i,m}}-{\hat{\varXi }}_{c_{i,m}})+\gamma _{c_{i,m}}{\hat{\varXi }}_{c_{i,m}}) \nonumber \\&\quad +{\tilde{\varXi }}_{f_{i,m}}^{T}(S_{f_{i,m}} z_{i,m}-\sigma _{i,m}{\hat{\varXi }}_{f_{i,m}})+z_{i,m}z_{i,m+1} \nonumber \\&\quad -\gamma _{c_{i,m}}{\tilde{\varXi }}_{c_{i,m}}^{T}S_{J_{i,m}}S_{J_{i,m}}^{T} {\hat{\varXi }} _{c_{i,m}} -\frac{7}{4} z_{i,m}^{2} . \end{aligned}$$

(62)

Using Young’s inequality, the following facts hold

$$\begin{aligned}&-\frac{1}{2}z_{i,m}{\hat{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}}\le \frac{1}{4} z_{i,m}^{2}+\frac{1}{4}({\hat{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}})^2,\\&\quad z_{i,m}z_{i,m+1}\le \frac{1}{2}z_{i,m}^{2}+\frac{1}{2} z_{i,m+1}^{2},\\&\quad z_{i,m}\varepsilon _{f_{i,m}}\le \frac{1}{2}z_{i,m}^{2}+\frac{1}{2} \varepsilon _{f_{i,m}}^{2}. \end{aligned}$$

Owing to the fact that ${\tilde{\varXi }}_{o_{i,m}}={\hat{\varXi }}_{o_{i,m}}-\varXi _{o_{i,m}}^{*}$ with $o=f,a,c$ and based on Young’s inequality, we have

$$\begin{aligned}&{\tilde{\varXi }}_{f_{i,m}}^{T}{\hat{\varXi }}_{f_{i,m}}\\&=\frac{1}{2}{\tilde{\varXi }} _{f_{i,m}}^{T}{\tilde{\varXi }}_{f_{i,m}}+\frac{1}{2}{\hat{\varXi }}_{f_{i,m}}^{T}{\hat{\varXi }}_{f_{i,m}} -\frac{1}{2}\varXi _{f_{i,m}}^{*T}\varXi _{f_{i,m}}^{*},\\&{\tilde{\varXi }}_{c_{i,m}}^{T}S_{J_{i,m}}S_{J_{i,m}}^{T}{\hat{\varXi }}_{c_{i,m}}\\&= \frac{1}{2}(({\tilde{\varXi }}_{c_{i,m}}^{T}S_{J_{i,m}})^2 +({\hat{\varXi }}_{c_{i,m}}^{T}S_{J_{i,m}})^2-( \varXi _{J_{i,m}}^{*T}S_{J_{i,m}}) ^{2}),\\&{\tilde{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}}S_{J_{i,m}}^{T}{\hat{\varXi }}_{a_{i,m}}\\&=\frac{1}{2}(({\tilde{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}})^2 +({\hat{\varXi }}_{a_{i,m}}^{T}S_{J_{i,m}})^2-( \varXi _{J_{i,m}}^{*T}S_{J_{i,m}}) ^{2}). \end{aligned}$$

By using Young’s inequality, one has

$$\begin{aligned} {\tilde{\varXi }}_{a_{i, m}}^{T}\!S_{J_{i, m}}\! S_{J_{i, m}}^{T} \!{\hat{\varXi }}_{c_{i, m}} \le \frac{({\tilde{\varXi }}_{a_{i, m}}^{T} \!S_{J_{i, m}} )^2}{2}\! \!+\!\!\frac{({\hat{\varXi }}_{c_{i, m}}^{T}\! S_{J_{i, m}})^2}{2}. \end{aligned}$$

Substituting the above equations into (62), it yields

$$\begin{aligned} {\dot{V}}_{i,m}\le&-\frac{\gamma _{c_{i,m}}}{2}({\tilde{\varXi }} _{a_{i,m}}^{T}S_{J_{i,m}})^2\!\!-\!(\frac{\gamma _{a_{i,m}}}{2}\!\!-\!\frac{1}{4})({\hat{\varXi }} _{a_{i,m}}^{T}\!S_{J_{i,m}})^2\nonumber \\&-\sum _{k=1}^{m-1}\frac{\gamma _{c_{i,k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{a_{i,k}}^{T} {\tilde{\varXi }}_{a_{i,k}}-\frac{\sigma _{i,m}}{2} {\tilde{\varXi }}_{f_{i,m}}^{T}{\tilde{\varXi }}_{f_{i,m}} \nonumber \\&-\frac{\gamma _{c_{i,m}}}{2}({\tilde{\varXi }} _{c_{i,m}}^{T}S_{J_{i,m}})^2+\frac{1}{2}z_{i,m+1}^{2}+c_{i,m} \nonumber \\&-\sum _{k=1}^{m}\beta _{i,k}z_{i,k}^{2l}-\sum _{k=1}^{m-1}\frac{\gamma _{c_{i, k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{c_{i,k}}^{T} {\tilde{\varXi }}_{c_{i,k}}\nonumber \\&-\sum _{k=1}^{m-1}\frac{\sigma _{i, k}}{2 \zeta _{\varGamma _{i, k}^{-1}}^{\max }} {\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}+\sum _{k=1}^{m-1}C_{i,k}\nonumber \\&-(\gamma _{c_{i,m}}-\frac{\gamma _{a_{i,m}}}{2})({\hat{\varXi }} _{c_{i,m}}^{T}S_{J_{i,m}})^2, \end{aligned}$$

(63)

where $c_{i,m}=\frac{\gamma _{c_{i,m}}}{2}( \varXi _{J_{i,m}}^{*T}S_{J_{i,m}}) ^{2}+\frac{\gamma _{a_{i,m}}}{2}( \varXi _{J_{i,m}}^{*T}S_{J_{i,m}}) ^{2}+\frac{\sigma _{i,m}}{2} \varXi _{f_{i,m}}^{*T}\varXi _{f_{i,m}}^{*}+\frac{1}{2}\varepsilon _{f_{i,m}}^{2}$ is a bounded function by a positive constant $C_{i,m}$, that is, $|c_{i,m}(t)|\le C_{i,m}$, $\zeta _{\varGamma _{i,m}^{-1}}^{\max }$ and $\zeta _{S_{J_{i,m}}}^{\min }$ are denoted as the maximum eigenvalue and the minimum eigenvalue of $\varGamma _{i,m}^{-1}$ and $S_{J_{i,m}}S_{J_{i,m}}^T$, respectively.

Similar to Step 1 and based on the condition (30), (63) becomes

$$\begin{aligned} {\dot{V}}_{i,m}\le&-\sum _{k=1}^{m}\beta _{i,k}z_{i,k}^{2l}-\sum _{k=1}^{m}\frac{\sigma _{i,k}}{2 \zeta _{\varGamma _{i, k}^{-1}}^{\max }} {\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}\\&-\sum _{k=1}^{m}\frac{\gamma _{c_{i,k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{a_{i,k}}^{T} {\tilde{\varXi }}_{a_{i,k}}+\frac{1}{2}z_{i,m+1}^{2}\\&-\sum _{k=1}^{m}\frac{\gamma _{c_{i, k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{c_{i,k}}^{T} {\tilde{\varXi }}_{c_{i,k}}+\sum _{k=1}^{m}C_{i,k}. \end{aligned}$$

Step n: In the last Step, we choose the following Lyapunov function

$$\begin{aligned}&V_{i,n}= V_{i,n-1}+\frac{1}{2}z_{i,n}^{2}+\frac{1}{2}{\tilde{\varXi }} _{f_{i,n}}^{T}\varGamma _{i,n}^{-1}{\tilde{\varXi }}_{f_{i,n}}+\frac{1}{2}\tilde{u}^2_{f_i}\\&\quad +\frac{1}{2}{\tilde{\varXi }}_{c_{i,n}}^{T}{\tilde{\varXi }}_{c_{i,n}}+\frac{1}{2}{\tilde{\varXi }} _{a_{i,n}}^{T}{\tilde{\varXi }}_{a_{i,n}}, \end{aligned}$$

where ${{\tilde{\varXi }}}_{f_{i,n}}$ $={{\hat{\varXi }}}_{f_{i,n}}-{\varXi }_{f_{i,n}}^{*} ,{\tilde{\varXi }}_{c_{i,n}}={\hat{\varXi }}_{c_{i,n}}-\varXi _{c_{i,n}}^{*}$, ${\tilde{\varXi }} _{a_{i,n}}={\hat{\varXi }}_{a_{i,n}}-\varXi _{a_{i,n}}^{*}$ and ${\tilde{u}}_{f_{i}}$ $={u}_{f_{i}}-{{\hat{u}}}_{f_{i}}$.

According to (47), the time derivative of $V_{i,n}$ is given by

$$\begin{aligned} \dot{V}_{i,n}=&\dot{V}_{i,n-1}+z_{i,n}({{\hat{u}}}_{i}^*+g_i{\tilde{u}}_{f_i}+f_{i,n}-\dot{{\hat{\alpha }}}_{i,n-1}^{*})\nonumber \\&+{\tilde{\varXi }} _{f_{i,n}}^{T}\varGamma _{i,n}^{-1}\dot{{\hat{\varXi }}}_{f_{i,n}} +{\tilde{\varXi }}_{c_{i,n}}^{T}\dot{{\hat{\varXi }}}_{c_{i,n}}+{\tilde{\varXi }} _{a_{i,n}}^{T}\dot{{\hat{\varXi }}}_{a_{i,n}}\nonumber \\&+\tilde{u}_{f_i}({{\dot{u}}}_{f_{i}}-\dot{{\hat{u}}}_{f_{i}}). \end{aligned}$$

(64)

Substituting (49) into (64), one has

$$\begin{aligned} \dot{V}_{i,n}=&\dot{V}_{i,n-1}+z_{i,n}\left( {{\hat{u}}}_{i}^*+F_{i,n}-\frac{5}{4}z_{i,n}\right) +{\tilde{\varXi }} _{a_{i,n}}^{T}\dot{{\hat{\varXi }}}_{a_{i,n}}\nonumber \\&+{\tilde{\varXi }} _{f_{i,n}}^{T}\varGamma _{i,n}^{-1}\dot{{\hat{\varXi }}}_{f_{i,n}} +{\tilde{\varXi }}_{c_{i,n}}^{T}\dot{{\hat{\varXi }}}_{c_{i,n}} +{{\dot{u}}}_{f_{i}}\tilde{u}_{f_i}\nonumber \\&+\gamma _{t_{i}}\tilde{u}_{f_i}{\hat{u}}_{f_i}, \end{aligned}$$

(65)

where $F_{i,n}=f_{i,n}-\dot{{\hat{\alpha }}}_{i,n-1}^{*}+(5/4)z_{i,n}$.

Then, according to Young’s inequality and Assumption 2, we can obtain

$$\begin{aligned} \gamma _{t_{i}}\tilde{u}_{f_i}{\hat{u}}_{f_i}&\le -\gamma _{t_{i}}\tilde{u}^2_{f_i}+\frac{\gamma _{t_{i}}}{2}\varpi ^2_{i,1}+\frac{\gamma _{t_{i}}}{2}\tilde{u}^2_{f_i}, \end{aligned}$$

(66)

$$\begin{aligned} {{\dot{u}}}_{f_{i}}\tilde{u}_{f_i}&\le \frac{1}{2}\varpi ^2_{i,2}+\frac{1}{2}\tilde{u}^2_{f_i}, \end{aligned}$$

(67)

where $\varpi _{i,1}$ and $\varpi _{i,2}$ are constants.

Combining (49) and (65)–(67), and similar to Step 1, (65) yields

$$\begin{aligned} {\dot{V}}_{i,n}\!\le \!&-\!\sum _{k=1}^{n}\beta _{i,k}z_{i,k}^{2l}\!-\!\!\sum _{k=1}^{n}\frac{\sigma _{i,k}}{2\zeta _{\varGamma _{i, k}^{-1}}^{\max }} {\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}\!\!+\!\sum _{k=1}^{n}C_{i,k}\\&-\sum _{k=1}^{n}\frac{\gamma _{c_{i,k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{a_{i,k}}^{T} {\tilde{\varXi }}_{a_{i,k}}+\frac{\gamma _{t_{i}}}{2}\varpi ^2_{i,1}+\frac{1}{2}\varpi ^2_{i,2}\\&-\sum _{k=1}^{n}\frac{\gamma _{c_{i, k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{c_{i,k}}^{T} {\tilde{\varXi }}_{c_{i,k}}-\left( \frac{\gamma _{t_i}-1}{2}\right) \tilde{u}^2_{f_i}. \end{aligned}$$

Next, the proof of Theorem 1 (1) is given as follows. The total Lyapunov function candidate is selected as

$$\begin{aligned} V=\sum _{i=1}^{Q}{V}_{i,n} . \end{aligned}$$

Then, the time derivative of V is given by

$$\begin{aligned} \dot{V} \le&\sum _{i=1}^{Q}\Bigg (-\sum _{k=1}^{n}\beta _{i,k}z_{i,k}^{2l}-\sum _{k=1}^{n}\frac{\sigma _{i,k}}{2\zeta _{\varGamma _{i, k}^{-1}}^{\max }} {\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}\nonumber \\&-\sum _{k=1}^{n}\frac{\gamma _{c_{i, k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{c_{i,k}}^{T} {\tilde{\varXi }}_{c_{i,k}}-\left( \frac{\gamma _{t_i}-1}{2}\right) \tilde{u}^2_{f_i}\nonumber \\&-\sum _{k=1}^{n}\frac{\gamma _{c_{i,k}}}{2} \zeta _{J_{i,k}}^{\min } {\tilde{\varXi }}_{a_{i,k}}^{T} {\tilde{\varXi }}_{a_{i,k}}+D_{i}\Bigg ), \end{aligned}$$

(68)

where $D_i=\sum _{k=1}^{n}C_{i,k}+\frac{\gamma _{t_{i}}}{2}\varpi ^2_{i,1}+\frac{1}{2}\varpi ^2_{i,2}$ is a constant.

Moreover, by Lemma 4 and denoting $p=1,q={\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}$, and $\imath =1-l$, $\jmath =l$, $\tau =l^{\frac{l}{ 1-l}}$, the following inequality holds

$$\begin{aligned} ({\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}) ^{l}\le \tau l+{\tilde{\varXi }}_{f_{i, k}}^{T} \varGamma _{i, k}^{-1} {\tilde{\varXi }}_{f_{i, k}}. \end{aligned}$$

(69)

Similar to (69), we can get

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&({\tilde{\varXi }}_{c_{i,k}}^{T}{\tilde{\varXi }}_{c_{i,k}}) ^{l}\le \ \tau l+\tilde{\varXi }_{c_{i,k}}^{T}{\tilde{\varXi }}_{c_{i,k}},\\&({\tilde{\varXi }}_{a_{i,k}}^{T}{\tilde{\varXi }}_{a_{i,k}}) ^{l}\le \ \tau l+\tilde{\varXi }_{a_{i,k}}^{T}{\tilde{\varXi }}_{a_{i,k}},\\&(\tilde{u}_{f_{i}}^2) ^{l}\le \ \tau l+\tilde{u }_{f_{i}}^2. \end{aligned}\right. \end{aligned} \end{aligned}$$

(70)

Remark 3

The above inequalities are based on Lemma 3 and Lemma 4, which are proposed in [40] and are common in the results related to finite-time control. These lemmas are used to ensure that the power of system weight vector errors and the fault observation error meets the requirement of finite-time control.

Further, (68) becomes

$$\begin{aligned} \dot{V}\le&\sum _{i=1}^{Q}\sum _{k=1}^{n}(-2^l\beta _{i,k}(\frac{z_{i,k}^{2}}{2})^l\!-\!\frac{\sigma _{i,k}}{2{ \zeta _{{\varGamma _{i,k}^{-1}}}^{\max }}}({\tilde{\varXi }}_{f_{i,k}}^{T}\varGamma _{i,k}^{-1}{\tilde{\varXi }}_{f_{i,k}})^{l} \\&+\frac{\sigma _{i,k}}{2{\zeta _{{\varGamma _{i,k}^{-1}}}^{\max }}}\tau l\!-\!\frac{ \gamma _{c_{i,k}}}{2}{\zeta _{{S_{{J_{i,k}}}}}^{\min }}({\tilde{\varXi }} _{a_{i,k}}^{T}{\tilde{\varXi }}_{a_{i,k}})^{l}\!+\!\frac{\gamma _{t_i}-1}{2}\tau l\\&-\frac{\gamma _{c_{i,k}}}{2}{\zeta _{{S_{{J_{i,k}}}}}^{\min }}({\tilde{\varXi }} _{c_{i,k}}^{T}{\tilde{\varXi }}_{c_{i,k}})^{l}+\frac{\gamma _{c_{i,k}}}{2}{\zeta _{ {S_{{J_{i,k}}}}}^{\min }}\tau l \\&+\frac{\gamma _{c_{i,k}}}{2}{\zeta _{{S_{{J_{i,k}}}}}^{\min }}\tau l+D_{i} -\frac{\gamma _{t_i}-1}{2}(\tilde{u}_{f_{i}}^2) ^{l})\\ \le&\ a{V^l}+b, \end{aligned}$$

where $a=\min \{-2^l\beta _{i,k},\frac{\sigma _{i,k}}{2{\zeta _{{\varGamma _{i,k}^{-1}}}^{\max }}}, \frac{\gamma _{c_{i,k}}}{2}{\zeta _{{S_{{J_{i,k}}}}}^{\min }},\frac{\gamma _{t_i}-1}{2}\}$ and $b=\min \{ \frac{\sigma _{i,k}}{2{\zeta _{{\varGamma _{i,k}^{-1}}}^{\max }}}\tau l, \frac{\gamma _{c_{i,k}}}{2}{\zeta _{{S_{{J_{i,k}}}}}^{\min }}\tau l, D_{i,k},\frac{\gamma _{t_i}-1}{2}\tau l\}$.

Define $T^R=[V^{1-l}z(0)-([(b/(1-p)a)])^{(1-l)/l}][1/(1-l)pa]$ and $p\in (0,1)$, then, on the basis of Lemma 2, $\forall t\ge T^R, V^l\le [b/(1-p)a]$ holds, i.e., all signals in the closed-loop system are SGPFS.

In addition, according to the definition of V in this section, one has

$$\begin{aligned} \Vert {\underline{z}}\Vert \le 2(\frac{b}{(1-p)a})^{\frac{1}{2l}},\ \forall t\ge T^R, \end{aligned}$$

where ${\underline{z}}=[z_{1,1},z_{2,1},\ldots ,z_{Q,1}]^{T}\!\in \! {\mathbb {R}}^{Q}$. Denote ${\underline{y}}\! =\! [y_{1},\ldots ,y_{Q}]^{T}\in {\mathbb {R}}^{N}$ and ${\underline{y}}_{d}=[y_{Q+1,d},\ldots ,y_{Q+P,d}]^{T}\in {\mathbb {R}}^{P}$, then, the following fact holds

$$\begin{aligned} {\underline{z}}={\mathbb {L}}_{1}{\underline{y}}+{\mathbb {L}}_{2}{\underline{y}}_{d} ={\mathbb {L}}_{1}({\underline{y}}+{\mathbb {L}}_{1}^{-1}{\mathbb {L}}_{2}{\underline{y}}_{d}). \end{aligned}$$

Therefore, one gets

$$\begin{aligned} \Vert {\underline{y}}+{\mathbb {L}}_{1}^{-1}{\mathbb {L}}_{2}{\underline{y}}_{d}\Vert \le \frac{\Vert {\underline{z}}\Vert }{ \lambda _{min}({\mathbb {L}}_{1})} \le \frac{2(\frac{b}{(1-p)a})^{\frac{1}{2l}}}{\lambda _{min}({\mathbb {L}}_{1})}, \end{aligned}$$

(71)

where $\lambda _{min}({\mathbb {L}}_{1})$ represents the minimum eigenvalue of the matrix ${\mathbb {L}}_{1}$. From formula (71), it means that all followers eventually converge to the convex hull space formed by multiple leaders, and the goal of containment control is achieved.

Thus, the proof of Theorem 1 is completed.