Appendix A
The derivations of (28) are given as follows:
From (15)–(24), the time derivative of e is described as
$$\begin{aligned} {\dot{e}}= & {} {\dot{x}}-{\dot{\eta }} \nonumber \\= & {} A\left( x-\eta \right) +\sum _{i=1}^{n}H_{i}\left( f\left( {\bar{\eta }} _{i}\right) +\varDelta F_{i}-{\hat{f}}_{i}\left( {\bar{\eta }}_{i}|\theta _{i}\right) \right) \nonumber \\= & {} Ae+\sum _{i=1}^{n}H_{i}\left( {\hat{f}}_{i}\left( {\bar{\eta }}_{i}|\theta _{i}^{*}\right) +\varepsilon _{i}+\varDelta F_{i}-{\hat{f}}_{i}\left( \bar{ \eta }_{i}|\theta _{i}\right) \right) \nonumber \\= & {} Ae+\sum _{i=1}^{n}H_{i}\left( {\tilde{\theta }}_{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) +\varepsilon _{i}+\varDelta F_{i}\right) . \end{aligned}$$
(54)
Then, the time derivative of \(V_{e}\) can be obtained.
$$\begin{aligned} {\dot{V}}_{e}= & {} e^{\mathrm{T}}P\left( Ae+\sum _{i=1}^{n}H_{i}^{\mathrm{T}}{\tilde{\theta }} _{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) +\varDelta F_{i}+\varepsilon _{i}\right) \nonumber \\\le & {} -e^{\mathrm{T}}Qe+e^{\mathrm{T}}P\sum _{i=1}^{n}H_{i}^{\mathrm{T}}{\tilde{\theta }}_{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) \nonumber \\&+\,e^{\mathrm{T}}P\left( \varepsilon _{i}+\varDelta F_{i}\right) . \end{aligned}$$
(55)
Combining Young’s inequality and \(\varphi _{i}^{\mathrm{T}}\left( {\bar{\eta }} _{i}\right) \varphi _{i}\left( {\bar{\eta }}_{i}\right) \le 1\) yields
$$\begin{aligned} e^{\mathrm{T}}P\sum _{i=1}^{n}H_{i}{\tilde{\theta }}_{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }} _{i}\right) \!\le \! \frac{1}{2}\lambda _{\max }\left( P\right) \left| \left| e\right| \right| ^{2}+\sum _{j=1}^{n}{\tilde{\theta }}_{j}^{\mathrm{T}} {\tilde{\theta }}_{j}. \nonumber \\ \end{aligned}$$
(56)
By applying the method of Young’s inequality and Assumption 1, (58)–(57) holds.
$$\begin{aligned}&e^{\mathrm{T}}P\varDelta F_{i}\le \frac{1}{2}\left| \left| e\right| \right| ^{2}+\frac{1}{2}\sum _{j=1}^{n}\left| \left| P\right| \right| ^{2}m_{j}^{2}\left| \left| e\right| \right| ^{2}, \end{aligned}$$
(57)
$$\begin{aligned}&e^{\mathrm{T}}P\varepsilon _{i}\le \frac{1}{2}\left| \left| e\right| \right| ^{2}+\frac{1}{2}\sum _{j=1}^{n}\left| \left| P\right| \right| ^{2}\varepsilon _{j}^{*2}. \end{aligned}$$
(58)
Substituting (56)–(58) into (55) produces
$$\begin{aligned} {\dot{V}}_{e}\le & {} -\lambda _{\min }\left( Q\right) \left| \left| e\right| \right| ^{2}+\left( \frac{1}{2}\lambda _{\max }\left( P\right) +1\right) \left| \left| e\right| \right| ^{2} \nonumber \\&+\,\frac{1}{2}\left| \left| P\right| \right| ^{2}\sum \limits _{j=1}^{n}m_{j}^{2}\left| \left| e\right| \right| ^{2}+\sum _{j=1}^{n}\tilde{\theta }_{j}^{\mathrm{T}}\tilde{\theta }_{j} \nonumber \\&+\,\frac{1}{2}\sum _{j=1}^{n}\left| \left| P\right| \right| ^{2}\varepsilon _{j}^{*2} \nonumber \\\le & {} -q_{0}\left| \left| e\right| \right| ^{2}+\sum _{j=1}^{n}\tilde{\theta }_{j}^{\mathrm{T}}\tilde{\theta }_{j}+\lambda _{0}. \end{aligned}$$
(59)
Appendix B
The specific design process are given as follows:
Step 1:
Consider the following Lyapunov function candidate:
$$\begin{aligned} V_{z1}=\frac{1}{2}z_{1}^{2}+\frac{1}{2r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}\tilde{ \theta }_{1}. \end{aligned}$$
(60)
The time-derivative of \(V_{z1}\) can be expressed as
$$\begin{aligned} {\dot{V}}_{z1}= & {} z_{1}{\dot{z}}_{1}-\frac{1}{r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}\dot{ \theta }_{1} \nonumber \\= & {} z_{1}\left( \varGamma +\digamma \left( \eta _{2}+e_{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +{\tilde{\theta }}_{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \right) \right) \nonumber \\&+\,z_{1}\digamma \left( \varDelta F_{1}-{\dot{y}}_{d}+\varepsilon _{1}\right) - \frac{1}{r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}{\dot{\theta }}_{1} \nonumber \\= & {} z_{1}(\varGamma +\digamma \left( \alpha _{1}+e_{2}-{\dot{y}}_{d}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +\varepsilon _{1}\right) ) \nonumber \\&+\,\digamma z_{1}z_{2}+\frac{1}{r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}\left( r_{1}z_{1}\digamma \varphi _{1}\left( \eta _{1}\right) -{\dot{\theta }} _{1}\right) . \end{aligned}$$
(61)
By applying the method of Young’s inequality, it is easily verified that
$$\begin{aligned} z_{1}\digamma \left( e_{2}+\varDelta F_{1}\right)\le & {} z_{1}^{2}\digamma ^{2}+ \frac{1+m_{1}^{2}}{2}\left\| e\right\| ^{2}, \end{aligned}$$
(62)
$$\begin{aligned} z_{1}\digamma \varepsilon _{1}\le & {} \frac{1}{2}z_{1}^{2}\digamma ^{2}+ \frac{1}{2}\varepsilon _{1}^{*2}. \end{aligned}$$
(63)
Then, substituting (62) and (63) into (61) gives
$$\begin{aligned} {\dot{V}}_{z1}\le & {} z_{1}\left( \varGamma +\digamma \left( \alpha _{1}+\frac{3}{ 2}\digamma z_{1}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) -{\dot{y}} _{d}\right) \right) \nonumber \\&+\,\digamma z_{1}z_{2}+\frac{1}{2}\varepsilon _{1}^{*2}+\frac{1+m_{1}^{2} }{2}\left\| e\right\| ^{2} \nonumber \\&+\,\frac{1}{r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}\left( r_{1}z_{1}\digamma \varphi _{1}\left( \eta _{1}\right) -{\dot{\theta }}_{1}\right) . \end{aligned}$$
(64)
Substituting (38) and (39) into (64), it yields
$$\begin{aligned} {\dot{V}}_{z1}\le & {} -c_{1}z_{1}^{2}+\frac{\kappa _{1}}{r_{1}}{\tilde{\theta }} _{1}^{\mathrm{T}}\theta _{1}+\digamma z_{1}z_{2} \nonumber \\&+\,\frac{1+m_{1}^{2}}{2}\left\| e\right\| ^{2}+\frac{1}{2}\varepsilon _{1}^{*2}. \end{aligned}$$
(65)
Based on Young’s inequality, it produces that
$$\begin{aligned} {\tilde{\theta }}_{1}^{\mathrm{T}}\theta _{1}\le \frac{1}{2}\left| \left| \theta _{1}^{*}\right| \right| ^{2}-\frac{1}{2}{\tilde{\theta }} _{1}^{\mathrm{T}}{\tilde{\theta }}_{1}. \end{aligned}$$
(66)
Therefore, (64) can be rewritten as
$$\begin{aligned} {\dot{V}}_{z1}\le & {} -c_{1}z_{1}^{2}-\frac{\kappa _{1}}{2r_{1}}{\tilde{\theta }} _{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\digamma z_{1}z_{2}+\frac{1+m_{1}^{2}}{2} \left\| e\right\| ^{2} \nonumber \\&+\,\frac{\kappa _{1}}{2r_{1}}\left| \left| \theta _{1}^{*}\right| \right| ^{2}+\frac{1}{2}\varepsilon _{1}^{*2}. \end{aligned}$$
(67)
Step 2:
Construct the Lyapunov function for the second subsystem as follows:
$$\begin{aligned} V_{z2}=V_{z1}+\frac{1}{2}z_{2}^{2}+\frac{1}{2r_{2}}{\tilde{\theta }}_{2}^{\mathrm{T}} {\tilde{\theta }}_{2}. \end{aligned}$$
(68)
The time-derivative of \(V_{z2}\) can be described as
$$\begin{aligned} {\dot{V}}_{z2}= & {} {\dot{V}}_{z1}+z_{2}{\dot{z}}_{2}-\frac{1}{r_{2}}{\tilde{\theta }} _{2}^{\mathrm{T}}{\dot{\theta }}_{2} \nonumber \\= & {} {\dot{V}}_{z1}+z_{2}\left( \eta _{3}+\theta _{2}^{\mathrm{T}}\varphi _{2}\left( \bar{ \eta }_{2}\right) +k_{2}e_{1}\right) \nonumber \\&-\,z_{2}{\dot{\alpha }}_{1}-\frac{1}{r_{2}}{\tilde{\theta }}_{2}^{\mathrm{T}}{\dot{\theta }} _{2}, \end{aligned}$$
(69)
where
$$\begin{aligned} {\dot{\alpha }}_{1}= & {} \frac{\partial \alpha _{1}}{\partial y}\left( \eta _{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +{\tilde{\theta }} _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +e_{2}+\varepsilon _{1}\right) \nonumber \\&+\,\frac{\partial \alpha _{1}}{\partial \eta _{1}}\left( \eta _{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +k_{1}e_{1}\right) \nonumber \\&+\,\sum _{j=1}^{2}\frac{\partial \alpha _{1}}{\partial y_{d}^{\left( j-1\right) }}y_{d}^{\left( j\right) }+\frac{\partial \alpha _{1}}{\partial \theta _{1}}{\dot{\theta }}_{1}+\frac{\partial \alpha _{1}}{\partial \nu }\nu ^{\left( 1\right) }. \end{aligned}$$
(70)
According to (70) , \({\dot{V}}_{z2}\) can be rewritten as
$$\begin{aligned} {\dot{V}}_{z2}= & {} {\dot{V}}_{z1}+z_{2}\left( z_{3}+\alpha _{2}+k_{2}e_{1}+\theta _{2}^{\mathrm{T}}\varphi _{2}\left( {\bar{\eta }}_{2}\right) \right) \nonumber \\&+\,z_{2}\left( {\tilde{\theta }}_{2}^{\mathrm{T}}\varphi _{2}\left( {\bar{\eta }} _{2}\right) -{\tilde{\theta }}_{2}^{\mathrm{T}}\varphi _{2}\left( {\bar{\eta }}_{2}\right) - \frac{\partial \alpha _{1}}{\partial y}\left( \varepsilon _{1}+e_{2}\right) \right) \nonumber \\&-\,z_{2}\left( \frac{\partial \alpha _{1}}{\partial y}({\tilde{\theta }} _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +\eta _{2})\right) \nonumber \\&-\,z_{2}\frac{\partial \alpha _{1}}{\partial \eta _{1}}\left( \eta _{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +k_{1}e_{1}\right) \nonumber \\&-\,z_{2}\left( \sum _{j=1}^{2}\frac{\partial \alpha _{1}}{\partial y_{d}^{\left( j-1\right) }}y_{d}^{\left( j\right) }+\frac{\partial \alpha _{1}}{\partial \theta _{1}}{\dot{\theta }}_{1}\right) \nonumber \\&-\,z_{2}\frac{\partial \alpha _{1}}{\partial \nu }\nu ^{\left( 1\right) }- \frac{1}{r_{2}}{\tilde{\theta }}_{2}^{\mathrm{T}}{\dot{\theta }}_{2}. \end{aligned}$$
(71)
By applying the method of Young’s inequality, it is verified that
$$\begin{aligned} -z_{2}{\tilde{\theta }}_{2}^{\mathrm{T}}\varphi _{2}\left( {\bar{\eta }}_{2}\right) \le \frac{1}{2}z_{2}^{2}+\frac{1}{2}{\tilde{\theta }}_{2}^{\mathrm{T}}{\tilde{\theta }}_{2}, \end{aligned}$$
(72)
$$\begin{aligned} -z_{2}\frac{\partial \alpha _{1}}{\partial y}{\tilde{\theta }}_{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \le \frac{1}{2}z_{2}^{2}\left( \frac{\partial \alpha _{1}}{\partial y}\right) ^{2}+\frac{1}{2}{\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}, \end{aligned}$$
(73)
$$\begin{aligned} -z_{2}\frac{\partial \alpha _{1}}{\partial y}\left( \varepsilon _{1}+e_{2}\right) \le z_{2}^{2}\left( \frac{\partial \alpha _{1}}{\partial y }\right) ^{2}\!+\!\frac{1}{2}\varepsilon _{1}^{*2}\!+\!\frac{1}{2}\left\| e\right\| ^{2}. \nonumber \\ \end{aligned}$$
(74)
Then, substituting (67), (72)–(74) into (71) yields
$$\begin{aligned} {\dot{V}}_{z2}\le & {} -c_{1}z_{1}^{2}+z_{2}\left( z_{3}+\alpha _{2}+D_{2}+\digamma z_{1}+\frac{1}{2}z_{2}\right) \nonumber \\&+\,\frac{3}{2}z_{2}^{2}\left( \frac{\partial \alpha _{1}}{\partial y}\right) ^{2}+\frac{1}{r_{2}}{\tilde{\theta }}_{2}^{\mathrm{T}}\left( r_{2}z_{2}\varphi _{2}\left( {\bar{\eta }}_{2}\right) -{\dot{\theta }}_{2}\right) \nonumber \\&+\,\sum _{j=1}^{2}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }}_{j}+\frac{ \kappa _{1}}{2r_{1}}\left| \left| \theta _{1}^{*}\right| \right| ^{2} \nonumber \\&-\,\frac{\kappa _{1}}{2r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }} _{1}+\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}+\frac{1}{2} m_{1}^{2}\left\| e\right\| ^{2}. \end{aligned}$$
(75)
According the design of controller, substituting (38) and (39) into (40) yields
$$\begin{aligned} {\dot{V}}_{z2}\le & {} -\sum _{j=1}^{2}c_{j}z_{j}^{2}+z_{2}z_{3}+\frac{\kappa _{2}}{r_{2}}{\tilde{\theta }}_{2}^{\mathrm{T}}\theta _{2}+\sum _{j=1}^{2}\frac{1}{2} {\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }}_{j} \nonumber \\&-\,\frac{\kappa _{1}}{2r_{1}}{\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\frac{ \kappa _{1}}{2r_{1}}\left| \left| \theta _{1}^{*}\right| \right| ^{2} \nonumber \\&+\,\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}+\frac{1}{2} m_{1}^{2}\left\| e\right\| ^{2} \nonumber \\\le & {} -\sum _{j=1}^{2}c_{j}z_{j}^{2}+z_{2}z_{3}-\sum _{j=1}^{2}\frac{\kappa _{j}}{2r_{j}}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }}_{j} \nonumber \\&+\,\sum _{j=1}^{2}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }} _{j}+\sum _{j=1}^{2}\frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2} \nonumber \\&+\,\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}+\frac{1}{2} m_{1}^{2}\left\| e\right\| ^{2}. \end{aligned}$$
(76)
Step i:
Construct the Lyapunov function for the i-th subsystem as follows:
$$\begin{aligned} V_{zi}=V_{zi-1}+\frac{1}{2}z_{i}^{2}+\frac{1}{2r_{i}}{\tilde{\theta }}_{i}^{\mathrm{T}} {\tilde{\theta }}_{i}. \end{aligned}$$
(77)
Subsequently, the time-derivative of \(V_{zi}\) can be described as
$$\begin{aligned} {\dot{V}}_{zi}= & {} {\dot{V}}_{zi-1}+z_{i}\left( \eta _{i+1}+\theta _{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) +k_{i}e_{1}\right) \nonumber \\&-\,z_{i}{\dot{\alpha }}_{i-1}-\frac{1}{r_{i}}{\tilde{\theta }}_{i}^{\mathrm{T}}{\dot{\theta }} _{i}, \end{aligned}$$
(78)
where
$$\begin{aligned} {\dot{\alpha }}_{i-1}= & {} \frac{\partial \alpha _{i-1}}{\partial y}\left( \eta _{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +{\tilde{\theta }} _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +e_{2}+\varepsilon _{1}\right) \nonumber \\&+\,\sum _{j=1}^{i}\frac{\partial \alpha _{j-1}}{\partial y_{d}^{\left( j-1\right) }}y_{d}^{\left( j\right) }+\sum _{j=1}^{i-1}\frac{\partial \alpha _{j}}{\partial \theta _{j}}\theta _{j}^{\left( j\right) } \nonumber \\&+\,\sum _{j=1}^{i-1}\frac{\partial \alpha _{j}}{\partial \eta _{j}}\left( \eta _{j+1}+\theta _{j}^{\mathrm{T}}\varphi _{j}\left( {\bar{\eta }}_{j}\right) +k_{j}e_{1}\right) \nonumber \\&+\,\sum _{j=1}^{i-1}\frac{\partial \alpha _{j-1}}{\partial \nu ^{\left( j-1\right) }}\nu ^{\left( j\right) }. \end{aligned}$$
(79)
According to (79) , \({\dot{V}}_{zi}\) can be expressed as
$$\begin{aligned} {\dot{V}}_{zi}= & {} {\dot{V}}_{zi-1}+z_{i}\left( z_{i+1}+\alpha _{i}+k_{i}e_{1}+\theta _{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) \right) \nonumber \\&+\,z_{i}\left( {\tilde{\theta }}_{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }} _{i}\right) -{\tilde{\theta }}_{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) - \frac{\partial \alpha _{i-1}}{\partial y}{\tilde{\theta }}_{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \right) \nonumber \\&-\,z_{i}\left( \frac{\partial \alpha _{i-1}}{\partial y}\left( e_{2}+\varepsilon _{1}+\eta _{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \right) \right) \nonumber \\&-\,z_{i}\left( \sum _{j=1}^{i-1}\frac{\partial \alpha _{j}}{\partial \eta _{j} }\left( \eta _{j+1}+\theta _{j}^{\mathrm{T}}\varphi _{j}\left( {\bar{\eta }}_{j}\right) +k_{j}e_{1}\right) \right) \nonumber \\&-\,z_{i}\left( \sum _{j=1}^{i-1}\frac{\partial \alpha _{j}}{\partial \theta _{j}}\theta _{j}^{\left( j\right) }+\sum _{j=1}^{i}\frac{\partial \alpha _{j-1}}{\partial y_{d}^{\left( j-1\right) }}y_{d}^{\left( j\right) }\right) \nonumber \\&-\,z_{i}\sum _{j=1}^{i-1}\frac{\partial \alpha _{j-1}}{\partial \nu ^{\left( j-1\right) }}\nu ^{\left( j\right) }-\frac{1}{r_{i}}{\tilde{\theta }}_{i}^{\mathrm{T}} {\dot{\theta }}_{i}. \end{aligned}$$
(80)
By applying the method of Young’s inequality, it is verified that
$$\begin{aligned}&-\,z_{i}{\tilde{\theta }}_{i}^{\mathrm{T}}\varphi _{i}\left( {\bar{\eta }}_{i}\right) \le \frac{1}{2}z_{i}^{2}+\frac{1}{2}{\tilde{\theta }}_{i}^{\mathrm{T}}{\tilde{\theta }}_{i}, \end{aligned}$$
(81)
$$\begin{aligned}&-\,z_{i}\frac{\partial \alpha _{i-1}}{\partial y}{\tilde{\theta }}_{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \le \frac{1}{2}z_{i}^{2}\left( \frac{\partial \alpha _{i-1}}{\partial y}\right) ^{2}+\frac{1}{2}{\tilde{\theta }}_{1}^{\mathrm{T}} {\tilde{\theta }}_{1}, \end{aligned}$$
(82)
$$\begin{aligned}&-\,z_{i}\frac{\partial \alpha _{i-1}}{\partial y}e_{2}\le \frac{1}{2} z_{i}^{2}\left( \frac{\partial \alpha _{i-1}}{\partial y}\right) ^{2}+\frac{1 }{2}\left\| e\right\| ^{2}, \end{aligned}$$
(83)
$$\begin{aligned}&-\,z_{i}\frac{\partial \alpha _{i-1}}{\partial y}\varepsilon _{1}\le \frac{1}{ 2}z_{i}^{2}\left( \frac{\partial \alpha _{i-1}}{\partial y}\right) ^{2}+ \frac{1}{2}\varepsilon _{1}^{*2}. \end{aligned}$$
(84)
Then, substituting (81)–(84) into (80) yields
$$\begin{aligned} {\dot{V}}_{zi}\le & {} -\sum _{j=1}^{i-1}c_{j}z_{j}^{2}+z_{i}\left( z_{i+1}+\alpha _{i}+D_{i}+\frac{1}{2}z_{i}\right) \nonumber \\&+\,\frac{3}{2}z_{i}^{2}\left( \frac{\partial \alpha _{i-1}}{\partial y} \right) ^{2}+\frac{1}{r_{i}}{\tilde{\theta }}_{i}^{\mathrm{T}}\left( r_{i}z_{i}\varphi _{i}\left( {\bar{\eta }}_{i}\right) -{\dot{\theta }}_{i}\right) \nonumber \\&-\,\sum _{j=1}^{i-1}\frac{\kappa _{j}}{2r_{j}}{\tilde{\theta }}_{j}^{\mathrm{T}}\tilde{ \theta }_{j}+\sum _{j=2}^{i}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }} _{j}+\sum _{j=1}^{i-1}\frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2} \nonumber \\&+\,\frac{i}{2}\left( {\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}\right) +\frac{1}{2} m_{1}^{2}\left\| e\right\| ^{2}. \end{aligned}$$
(85)
Substituting (38) and (39) into (40 , \({\dot{V}}_{zi}\) is expressed as
$$\begin{aligned} {\dot{V}}_{zi}\le & {} -\sum _{j=1}^{i}c_{j}z_{j}^{2}+z_{i}z_{i+1}+\frac{i}{2} \left( {\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}\right) \nonumber \\&-\,\sum _{j=1}^{i}\frac{\kappa _{j}}{2r_{j}}{\tilde{\theta }}_{j}^{\mathrm{T}}\tilde{ \theta }_{j}+\sum _{j=1}^{i}\frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2} \nonumber \\&+\,\sum _{j=2}^{i}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }}_{j}+\frac{1 }{2}m_{1}^{2}\left\| e\right\| ^{2}. \end{aligned}$$
(86)
Step n:
Differentiating \(z_{n}=\eta _{n}-\alpha _{n-1}\) with respect to time yields
$$\begin{aligned} {\dot{z}}_{n}= & {} u+\theta _{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }}_{n}\right) +k_{n}e_{1}-{\dot{\alpha }}_{n-1} \nonumber \\= & {} h\left( v\right) +d\left( v\right) +\theta _{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }}_{n}\right) +k_{n}e_{1}-{\dot{\alpha }}_{n-1} \end{aligned}$$
(87)
where
$$\begin{aligned} {\dot{\alpha }}_{n-1}= & {} \sum _{j=1}^{n-1}\frac{\partial \alpha _{j}}{\partial \eta _{j}}\left( \eta _{j+1}+\theta _{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }} _{n}\right) +k_{j}e_{1}\right) \nonumber \\&+\,\frac{\partial \alpha _{n-1}}{\partial y}\left( \eta _{2}+\theta _{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }}_{n}\right) +{\tilde{\theta }} _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) +e_{2}\right) \nonumber \\&+\,\frac{\partial \alpha _{n-1}}{\partial y}\varepsilon _{1}+\sum _{j=1}^{n} \frac{\partial \alpha _{j-1}}{\partial y_{d}^{\left( j-1\right) }} y_{d}^{\left( j\right) } \nonumber \\&+\,\sum _{j=1}^{n-1}\frac{\partial \alpha _{j}}{\partial \theta _{j}}\theta _{j}^{\left( j\right) }+\sum _{j=1}^{n-1}\frac{\partial \alpha _{j-1}}{ \partial \nu ^{\left( j-1\right) }}\nu ^{\left( j\right) }. \end{aligned}$$
(88)
Construct the Lyapunov function for the n-th subsystem as follows:
$$\begin{aligned} V_{zn}=V_{zn-1}+\frac{1}{2}z_{n}^{2}+\frac{1}{2r_{n}}{\tilde{\theta }}_{n}^{\mathrm{T}} {\tilde{\theta }}_{n}. \end{aligned}$$
(89)
From (87), the time-derivative of \(V_{zn}\) can be described as
$$\begin{aligned} {\dot{V}}_{zn}= & {} {\dot{V}}_{zn-1}+z_{n}{\dot{z}}_{n}-\frac{1}{r_{n}}{\tilde{\theta }} _{n}^{\mathrm{T}}{\dot{\theta }}_{n} \nonumber \\= & {} {\dot{V}}_{zn-1}+z_{n}\left( d\left( v\right) +h\left( v\right) +\theta _{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }}_{n}\right) +k_{n}e_{1}\right) \nonumber \\&-\,z_{n}{\dot{\alpha }}_{n-1}-\frac{1}{r_{n}}{\tilde{\theta }}_{n}^{\mathrm{T}}{\dot{\theta }} _{n}. \end{aligned}$$
(90)
Then, substituting (13) into (90), \({\dot{V}}_{zn}\) can be rewritten as
$$\begin{aligned} {\dot{V}}_{zn}= & {} {\dot{V}}_{zn-1}+z_{n}\left( {\dot{d}}\left( v_{u}\right) v+h\left( v\right) +k_{n}e_{1}+\theta _{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }} _{n}\right) \right) \nonumber \\&+\,z_{n}\left( {\tilde{\theta }}_{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }} _{n}\right) -\sum _{j=1}^{n-1}\frac{\partial \alpha _{j}}{\partial \eta _{j}} \left( \eta _{j+1}+k_{j}e_{1}\right) \right) \nonumber \\&-\,z_{n}\left( \frac{\partial \alpha _{n-1}}{\partial y}\left( e_{2}+\varepsilon _{1}+{\tilde{\theta }}_{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \right) \right) \nonumber \\&-\,z_{n}\left( \frac{\partial \alpha _{n-1}}{\partial y}\left( \eta _{2}+\theta _{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \right) \right) \nonumber \\&-\,z_{n}\left( \sum _{j=1}^{n-1}\frac{\partial \alpha _{j}}{\partial \eta _{j} }\theta _{j}^{\mathrm{T}}\varphi _{j}\left( {\bar{\eta }}_{j}\right) +\sum _{j=1}^{n} \frac{\partial \alpha _{j-1}}{\partial y_{d}^{\left( j-1\right) }} y_{d}^{\left( j\right) }\right) \nonumber \\&-\,z_{n}\left( \sum _{j=1}^{n-1}\frac{\partial \alpha _{j-1}}{\partial \nu ^{\left( j-1\right) }}\nu ^{\left( j\right) }+\sum _{j=1}^{n-1}\frac{\partial \alpha _{j}}{\partial \theta _{j}}\theta _{j}^{\left( j\right) }\right) \nonumber \\&+\,\frac{1}{r_{n}}{\tilde{\theta }}_{n}^{\mathrm{T}}\left( r_{n}z_{n}\varphi _{n}\left( {\bar{\eta }}_{n}\right) -{\dot{\theta }}_{n}\right) . \end{aligned}$$
(91)
By applying the method of Young’s inequality, the following results hold.
$$\begin{aligned}&-\,z_{n}{\tilde{\theta }}_{n}^{\mathrm{T}}\varphi _{n}\left( {\bar{\eta }}_{n}\right) \le \frac{1}{2}z_{n}^{2}+\frac{1}{2}{\tilde{\theta }}_{n}^{\mathrm{T}}{\tilde{\theta }}_{n}, \end{aligned}$$
(92)
$$\begin{aligned}&-\,z_{n}\frac{\partial \alpha _{n-1}}{\partial y}{\tilde{\theta }}_{1}^{\mathrm{T}}\varphi _{1}\left( \eta _{1}\right) \le \frac{1}{2}z_{n}^{2}\left( \frac{\partial \alpha _{n-1}}{\partial y}\right) ^{2} \nonumber \\&\quad +\frac{1}{2}{\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}, \end{aligned}$$
(93)
$$\begin{aligned}&-\,z_{n}\frac{\partial \alpha _{n-1}}{\partial y}\left( \varepsilon _{1}+e_{2}\right) \le z_{n}^{2}\left( \frac{\partial \alpha _{n-1}}{ \partial y}\right) ^{2}+\frac{1}{2}\varepsilon _{1}^{*2} \nonumber \\&\quad +\frac{1}{2}\left\| e\right\| ^{2}. \end{aligned}$$
(94)
Then, substituting (92)–(94) into (91) yields
$$\begin{aligned} {\dot{V}}_{zn}\le & {} -\sum _{j=1}^{n-1}c_{j}z_{j}^{2}+z_{n}\left( {\dot{d}} \left( v_{u}\right) v+h\left( v\right) +D_{n}+\frac{1}{2}z_{n}\right) \nonumber \\&+\,\frac{3}{2}z_{n}^{2}\left( \frac{\partial \alpha _{n-1}}{\partial y} \right) ^{2}+\frac{1}{r_{n}}{\tilde{\theta }}_{n}^{\mathrm{T}}\left( r_{n}z_{n}\varphi _{n}\left( {\bar{\eta }}_{n}\right) -{\dot{\theta }}_{n}\right) \nonumber \\&-\,\sum _{j=1}^{n-1}\frac{\kappa _{j}}{2r_{j}}{\tilde{\theta }}_{j}^{\mathrm{T}}\tilde{ \theta }_{j}+\sum _{j=2}^{n}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }} _{j}+\sum _{j=1}^{n-1}\frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2} \nonumber \\&+\,\frac{n}{2}\left( {\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}\right) +\frac{1}{2} m_{1}^{2}\left\| e\right\| ^{2}. \end{aligned}$$
(95)
Based on Assumption 4 and Young’s inequality, the following inequality can be obtained.
$$\begin{aligned} z_{n}h\left( v\right) \le \frac{1}{2}z_{n}^{2}+\frac{1}{2}q^{2}. \end{aligned}$$
(96)
Note that in (95), \({\dot{d}}\left( v_{u}\right) >0\) satisfies Assumption 5, it is similar to the processing method in [38]. Therefore, consider the control laws (41) and adaptive law (39), (95) can be rewritten as
$$\begin{aligned} {\dot{V}}_{zn}\le & {} -\sum _{j=1}^{n-1}c_{j}z_{j}^{2}+z_{n}\left( {\dot{d}} \left( v_{u}\right) N\left( \xi \right) {\bar{v}}\left( t\right) +D_{n}+z_{n}\right) \nonumber \\&+\,\frac{3}{2}z_{n}^{2}\left( \frac{\partial \alpha _{n-1}}{\partial y} \right) ^{2}+\sum _{j=1}^{n}\frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2} \nonumber \\&-\,\sum _{j=1}^{n}\frac{\kappa _{j}}{2r_{j}}{\tilde{\theta }}_{j}^{\mathrm{T}}\tilde{ \theta }_{j}+\sum _{j=2}^{n}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }}_{j} \nonumber \\&+\,\frac{n}{2}\left( {\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}\right) +\frac{1}{2} m_{1}^{2}\left\| e\right\| ^{2}+\frac{1}{2}q^{2} \nonumber \\= & {} \left( {\dot{d}}\left( v_{u}\right) N\left( \xi \right) +1\right) {\dot{\xi }} -\sum _{j=1}^{n}c_{j}z_{j}^{2}-\sum _{j=1}^{n}\frac{\kappa _{j}}{2r_{j}}\tilde{ \theta }_{j}^{\mathrm{T}}{\tilde{\theta }}_{j} \nonumber \\&+\,\sum _{j=2}^{n}\frac{1}{2}{\tilde{\theta }}_{j}^{\mathrm{T}}{\tilde{\theta }}_{j}+\frac{n }{2}\left( {\tilde{\theta }}_{1}^{\mathrm{T}}{\tilde{\theta }}_{1}+\left\| e\right\| ^{2}+\varepsilon _{1}^{*2}\right) \nonumber \\&+\,\sum _{j=1}^{n}\frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2}+\frac{1}{2}m_{1}^{2}\left\| e\right\| ^{2}+\frac{1}{2 }q^{2}. \end{aligned}$$
(97)
Consider the Lyapunov function candidate as \(V=V_{e}+V_{zn}\), and the time-derivative of V can be described as
$$\begin{aligned} {\dot{V}}= & {} {\dot{V}}_{e}+{\dot{V}}_{zn} \nonumber \\\le & {} -aV+\left( {\dot{d}}\left( v_{u}\right) N\left( \xi \right) +1\right) {\dot{\xi }}+b_{0}. \end{aligned}$$
(98)
where \(a=\min \left\{ \frac{2a_{0}}{\lambda _{\max }\left( P\right) } ,2c_{j},2\beta r_{j},\kappa _{j}\right\} \) \(\left( j=1,\cdots ,n\right) \) with \(a_{0}=q_{0}-\frac{n}{2}-\frac{1}{2}m_{1}^{2}\), \(\beta =\frac{\kappa _{j}}{2r_{j}}-\frac{n+1}{2}\) and \(b_{0}=\lambda _{0}+\sum \limits _{j=1}^{n} \frac{\kappa _{j}}{2r_{j}}\left\| \theta _{j}^{*}\right\| ^{2}+ \frac{1}{2}q^{2}+\frac{n}{2}\varepsilon _{1}^{*2}\).
