Observer-based adaptive finite-time prescribed performance NN control for nonstrict-feedback nonlinear systems

Abstract

This article focuses on an adaptive neural network (NN) finite-time prescribed performance control problem for nonstrict-feedback nonlinear systems subject to full-state constraints. Specifically, a finite-time performance function is employed, which can guarantee that the tracking error converges to a prescribed region within a finite-time. Neural networks (NNs) are used to approximate the unknown nonlinear function. The unmeasurable states are estimated via constructing a state observer. By using the dynamic surface control (DSC) technique, the complexity problem has been avoided in traditional backstepping control. In order to satisfy the state constraint condition, the barrier Lyapunov function (BLF) is incorporated in the process of backstepping. The developed adaptive finite-time NN backstepping control strategy can make that the closed-loop system is semiglobally practical finite-time stability (SGPFS). Meanwhile, all states can be guaranteed to remain in the constrained space. Simulation results demonstrate the validity of the control method.

Appendices

Appendix 1

By (15), ${\dot{V}}_0$ is calculated as

$$\begin{aligned} \begin{aligned} {\dot{V}}_0(t)=&-e^\mathrm {T}(t)Qe(t)+2e^\mathrm {T}(t)W\sum _{i=1}^{n}{\tilde{\varGamma }}_i\phi _i({\hat{\chi }}(t))\\ {}&+2e^\mathrm {T}(t)W\big (\zeta +\triangle f\big ). \end{aligned} \end{aligned}$$

(46)

According to Lemma 1, Assumption 2 and $\phi _i^\mathrm {T}(X)\phi _i(X)\le \mathfrak {I}$, the following inequalities hold

$$\begin{aligned}&\begin{aligned} 2e(t)^\mathrm {T}W\sum _{i=1}^{n}{\tilde{\varGamma }}_i\phi _i({\hat{\chi }})\le \mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i+\Vert e\Vert ^2, \end{aligned} \end{aligned}$$

(47)

$$\begin{aligned}&\begin{aligned} 2e(t)^\mathrm {T}W\zeta \le \Vert W\Vert ^2\Vert \zeta ^*\Vert ^2+\Vert e\Vert ^2, \end{aligned} \end{aligned}$$

(48)

$$\begin{aligned}&\begin{aligned} 2e(t)^\mathrm {T}W\triangle f\le \left(\Vert W\Vert ^2\sum _{i=1}^{n}l_i^2+1\right)\Vert e\Vert ^2, \end{aligned} \end{aligned}$$

(49)

where $\zeta ^*=[\zeta _1^*,\dots ,\zeta _n^*]^\mathrm {T}$.

According to (47, 48 and 49), it yields

$$\begin{aligned} \begin{aligned} {\dot{V}}_0\le&-e(t)^\mathrm {T}Qe(t)+\left(\Vert W\Vert ^2\sum _{i=1}^{n}l_i^2+3\right)\Vert e\Vert ^2\\ {}&+\Vert W\Vert ^2\Vert \zeta ^*\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i\\ \le&-q_0\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i+M_0.\\ \end{aligned} \end{aligned}$$

(50)

Appendix 2

Step 1: From (1), (5) and (17), ${\dot{z}}_1$ can be given as

$$\begin{aligned} \begin{aligned} {\dot{z}}_1=&\frac{\pi (1+\psi ^2)}{2\rho }({\dot{\chi }}_1-{\dot{y}}_d -\frac{2}{\pi }{\dot{\rho }}\arctan (\psi ))\\ =&\frac{\pi (1+\psi ^2)}{2\rho }(z_2+\alpha _1+\varsigma _2+\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }}) -{\dot{y}}_d\\ {}&+e_2+\zeta _1-\frac{2}{\pi }{\dot{\rho }}\arctan (\psi ))\\ =&\frac{\pi (1+\psi ^2)}{2\rho }(z_2+\alpha _1+\varsigma _2+\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }})\\ {}&-\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }}_1) +\varGamma _1^{\mathrm {T}}\phi _1({\hat{\chi }}_1)+{\tilde{\varGamma }}_1^{\mathrm {T}}\phi _1({\hat{\chi }}_1)+\zeta _1\\ {}&+e_2-{\dot{y}}_d-\frac{2}{\pi }{\dot{\rho }}\arctan (\psi )). \end{aligned} \end{aligned}$$

(51)

Choose a Lyapunov function as

$$\begin{aligned} \begin{aligned} V_1=V_0+\frac{1}{2}\log \frac{k_{b1}^2}{k_{b_1}^2-z_1^2}+\frac{1}{2\delta _1}{\tilde{\varGamma }}_1^{\mathrm {T}}{\tilde{\varGamma }}_1, \end{aligned} \end{aligned}$$

(52)

where $k_{b_1}=k_{c_1}-A_0$, and $\delta _1$ is the design parameter. Obviously, $V_0\ge 0$ and ${\tilde{\varGamma }}_1^{\mathrm {T}}{\tilde{\varGamma }}_1\ge 0$. In addition, $k_{b1}^2/(k_{b1}^2-z_1^2)\ge 1$, that is , $\frac{1}{2}\log \frac{k_{b1}^2}{k_{b_1}^2-z_1^2}\ge 0$. Then, $V_1\ge 0$ can be obtained.

According to (51), the derivative of $V_1$ yields

$$\begin{aligned} \begin{aligned} {\dot{V}}_1=\,&{\dot{V}}_0+\frac{z_1{\dot{z}}_1}{k_{b_1}^2-z_1^2} -\frac{1}{\gamma _1}{\tilde{\varGamma }}_1^{\mathrm {T}}{\dot{\varGamma }}_1\\ =\,&{\dot{V}}_0+\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1(\alpha _1-{\dot{y}}_d+z_2+\varsigma _2\\ {}&+e_2+\zeta _1+\varGamma _1^{\mathrm {T}}\phi _1({\hat{\chi }}_1) -\frac{2}{\pi }{\dot{\rho }}\arctan (\psi ))\\ {}&+\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1 (\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }})-\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }}_1))\\&-\frac{1}{\delta _1}{\tilde{\varGamma }}_1^{\mathrm {T}}({\dot{\varGamma }}_1 -\delta _1\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1\phi _1({\hat{\chi }}_1)). \end{aligned} \end{aligned}$$

(53)

According to (53), Lemma 1 and $\phi _i^\mathrm {T}(X)\phi _i(X)\le \mathfrak {I}$, we can obtain

$$\begin{aligned}&\begin{aligned} \frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1\zeta _1\le \frac{1}{2}(\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)})^2z_1^2+\frac{1}{2}\zeta _1^{*2}, \end{aligned} \end{aligned}$$

(54)

$$\begin{aligned}&\begin{aligned} \frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1\varsigma _2\le \frac{1}{2}(\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)})^2z_1^2+\frac{1}{2}\varsigma _2^2, \end{aligned} \end{aligned}$$

(55)

$$\begin{aligned}&\begin{aligned} \frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1e_2\le \frac{1}{2}(\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)})^2z_1^2+\frac{1}{2}\Vert e\Vert ^2, \end{aligned} \end{aligned}$$

(56)

$$\begin{aligned}&\begin{aligned} \frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1&(\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }})-\varGamma _1^{*\mathrm {T}}\phi _1({\hat{\chi }}_1)) \\\le {}&\frac{\tau }{2}(\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)})^2z_1^2+\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _1^*\Vert ^2, \end{aligned} \end{aligned}$$

(57)

From (54)−(57), (53) can be written as

$$\begin{aligned} \begin{aligned} {\dot{V}}_1\le&{\dot{V}}_0+\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1 (\alpha _1-{\dot{y}}_d+\varGamma _1^{\mathrm {T}}\phi _1({\hat{\chi }}_1)\\ {}&+\frac{(3+\tau )\pi (1+\psi ^2)}{4\rho (k_{b_1}^2-z_1^2)}z_1 -\frac{2}{\pi }{\dot{\rho }}\arctan (\psi ))+\frac{1}{2}\Vert e\Vert ^2\\ {}&+\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _1^*\Vert ^2 +\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1z_2 +\frac{1}{2}\zeta _1^{*2}+\frac{1}{2}\varsigma _2^2\\ {}&-\frac{1}{\delta _1}{\tilde{\varGamma }}_1^{\mathrm {T}}({\dot{\varGamma }}_1 -\delta _1\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1\phi _1({\hat{\chi }}_1)). \end{aligned} \end{aligned}$$

(58)

Considering (18), it follows that

$$\begin{aligned} \begin{aligned} {\dot{V}}_1\le&-q_1\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i+M_1\\ {}&-\frac{c_1z_1^{2\sigma }}{(k_{b_1}^2-z_1^2)^\sigma }+\frac{\pi (1+\psi ^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1z_2\\ {}&+\frac{\mu _1}{\delta _1}{\tilde{\varGamma }}_1^\mathrm {T}\varGamma _1+\frac{1}{2}\varsigma _2^2, \end{aligned} \end{aligned}$$

(59)

where $q_1=q_0-\frac{1}{2}$ and $M_1=M_0+\frac{1}{2}\zeta _1^{*2}+\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _1^*\Vert ^2$.

The following first-order low-pass filter will be utilized to filter $\alpha _1$ and to obtain $\gamma _2$

$$\begin{aligned} \begin{aligned} \varpi _2{\dot{\gamma }}_2+\gamma _2=\alpha _1, \gamma _2(0)=\alpha _1(0), \end{aligned} \end{aligned}$$

(60)

where $\varpi _2$ is a positive constant.

Define $\varsigma _2=\gamma _2-\alpha _1$. According to (60), it is easy to obtain ${\dot{\gamma }}_2=-\frac{\varsigma _2}{\varpi _2}$, and then one gets

$$\begin{aligned} \begin{aligned} {\dot{\varsigma }}_2={\dot{\gamma }}_2-{\dot{\alpha }}_1=-\frac{\varsigma _2}{\varpi _2}+Y_2(\cdot ), \end{aligned} \end{aligned}$$

(61)

where

$$\begin{aligned} \begin{aligned} Y_2(\cdot )\,=\,\frac{\partial \alpha _1}{\partial {\hat{\chi }}_1}\dot{{\hat{\chi }}}_1 -\frac{\partial \alpha _1}{\partial \varGamma _1}{\dot{\varGamma }}_1 -\frac{\partial \alpha _1}{\partial y_d}{\dot{y}}_d -\frac{\partial \alpha _1}{\partial {\dot{y}}_d}\ddot{y}_d. \end{aligned} \end{aligned}$$

Step 2: According to (1), (5) and (17), ${\dot{z}}_2$ can be calculated as

$$\begin{aligned} \begin{aligned} {\dot{z}}_2=\,&\dot{{\hat{\chi }}}_2-{\dot{\gamma }}_2\\ =&{\hat{\chi }}_3+k_2(y-{\hat{\chi }}_1)+\varGamma _2^\mathrm {T}\phi _2({\hat{\chi }})-{\dot{\gamma }}_2\\ =\,&z_3+\varsigma _3+\alpha _2+k_2e_1+\varGamma _2^{*\mathrm {T}}\phi _2({\hat{\chi }}) -{\tilde{\varGamma }}_2^\mathrm {T}\phi _2({\hat{\chi }})\\ {}&-\varGamma _2^{*\mathrm {T}}\phi _2(\hat{{\bar{\chi }}}_2) +\varGamma _2^{\mathrm {T}}\phi _2(\hat{{\bar{\chi }}}_2) +{\tilde{\varGamma }}_2^{\mathrm {T}}\phi _2(\hat{{\bar{\chi }}}_2)-{\dot{\gamma }}_2, \end{aligned} \end{aligned}$$

(62)

where $\hat{{\bar{\chi }}}_2=({\hat{\chi }}_1,{\hat{\chi }}_2)^\mathrm {T}$.

Choose a Lyapunov function as

$$\begin{aligned} \begin{aligned} V_2=V_1+\frac{1}{2}\log \frac{k_{b_2}^2}{k_{b_2}^2-z_2^2}+\frac{1}{2}\varsigma _2^2 +\frac{1}{2\delta _2}{\tilde{\varGamma }}_2^\mathrm {T}{\tilde{\varGamma }}_2, \end{aligned} \end{aligned}$$

(63)

where $\delta _2$ is the positive designed parameter, and $k_{b_2}$ will be given later. Similar to (52), $V_2\ge 0$ can be obtained.

From (62) and (63), one has

$$\begin{aligned} \begin{aligned} {\dot{V}}_2=\,&{\dot{V}}_1+\frac{z_2}{k_{b_2}^2-z_2^2}(\alpha _2+k_2e_1+z_3+\varsigma _3 -{\dot{\gamma }}_2\\ {}&+\varGamma _i^\mathrm {T}\phi _i(\hat{{\bar{\chi }}}_2))-\frac{z_2}{k_{b_2}^2-z_2^2}{\tilde{\varGamma }}_2^\mathrm {T}\phi _2({\hat{\chi }})\\ {}&+\frac{z_2}{k_{b_2}^2-z_2^2}(\varGamma _2^{*\mathrm {T}}\phi _2({\hat{\chi }}) -\varGamma _2^{*\mathrm {T}}\phi _2(\hat{{\bar{\chi }}}_2))\\ {}&+\varsigma _2{\dot{\varsigma }}_2 -\frac{1}{\delta _2}{\tilde{\varGamma }}_2^\mathrm {T}({\dot{\varGamma }}_2-\frac{\delta _2}{k_{b_2}^2 -z_2^2}z_2\phi _2(\hat{{\bar{\chi }}}_2)). \end{aligned} \end{aligned}$$

(64)

By Lemma 1, we can obtain the following inequalities

$$\begin{aligned}&\begin{aligned} \frac{z_2}{k_{b_2}^2-z_2^2}(\varGamma _2^{*\mathrm {T}}\phi _2({\hat{\chi }})&-\varGamma _2^{*\mathrm {T}}\phi (\hat{{\bar{\chi }}}_2)) \\\le {}&\frac{\tau z_2^2}{2(k_{b_2}^2-z_2^2)}+\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _2^*\Vert ^2, \end{aligned} \end{aligned}$$

(65)

$$\begin{aligned}&\begin{aligned} \frac{z_2}{k_{b_2}^2-z_2^2}\varsigma _3\le \frac{z_2^2}{2(k_{b_2}^2-z_2^2)^2} +\frac{1}{2}\varsigma _3^2, \end{aligned} \end{aligned}$$

(66)

$$\begin{aligned}&\begin{aligned} -\frac{z_2}{k_{b_2}^2-z_2^2}{\tilde{\varGamma }}_2^\mathrm {T}\phi _2({\hat{\chi }})\le \frac{z_2^2}{2(k_{b_2}^2-z_2^2)^2}+\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_2^\mathrm {T}{\tilde{\varGamma }}_2, \end{aligned} \end{aligned}$$

(67)

According (65)-(67), one has

$$\begin{aligned} \begin{aligned} {\dot{V}}_2\le&{\dot{V}}_1+\frac{z_2}{k_{b_2}^2-z_2^2}(\alpha _2+k_2e_1 -{\dot{\gamma }}_2+\varGamma _2^\mathrm {T}\phi _2(\hat{{\bar{\chi }}}_2)\\ {}&+\frac{(2+\tau )z_2}{k_{b_2}^2-z_2^2}) +\frac{1}{2}\varsigma _3^2+\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_2^\mathrm {T}{\tilde{\varGamma }}_2 +\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _2^*\Vert ^2\\ {}&+\varsigma _2{\dot{\varsigma }}_2 -\frac{1}{\delta _2}{\tilde{\varGamma }}_2^\mathrm {T}({\dot{\varGamma }}_2 -\frac{\delta _2}{k_{b_2}^2-z_2^2}z_2\phi _2(\hat{{\bar{\chi }}}_2))\\ {}&+\frac{z_2z_3}{k_{b_2}^2-z_2^2},\\ \end{aligned} \end{aligned}$$

(68)

where $M_2=M_1+\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _2^*\Vert ^2$.

And then, we can obtain

$$\begin{aligned} \begin{aligned} {\dot{V}}_2\le&-q_1\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i+M_2 \\&+\frac{z_2z_3}{k_{b_2}^2-z_2^2} +\frac{z_2}{k_{b_2}^2-z_2^2}\Big (\alpha _2+k_2e_1 -{\dot{\gamma }}_2\\ {}&+\varGamma _i^\mathrm {T}\phi _2(\hat{{\bar{\chi }}}_2) +\frac{(2+\tau )z_2}{k_{b_2}^2-z_2^2}\\&+\frac{\pi (1+v^2)(k_{b_2}^2-z_2^2)}{2\rho (k_{b_1}^2-z_1^2)}z_1\Big ) +\frac{1}{2}\varsigma _3^2\\ {}&+\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_2^\mathrm {T}{\tilde{\varGamma }}_2 +\frac{\mu _1}{\delta _1}{\tilde{\varGamma }}_1^\mathrm {T}\varGamma _1+\frac{1}{2}\varsigma _2^2 -\frac{c_1z_1^{2\sigma }}{(k_{b_1}^2-z_1^2)^\sigma } \\ {}&+\varsigma _2{\dot{\varsigma }}_2 -\frac{1}{\delta _2}{\tilde{\varGamma }}_2^\mathrm {T}({\dot{\varGamma }}_2 -\frac{\delta _2}{k_{b_2}^2-z_2^2}z_2\phi _2(\hat{{\bar{\chi }}}_2)). \end{aligned} \end{aligned}$$

(69)

Substituting (19) into (69), one has

$$\begin{aligned} \begin{aligned} {\dot{V}}_2\le&-q_1\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i+M_2\\ {}&-\frac{c_1z_1^{2\sigma }}{(k_{b_1}^2-z_1^2)^\sigma } -\frac{c_2z_2^{2\sigma }}{(k_{b_2}^2-z_2^2)^\sigma } +\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_2^\mathrm {T}{\tilde{\varGamma }}_2\\ {}&+\frac{\mu _1}{\delta _1}{\tilde{\varGamma }}_1^\mathrm {T}\varGamma _1 +\frac{\mu _2}{\delta _2}{\tilde{\varGamma }}_2^\mathrm {T}\varGamma _2+\frac{1}{2}\varsigma _2^2+\frac{1}{2}\varsigma _3^2\\ {}&+\varsigma _2\left(-\frac{\varsigma _2}{\varpi _2}+Y_2(\cdot )\right)+\frac{z_2z_3}{k_{b_2}^2-z_2^2}. \end{aligned} \end{aligned}$$

(70)

The first-order filter is designed as

$$\begin{aligned} \begin{aligned} \varpi _3{\dot{\gamma }}_3+\gamma _3=\alpha _2, \gamma _3(0)=\alpha _2(0). \end{aligned} \end{aligned}$$

(71)

Define $\varsigma _3=\gamma _3-\alpha _2$. According to (71), we can obtain ${\dot{\gamma }}_3=-\frac{\varsigma _3}{\varpi _3}$, which implies

$$\begin{aligned} \begin{aligned} {\dot{\varsigma }}_3= {\dot{\gamma }}_3-{\dot{\alpha }}_2=-\frac{\varsigma _3}{\varpi _3}+Y_3(\cdot ), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} Y_3(\cdot )=&-\frac{\partial \alpha _2}{\partial {\hat{\chi }}_1}\dot{{\hat{\chi }}}_1 -\frac{\partial \alpha _2}{\partial {\hat{\chi }}_2}\dot{{\hat{\chi }}}_2 -\frac{\partial \alpha _2}{\partial \varGamma _1}{\dot{\varGamma }}_1 -\frac{\partial \alpha _2}{\partial \varGamma _2}{\dot{\varGamma }}_2 -\frac{\partial \alpha _2}{\partial \gamma _2}{\dot{\gamma }}_2. \end{aligned} \end{aligned}$$

Step i ($3\le i \le n - 1$): By (17), ${\dot{z}}_i$ is

$$\begin{aligned} \begin{aligned} {\dot{z}}_i=\,&\dot{{\hat{\chi }}}_i-{\dot{\gamma }}_i\\ =\,&{\hat{\chi }}_{i+1}+k_i(y-{\hat{\chi }}_1)+\varGamma _i^\mathrm {T}\phi _i({\hat{\chi }})-{\dot{\gamma }}_i\\ =\,&z_{i+1}+\varsigma _{i+1}+\alpha _i+k_ie_1+\varGamma _i^{*\mathrm {T}}\phi _i({\hat{\chi }}) \\ {}&-{\tilde{\varGamma }}_i^\mathrm {T}\phi _i({\hat{\chi }})-\varGamma _i^{*\mathrm {T}}\phi _i(\hat{{\bar{\chi }}}_i) +\varGamma _i^{\mathrm {T}}\phi _i(\hat{{\bar{\chi }}}_i) \\ {}&+{\tilde{\varGamma }}_i^{\mathrm {T}}\phi _i(\hat{{\bar{\chi }}}_i)-{\dot{\gamma }}_i. \end{aligned} \end{aligned}$$

(72)

Choose a Lyapunov function as

$$\begin{aligned} \begin{aligned} V_i=V_{i-1}+\frac{1}{2}\log \frac{k_{b_i}^2}{k_{b_i}^2-z_i^2}+\frac{1}{2}\varsigma _i^2 +\frac{1}{2\delta _i}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i, \end{aligned} \end{aligned}$$

(73)

where $\delta _i$ is the positive parameter, and $k_{b_i}$ will be given later. Similar to (63), it can be seen $V_i\ge 0$.

From (72) and (73), one has

$$\begin{aligned} \begin{aligned} {\dot{V}}_i=\,&{\dot{V}}_{i-1}+\frac{z_i}{k_{b_i}^2-z_i^2}(\alpha _i+k_2e_1+z_{i+1}+\varsigma _{i+1}\\ {}&-{\dot{\gamma }}_{i+1} +\varGamma _i^\mathrm {T}\phi _i(\hat{{\bar{\chi }}}_i))-\frac{z_i}{k_{b_i}^2-z_i^2}{\tilde{\varGamma }}_i^\mathrm {T}\phi _i({\hat{\chi }})\\ {}&+\frac{z_i}{k_{b_i}^2-z_i^2}(\varGamma _i^{*\mathrm {T}}\phi _i({\hat{\chi }}) -\varGamma _i^{*\mathrm {T}}\phi _i(\hat{{\bar{\chi }}}_i))\\&+\varsigma _i{\dot{\varsigma }}_i -\frac{1}{\delta _i}{\tilde{\varGamma }}_i^\mathrm {T}\left({\dot{\varGamma }}_i-\frac{\delta _i}{k_{b_i}^2 -z_i^2}z_i\phi _i(\hat{{\bar{\chi }}}_i)\right). \end{aligned} \end{aligned}$$

(74)

Similarly, it yields

$$\begin{aligned}&\begin{aligned} \frac{z_i}{k_{b_i}^2-z_i^2}(\varGamma _i^{*\mathrm {T}}\phi _i({\hat{\chi }})&-\varGamma _i^{*\mathrm {T}}\phi (\hat{{\bar{\chi }}}_i)) \\\le {}\frac{\tau z_i^2}{2(k_{b_i}^2-z_i^2)} +\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _i^*\Vert ^2, \end{aligned} \end{aligned}$$

(75)

$$\begin{aligned}&\begin{aligned} \frac{z_i}{k_{b_i}^2-z_i^2}\varsigma _{i+1}\le \frac{z_i^2}{2(k_{b_i}^2-z_i^2)^2} +\frac{1}{2}\varsigma _{i+1}, \end{aligned} \end{aligned}$$

(76)

$$\begin{aligned}&\begin{aligned} -\frac{z_i}{k_{b_i}^2-z_i^2}{\tilde{\varGamma }}_i^\mathrm {T}\phi _i({\hat{\chi }})\le \frac{z_i^2}{2(k_{b_i}^2-z_i^2)^2}+\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i. \end{aligned} \end{aligned}$$

(77)

The first-order filter is designed as

$$\begin{aligned} \begin{aligned} \varpi _{i+1}{\dot{\gamma }}_{i+1}+\gamma _{i+1}=\alpha _i, \gamma _{i+1}(0)=\alpha _i(0). \end{aligned} \end{aligned}$$

(78)

Define $\varsigma _{i+1}=\gamma _{i+1}-\alpha _i$. According to (78), we can obtain ${\dot{\gamma }}_{i+1}=-\frac{\varsigma _{i+1}}{\varpi _{i+1}}$, which implies

$$\begin{aligned} \begin{aligned} {\dot{\varsigma }}_{i+1}= {\dot{\gamma }}_{i+1}-{\dot{\alpha }}_i=-\frac{\varsigma _{i+1}}{\varpi _{i+1}}+Y_{i+1}(\cdot ), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} Y_{i+1}(\cdot )=&-\sum _{j=1}^{i}\frac{\partial \alpha _{i}}{\partial {\hat{\chi }}_j}\dot{{\hat{\chi }}}_j -\sum _{j=1}^{i}\frac{\partial \alpha _{i}}{\partial \varGamma _j}{\dot{\varGamma }}_j -\sum _{j=1}^{i}\frac{\partial \alpha _{i}}{\partial \gamma _i}{\dot{\gamma }}_i. \end{aligned} \end{aligned}$$

Substituting (75)−(77) and (19) into (53), it yields

$$\begin{aligned} \begin{aligned} {\dot{V}}_i\le&-q_1\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i +\sum _{j=1}^{i}\frac{1}{2}\varsigma _{j+1}^2\\&-\sum _{j=1}^{i}\frac{c_jz_j^{2\sigma }}{(k_{b_j}^2-z_j^2)^\sigma }-\sum _{j=1}^{i}\frac{\mu _j}{\delta _j}{\tilde{\varGamma }}_j^\mathrm {T}\varGamma _j +\frac{z_iz_{i+1}}{k_{b_i}^2-z_i^2}\\ {}&+\sum _{j=2}^{i}\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_j^\mathrm {T}{\tilde{\varGamma }}_j +\sum _{j=2}^{i}\varsigma _j(-\frac{\varsigma _j}{\varpi _j}+Y_j(\cdot ))+M_i, \end{aligned} \end{aligned}$$

(79)

where $M_i=M_{i-1}+\frac{2\mathfrak {I}}{\tau }\Vert \varGamma _i^*\Vert ^2$.

Step n: From (1) and (17), the derivation of $z_n$ is given as

$$\begin{aligned} \begin{aligned} {\dot{z}}_n=\,&u+k_ne_1+\varGamma _n^\mathrm {T}\phi _n({\hat{\chi }}) -{\tilde{\varGamma }}_n^\mathrm {T}\phi _n({\hat{\chi }}) \\ {}&+{\tilde{\varGamma }}_n^\mathrm {T}\phi _n({\hat{\chi }})-{\dot{\gamma }}_i. \end{aligned} \end{aligned}$$

(80)

The Lyapunov function is given as

$$\begin{aligned} \begin{aligned} V_n=V_{n-1}+\frac{1}{2}\log \frac{k_{b_n}^2}{k_{b_n}^2-z_n^2}+\frac{1}{2}\varsigma _n^2 +\frac{1}{2\delta _n}{\tilde{\varGamma }}_n^\mathrm {T}{\tilde{\varGamma }}_n, \end{aligned} \end{aligned}$$

(81)

where $\delta _n>0$ is a designed parameter. Similar to (73), $V_n\ge 0$ can be obtained.

According to (80) and (81), one has

$$\begin{aligned} \begin{aligned} {\dot{V}}_n=\,&{\dot{V}}_{n-1}+\frac{z_n}{k_{b_n}^2-z_n^2}(u+k_ne_1 -{\dot{\gamma }}_n+\varGamma _n^\mathrm {T}\phi _n(\hat{{\bar{\chi }}}))\\ {}&-\frac{z_n}{k_{b_n}^2-z_n^2}{\tilde{\varGamma }}_n^\mathrm {T}\phi _n({\hat{\chi }}) +\varsigma _n{\dot{\varsigma }}_n \\ {}&-\frac{1}{\delta _n}{\tilde{\varGamma }}_n^\mathrm {T}({\dot{\varGamma }}_n-\frac{\delta _n}{k_{b_n}^2 -z_n^2}z_n\phi _n({\hat{\chi }})). \end{aligned} \end{aligned}$$

(82)

From Lemma 1, it yields

$$\begin{aligned} \begin{aligned} -\frac{{\tilde{\varGamma }}_n^\mathrm {T}\phi _n({\hat{\chi }})}{k_{b_n}^2-z_n^2}z_n\le \frac{z_n^2}{2(k_{b_n}^2-z_n^2)^2}+\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_n^\mathrm {T}{\tilde{\varGamma }}_n. \end{aligned} \end{aligned}$$

(83)

From (79), (82) and (83), one gets

$$\begin{aligned} \begin{aligned} {\dot{V}}_n\le&-q_1\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i +M_{n-1}\\ {}&-\sum _{i=1}^{n-1}\frac{c_iz_i^{2\sigma }}{(k_{b_i}^2-z_i^2)^\sigma } +\frac{z_n}{k_{b_n}^2-z_n^2}(u+k_ne_1 -{\dot{\gamma }}_n\\ {}&+\frac{z_n}{2(k_{b_n}^2-z_n^2)} +\frac{(k_{b_n}^2-z_n^2)z_{n-1}}{k_{b_{n-1}}^2-z_{n-1}^2} +\varGamma _n^\mathrm {T}\phi _n(\hat{{\bar{\chi }}}))\\ {}&-\sum _{i=1}^{n-1}\frac{\mu _i}{\delta _i}{\tilde{\varGamma }}_i^\mathrm {T}\varGamma _i +\sum _{i=1}^{n-1}\frac{1}{2}\varsigma _{i+1}^2 +\sum _{i=2}^{n}\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i\\ {}&+\sum _{i=2}^{n}\varsigma _i(-\frac{\varsigma _i}{\varpi _i}+Y_i(\cdot ))\\ {}&-\frac{1}{\delta _n}{\tilde{\varGamma }}_n^\mathrm {T}({\dot{\varGamma }}_n-\frac{\delta _n}{k_{b_n}^2 -z_n^2}z_n\phi _n({\hat{\chi }})). \end{aligned} \end{aligned}$$

(84)

Considering (21), it yields

$$\begin{aligned} \begin{aligned} {\dot{V}}_n\le&-q_1\Vert e\Vert ^2+\mathfrak {I}\Vert W\Vert ^2\sum _{i=1}^{n}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i +\sum _{i=1}^{n}\frac{1}{2}\varsigma _{i}^2\\ {}&-\sum _{i=1}^{n}\frac{\mu _i}{\delta _i}{\tilde{\varGamma }}_i^\mathrm {T}\varGamma _i -\sum _{i=1}^{n}\frac{c_iz_i^{2\sigma }}{(k_{b_i}^2-z_i^2)^\sigma }\\ {}&+\sum _{i=2}^{n}\varsigma _i\left(-\frac{\varsigma _i}{\varpi _i}+Y_i(\cdot )\right)+\sum _{i=2}^{n}\frac{\mathfrak {I}}{2}{\tilde{\varGamma }}_i^\mathrm {T}{\tilde{\varGamma }}_i+M_i. \end{aligned} \end{aligned}$$

(85)