A bounded-mapping-based prescribed constraint tracking control method without initial condition

Abstract

Based on a backstepping technique, a prescribed constraint tracking control method without initial conditions is investigated for a class of strict-feedback nonlinear systems with actuator saturation and external disturbances. Unlike the existing constraint control method without initial conditions, the proposed method gives another refreshing solution by means of a bounded nonlinear mapping function, as well as two proposed prescribed performance constraint functions whose design is unrelated to the initial tracking conditions. The setting time when the constrained tracking error enters into a prescribed region is a design parameter that can be set according to any reasonable requirements. A prescribed performance constraint tracking controller is designed, and it guarantees that the tracking error of the nonlinear system gets into a prescribed constraint region from different initial values no later than a setting time, and both the transient and steady-state performance of the system is ensured. A comparison with the existing method is given, and the effectiveness and superiority of the proposed method are demonstrated using two practical examples.

Appendices

Appendix I: Proof of $\Lambda \le 0$

Proof

$\Lambda$ in (48) can be rewritten as

$$\begin{aligned} \Lambda&= b_{m}\sum _{i=2}^{n-1}\eta _{i}\left( \xi _{i}({\varvec{Z}}_{i})-\frac{ \partial \alpha _{i-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}\right) \nonumber \\&\quad +b_{m}\eta _{n}\frac{1}{\lambda }\left( \lambda \xi _{n}({\varvec{Z}}_{n}) -\frac{\partial \alpha _{n-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}\right) \nonumber \\&= b_{m}\sum \limits _{i=2}^{n}\eta _{i}\xi _{i}({\varvec{Z}}_{i})-b_{m}\sum \limits _{i=2}^{n-1}\eta _{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\beta }}\dot{{\hat{\beta }}}\nonumber \\&\quad -b_{m}\frac{\eta _{n}}{\lambda }\frac{\partial \alpha _{n-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}-b_{m}\frac{1}{\lambda } \sum \limits _{i=2}^{n-1}\eta _{i}\frac{\partial \alpha _{i-1}}{\partial \hat{ \beta }}\dot{{\hat{\beta }}}\nonumber \\&\quad +b_{m}\frac{1}{\lambda }\sum \limits _{i=2}^{n-1}\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}} \nonumber \\&= b_{m}\sum \limits _{i=2}^{n}\eta _{i}\xi _{i}({\varvec{Z}}_{i})-b_{m}\frac{1}{ \lambda }\sum \limits _{i=2}^{n}\eta _{i}\frac{\partial \alpha _{i-1}}{ \partial {\hat{\beta }}}\dot{{\hat{\beta }}}\nonumber \\&\quad -b_{m}\frac{1}{\lambda }(\lambda -1)\sum \limits _{i=2}^{n-1}\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}, \end{aligned}$$

(61)

where the terms $-b_{m}\frac{1}{\lambda }\sum _{i=2}^{n}\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}$ and $-b_{m}\frac{1}{\lambda }(\lambda -1)\sum _{i=2}^{n-1}\eta _{i} \frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}$ need to be handled. Employing (25) and [45, Lemma 1] obtains

$$\begin{aligned}&-\frac{b_{m}}{\lambda }\sum _{i=2}^{n}\eta _{i}\frac{\partial \alpha _{i-1} }{\partial {\hat{\beta }}}\dot{{\hat{\beta }}} \nonumber \\&\quad = -\frac{b_{m}}{\lambda }\sum _{i=2}^{n}\left( -\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}{\hat{\beta }}+\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}} \right. \nonumber \\&\quad \quad \left. \left( \sum \limits _{k=1}^{i-1}\frac{1}{ 2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}({\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}} _{k})+\sum \limits _{k=i}^{n}\frac{1}{2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}( {\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k})\right) \right) \nonumber \\&\quad \le -\frac{b_{m}}{\lambda }\sum _{i=2}^{n}\left( -\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}{\hat{\beta }}+\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\right. \nonumber \\&\quad \quad \left. \sum _{k=1}^{i-1}\frac{1}{2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}({\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k})-\frac{1}{ 2}\eta _{i}^{2}{\varvec{S}}_{i}^{\text {T}}({\varvec{Z}}_{i}){\varvec{S}}_{i}({\varvec{Z}} _{i}) \right. \nonumber \\&\quad \quad \left. \sum _{k=2}^{i}\left| \eta _{k}\frac{\partial \alpha _{k-1}}{ \partial {\hat{\beta }}}\right| \right) \nonumber \\&\quad \le -\frac{b_{m}}{\lambda }\sum _{i=2}^{n}\left( -\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}{\hat{\beta }}+\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\right. \nonumber \\&\quad \quad \left. \sum _{k=1}^{i-1}\frac{1}{2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}({\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k})-\frac{1}{ 2}\eta _{i}^{2}s^{2}\sum _{k=2}^{i}\left| \eta _{k}\frac{\partial \alpha _{k-1}}{\partial {\hat{\beta }}}\right| \right) \nonumber \\&\quad = -b_{m}\sum _{i=2}^{n}\eta _{i}\Xi _{i}, \end{aligned}$$

(62)

where

$$\begin{aligned} \Xi _{i}= & {} \frac{1}{\lambda }\left( -\frac{\partial \alpha _{i-1}}{\partial \hat{ \beta }}{\hat{\beta }}+\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}} \sum _{k=1}^{i-1}\frac{1}{2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}({\varvec{Z}} _{k}){\varvec{S}}_{k}({\varvec{Z}}_{k}) \right. \\{} & {} \left. -\frac{1}{2}\eta _{i}s^{2}\sum _{k=2}^{i}\left| \eta _{k}\frac{\partial \alpha _{k-1}}{ \partial {\hat{\beta }}}\right| \right) ,\quad 2\le i \le n, \end{aligned}$$

s is the upper bound of $\left\| {\varvec{S}}_{i}({\varvec{Z}}_{i})\right\|$, and

$$\begin{aligned}&-\frac{b_{m}(\lambda -1)}{\lambda }\sum \limits _{i=2}^{n-1}\eta _{i}\frac{ \partial \alpha _{i-1}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}} \nonumber \\&\quad = -\frac{b_{m}(\lambda -1)}{\lambda }\sum \limits _{i=2}^{n-1}\eta _{i}\frac{ \partial \alpha _{i-1}}{\partial {\hat{\beta }}}\nonumber \\&\qquad \left( \sum \limits _{i=1}^{n} \frac{1}{2}\eta _{i}^{2}{\varvec{S}}_{i}^{\text {T}}({\varvec{Z}}_{i}){\varvec{S}}_{i}( {\varvec{Z}}_{i})-{\hat{\beta }}\right) \nonumber \\&\quad = -\frac{b_{m}(\lambda -1)}{\lambda }\sum \limits _{i=2}^{n-1}\eta _{i}\frac{ \partial \alpha _{i-1}}{\partial {\hat{\beta }}}\nonumber \\&\qquad \left( -{\hat{\beta }} +\sum \limits _{k=1}^{i-1}\frac{1}{2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}( {\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k}) \right. \nonumber \\&\quad \left. +\sum \limits _{k=i}^{n}\frac{1}{ 2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}({\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k})\right) \nonumber \\&\quad \le -b_{m}\sum \limits _{i=2}^{n-1}\left( -\frac{(\lambda -1)}{\lambda } \eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}{\hat{\beta }} \right. \nonumber \\&\qquad +\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\frac{ (\lambda -1)}{\lambda }\sum \limits _{k=1}^{i-1}\frac{1}{2}\eta _{k}^{2} {\varvec{S}}_{k}^{\text {T}}({\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k}) \nonumber \\&\qquad \left. -\frac{\left| \lambda -1\right| }{\lambda }\left| \eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\right| \right. \nonumber \\&\quad \left. \sum \limits _{k=i}^{n}\frac{1}{2}\eta _{k}^{2}{\varvec{S}}_{k}^{\text {T}}( {\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k})\right) \nonumber \\&\quad \le -b_{m}\sum _{i=2}^{n-1}\left( -\frac{(\lambda -1)}{\lambda }\eta _{i} \frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}{\hat{\beta }} \right. \nonumber \\&\qquad \left. +\eta _{i}\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\frac{ (\lambda -1)}{\lambda }\sum _{k=1}^{i-1}\frac{1}{2}{\varvec{S}}_{k}^{\text {T}}( {\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k})\right) \nonumber \\&\qquad +b_{m}\frac{\left| \lambda -1\right| }{\lambda }\sum \limits _{i=2}^{n}\frac{1}{2}\eta _{i}^{2}s^{2}\sum \limits _{k=2}^{i}\left| \eta _{k}\frac{\partial \alpha _{k-1}}{\partial {\hat{\beta }}}\right| \nonumber \\&\quad = -b_{m}\sum \limits _{i=2}^{n}\eta _{i}\Upsilon _{i}, \end{aligned}$$

(63)

where

$$\begin{aligned} \Upsilon _{i}=&-\frac{(\lambda -1)}{\lambda } \frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}{\hat{\beta }}+\frac{(\lambda -1)}{ \lambda }\frac{\partial \alpha _{i-1}}{\partial {\hat{\beta }}}\nonumber \\&\quad \sum _{k=1}^{i-1} \eta _{k}^{2}\frac{1}{2} {\varvec{S}}_{k}^{\text {T}}({\varvec{Z}}_{k}){\varvec{S}}_{k}({\varvec{Z}}_{k}) - \frac{1}{2}\frac{\left| \lambda -1\right| }{\lambda }\eta _{i}s^{2}\nonumber \\&\quad \sum _{k=2}^{i}\left| \eta _{k}\frac{\partial \alpha _{k-1}}{ \partial {\hat{\beta }}}\right| ,i=2,\ldots ,n-1, \end{aligned}$$

(64)

$$\begin{aligned} \Upsilon _{n}=&- \frac{1}{2}\frac{\left| \lambda -1\right| }{\lambda }\eta _{n}s^{2}\sum _{k=2}^{n}\left| \eta _{k}\frac{\partial \alpha _{k-1}}{\partial {\hat{\beta }}}\right| . \end{aligned}$$

(65)

Substituting (62) and (63) into (61), we have

$$\begin{aligned} \Lambda&\le b_{m}\sum \limits _{i=2}^{n}\eta _{i}\xi _{i}({\varvec{Z}} _{i})-b_{m}\sum _{i=2}^{n}\eta _{i}\Xi _{i}-b_{m}\sum \limits _{i=2}^{n}\eta _{i}\Upsilon _{i} \nonumber \\&= b_{m}\sum \limits _{i=2}^{n}\eta _{i}\left( \xi _{i}({\varvec{Z}}_{i})-\Xi _{i}-\Upsilon _{i}\right) . \end{aligned}$$

(66)

If the auxiliary functions $\xi _{i}({\varvec{Z}}_{i})$, $i=2,$ $\ldots$ ,  n, are chosen as $\xi _{i}({\varvec{Z}}_{i})=\Xi _{i}+\Upsilon _{i}$, then $\Lambda \le 0.$

Appendix II: Controller design based on method in [30]

The design process is presented as follows. From (5), the rigid robot manipulator in (58) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{x}}_{1}=x_{2},\\ {\dot{x}}_{2}=f_{2}+g_{2}(\lambda v+\Delta (v))+d_{2}, \\ y=x_{1}. \end{array}\right. \end{aligned}$$

(67)

Define the coordinate transformation

$$\begin{aligned} e_{1}=x_{1}-y_{r},\quad e_{2}=x_{2}-\alpha _{1}, \end{aligned}$$

(68)

where $e_{1}$ is the tracking error, $y_{r}$ is the reference signal and $\alpha _{1}$ is the virtual control. The shifting function is chosen as

$$\begin{aligned} \varphi (t)=\left\{ \begin{array}{ll} 1-\left( {{\frac{T-t}{T}}}\right) {^{4}},&{}\quad 0\le t<T \\ 1,&{}\quad t\ge T\end{array}\right. \end{aligned}$$

(69)

and

$$\begin{aligned} {{\zeta }_{1}}(t)=\varphi (t){e_{1}}(t). \end{aligned}$$

(70)

Step 1. Choose a barrier Lyapunov function candidate as

$$\begin{aligned} {V}_{{1}}=\frac{{{b}_{m}}\zeta _{1}^{2}}{\left( {{F}_{1}}+{{\zeta }_{1}} \right) \left( {{F}_{2}}-{{\zeta }_{1}}\right) }+\frac{1}{2}b_{m}^{2}{{ {\tilde{\beta }}}^{2},} \end{aligned}$$

(71)

where ${{b}_{m}}$ and ${{{\tilde{\beta }}}}$ are the same as the definition in this paper, and ${{F}_{1}}$ and ${{F}_{2}}$ are the prescribed constraint functions of ${{\zeta }_{1}}$.

Then, the time derivative of ${V}_{{1}}$ is

$$\begin{aligned} {{{\dot{V}}}_{1}}&= {{b}_{m}}{M}_{1}{{\zeta }_{1}}({\dot{\varphi }}{e_{1}} +\varphi {{\alpha }_{1}}+\varphi {e}_{{2}} -\varphi {{{{\dot{y}}}}_{r}} +\varphi {e_{1}{\eta }_{1}})\nonumber \\&\quad -b_{m}^{2}{\tilde{\beta }}\dot{{\hat{\beta }}}, \end{aligned}$$

(72)

where

$$\begin{aligned} {M}_{1}= & {} \frac{2{{F}_{1}F}_{{2}}+{{\zeta }_{1}F}_{{2}}-F_{1}{{\zeta }_{1}}}{ \left( {{F}_{1}}(t)+{{\zeta }_{1}}(t)\right) ^{2}\left( {{F}_{2}}(t)-{{\zeta }_{1}}(t)\right) ^{2}}, \nonumber \\ {{\eta }_{1}}= & {} \frac{\left( -{{{{\dot{F}}}}_{1}}{{F}_{2} }-{{F}_{1}}{{{{\dot{F}}}}_{2}}+\left( {{{{\dot{F}}}}_{1}}-{{{{\dot{F}}}}_{2}} \right) {{\zeta }_{1}}\right) }{\left( 2{{F}_{1}}{{F}_{2}}+{{\zeta }_{1}}{{F} _{2}}-{{F}_{1}}{{\zeta }_{1}}\right) }. \end{aligned}$$

(73)

From Young’s inequality, we can derive that

$$\begin{aligned} -{{b}_{m}}{{M}_{1}{\zeta }_{1}}{\dot{\varphi }}{{e}_{1}}&\le \frac{1}{2}b_{m}^{2}{M}_{1}^{2}{{{\dot{\varphi }}}^{2}}\zeta _{1}^{2}e_{1}^{2}+\frac{1 }{2} \nonumber \\&=\frac{1}{2}b_{m}^{2}{M}_{1}^{2}{{{\dot{\varphi }}}^{2}{ \zeta }_{1}}\varphi e_{1}^{3}+\frac{1}{2}. \end{aligned}$$

(74)

Substituting (74) into (72) gives

$$\begin{aligned} {{{\dot{V}}}_{1}}&\le {b}_{m}{M}_{1}{{\zeta }_{1}}\varphi (\alpha _{1}+{{ {\bar{f}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) )+{{b}_{m}}{M}_{1}{{\zeta }_{1}} \varphi e_{2} \nonumber \\&\quad -\frac{1}{2}{{\left( {{b}_{m}}{M}_{{1}}{{\zeta }_{1}}\varphi \right) }^{2}}+\frac{1}{{2}}, \end{aligned}$$

(75)

where

$$\begin{aligned}{} & {} {\varvec{Z}}_{1}=[x_{1},y_{r},{\dot{y}}_{r},F_{1}, {\dot{F}}_{1},F_{2},{\dot{F}}_{2}]^{\text {T}}, \end{aligned}$$

(76)

$$\begin{aligned}{} & {} {{{\bar{f}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) =\frac{1}{2}{{b}_{m}{M} _{1}{{\dot{\varphi }}}^{2}}z_{1}^{3}-{{{{\dot{y}}}}_{r}}+{{e}_{1}{\eta }_{1}}\nonumber \\{} & {} \quad +\frac{1}{2}{{b}_{m}{M}_{1}{\zeta }_{1}}\varphi . \end{aligned}$$

(77)

Step 2. Consider a Lyapunov function candidate as

$$\begin{aligned} {V}_{{2}}=\frac{1}{2\lambda }b_{m}e_{2}^{2}. \end{aligned}$$

(78)

Then

$$\begin{aligned} {{{\dot{V}}}_{2}}&=\frac{1}{\lambda }{{b}_{m}e_{2}}\left( {{g}_{2}}\lambda v+\Delta v+{{f}_{2}}+{{d}_{2}}-{{{{\dot{\alpha }}}}_{1}}\right) \nonumber \\&=\frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{g}_{2}}\lambda v+\Delta v+{{f} _{2}}+{{d}_{2}}-{{\xi }_{1}}-\frac{\partial {{\alpha }_{1}}}{\partial \hat{ \beta }}\dot{{\hat{\beta }}}\right) , \end{aligned}$$

(79)

where

$$\begin{aligned} {{\xi }_{1}}&=\frac{\partial {{\alpha }_{1}}}{\partial {{x}_{1}}}{{x}_{2}} +\sum \limits _{j=0}^{i-1}{\frac{\partial {{\alpha }_{1}}}{\partial y_{r}^{(j)} }}y_{r}^{(j+1)}+\sum \limits _{j=0}^{1}{\frac{\partial {{\alpha }_{1}}}{ \partial {{\varphi }^{(j)}}}}{{\varphi }^{(j+1)}} \nonumber \\&\quad +\sum \limits _{k=1}^{2}\sum \limits _{j=0}^{1}{\frac{\partial {{\alpha }_{1}}}{\partial F_{k}^{(j)}}}F_{k}^{(j+1)}. \end{aligned}$$

(80)

Applying Young’s inequality, we obtain

$$\begin{aligned}{} & {} \frac{1}{\lambda }{{b}_{m}}{{e}_{2}}{{d}_{2}}\le \frac{1}{2\lambda ^{2}}{{b} _{m}^{2}}e_{2}^{2}+\frac{1}{2}d_{2}^{2}, \end{aligned}$$

(81)

$$\begin{aligned}{} & {} \frac{1}{\lambda }{{b}_{m}{e}_{2}}\Delta v\le \frac{1}{2\lambda ^{2}}{{b} _{m}^{2}{e}_{2}^{2}}+\frac{1}{2}\Delta _{\max }^{2}. \end{aligned}$$

(82)

Substituting (81)−(82) into (79), it follows that

$$\begin{aligned} {{{\dot{V}}}_{2}}&\le \frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{g}_{2}} \lambda v+{{{{\bar{f}}}}_{2}}\left( {{{\varvec{Z}}}_{2}}\right) \right) +\frac{1}{2}d_{2}^{2}+\frac{1}{2}\Delta _{\max }^{2}\nonumber \\&\quad -{{b}_{m}}{M}_{1}{{\zeta }_{1}}\varphi {{e}_{2} } -\frac{1}{2\lambda ^{2}}{{b} _{m}^{2}}e_{2}^{2}+\frac{1}{\lambda }{{b}_{m}{ e}_{2}}\left( {{\xi }_{2}} \right. \nonumber \\&\quad \left. \left( {{{\varvec{Z}}}_{2}}\right) -\frac{\partial {{ \alpha }_{1}}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}\right) , \end{aligned}$$

(83)

where

$$\begin{aligned} {\varvec{Z}}_{2}= & {} [x_{1}, x_{2},{\hat{\beta }},y_{r}, {\dot{y}}_{r},\ddot{y}_{r},F_{1},{\dot{F}}_{1},\nonumber \\{} & {} {\ddot{F}}_{1},F_{2},{\dot{F}}_{2},{\ddot{F}}_{2}]^{\text {T}},\end{aligned}$$

(84)

$$\begin{aligned} {{{\bar{f}}}_{2}}\left( {{{\varvec{Z}}}_{2}}\right)= & {} {{f}_{2}}-{{\xi }_{1}}+\frac{3}{ 2\lambda }{{b}_{m}e}{_{2}}+\lambda {{M}_{1}}{{\zeta }_{1}}\varphi \nonumber \\{} & {} -{{\xi } _{2}}\left( {{{\varvec{Z}}}_{2}}\right) ,\end{aligned}$$

(85)

$$\begin{aligned} {{\xi }_{2}}\left( {{{\varvec{Z}}}_{2}}\right)= & {} \frac{\partial {{\alpha }_{1}}}{ \partial {\hat{\beta }}}\frac{1}{2}{({{M}_{1}{\zeta }_{1}\varphi ) ^{2}}} {\varvec{S}}_{1}^{\text {T}}\left( {{{\varvec{Z}}}_{1}}\right) {{\varvec{S}}_{1}}\left( {{{\varvec{Z}} }_{1}}\right) \nonumber \\{} & {} -\left| \frac{\partial {{\alpha }_{1}}}{\partial \hat{\beta }} e_{2}\right| \frac{1}{2}{{e}_{2}}s^{2}-\frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}{\hat{\beta }}, \end{aligned}$$

(86)

where s is the upper bound of $\left\| {\varvec{S}}_{2}({\varvec{Z}}_{2})\right\|$. The unknown function ${{{{\bar{f}}}}_{i}}\left( {{{\varvec{Z}}}_{i}}\right)$ can be approximated by a neural network, i.e., ${{{{\bar{f}}}}_{i}}\left( {{ {\varvec{Z}}}_{i}}\right) ={\varvec{W}}_{i}^{*\text {T}}{{\varvec{S}}_{i}}\left( {{{\varvec{Z}}} _{i}}\right) +{{\delta }_{i}}\left( {{{\varvec{Z}}}_{i,}}\right) ,$ $i=1,$ 2. Applying Young’s inequality, we obtain

$$\begin{aligned}&{{b}_{m}{M}}_{{1}}{{\zeta }_{1}}\varphi {{{{\bar{f}}}}_{1}}\left( {{{\varvec{Z}}}_{1} }\right) ={{b}_{m}}{M}_{{1}}{{\zeta }_{1}}\varphi \left( {\varvec{W}}_{1}^{*\text {T}}{{{\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) +{{\delta }_{1}}\left( {{ {\varvec{Z}}}_{1}}\right) \right) \nonumber \\&\quad = {{b}_{m}}{M}_{1}{{\zeta }_{1}}\varphi {\varvec{W}}_{1}^{*\text {T}}{{{\varvec{S}}} _{1}}\left( {{{\varvec{Z}}}_{1}}\right) +{{b}_{m}}{M}_{{1}}{{\zeta }_{1}}\varphi {{ \delta }_{1}}\left( {{{\varvec{Z}}}_{1}}\right) \nonumber \\&\quad \le \frac{1}{2}{{\left( {{b}_{m}{M}}_{1}{{\zeta }_{1}}\varphi \right) }^{2}}{\varvec{W}}_{1}^{*\text {T}}{{\varvec{W}}}_{1}^{*}{\varvec{S}}_{1}^{ \text {T}}\left( {{{\varvec{Z}}}_{1}}\right) {{{\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}} \right) \nonumber \\&\quad +\frac{1}{2} +\frac{1}{2}{{\left( {{b}_{m}{M}}_{1}{{\zeta } _{1}}\varphi \right) }^{2}}+\frac{1}{2}\varepsilon _{1}^{2} \nonumber \\&\quad \le \frac{1}{2}{{\left( {{b}_{m}}{{M}_{{1}}{\zeta }_{1}}\varphi \right) }^{2}}\beta {\varvec{S}}_{1}^{\text {T}}\left( {{{\varvec{Z}}}_{1}}\right) {{ {\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) \nonumber \\&\quad +\frac{1}{2} +\frac{1}{2}{{ \left( {{b}_{m}}{M}_{{1}}{{\zeta }_{1}}\varphi \right) }^{2}}+\frac{1}{2}\varepsilon _{1}^{2}, \end{aligned}$$

(87)

$$\begin{aligned}&\quad \frac{1}{\lambda }{{b}_{m}{e}_{2}}{{{{\bar{f}}}}_{2}}\left( {{\varvec{Z}}_{2}} \right) =\frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {\varvec{W}}_{2}^{*\text {T}} {{{\varvec{S}}}_{2}}\left( {{{\varvec{Z}}}_{2}}\right) +{{\delta }_{2}}\left( {{{\varvec{Z}}} _{2}}\right) \right) \nonumber \\&\quad =\frac{1}{\lambda }{{b}_{m}{e}_{2}}{\varvec{W}}_{2}^{*\text {T}}{{{\varvec{S}}}_{2}} \left( {{{\varvec{Z}}}_{2}}\right) +\frac{1}{\lambda }{{b}_{m}{e}_{2}}{{\delta } _{2}}\left( {{{\varvec{Z}}}_{2}}\right) \nonumber \\&\quad \le \frac{b_{m}^{2}}{2}{{{{e}_{2}^{2}}}}{\varvec{W}}_{2}^{*\text {T}} {{{\varvec{W}}}_{2}}^{*}{\varvec{S}}_{2}^\text {T}\left( {{{\varvec{Z}}}_{2}}\right) {{{\varvec{S}}} _{2}}\left( {{{\varvec{Z}}}_{2}}\right) \nonumber \\&\quad +\frac{1}{2{{\lambda }^{2}}}+\frac{ 1}{2\lambda ^{2}}{{b}_{m}^{2}{e}_{2}^{2}}+\frac{1}{2}\varepsilon _{2}^{2} \nonumber \\&\quad \le \frac{b_{m}^{2}}{2}{{{{e}_{2}^{2}}}}\beta {\varvec{S}}_{2}^{\text {T} }\left( {{{\varvec{Z}}}_{2}}\right) {{{\varvec{S}}}_{2}}\left( {{{\varvec{Z}}}_{2}}\right) + \frac{1}{2{{\lambda }^{2}}}\nonumber \\&\quad +\frac{1}{2\lambda ^{2}}{{b}_{m}^{2}{e} _{2}^{2}}+\frac{1}{2}\varepsilon _{2}^{2}. \end{aligned}$$

(88)

The total Lyapunov function of system (67) is

$$\begin{aligned} V=V_{1}+V_{2}. \end{aligned}$$

(89)

Based on (75), (83), (87) and (88), we obtain

$$\begin{aligned} {\dot{V}}&= {{{{\dot{V}}}}_{1}}+{{{{\dot{V}}}}_{2}} \le {{b}_{m}{M}_{1}{\zeta }_{1}}\varphi \left( {{\alpha }_{1}}+{{{{\bar{f}}} }_{1}}\left( {{{\varvec{Z}}}_{1}}\right) \right) \nonumber \\&\quad +\frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{g}_{2}}\lambda v+{{{{\bar{f}}}}_{2}}\left( {{{\varvec{Z}}}_{2}}\right) \right) +1\nonumber \\&\quad +\frac{1}{2}d_{2}^{2}+\frac{1}{2}\Delta _{\max }^{2}+\frac{1}{2\lambda ^2}{+} \frac{1}{2}\varepsilon _{1}^{2}+\frac{1}{2}\varepsilon _{2}^{2} \nonumber \\&\quad +\frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{\xi }_{2}}\left( {{{\varvec{Z}}}_{2}}\right) -\frac{\partial {{\alpha } _{1}}}{\partial {\hat{\beta }}}\dot{{\hat{\beta }}}\right) -b_{m}^{2}{\tilde{\beta }} \dot{{\hat{\beta }}}. \end{aligned}$$

(90)

According to (90), we design the virtual control

$$\begin{aligned} {{\alpha }_{1}}=-{{c}_{1}{e}_{1}}-\frac{1}{2}{{\zeta }_{1}}{M} _{1}\varphi {\hat{\beta }}{\varvec{S}}_{1}^{\text {T}}\left( {{{\varvec{Z}}}_{1}}\right) {{ {\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) , \end{aligned}$$

(91)

control input

$$\begin{aligned} v=-{{c}_{2}{e}_{2}}-\frac{1}{2}{{e}_{2}}{\hat{\beta }}{\varvec{S}}_{2}^{\text { T}}\left( {{{\varvec{Z}}}_{2}}\right) {{{\varvec{S}}}_{2}}\left( {{{\varvec{Z}}}_{2}}\right) , \end{aligned}$$

(92)

and adaptive law

$$\begin{aligned} \dot{{\hat{\beta }}}= & {} \frac{{1}}{2}{{\left( {M}_{1}{{\zeta }_{1}} \varphi \right) }^{2}}{\varvec{S}}_{1}^{\text {T}}\left( {{{\varvec{Z}}}_{1}}\right) {{ {\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) \nonumber \\{} & {} +\frac{{1}}{2}{{e}_{2}^{2}} {\varvec{S}}_{2}^{\text {T}}\left( {{{\varvec{Z}}}_{2}}\right) {{{\varvec{S}}}_{2}}\left( {{ {\varvec{Z}}}_{2}}\right) -{\hat{\beta }}, \end{aligned}$$

(93)

where ${{c}_{1}}$ and ${{c}_{2}}$ are positive design parameters. It can be deduced from (86) and (93) and [45, Lemma 1] that

$$\begin{aligned}&\frac{1}{\lambda }{b}_{m}{e}_{2}\left( {{\xi }_{2}}\left( {{{\varvec{Z}}}_{2}} \right) -\frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}\dot{\hat{\beta }}\right) \nonumber \\&\quad = \frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{\xi }_{2}}\left( {{{\varvec{Z}}}_{2}} \right) -\frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}\frac{{1}}{2}{{\left( {M}_{1}{{\zeta }_{1}}\varphi \right) }^{2}}\right. \nonumber \\&\quad \quad \left. {\varvec{S}}_{1}^{\text { T}}\left( {{{\varvec{Z}}}_{1}}\right) {{{\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) \right. \nonumber \\&\quad \quad \left. - \frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}\frac{{1}}{2}{{ e}_{2}^{2}}{\varvec{S}}_{2}^{\text {T}}\left( {{{\varvec{{\varvec{Z}}}}}_{2}}\right) {{{\varvec{S}}} _{2}}\left( {{{\varvec{Z}}}_{2}}\right) +\frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}{{\hat{\beta }}}\right) \nonumber \\&\quad \le \frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{\xi }_{2}}\left( {{{\varvec{Z}}} _{2}}\right) -\frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}\frac{{1} }{2}({{{{M}_{1}{\zeta }_{1}\varphi })^{2}}} \right. \nonumber \\&\quad \left. {\varvec{S}}_{1}^{\text {T}}\left( {{{\varvec{Z}}}_{1}}\right) {{{\varvec{S}}}_{1}}\left( {{{\varvec{Z}}}_{1}}\right) +\left| \frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}e_{2}\right| \frac{{1}}{2}{{e}_{2}}s^{2}+\frac{\partial {{\alpha }_{1}}}{\partial \hat{ \beta }}{{\hat{\beta }}}\right) \nonumber \\&\quad =\ 0, \end{aligned}$$

(94)

i.e.,

$$\begin{aligned} \frac{1}{\lambda }{{b}_{m}{e}_{2}}\left( {{\xi }_{2}}\left( {{{\varvec{Z}}}_{2}} \right) -\frac{\partial {{\alpha }_{1}}}{\partial {\hat{\beta }}}\dot{\hat{\beta }}\right) \le 0. \end{aligned}$$

(95)

Substituting (91)−(95) into (90), it is derived that

$$\begin{aligned} {\dot{V}}&\le -{{c}_{1}}b_{m}^{2}{{M}_{1}}\zeta _{1}^{2}-{{c}_{2}}b_{m}^{2}{e }_{2}^{2}+b_{m}^{2}{\tilde{\beta }}{\hat{\beta }}+\frac{1}{2}+\frac{1}{2}d_{2}^{2}\nonumber \\&\quad +\frac{1}{2}\Delta _{\max }^{2}+\frac{1}{2}+\frac{1}{2{{\lambda }^{2}}}{+}\frac{1}{2}\varepsilon _{1}^{2}+\frac{1}{2}\varepsilon _{2}^{2} \nonumber \\&\le -{{c}_{1}}b_{m}^{2}\frac{\zeta _{1}^{2}}{\left( {{F}_{1}}+{{\zeta } _{1}}\right) \left( {{F}_{2}}-{{\zeta }_{1}}\right) }-{{c}_{2}}b_{m}^{2}{e} _{2}^{2}\nonumber \\&\quad -\frac{1}{2}b_{m}^{2}{\tilde{\beta }}^{2} +\frac{1}{2}b_{m}^{2}\beta ^{2}+1+\frac{1}{2}d_{2}^{2}+\frac{1}{2}\Delta _{\max }^{2}\nonumber \\&\quad +\frac{1}{2{{\lambda }^{2}}}{+}\frac{1}{2}\varepsilon _{1}^{2}+\frac{1}{2}\varepsilon _{2}^{2} \nonumber \\&\le -aV+b\rho (t), \end{aligned}$$

(96)

where $a=\min \{{{c}_{1}}b_{m},\lambda {{c}_{2}}b_{m}$}, $b=1$, $\rho (t)=\frac{1}{2} b_{m}^{2}\beta ^{2}+1+\frac{1}{2}d_{2}^{2}+\frac{1}{2}\Delta _{\max }^{2}+\frac{1}{2{{\lambda }^{2}}}{+}\frac{1}{2}\varepsilon _{1}^{2}+\frac{1}{2}\varepsilon _{2_{}}^{2^{}}$. According to Lemma 1, V is bounded, and furthermore,

$$\begin{aligned} -F_{1}(t)<{{\zeta }_{1}(t)<{F}_{2}}(t). \end{aligned}$$

(97)

When $t\ge T,$ it follows from (69) and (70) that

$$\begin{aligned} -F_{1}(t)<{e_{1}(t)<{F}_{2}}(t). \end{aligned}$$

(98)

Hence, $F_{1}$ and $F_{2}$ will become the prescribed performance functions of the tracking error ${e_{1}.}$ The constraint control for $e_1(t)$ is achieved.