Dead-Zone Model-Based Adaptive Fuzzy Wavelet Control for Nonlinear Systems Including Input Saturation and Dynamic Uncertainties

Abstract

In this study, the problem of adaptive fuzzy wavelet network (FWN) control is investigated for nonlinear strict-feedback systems with unknown functions, unknown virtual control gains and unknown input saturation. An adaptive FWN as an adaptive nonlinear-in-parameter approximator is proposed to represent the model of the unknown functions. Saturation nonlinearity is described by the dead-zone operator-based model which does not require the bound of the saturated input to be known. Then, a novel control scheme is designed based on the adaptive FWN, the saturation model and the dynamic surface control approach. The proposed control scheme does not require any prior knowledge about input saturation, unknown dynamics and unknown virtual control gains. It simultaneously eliminates the “explosion of complexity” and “curse of dimensionality” problems; also, the design approach avoids the controller singularity problem completely without using projection algorithm. The stability analysis is studied using Lyapunov theorem; it shows that all signals of the resulting closed-loop system are uniformly ultimately bounded and the tracking error can be made small by proper selection of the design parameters. Comparing the simulation results of the proposed scheme with other control methods demonstrates the effectiveness and superior performance of the proposed scheme.

Appendix A: Proof of Theorem 1

In this section, proof of Theorem 1 is presented. To analysis the stability, the following Lyapunov function candidate is considered:

$$ V = \sum\limits_{i = 1}^{n - 1} {\left( {\frac{1}{{2g_{i} }}e_{i}^{2} + \frac{1}{2}\eta_{i + 1}^{2} } \right)} + \sum\limits_{i = 1}^{n} {\left( {\frac{1}{{2\gamma_{1} }}{\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\tilde{\varvec{\theta }}}_{i} + \frac{1}{{2\gamma_{2} }}{\tilde{\varvec{c}}}_{i}^{\text{T}} {\tilde{\varvec{c}}}_{i} + \frac{1}{{2\gamma_{3} }}{\tilde{\varvec{\omega }}}_{i}^{\text{T}} {\tilde{\varvec{\omega }}}_{i} } \right)} + \frac{1}{2\beta }e_{n}^{2} + \frac{1}{{2\gamma_{\rho } }}\int\limits_{0}^{R} {\tilde{\rho }_{\lambda }^{2} (r,t){\text{d}}r} $$

(A1)

where $\tilde{\rho }_{\lambda } (r,t) = \rho_{\lambda } (r) - \hat{\rho }_{\lambda } (r,t)$, ${\tilde{\varvec{\theta }}}_{i} = {\varvec{\theta}}_{i}^{*} - {\hat{\varvec{\theta }}}_{i}$, ${\tilde{\varvec{c}}}_{i} = {\varvec{c}}_{i}^{*} - {\hat{\varvec{c}}}_{i}$ and ${\tilde{\varvec{\omega }}}_{i} = {\varvec{\omega}}_{i}^{*} - {\hat{\varvec{\omega }}}_{i}$. Differentiating (A1) with respect to time results in:

$$ \begin{aligned} \dot{V} & = \sum\limits_{{i = 1}}^{{n - 1}} {\left( {\frac{1}{{g_{i} }}e_{i} \dot{e}_{i} - \frac{{\dot{g}_{i} e_{i}^{2} }}{{2g_{i}^{2} }} + \eta _{{i + 1}} \dot{\eta }_{{i + 1}} } \right)} + \frac{1}{\beta }e_{n} \dot{e}_{n} - \sum\limits_{{i = 1}}^{n} {\left( {\frac{1}{{\gamma _{1} }}{\tilde{{\varvec{\theta }}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{\theta }}}}}_{i} + \frac{1}{{\gamma _{2} }}{\tilde{{\varvec{c}}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{c}}}}}_{i} + \frac{1}{{\gamma _{3} }}{\tilde{{\varvec{\omega }}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{\omega }}}}}_{i} } \right)} - \frac{{\dot{\beta }}}{{2\beta ^{2} }}e_{n}^{2} \\ & \quad - \frac{1}{{\gamma _{\rho } }}\int\limits_{0}^{R} {\tilde{\rho }_{\lambda } (r,t)\frac{\partial }{{\partial t}}\hat{\rho }_{\lambda } (r,t){\text{d}}r} \\ \end{aligned} $$

(A2)

Substituting (22), (31) and (39) into (A2) results in:

$$ \begin{aligned} \dot{V} & = \sum\limits_{{i = 1}}^{{n - 1}} {\left( {e_{i} (e_{{i + 1}} + \eta _{{i + 1}} - k_{i} e_{i} + \tilde{h}_{i} ) - \frac{{\dot{g}_{i} e_{i}^{2} }}{{2g_{i}^{2} }} + \eta _{{i + 1}} \dot{\eta }_{{i + 1}} } \right)} + e_{n} \left( { - \,k_{n} e_{n} + \int\limits_{0}^{R} {\left( {\hat{\rho }_{\lambda } (r,t) - \rho _{\lambda } (r)} \right){\text{d}}z_{r} (v){\text{d}}r + \tilde{h}_{n} } } \right) \\ & \quad - \sum\limits_{{i = 1}}^{n} {\left( {\frac{1}{{\gamma _{1} }}{\tilde{{\varvec{\theta }}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{\theta }}}}}_{i} + \frac{1}{{\gamma _{2} }}{\tilde{{\varvec{c}}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{c}}}}}_{i} + \frac{1}{{\gamma _{3} }}{\tilde{{\varvec{\omega }}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{\omega }}}}}_{i} } \right)} - \frac{{\dot{\beta }}}{{2\beta ^{2} }}e_{n}^{2} - \frac{1}{{\gamma _{\rho } }}\int\limits_{0}^{R} {\tilde{\rho }_{\lambda } (r,t)\frac{\partial }{{\partial t}}\hat{\rho }_{\lambda } (r,t){\text{d}}r} \\ \end{aligned} $$

(A3)

Let us define ${\tilde{\varvec{\psi }}}_{i} = {\varvec{\psi}}_{i}^{*} - {\hat{\varvec{\psi }}}_{i}$. Now, $\tilde{h}_{i}$ can be written as [50]

$$ \tilde{h}_{i} = {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\hat{\varvec{\psi }}}_{i} + {\hat{\varvec{\theta }}}_{i}^{\text{T}} {\tilde{\varvec{\psi }}}_{i} + {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\tilde{\varvec{\psi }}}_{i} + \delta_{i}^{*} $$

(A4)

where $i = 1, \ldots ,n$. Adaptive FWN is used as a NIP approximator, so the basis function ${\varvec{\psi}}_{i}$ has nonlinear dependencies to the adjustable parameters of the network. Therefore, to develop the adaptive learning laws for tuning the network parameters, the Taylor expansion linearization technique is employed to transform the nonlinear function into a partially linear form [48, 50]. The result is obtained as

$$ {\tilde{\varvec{\psi }}}_{i} = A_{i}^{\text{T}} {\tilde{\varvec{\omega }}}_{i} + B_{i}^{\text{T}} {\tilde{\varvec{c}}}_{i} + o_{i} $$

(A5)

where $i = 1, \ldots ,n$, $A_{i} = \left. {\left( {\frac{{\partial {\varvec{\psi}}_{i} }}{{\partial {\varvec{\omega}}_{i} }}} \right)} \right|_{{{\varvec{\omega}}_{i} = {\hat{\varvec{\omega }}}_{i} }}$, $B_{i} = \left. {\left( {\frac{{\partial {\varvec{\psi}}_{i} }}{{\partial {\varvec{c}}_{i} }}} \right)} \right|_{{{\varvec{c}}_{i} = {\hat{\varvec{c}}}_{i} }}$ and $o_{i}$ is the high-order terms of expansion. Substituting (A5) into (A4) gives:

$$ \tilde{h}_{i} = {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\hat{\varvec{\psi }}}_{i} + {\hat{\varvec{\theta }}}_{i}^{\text{T}} \left( {A_{i}^{\text{T}} {\tilde{\varvec{\omega }}}_{i} + B_{i}^{\text{T}} {\tilde{\varvec{c}}}_{i} } \right) + {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\tilde{\varvec{\psi }}}_{i} + {\hat{\varvec{\theta }}}_{i}^{\text{T}} o_{i} + \delta_{i}^{*} $$

(A6)

Also, substituting (23), (32) and (A6) into (A3) results in:

$$ \begin{aligned} \dot{V} & = - \sum\limits_{{i = 1}}^{{n - 1}} {\left( {k_{i} + \frac{{\dot{g}_{i} }}{{2g_{i}^{2} }}} \right)e_{i}^{2} } - \left( {k_{n} + \frac{{\dot{\beta }}}{{2\beta ^{2} }}} \right)e_{n}^{2} + \sum\limits_{{i = 1}}^{{n - 1}} {e_{i} \left( {e_{{i + 1}} + \eta _{{i + 1}} } \right)} + \sum\limits_{{i = 1}}^{n} {e_{i} \left( {\tilde{\theta }_{i}^{{\text{T}}} \left( {A_{i}^{{\text{T}}} {\tilde{{\varvec{\omega }}}}_{i} + B_{i}^{{\text{T}}} {\tilde{{\varvec{c}}}}_{i} + o_{i} } \right)} \right)} \\ & \quad + \sum\limits_{{i = 1}}^{n} {e_{i} \left( {{\tilde{{\varvec{\theta }}}}_{i}^{{\text{T}}} {\hat{{\varvec{\psi }}}}_{i} + {\hat{{\varvec{\theta }}}}_{i}^{{\text{T}}} \left( {A_{i}^{{\text{T}}} {\tilde{{\varvec{\omega }}}}_{i} + B_{i}^{{\text{T}}} {\tilde{{\varvec{c}}}}_{i} } \right) + {\hat{{\varvec{\theta }}}}_{i}^{{\text{T}}} o_{i} + \delta _{i}^{*} } \right)} + \sum\limits_{{i = 1}}^{{n - 1}} {\left( { - \frac{{\eta _{{i + 1}}^{2} }}{{\tau _{{i + 1}} }} + \eta _{{i + 1}} M_{{i + 1}} } \right)} \\ & \quad - \sum\limits_{{i = 1}}^{n} {\left( {\frac{1}{{\gamma _{1} }}{\tilde{{\varvec{\theta }}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{\theta }}}}}_{i} + \frac{1}{{\gamma _{2} }}{\tilde{{\varvec{c}}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{c}}}}}_{i} + \frac{1}{{\gamma _{3} }}{\tilde{{\varvec{\omega }}}}_{i}^{{\text{T}}} {\dot{\hat{{\varvec{\omega }}}}}_{i} } \right)} + e_{n} \left( {\int\limits_{0}^{R} {\left( {\hat{\rho }_{\lambda } (r,t) - \rho _{\lambda } (r)} \right){\text{d}}z_{r} (v){\text{d}}r} } \right) \\ & \quad - \frac{1}{{\gamma _{\rho } }}\int\limits_{0}^{R} {\tilde{\rho }_{\lambda } (r,t)\frac{\partial }{{\partial t}}\hat{\rho }_{\lambda } (r,t){\text{d}}r} \\ \end{aligned} $$

(A7)

Since ${\hat{\varvec{\theta }}}_{i}^{\text{T}} A_{i}^{\text{T}} {\tilde{\varvec{\omega }}}_{i} = {\tilde{\varvec{\omega }}}_{i}^{\text{T}} A_{i} {\hat{\varvec{\theta }}}_{i}$ and ${\hat{\varvec{\theta }}}_{i}^{\text{T}} B_{i}^{\text{T}} {\tilde{\varvec{c}}}_{i} = {\tilde{\varvec{c}}}_{i}^{\text{T}} B_{i} {\hat{\varvec{\theta }}}_{i}$, (A7) is rewritten as:

$$ \begin{aligned} \dot{V} & = - \sum\limits_{i = 1}^{n - 1} {\left( {k_{i} + \frac{{\dot{g}_{i} }}{{2g_{i}^{2} }}} \right)e_{i}^{2} } - \left( {k_{n} + \frac{{\dot{\beta }}}{{2\beta^{2} }}} \right)e_{n}^{2} + \sum\limits_{i = 1}^{n - 1} {e_{i} \left( {e_{i + 1} + \eta_{i + 1} } \right)} + \sum\limits_{i = 1}^{n - 1} {\left( { - \frac{{\eta_{i + 1}^{2} }}{{\tau_{i + 1} }} + \eta_{i + 1} M_{i + 1} } \right)} \\ & \quad + \sum\limits_{i = 1}^{n} {{\tilde{\varvec{\theta }}}_{i}^{\text{T}} \left( {\left( {{\hat{\varvec{\psi }}}_{i} - A_{i}^{\text{T}} {\hat{\varvec{\omega }}}_{i} - B_{i}^{\text{T}} {\hat{\varvec{c}}}_{i} } \right)e_{i} - \frac{1}{{\gamma_{1} }}{\varvec{\dot{\hat{\theta }}}}_{i} } \right)} + \sum\limits_{i = 1}^{n} {{\tilde{\varvec{\omega }}}_{i}^{\text{T}} \left( {A_{i} {\hat{\varvec{\theta }}}_{i} e_{i} - \frac{1}{{\gamma_{3} }}{\varvec{\dot{\hat{\omega }}}}_{i} } \right) + \sum\limits_{i = 1}^{n} {{\tilde{\varvec{c}}}_{i}^{\text{T}} \left( {B_{i} {\hat{\varvec{\theta }}}_{i} e_{i} - \frac{1}{{\gamma_{2} }}{\hat{\varvec{c}}}_{i} } \right)} } \\ & \quad + \sum\limits_{i = 1}^{n} {e_{i} \left( {{\tilde{\varvec{\theta }}}_{i}^{\text{T}} \left( {A_{i}^{\text{T}} {\varvec{\omega}}_{i}^{*} + B_{i}^{\text{T}} {\varvec{c}}_{i}^{*} } \right) + {\varvec{\theta}}_{i}^{{*{\text{T}}}} o_{i} + \delta_{i}^{*} } \right)} - \frac{1}{{\gamma_{\rho } }}\int\limits_{0}^{R} {\tilde{\rho }_{\lambda } (r,t)\left( {\frac{\partial }{\partial t}\hat{\rho }_{\lambda } (r,t) + \gamma_{\rho } e_{n} {\text{d}}z_{r} (v)} \right)} {\text{d}}r \\ \end{aligned} $$

(A8)

Let us define $\Delta_{i} = {\tilde{\varvec{\theta }}}_{i}^{T} \left( {A_{i}^{T} {\varvec{\omega}}_{i}^{*} + B_{i}^{T} {\varvec{c}}_{i}^{*} } \right) + {\varvec{\theta}}_{i}^{*T} o_{i} + \delta_{i}^{*}$ for $i = 1, \ldots ,n$. Substituting adaptive laws (20), (29) and (38) into (A8) results in:

$$ \begin{aligned} \dot{V} & = - \sum\limits_{i = 1}^{n - 1} {\left( {k_{i} + \frac{{\dot{g}_{i} }}{{2g_{i}^{2} }}} \right)e_{i}^{2} } - \left( {k_{n} + \frac{{\dot{\beta }}}{{2\beta^{2} }}} \right)e_{n}^{2} + \sum\limits_{i = 1}^{n - 1} {e_{i} \left( {e_{i + 1} + \eta_{i + 1} } \right)} + \sum\limits_{i = 1}^{n - 1} {\left( { - \frac{{\eta_{i + 1}^{2} }}{{\tau_{i + 1} }} + \eta_{i + 1} M_{i + 1} } \right)} \\ & \quad + \sum\limits_{i = 1}^{n} {\sigma {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\hat{\varvec{\theta }}}_{i} } + \sum\limits_{i = 1}^{n} {\sigma {\tilde{\varvec{\omega }}}_{i}^{\text{T}} {\hat{\varvec{\omega }}}_{i} + \sum\limits_{i = 1}^{n} {\sigma {\tilde{\varvec{c}}}_{i}^{\text{T}} {\hat{\varvec{c}}}_{i} } } + \sum\limits_{i = 1}^{n} {e_{i} \Delta_{i} } + \sigma_{\rho } \int\limits_{0}^{R} {\hat{\rho }_{\lambda } (t,r)\tilde{\rho }_{\lambda } (r,t)} {\text{d}}r \\ \end{aligned} $$

(A9)

Considering the following facts

$$ \begin{aligned} & e_{i} e_{i + 1} \le 0.5\left( {e_{i}^{2} + e_{i + 1}^{2} } \right),\quad i = 1,2, \ldots ,n \\ & e_{i} \eta_{i + 1} \le 0.5\left( {e_{i}^{2} + \eta_{i + 1}^{2} } \right),\quad i = 1,2, \ldots ,n - 1 \\ & \left| {\eta_{i + 1} M_{i + 1} } \right| \le 0.5\varepsilon \eta_{i + 1}^{2} + 0.5\varepsilon^{ - 1} \bar{M}_{i + 1}^{2} ,\quad i = 1,2, \ldots ,n - 1 \\ & e_{i} \Delta_{i} \le 0.5\left( {e_{i}^{2} + \bar{\Delta }_{i}^{2} } \right),\quad i = 1,2, \ldots n \\ & \tilde{\rho }_{\lambda } (r,t)p \le 0.5\tilde{\rho }_{\lambda }^{2} (r,t) + 0.5p_{{\lambda_{\hbox{max} } }}^{2} \\ \end{aligned} $$

(A10)

where $\rho_{\lambda } (r) \le \rho_{\lambda \hbox{max} }$, and $\varepsilon$ is a positive constant. Also, considering the following inequalities

$$ \begin{aligned} & {\tilde{\varvec{\omega }}}_{i}^{\text{T}} {\hat{\varvec{\omega }}}_{i} \le 0.5\left( {{\varvec{\omega}}_{i}^{{*{\text{T}}}} {\varvec{\omega}}_{i}^{*} - {\tilde{\varvec{\omega }}}_{i}^{\text{T}} {\tilde{\varvec{\omega }}}_{i} } \right),\quad i = 1,2, \ldots ,n \\ & {\tilde{\varvec{c}}}_{i}^{\text{T}} {\hat{\varvec{c}}}_{i} \le 0.5\left( {{\varvec{c}}_{i}^{{*{\text{T}}}} {\varvec{c}}_{i}^{*} - {\tilde{\varvec{c}}}_{i}^{\text{T}} {\tilde{\varvec{c}}}_{i} } \right),\quad i = 1,2, \ldots ,n \\ & {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\hat{\varvec{\theta }}}_{i} \le 0.5\left( {{\varvec{\theta}}_{i}^{{*{\text{T}}}} {\varvec{\theta}}_{i}^{*} - {\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\tilde{\varvec{\theta }}}_{i} } \right),\quad i = 1,2, \ldots ,n \\ \end{aligned} $$

(A11)

Now, using (A10) and (A11), (A9) is written as

$$ \begin{aligned} \dot{V} & \le - \,\left( {k_{1} - \frac{{g_{1}^{d} }}{{2g_{l1}^{2} }} - 1.5} \right)e_{1}^{2} - \sum\limits_{i = 2}^{n - 1} {\left( {k_{i} - \frac{{g_{i}^{d} }}{{2g_{li}^{2} }} - 2} \right)e_{i}^{2} } - \left( {k_{n} - \frac{{\beta^{d} }}{{2\beta_{l}^{2} }} - 1} \right)e_{n}^{2} - \sum\limits_{i = 1}^{n - 1} {\left( {\frac{1}{{\tau_{i + 1} }} - 0.5\varepsilon - 0.5} \right)} \eta_{i + 1}^{2} \\ & \quad + 0.5\sigma \sum\limits_{i = 1}^{n} {\left( {\bar{\theta }_{i}^{2} + \bar{c}_{i}^{2} + \,\bar{\omega }_{i}^{2} } \right) + 0.5\sum\limits_{i = 1}^{n - 1} {\left( {\bar{\Delta }_{i}^{2} + \varepsilon^{ - 1} \bar{M}_{i + 1}^{2} } \right)} - 0.5\sigma \sum\limits_{i = 1}^{n} {\left( {{\tilde{\varvec{\theta }}}_{i}^{\text{T}} {\tilde{\varvec{\theta }}}_{i} + {\tilde{\varvec{c}}}_{i}^{\text{T}} {\tilde{\varvec{c}}}_{i} + {\tilde{\varvec{\omega }}}_{i}^{\text{T}} {\tilde{\varvec{\omega }}}_{i} } \right)} } \\ & \quad - \sigma_{\rho } \int\limits_{0}^{R} {\left( {0.5\tilde{\rho }_{\lambda }^{2} (r,t) - 0.5p_{{\lambda_{\hbox{max} } }}^{2} } \right)} {\text{d}}r \\ \end{aligned} $$

(A12)

Choose the deign parameters $k_{i}$, $\tau_{i + 1}$, $\sigma$ and $\sigma_{\rho }$ such that $k_{1} - \frac{{g_{1}^{d} }}{{2g_{l1}^{2} }} - 1.5 > 0$, $k_{i} - \frac{{g_{i}^{d} }}{{2g_{li}^{2} }} - 2 > 0,\quad i = 2, \ldots ,n - 1$, $k_{n} - \frac{{\beta^{d} }}{{2\beta_{l}^{2} }} - 1 > 0$, $\frac{1}{{\tau_{i + 1} }} - 0.5\varepsilon - 0.5 > 0$, $\sigma > 0$ and $\sigma_{\rho } > 0$, respectively.

Considering the design parameters $k_{i}$, $\tau_{i + 1}$, $\sigma$ and $\sigma_{\rho }$, comparing (A12) with (A1) reveals that (A12) satisfies the following inequality

$$ \dot{V} \le - \alpha V + \beta $$

(A13)

where $\alpha$ and $\beta$ are as follows:

$$ \alpha \le \min \left( {\begin{array}{*{20}l} {2g_{{h1}} \left( {k_{1} - 1.5 - \frac{{g_{1}^{d} }}{{2g_{{l1}}^{2} }}} \right)} \hfill \\ {2g_{{hi}} \left( {k_{i} - 2 - \frac{{g_{i}^{d} }}{{2g_{{li}}^{2} }}} \right)} \hfill \\ {2\beta _{h} \left( {k_{n} - 1 - \frac{{\beta ^{d} }}{{2\beta _{l}^{2} }}} \right)} \hfill \\ {\frac{2}{{\tau _{{i + 1}} }} - \varepsilon - 1} \hfill \\ {\sigma \gamma _{1} ,\sigma \gamma _{2} ,\sigma \gamma _{2} } \hfill \\ {\sigma _{\rho } \gamma _{\rho } } \hfill \\ \end{array} } \right) $$

$$ \beta = 0.5\sum\limits_{i = 1}^{n - 1} {\left( {\bar{\Delta }_{i}^{2} + \varepsilon^{ - 1} \bar{M}_{i + 1}^{2} } \right)} + 0.5\sigma \sum\limits_{i = 1}^{n} {\left( {\bar{\theta }_{i}^{2} + \bar{c}_{i}^{2} + \bar{\omega }_{i}^{2} } \right)} + 0.5\sigma_{\rho } Rp_{{\lambda_{\hbox{max} } }}^{2} $$

Solving inequality (A13) results in:

$$ 0 \le V \le \frac{\beta }{\alpha } + \left( {V(0) - \frac{\beta }{\alpha }} \right)e^{ - \alpha t} $$

(A14)

From (A14), it is obtained that $\lim_{t \to \infty } V = \frac{\beta }{\alpha }$. So, $V$ is bounded by $\frac{\beta }{\alpha }$. Therefore, all signals of the closed-loop system, i.e. $e_{i} ,\,\eta_{i + 1} ,\,{\tilde{\varvec{\theta }}}_{i} ,\,{\tilde{\varvec{c}}}_{i} ,\,{\tilde{\varvec{\omega }}}_{i}$ and $\tilde{\rho }_{\lambda }$ are uniformly ultimately bounded. From the considered Lyapunov function in (A1), it can be inferred that

$$ \frac{1}{{2g_{1} }}\,e_{1}^{2} \le V $$

(A15)

Considering Assumption 2, Eq. (A15) is written as

$$ e_{1}^{2} \le 2g_{h1} V $$

(A16)

which results in the following bound

$$ \left| {e_{1} (t)} \right| \le \sqrt {2g_{h1} V} $$

(A17)

Now, considering (A14), the following bound is obtained for the tracking error

$$ \left| {e_{1} } \right| \le \sqrt {2g_{h1} \left( {e^{ - \alpha \,t} V(0) + \frac{\beta }{\alpha }(1 - e^{ - \alpha t} )} \right)} $$

(A18)

It is seen form (A18) that the bound of the tracking error can be made arbitrarily small by proper selection of the design parameters.