Disturbance rejection control of morphing aircraft based on switched nonlinear systems

Abstract

The problem of disturbance rejection control for the longitudinal dynamics of a morphing aircraft is investigated based on switched nonlinear systems. A disturbance observer with nonlinear gains composed of two distinct linear regions is first designed, which adopts the idea of the extended state observer to provide the estimations of disturbances and can reduce the peaking value problem. To achieve disturbance rejection via the designed disturbance observer, two separate controllers are then designed for the velocity and altitude subsystem of the longitudinal motion model, where the altitude subsystem is modeled as switched nonlinear systems in lower triangular form and the backstepping design technique is applied. Furthermore, a modified dynamic surface is introduced at each step of the backstepping method to avoid the ‘explosion of complexity’ problem. It is proven rigorously that all signals of the closed-loop system remain bounded by the developed control scheme. Finally, comparative simulations validate the effectiveness of the proposed approach.

Appendix: Proof of Theorem 1

Proof

Define $\eta _{\varepsilon _1}=\frac{x-{\hat{x}}}{\varepsilon _1}$, $\eta _{\varepsilon _2}=\frac{x-{\hat{x}}}{\varepsilon _2}$, $\eta _2=d-{\hat{d}}$, $\eta =[\eta _{\varepsilon _1},\eta _2]^{T}$, ${\bar{\eta }}=[\eta _{\varepsilon _2},\eta _2]^{T}$, the dynamics of $\eta _{\varepsilon _1}$ and $\eta _2$ can be written as

$$\begin{aligned}&\left\{ \begin{aligned}&{\dot{\eta }}_{\varepsilon _1}=\frac{1}{\varepsilon _1}\left[ \eta _2-l_1\eta _{\varepsilon _1}-l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) \mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right] ,\\&{\dot{\eta }}_2=\frac{1}{\varepsilon _1}\left[ -l_2\eta _{\varepsilon _1}-l_2l_3\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) \mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right] +{\dot{d}},\\ \end{aligned}\right. \end{aligned}$$

(60)

Choose the following Lyapunov function candidate

$$\begin{aligned} W_1&=\frac{1}{2}k_1\eta _{\varepsilon _1}^2+\frac{1}{2}k_2\eta _2^2- k_3\eta _{\varepsilon _1}\eta _2 \nonumber \\&\quad +\, k_4\int _{0}^{\eta _{\varepsilon _1}}\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \mathrm {d}\eta _{\varepsilon _1} \end{aligned}$$

(61)

where $k_\iota (\iota =1,2,3,4)$ are positive constants, $k_1>k_3$, $k_2>k_3$. For the first three terms on the right-hand side of (61), it can be verified that

$$\begin{aligned}&\frac{1}{2}k_1\eta _{\varepsilon _1}^2+\frac{1}{2}k_2\eta _2^2- k_3\eta _{\varepsilon _1}\eta _2\geqslant \frac{1}{2}(k_1-k_3)\eta _{\varepsilon _1}^2 \nonumber \\&\quad +\,\frac{1}{2}(k_2-k_3)\eta _2^2>0 \end{aligned}$$

(62)

Besides, for the last term on the right-hand side of (61), $\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right)$ is an odd function of $\eta _{\varepsilon _1}$. It follows that $k_4\int _{0}^{\eta _{\varepsilon _1}}\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \mathrm {d}\eta _{\varepsilon _1}=0$ for $\eta _{\varepsilon _1}=0$, and $k_4\int _{0}^{\eta _{\varepsilon _1}}\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \mathrm {d}\eta _{\varepsilon _1}>0$ for $\eta _{\varepsilon _1}\ne 0$. With (62), we can know that $W_1$ is a positive definite function.

The time derivative of $W_1$ along the trajectories of (60) is given by

$$\begin{aligned} {\dot{W}}_1&=\frac{1}{\varepsilon _1}\left\{ (k_1\eta _{\varepsilon _1}-k_3\eta _2) \left[ \eta _2-l_1\eta _{\varepsilon _1}-l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1 \right) \right. \right. \nonumber \\&\quad \times \left. \mathrm {sat}\left( \frac{\eta _{ \varepsilon _1}}{l_3}\right) \right] +(k_2\eta _2-k_3\eta _{\varepsilon _1}) \nonumber \\&\quad \times \left[ -l_2\eta _{\varepsilon _1}-l_2l_3\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) \right. \nonumber \\&\quad \times \left. \left. \mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right] \right\} +(k_2\eta _2-k_3\eta _{\varepsilon _1}){\dot{d}}\nonumber \\&\quad +\, \frac{1}{\varepsilon _1}\left\{ k_4\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right. \nonumber \\&\quad \times \eta _2-\left. k_4l_1\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \eta _{\varepsilon _1}\right. \nonumber \\&\left. \quad -\, k_4l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) \mathrm {sat}^2\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right\} \nonumber \\&=\frac{1}{\varepsilon _1}\left\{ -(k_1l_1-k_3l_2)\eta _{\varepsilon _1}^2-k_3\eta _2^2 \right. \nonumber \\&\quad \left. +\, (k_1-k_2l_2+k_3l_1)\nonumber \right. \\&\quad \times \left. \eta _{\varepsilon _1}\eta _2+\right. \left[ k_3l_1\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) - k_2l_2\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) \right] l_3\nonumber \\&\quad \times \mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \eta _2 +\left[ k_3l_2\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) \right. \nonumber \\&\left. \quad -\,k_1l_1\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) \right] \nonumber \\&\quad \times \left. l_3\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \eta _{\varepsilon _1} \right\} +(k_2\eta _2-k_3\eta _{\varepsilon _1}){\dot{d}}+\frac{1}{\varepsilon _1}\nonumber \\&\quad \times \left\{ k_4\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \eta _2- k_4l_1\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \eta _{\varepsilon _1}\right. \nonumber \\&\quad -\,\left. k_4l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) \mathrm {sat}^2\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right\} \end{aligned}$$

(63)

Let $k_\iota (\iota =1,2,3,4)$ be chosen to satisfy

$$\begin{aligned}&\left\{ \begin{aligned}&k_1>k_3,\\&k_2>k_3,\\&k_1l_1-k_3l_2>0,\\&k_1-k_2l_2+k_3l_1=0,\\&k_4=\left[ k_2l_2\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) - k_3l_1\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) \right] l_3 \end{aligned}\right. \end{aligned}$$

then we have

$$\begin{aligned} {\dot{W}}_1&=\frac{1}{\varepsilon _1}\left\{ -(k_1l_1-k_3l_2) \eta _{\varepsilon _1}^2-k_3\eta _2^2\right. \nonumber \\&\quad -\,\left[ k_4l_1-k_3l_2l_3\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) + k_1l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) \right] \nonumber \\&\quad \times \left. \mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \eta _{\varepsilon _1}- k_4l_1l_3\left( \frac{\varepsilon _1}{ \varepsilon _2}-1\right) \mathrm {sat}^2\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) \right\} \nonumber \\&\quad +\,(k_2\eta _2-k_3\eta _{\varepsilon _1}){\dot{d}} \end{aligned}$$

(64)

Since $\varepsilon _2<\varepsilon _1$ and $k_1l_1-k_3l_2>0$, it can be verified that

$$\begin{aligned}&k_4l_1-k_3l_2l_3\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) + k_1l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right) >0\nonumber \\ \end{aligned}$$

(65)

Define two compact sets ${\mathcal {D}}_0=\{\varvec{\eta }\in {\mathbb {R}}^2|W_1(\varvec{\eta })\leqslant A_0\}$ and ${\mathcal {D}}_1=\{\varvec{\eta }\in {\mathbb {R}}^2|W_1(\varvec{\eta })\leqslant A_1\}$, where $A_1$ is a constant such that the maximum value of $|\eta _{\varepsilon _1}|$ in the compact set ${\mathcal {D}}_1$ is $l_3$, $A_0=\max \{W_1(\varvec{\eta }(0)), A_1\}$. The proof is then divided into the following two steps.

Step 1 We show that if $\varvec{\eta }\in {\mathcal {D}}_0-{\mathcal {D}}_1$, then there exists a $t_{\varepsilon _1}\geqslant 1$ such that $\varvec{\eta }\in {\mathcal {D}}_1$ when $t\geqslant t_{\varepsilon _1}$. The analysis is proceeded in view of the value of $|\eta _{\varepsilon _1}|$.

Case 1$|\eta _{\varepsilon _1}|\leqslant l_3$ In this case, $\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) =\frac{\eta _{\varepsilon _1}}{l_3}$. For $W_1$ and ${\dot{W}}_1$, one can further obtain

$$\begin{aligned} W_1&=\frac{1}{2}\left( k_1+\frac{k_4}{l_3}\right) \eta _{\varepsilon _1}^2+\frac{1}{2}k_2\eta _2^2- k_3\eta _{\varepsilon _1}\eta _2\nonumber \\&\leqslant \frac{1}{2}\left( k_1+k_3+\frac{k_4}{l_3}\right) \eta _{\varepsilon _1}^2+\frac{1}{2}(k_2+k_3)\eta _2^2 \end{aligned}$$

(66)

$$\begin{aligned} {\dot{W}}_1&=-\frac{1}{\varepsilon _1}\left( \varLambda _1\eta _{\varepsilon _1}^2+k_3\eta _2^2\right) + \left( k_2\eta _2-k_3\eta _{\varepsilon _1}\right) {\dot{d}}\nonumber \\&\leqslant -\frac{1}{2\varepsilon _1}\left( \varLambda _1\eta _{\varepsilon _1}^2+k_3\eta _2^2\right) - \frac{\kappa _1}{2\varepsilon _1}\Big \Vert \varvec{\eta }\Big \Vert ^2\nonumber \\&\quad +\, \sqrt{k_2^2+k_3^2}N_1\Big \Vert \varvec{\eta }\Big \Vert \end{aligned}$$

(67)

where $\varLambda _1=\frac{\varepsilon _1}{\varepsilon _2}k_1l_1-\frac{\varepsilon _1^2}{\varepsilon _2^2}k_3l_2+\frac{\varepsilon _1}{\varepsilon _2l_3}k_4l_1$, $\kappa _1=\min \{\varLambda _1,k_3\}$.

By setting $\varepsilon _1\leqslant \varepsilon _{\mathrm {c}1}\triangleq \frac{\kappa _1\displaystyle \min _{\varvec{\eta }\in {\mathcal {D}}_0-{\mathcal {D}}_1} \Big \Vert \varvec{\eta }\Big \Vert }{2\sqrt{k_2^2+k_3^2}N_1}$, we arrive at

$$\begin{aligned} {\dot{W}}_1&\leqslant -\frac{1}{2\varepsilon _1}\left( \varLambda _1\eta _{\varepsilon _1}^2+k_3\eta _2^2\right) \nonumber \\&\quad -\, \Big \Vert \varvec{\eta }\Big \Vert \left( \frac{\kappa _1}{2\varepsilon _1}\Big \Vert \varvec{\eta }\Big \Vert - \sqrt{k_2^2+k_3^2}N_1\right) \nonumber \\&\leqslant -\frac{1}{2\varepsilon _1}\left( \varLambda _1\eta _{\varepsilon _1}^2+k_3\eta _2^2\right) \leqslant -\frac{\lambda _1}{\varepsilon _1}W_1<0 \end{aligned}$$

(68)

where $\lambda _1=\min \left\{ \frac{\varLambda _1}{k_1+k_3+\frac{k_4}{l_3}},\frac{k_3}{k_2+k_3}\right\}$.

Case 2$|\eta _{\varepsilon _1}|> l_3$ In this case, $\mathrm {sat}\left( \frac{\eta _{\varepsilon _1}}{l_3}\right) =\mathrm {sign}(\eta _{\varepsilon _1})$, and $W_1$ satisfies

$$\begin{aligned} W_1&=\frac{1}{2}k_1\eta _{\varepsilon _1}^2+\frac{1}{2}k_2\eta _2^2- k_3\eta _{\varepsilon _1}\eta _2+ k_4\left( |\eta _{\varepsilon _1}|-\frac{l_3}{2}\right) \nonumber \\&\leqslant \frac{1}{2}\left( k_1+k_3\right) \eta _{\varepsilon _1}^2+\frac{1}{2}(k_2+k_3)\eta _2^2+ k_4\left( |\eta _{\varepsilon _1}|-\frac{l_3}{2}\right) \end{aligned}$$

(69)

The corresponding derivative ${\dot{W}}_1$ now becomes

$$\begin{aligned} {\dot{W}}_1=&-\frac{1}{\varepsilon _1}\left[ (k_1l_1-k_3l_2)\eta _{\varepsilon _1}^2+k_3\eta _2^2\right. \nonumber \\&\quad \left. +\, \varLambda _2\left( |\eta _{\varepsilon _1}|-\frac{l_3}{2}\right) +\varLambda _3\right] +(k_2\eta _2-k_3\eta _{\varepsilon _1}){\dot{d}}\nonumber \\ \leqslant&-\frac{1}{\varepsilon _1}\left[ \frac{1}{2}(k_1l_1-k_3l_2) \eta _{\varepsilon _1}^2+\frac{k_3}{2}\eta _2^2\right. \nonumber \\&\quad \left. +\, \varLambda _2\left( |\eta _{\varepsilon _1}|-\frac{l_3}{2}\right) \right] \nonumber \\&\quad -\,\frac{\kappa _2}{2\varepsilon _1}\Big \Vert \varvec{\eta } \Big \Vert ^2-\frac{\Lambda _3}{\varepsilon _1}+ \sqrt{k_2^2+k_3^2}N_1\Big \Vert \varvec{\eta }\Big \Vert \end{aligned}$$

(70)

where $\varLambda _2=k_4l_1-k_3l_2l_3\left( \frac{\varepsilon _1^2}{\varepsilon _2^2}-1\right) + k_1l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right)$, $\varLambda _3=\frac{l_3}{2}\varLambda _2+k_4l_1l_3\left( \frac{\varepsilon _1}{\varepsilon _2}-1\right)$, $\kappa _2=\min \{k_1l_1-k_3l_2,k_3\}$.

Choosing $\varepsilon _1\leqslant \varepsilon _{\mathrm {c}2}\triangleq \sqrt{\frac{2\kappa _2\varLambda _3}{k_2^2+k_3^2}}\frac{1}{N_1}$, then by the Young’s inequality [40], one gets

$$\begin{aligned}&\frac{\kappa _2}{2\varepsilon _1}\Big \Vert \varvec{\eta } \Big \Vert ^2+\frac{\Lambda _3}{\varepsilon _1} \geqslant 2\sqrt{\frac{\kappa _2}{2\varepsilon _1} \frac{\Lambda _3}{\varepsilon _1}}\Big \Vert \varvec{\eta }\Big \Vert \geqslant \sqrt{k_2^2+k_3^2}N_1\Big \Vert \varvec{\eta }\Big \Vert \end{aligned}$$

(71)

By (70) and (71), it follows that

$$\begin{aligned} {\dot{W}}_1&\leqslant -\frac{1}{\varepsilon _1}[\frac{1}{2}(k_1l_1-k_3l_2)\eta _{\varepsilon _1}^2+\frac{k_3}{2}\eta _2^2\nonumber \\&\quad +\, \varLambda _2(|\eta _{\varepsilon _1}|-\frac{l_3}{2})] \leqslant -\frac{\lambda _2}{\varepsilon _1}W_1<0 \end{aligned}$$

(72)

where $\lambda _2=\min \left\{ \frac{k_1l_1-k_3l_2}{k_1+k_3},\frac{k_3}{k_2+k_3},\frac{\Lambda _2}{k_4}\right\}$.

With the above two cases, we know that $W_1$ is strictly decreasing in ${\mathcal {D}}_0-{\mathcal {D}}_1$. We can then compute the specific bound on $W_1$. Define ${\bar{\lambda }}_1=\min \{\lambda _1,\lambda _2\}$ and let $\varepsilon _1\leqslant \min \{\varepsilon _{\mathrm {c}1},$$\varepsilon _{\mathrm {c}2}\}$, then applying the comparison principle of ordinary differential equations in (68) and (72) implies

$$\begin{aligned}&W_1(\varvec{\eta }(t))\leqslant \mathrm {e}^{-\frac{{\bar{\lambda }}_1}{\varepsilon _1}t}W_1(\varvec{\eta }(0)) \end{aligned}$$

(73)

Let $t_{\varepsilon _1}=\frac{\varepsilon _1}{{\bar{\lambda }}_1}\ln (\frac{A_0}{A_1})$, it can be seen from (73) that $\varvec{\eta }\in {\mathcal {D}}_1$ when $t\geqslant t_{\varepsilon _1}$.

Step 2 We show that if $\varvec{\eta }\in {\mathcal {D}}_1$, then there exists a $t_{\varepsilon _2}\geqslant 0$ such that $|x(t)-{\hat{x}}(t)|=O(\varepsilon _2^2)$ and $|d(t)-{\hat{d}}(t)|=O(\varepsilon _2)$ when $t\geqslant t_{\varepsilon _1}+t_{\varepsilon _2}$. Since $|\eta _{\varepsilon _1}|\leqslant l_3$ always holds by the definition of ${\mathcal {D}}_1$, the dynamics of estimation errors can then be written as

$$\begin{aligned}&\left\{ \begin{aligned}&{\dot{\eta }}_{\varepsilon _2}=\frac{1}{\varepsilon _2}(\eta _2-l_1\eta _{\varepsilon _2}),\\&{\dot{\eta }}_2=-\frac{l_2}{\varepsilon _2}\eta _{\varepsilon _2}+{\dot{d}},\\ \end{aligned}\right. \end{aligned}$$

(74)

Choose the Lyapunov function candidate as follows

$$\begin{aligned}&W_2=\frac{1}{2}k_1\eta _{\varepsilon _2}^2+\frac{1}{2}k_2\eta _2^2- k_3\eta _{\varepsilon _2}\eta _2 \end{aligned}$$

(75)

It holds that

$$\begin{aligned}&\frac{1}{2}(k_1-k_3)\eta _{\varepsilon _2}^2+\frac{1}{2}(k_2-k_3)\eta _2^2\nonumber \\&\quad \leqslant W_2\leqslant \frac{1}{2}(k_1+k_3)\eta _{\varepsilon _2}^2+\frac{1}{2}(k_2+k_3)\eta _2^2 \end{aligned}$$

(76)

By the relationship between $k_i(i=1,2,3)$, the derivative of $W_2$ along the trajectories of (74) is given by

$$\begin{aligned} {\dot{W}}_2&=\frac{1}{\varepsilon _2}\left[ -(k_1l_1-k_3l_2)\eta _{\varepsilon _2}^2- k_3\eta _2^2\right] \nonumber \\&\quad +\,\left( k_2\eta _2-k_3\eta _{\varepsilon _2}\right) {\dot{d}} \end{aligned}$$

(77)

Define $\lambda _3=\min \left\{ \frac{k_1-k_3}{2},\frac{k_2-k_3}{2}\right\}$, $\lambda _4=\min \left\{ \frac{2(k_1l_1-k_3l_2)}{k_1+k_3},\right.$$\left. \frac{k_3}{k_2+k_3}\right\}$, we further obtain

$$\begin{aligned}&{\dot{W}}_2\leqslant -\frac{\lambda _4}{\varepsilon _2}W_2+\sqrt{k_2^2+k_3^2}N_1\sqrt{\frac{W_2}{\lambda _3}} \end{aligned}$$

(78)

With (78), we see that if $W_2>\frac{4\varepsilon _2^2(k_2^2+k_3^2)N_1^2}{\lambda _3\lambda _4^2}$, then

$$\begin{aligned}&{\dot{W}}_2\leqslant -\frac{\lambda _4}{2\varepsilon _2}W_2<0 \end{aligned}$$

(79)

By the comparison principle of the ordinary differential equations, we further have

$$\begin{aligned}&W_2(\bar{\varvec{\eta }}(t))\leqslant \mathrm {e}^{-\frac{\lambda _4}{2\varepsilon _2}(t-t_{\varepsilon _1})} W_2(\bar{\varvec{\eta }}(t_{\varepsilon _1})) \end{aligned}$$

(80)

In view of the relationship between $\varvec{\eta }$ and $\bar{\varvec{\eta }}$, since $\varvec{\eta }\in {\mathcal {D}}_1$ when $t\geqslant t_{\varepsilon _1}$, there exists a constant $A_2$ such that $W_2(\bar{\varvec{\eta }}(t_{\varepsilon _1}))\leqslant A_2$. Let $t_{\varepsilon _2}=\frac{2\varepsilon _2}{{\bar{\lambda }}_4}\ln (\frac{A_2}{A_3})$, where $A_3=\frac{4\varepsilon _2^2(k_2^2+k_3^2)N_1^2}{\lambda _3\lambda _4^2}$, $A_2=\max \{{\bar{A}}_2,A_3\}$, then for all $t\geqslant t_{\varepsilon _1}+t_{\varepsilon _2}$,

$$\begin{aligned}&W_2(\bar{\varvec{\eta }}(t))\leqslant \frac{4\varepsilon _2^2\left( k_2^2+k_3^2\right) N_1^2}{\lambda _3\lambda _4^2} \end{aligned}$$

(81)

By the definition of $W_2(\bar{\varvec{\eta }}(t))$, one finally reaches

$$\begin{aligned} \Big \Vert \bar{\varvec{\eta }}(t)\Big \Vert&\leqslant \sqrt{\frac{W_2(\bar{\varvec{\eta }}(t))}{\lambda _3}} \nonumber \\&\quad \leqslant \frac{2\varepsilon _2 N_1 \sqrt{k_2^2+k_3^2}}{\lambda _3\lambda _4} \end{aligned}$$

(82)

$$\begin{aligned} |x(t)-{\hat{x}}(t)|=\varepsilon _2|\eta _{\varepsilon _2}(t)|&\leqslant \varepsilon _2\Big \Vert \bar{\varvec{\eta }}(t)\Big \Vert \nonumber \\&\quad \leqslant \frac{2\varepsilon _2^2 N_1 \sqrt{k_2^2+k_3^2}}{\lambda _3\lambda _4} \end{aligned}$$

(83)

$$\begin{aligned} |d(t)-{\hat{d}}(t)|=|\eta _2(t)|&\leqslant \Big \Vert \bar{\varvec{\eta }}(t)\Big \Vert \nonumber \\&\quad \leqslant \frac{2\varepsilon _2 N_1 \sqrt{k_2^2+k_3^2}}{\lambda _3\lambda _4} \end{aligned}$$

(84)

We concluded from (83) and (84) that $|x(t)-{\hat{x}}(t)|=O(\varepsilon _2^2)$ and $|d(t)-{\hat{d}}(t)|=O(\varepsilon _2)$ when $t\geqslant t_{\varepsilon _1}+t_{\varepsilon _2}$, which completes the proof of Theorem 1. $\square$

Remark 10

We note that the utilization of $|\eta _{\varepsilon _1}|$ in (69) is for the convenience of the analysis in Step 1 to show that $W_1$ is strictly decreasing in ${\mathcal {D}}_0-{\mathcal {D}}_1$. In fact, it can be verified by the definition of $\eta _{\varepsilon _1}$ and $W_1$ that $W_1$ is a positive definite and continuously differentiable function. Moreover, the appearance of $|\eta _{\varepsilon _1}|$ in (69) is only for the case that $|\eta _{\varepsilon _1}|>l_3$, which does not contain the origin and the derivative of $W_1$ can then be computed without suffering singularity problems. Similar approaches can also be found in [48, 49].