Command filtered adaptive fuzzy tracking control for uncertain stochastic nonlinear systems with event-triggered input

Abstract

This paper addressed an issue regarding adaptive fuzzy backstepping for a class of stochastic nonlinear system with an event-triggered input. The presence of unknown nonlinear functions can be evaluated through fuzzy logic systems. With the aid of dynamic surface control, an adaptive tracking scheme is suggested, which can eliminate the “explosion of complexity.”An improved event-triggered co-design control scheme is proposed, whose threshold is a variable function combining tracking error, so that resources are saved. A tracking controller can guarantee that the signals of the closed-loop system are uniform. The tracking errors toward a small neighborhood of the origin. Simulations are conducted to validate the proposed method.

Appendix A

The design process is given as follows.

Step 1: According to (15), its derivative is

$$\begin{aligned} \text {d}z_{1}= & {} \left( g_{1}x_{2}+f_{1}-\dot{y}_{d}\right) \text {d}t+\varphi _{1}^{T}\text {d}\omega \nonumber \\= & {} \left( g_{1}\left( z_{2}+\eta _{2}+\alpha _{1}\right) +f_{1}-\dot{y} _{d}\right) \text {d}t+\varphi _{1}^{T}\text {d}\omega \nonumber \\= & {} \left( g_{1}\left( z_{2}+\eta _{2}+\alpha _{1}\right) +\delta _{1}+\theta _{1}^{*}\xi _{1}(x_{1})-\dot{y}_{d}\right) \text {d}t \nonumber \\{} & {} +\varphi _{1}^{T}\text {d}\omega \end{aligned}$$

(41)

We choose a Lyapunov function,

$$\begin{aligned} V_{1}=\frac{1}{4\underline{g}_{1}}z_{1}^{4}+\frac{1}{2\gamma _{1}}\tilde{ \omega }_{1}^{T}\tilde{\omega }_{1}+\frac{1}{2a_{1}}\tilde{\varepsilon }_{1}^{T} \tilde{\varepsilon }_{1}+\frac{1}{2b_{1}}\tilde{\epsilon }_{1}^{T}\tilde{ \epsilon }_{1}\nonumber \\ \end{aligned}$$

(42)

with parameters $\gamma _{1}>0,a_{1}>0$, and $b_{1}>0$. $\tilde{\omega }_{1}$ is the estimation error and $\omega _{1}$ is the estimations of $\omega _{1}^{*}$ and $\tilde{\omega }_{1}=\omega _{1}^{*}-\omega _{1}$.

According to (23) and (41),

$$\begin{aligned} LV_{1}\le & {} \frac{1}{\underline{g}_{1}}z_{1}^{3}\left( g_{1}(z_{2}+\eta _{2}+\alpha _{1})-\overset{\cdot }{y}_{d}\right) +\omega _{1}^{*}z_{1}^{3}\xi _{1}(x_{1}) \nonumber \\{} & {} +\varepsilon _{1}^{*}z_{1}^{3}+\frac{3}{2\underline{g}_{1}} z_{1}^{2}\left\| \varphi _{1}(x_{1})\right\| ^{2}+\frac{1}{\gamma _{1}} \tilde{\omega }_{1}^{T}\overset{.}{\tilde{\omega }}_{1} \nonumber \\{} & {} +\frac{1}{a_{1}}\tilde{\varepsilon }_{1}^{T}\overset{.}{\tilde{\varepsilon }} _{1}+\frac{1}{b_{1}}\tilde{\epsilon }_{1}^{T}\overset{.}{\tilde{\epsilon }}_{1} \end{aligned}$$

(43)

Using Young’s inequality, we get

$$\begin{aligned}{} & {} \frac{3}{2\underline{g}_{1}}z_{1}^{2}\left\| \varphi _{1}(x_{1})\right\| ^{2}\le \frac{3}{4}l_{1}^{2}+\frac{3}{4\underline{g} _{1}^{2}l_{1}^{2}}z_{1}^{4}\left\| \varphi _{1}(x_{1})\right\| ^{4} \end{aligned}$$

(44)

$$\begin{aligned}{} & {} \frac{g_{1}}{\underline{g}_{1}}z_{1}^{3}\eta _{2}\le \frac{1}{2}\frac{ g_{1}^{2}}{\underline{g}_{1}^{2}}z_{1}^{6}+\frac{1}{2}\eta _{2}^{2} \end{aligned}$$

(45)

$$\begin{aligned}{} & {} z_{1}^{3}z_{2}\le \frac{3}{4}z_{1}^{4}+\frac{1}{4}z_{2}^{4} \end{aligned}$$

(46)

Substituting (44)–(46) in (43) yields

$$\begin{aligned} LV_{1}\le & {} \frac{1}{\underline{g}_{1}}z_{1}^{3}\left( g_{1}\alpha _{1}+ \frac{3}{4\underline{g}_{1}l_{1}^{2}}z_{1}\left\| \varphi _{1}(x_{1})\right\| ^{4}-\overset{\cdot }{y}_{d}\right) \nonumber \\{} & {} +\frac{1}{\underline{g}_{1}}z_{1}^{3}\left( \frac{3}{4}g_{1}z_{1}+\frac{1}{ 2}\frac{g_{1}^{2}}{\underline{g}_{1}}z_{1}^{3}\right) +\frac{1}{4}\frac{g_{1} }{\underline{g}_{1}}z_{2}^{4} \nonumber \\{} & {} +\omega _{1}z_{1}^{3}\xi _{1}(x_{1})+\frac{1}{\gamma _{1}}\tilde{\omega } _{1}^{T}(\gamma _{1}z_{1}^{3}\xi _{1}(x_{1})-\dot{\omega }_{1}) \nonumber \\{} & {} +\frac{1}{a_{1}}\tilde{\varepsilon }_{1}^{T}(a_{1}z_{1}^{3}-\dot{\varepsilon }_{1})+\varepsilon _{1}z_{1}^{3}+\frac{3}{4}l_{1}^{2} \nonumber \\{} & {} +\frac{1}{2}\eta _{2}^{2}-\frac{1}{b_{1}}\tilde{\epsilon }_{1}^{T}\dot{ \epsilon }_{1} \end{aligned}$$

(47)

where $v_{1}^{T}=\left[ \left\| \varphi _{1}(x_{1})\right\| ^{4}/ \underline{g}_{1}^{2}l_{1}^{2},-1/\underline{g}_{1},g_{1}/\underline{g} _{1},g_{1}^{2}/\underline{g}_{1}^{2}\right]$ and $S_{1}^{T}=\left[ 3z_{1}/4, \overset{\cdot }{y}_{d},3z_{1}/4,z_{1}^{3}/2\right] .$ Then, we can write

$$\begin{aligned} LV_{1}\le & {} \frac{g_{1}}{\underline{g}_{1}}z_{1}^{3}\alpha _{1}+z_{1}^{3}v_{1}^{T}S_{1}+\omega _{1}z_{1}^{3}\xi _{1}(x_{1})+\varepsilon _{1}z_{1}^{3} \nonumber \\{} & {} +\frac{1}{2}\eta _{2}^{2}+\frac{1}{4}\frac{g_{1}}{\underline{g}_{1}} z_{2}^{4}+\frac{3}{4}l_{1}^{2}+\frac{1}{a_{1}}\tilde{\varepsilon } _{1}^{T}(a_{1}z_{1}^{3}-\dot{\varepsilon }_{1}) \nonumber \\{} & {} +\frac{1}{\gamma _{1}}\tilde{\omega }_{1}^{T}(\gamma _{1}z_{1}^{3}\xi _{1}(x_{1})-\dot{\omega }_{1})-\frac{1}{b_{1}}\tilde{\epsilon }_{1}^{T}\dot{ \epsilon }_{1} \end{aligned}$$

(48)

Using (17), we obtain

$$\begin{aligned} LV_{1}\le & {} \frac{g_{1}}{\underline{g}_{1}}z_{1}^{3}\alpha _{1}+z_{1}^{3}\epsilon _{1}^{*}\tau _{1}+\frac{1}{\gamma _{1}}\tilde{ \omega }_{1}^{T}(\gamma _{1}z_{1}^{3}\xi _{1}(x_{1})-\dot{\omega }_{1}) \nonumber \\{} & {} +\frac{1}{a_{1}}\tilde{\varepsilon }_{1}^{T}(a_{1}z_{1}^{3}-\dot{\varepsilon }_{1})+\rho _{1}\epsilon _{1}^{*}+\frac{3}{4}l_{1}^{2}+z_{1}^{3}\bar{ \alpha }_{1} \nonumber \\{} & {} -z_{1}^{3}\bar{\alpha }_{1}+\frac{1}{2}\eta _{2}^{2}+\frac{1}{4}\frac{g_{1} }{\underline{g}_{1}}z_{2}^{4}+\omega _{1}z_{1}^{3}\xi _{1}(x_{1}) \nonumber \\{} & {} +\varepsilon _{1}z_{1}^{3}-\frac{1}{b_{1}}\tilde{\epsilon }_{1}^{T}\dot{ \epsilon }_{1} \end{aligned}$$

(49)

where $\bar{\alpha }_{1}=-c_{1}z_{1}+\epsilon _{1}\tau _{1}+\omega _{1}\xi _{1}(x_{1})+\varepsilon _{1}, c_{1}>0,$ and we get

$$\begin{aligned} LV_{1}\le & {} -c_{1}z_{1}^{4}+\frac{g_{1}}{\underline{g}_{1}}z_{1}^{3}\alpha _{1}+\frac{1}{\gamma _{1}}\tilde{\omega }_{1}^{T}(\gamma _{1}z_{1}^{3}\xi _{1}(x_{1})-\dot{\omega }_{1}) \nonumber \\{} & {} +\frac{1}{a_{1}}\tilde{\varepsilon }_{1}^{T}(a_{1}z_{1}^{3}-\dot{\varepsilon }_{1})+\frac{1}{b_{1}}\tilde{\epsilon }_{1}^{T}(b_{1}z_{1}^{3}\tau _{1}-\dot{ \epsilon }_{1}) \nonumber \\{} & {} +\rho _{1}\epsilon _{1}^{*}+\frac{3}{4}l_{1}^{2}+z_{1}^{3}\bar{\alpha } _{1}+\frac{1}{2}\eta _{2}^{2}+\frac{1}{4}\frac{g_{1}}{\underline{g}_{1}} z_{2}^{4} \end{aligned}$$

(50)

We design the first virtual control input $\alpha _{1}$ and the first turning function as

$$\begin{aligned}{} & {} \alpha _{1}=-\frac{z_{1}^{3}\bar{\alpha }_{1}^{2}}{\sqrt{z_{1}^{6}\bar{\alpha } _{1}^{2}+\rho _{1}}} \end{aligned}$$

(51)

$$\begin{aligned}{} & {} \dot{\omega }_{1}=\gamma _{1}z_{1}^{3}\xi _{1}(x_{1})-m_{1}\omega _{1} \end{aligned}$$

(52)

$$\begin{aligned}{} & {} \dot{\varepsilon }_{1}=a_{1}z_{1}^{3}-n_{1}\varepsilon _{1} \end{aligned}$$

(53)

$$\begin{aligned} \dot{\epsilon }_{1}=b_{1}z_{1}^{3}\tau _{1}-h_{1}\epsilon _{1} \end{aligned}$$

(54)

with parameters $m_{1}>0,n_{1}>0$, and $h_{1}>0$.

From (51),

$$\begin{aligned} \frac{g_{1}}{\underline{g}_{1}}z_{1}^{3}\alpha _{1}+z_{1}^{3}\bar{\alpha } _{1}\le \rho _{1} \end{aligned}$$

(55)

Substituting (51)–(55) in (50), we can get

$$\begin{aligned} LV_{1}\le & {} -c_{1}z_{1}^{4}+\frac{m_{1}}{\gamma _{1}}\tilde{\omega } _{1}^{T}\omega _{1}+\frac{n_{1}}{a_{1}}\tilde{\varepsilon }\varepsilon _{1}+ \frac{h_{1}}{b_{1}}\tilde{\epsilon }_{1}^{T}\epsilon _{1} \nonumber \\{} & {} +\rho _{1}(1+\epsilon _{1}^{*})+\frac{3}{4}l_{1}^{2}+\frac{1}{2}\eta _{2}^{2}+\frac{1}{4}\frac{g_{1}}{\underline{g}_{1}}z_{2}^{4} \end{aligned}$$

(56)

To avoid continuously differentiating $\alpha _{1}$, we describe the first-order filter as

$$\begin{aligned} \chi _{2}\overset{.}{d}_{2}+d_{2}=\alpha _{1},d_{2}(0)=\alpha _{1}(0) \end{aligned}$$

(57)

where $\chi _{2}>0$ is a constant.

Using the definition $\eta _{2}=d_{2}-\alpha _{1},$ we have

$$\begin{aligned} \dot{\eta }_{2}=\overset{.}{d}_{2}-\dot{\alpha }_{1}=-\frac{\eta _{2}}{\chi _{2}}+M_{2}(\cdot ) \end{aligned}$$

(58)

where $M_{2}(\cdot )$ is a continuous function described as

$$\begin{aligned} M_{2}(\cdot )= & {} \frac{\partial \alpha _{1}}{\partial x_{1}} (g_{1}x_{2}+f_{1})+\sum _{i=0}^{1}\frac{\partial \alpha _{1}}{\partial y_{r}^{(i)}}y_{r}^{(i+1)} \nonumber \\{} & {} +\frac{\partial \alpha _{1}}{\partial \hat{\omega }_{1}}\dot{\omega }_{1}+ \frac{\partial \alpha _{1}}{\partial \hat{\varepsilon }_{1}}\dot{\varepsilon } _{1}+\frac{\partial \alpha _{1}}{\partial \hat{\epsilon }_{1}}\dot{\epsilon } _{1} \nonumber \\{} & {} +\frac{\partial \alpha _{1}}{\partial \hat{\theta }_{1}}\overset{.}{\hat{ \theta }}_{1}+\frac{1}{2}\frac{\partial ^{2}\alpha _{1}}{\partial x_{1}^{2}} \varphi _{1}^{T}\varphi _{1} \end{aligned}$$

(59)

Step i ($i=2$, ..., $n-1$)$:$ According to (15), its derivative is

$$\begin{aligned} \text {d}z_{i}= & {} \left( g_{i}x_{i+1}+f_{i}-\dot{d}_{i}\right) \text {d}t+\varphi _{i}^{T}\text {d}\omega \nonumber \\= & {} \left( g_{i}\left( z_{i+1}+\eta _{i+1}+\alpha _{i}\right) +f_{i}-\dot{d} _{i}\right) \text {d}t+\varphi _{i}^{T}\text {d}\omega \nonumber \\= & {} \left( g_{i}\left( z_{i+1}+\eta _{i+1}+\alpha _{i}\right) +\delta _{i}\right) \text {d}t \nonumber \\{} & {} \left( +\theta _{i}^{*}\xi _{i}(x_{i})-\dot{d}_{i}\right) \text {d}t+\varphi _{i}^{T}\text {d}\omega \end{aligned}$$

(60)

Choose a Lyapunov function as

$$\begin{aligned} V_{i}= & {} V_{i-1}+\frac{1}{4\underline{g}_{i}}z_{i}^{4}+\frac{1}{2\gamma _{i}} \tilde{\omega }_{i}^{T}\tilde{\omega }_{i}+\frac{1}{2a_{i}}\tilde{\varepsilon } _{i}^{T}\tilde{\varepsilon }_{i} \nonumber \\{} & {} +\frac{1}{2b_{i}}\tilde{\epsilon }_{i}^{T}\tilde{\epsilon }_{i}+\frac{1}{2} \eta _{i}^{2} \end{aligned}$$

(61)

with parameters $\gamma _{i}>0, a_{i}>0$, and $b_{i}>0$. $\tilde{\omega }_{i}$ is the estimation error and $\omega _{i}$ is the estimations of $\omega _{i}^{*} \tilde{\omega }_{i}=\omega _{i}^{*}-\omega _{i}$.

According to (23) and (60),

$$\begin{aligned} LV_{i}\le & {} LV_{i-1}+\frac{1}{\underline{g}_{i}}z_{i}^{3}\left( g_{i}(z_{i+1}+\eta _{i+1}+\alpha _{i})-\dot{d}_{i}\right) \nonumber \\{} & {} +\omega _{i}^{*}z_{i}^{3}\xi _{i}(x_{i})+\eta _{i}\dot{\eta }_{i}+\frac{ 1}{\gamma _{i}}\tilde{\omega }_{i}^{T}\overset{.}{\tilde{\omega }_{i}}+\frac{1 }{a_{i}}\tilde{\varepsilon }_{i}^{T}\overset{.}{\tilde{\varepsilon }}_{i} \nonumber \\{} & {} +\frac{3}{2\underline{g}_{i}}z_{i}^{2}\left\| \varphi _{i}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\varphi _{j}\right\| ^{2}+\varepsilon _{i}^{*}z_{i}^{3} \nonumber \\{} & {} +\frac{1}{b_{i}}\tilde{\epsilon }_{i}^{T}\overset{.}{\tilde{\epsilon }}_{i} \end{aligned}$$

(62)

Using Young’s inequality, we obtain

$$\begin{aligned}{} & {} \frac{3}{2\underline{g}_{i}}z_{i}^{2}\left\| \varphi _{i}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\varphi _{j}\right\| ^{2} \nonumber \\\le & {} \frac{3}{4}l_{i}^{2}+\frac{3}{4\underline{g}_{i}^{2}l_{i}^{2}} z_{i}^{4}\left\| \varphi _{i}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{i-1} }{\partial x_{j}}\varphi _{j}\right\| ^{4} \end{aligned}$$

(63)

$$\begin{aligned}{} & {} \frac{g_{i}}{\underline{g}_{i}}z_{i}^{3}\eta _{i+1}\le \frac{1}{2}\frac{ g_{i}^{2}}{\underline{g}_{i}^{2}}z_{i}^{6}+\frac{1}{2}\eta _{i+1}^{2} \end{aligned}$$

(64)

$$\begin{aligned}{} & {} \frac{g_{i}}{\underline{g}_{i}}z_{i}^{3}z_{i+1}\le \frac{3}{4}\frac{g_{i}}{ \underline{g}_{i}}z_{i}^{4}+\frac{1}{4}\frac{g_{i}}{\underline{g}_{i}} z_{i+1}^{4} \end{aligned}$$

(65)

Substituting (63)–(65) in (62) yields

$$\begin{aligned} LV_{i}\le & {} -\sum _{j=1}^{i-1}c_{j}z_{j}^{4}+\sum _{j=1}^{i-1}\frac{m_{j}}{ \gamma _{j}}\tilde{\omega }_{j}^{T}\omega _{j}+\sum _{j=1}^{i-1}\frac{n_{j}}{ a_{j}}\tilde{\varepsilon }_{j}^{T}\varepsilon _{j} \nonumber \\{} & {} +\sum _{j=1}^{i-1}\frac{h_{j}}{b_{j}}\tilde{\epsilon }_{j}^{T}\epsilon _{j}+\sum _{j=1}^{i-1}\rho _{j}(1+\epsilon _{j}^{*})+\frac{1}{4}\frac{ g_{i}}{\underline{g}_{i}}z_{i+1}^{4} \nonumber \\{} & {} +\frac{1}{\underline{g}_{i}}z_{i}^{3}\left( g_{i}\alpha _{i}-Ld_{i}+\frac{1 }{4}\frac{g_{i-1}\underline{g}_{i}}{\underline{g}_{i-1}}z_{i}\right) -\frac{1 }{b_{j}}\tilde{\epsilon }_{j}^{T}\dot{\epsilon }_{j} \nonumber \\{} & {} +\frac{1}{\underline{g}_{i}}z_{i}^{3}\left( \frac{3}{4\underline{g} _{i}l_{i}^{2}}z_{i}\left\| \varphi _{i}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\varphi _{j}\right\| ^{4}\right) \nonumber \\{} & {} +\frac{1}{\underline{g}_{i}}z_{i}^{3}\left( \frac{1}{2}\frac{g_{i}^{2}}{ \underline{g}_{i}}z_{i}^{3}+\frac{3}{4}g_{i}z_{i}\right) +\varepsilon _{i}z_{i}^{3}+\frac{1}{2}\eta _{i+1}^{2} \nonumber \\{} & {} +\frac{1}{a_{i}}\tilde{\varepsilon }_{i}^{T}(a_{i}z_{i}^{3}-\dot{\varepsilon }_{i})+\frac{1}{\gamma _{i}}\tilde{\omega }_{i}^{T}(\gamma _{i}z_{i}^{3}\xi _{i}(x_{i})-\dot{\omega }_{i}) \nonumber \\{} & {} +\sum _{j=1}^{i}\frac{3}{4}l_{j}^{2}+\omega _{i}z_{i}^{3}\xi _{i}(x_{i})+2(i-2)\iota +\eta _{i}\dot{\eta }_{i} \nonumber \\{} & {} +\sum _{j=2}^{i}\left( \frac{1}{2}-\frac{1}{\chi _{j}}+\frac{1}{2\iota } M_{j}^{2}(\cdot )\right) \eta _{j}^{2} \end{aligned}$$

(66)

where $v_{i}^{T}=[\left\| \varphi _{i}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\varphi _{j}\right\| ^{4}/\underline{g} _{i}^{2}l_{i}^{2}$,$-1/\underline{g}_{i}$, $g_{i}/\underline{g}_{i}$, $g_{i}^{2}/\underline{g}_{i}^{2}$, $g_{i-1} \underline{g}_{i}/\underline{g}_{i-1}]$, and $S_{i}^{T}=[3z_{1}/4$,$Ld_{i}$, $3z_{i}/4$,$z_{i}^{3}/2$,$z_{i}/4]$

Then we can get

$$\begin{aligned} LV_{i}\le & {} -\sum _{j=1}^{i-1}c_{j}z_{j}^{4}+\sum _{j=1}^{i-1}\frac{m_{j}}{ \gamma _{j}}\tilde{\omega }_{j}^{T}\omega _{j}+\sum _{j=1}^{i-1}\frac{n_{j}}{ a_{j}}\tilde{\varepsilon }_{j}^{T}\varepsilon _{j} \nonumber \\{} & {} +\sum _{j=1}^{i-1}\frac{h_{j}}{b_{j}}\tilde{\epsilon }_{j}^{T}\epsilon _{j}+\sum _{j=1}^{i-1}\rho _{j}(1+\epsilon _{j}^{*})+\frac{g_{i}}{ \underline{g}_{i}}z_{i}^{3}\alpha _{i} \nonumber \\{} & {} +z_{i}^{3}v_{i}^{T}S_{i1}+\omega _{i}z_{i}^{3}\xi _{i}(x_{i})+\varepsilon _{i}z_{i}^{3}+\sum _{j=2}^{i+1}\frac{1}{2}\eta _{i}^{2} \nonumber \\{} & {} +\eta _{i}\dot{\eta }_{i}+2(i-2)\iota +\frac{1}{a_{i}}\tilde{\varepsilon } _{i}^{T}(a_{i}z_{i}^{3}-\dot{\varepsilon }_{i}) \nonumber \\{} & {} +\sum _{j=2}^{i-1}\left( -\frac{1}{\chi _{j}}+\frac{1}{2\iota } M_{j}^{2}(\cdot )\right) \eta _{j}^{2}+\frac{1}{4}\frac{g_{i}}{\underline{g} _{i}}z_{i+1}^{4} \nonumber \\{} & {} +\sum _{j=1}^{i}\frac{3}{4}l_{j}^{2}\frac{1}{\gamma _{i}}\tilde{\omega } _{i}^{T}(\gamma _{i}z_{i}^{3}\xi _{i}(x_{i})-\dot{\omega }_{i})-\frac{1}{b_{i} }\tilde{\epsilon }_{i}^{T}\dot{\epsilon }_{i}\nonumber \\ \end{aligned}$$

(67)

From (17),

$$\begin{aligned} LV_{i}\le & {} -\sum _{i=1}^{i-1}c_{i}z_{i}^{4}++\sum _{j=1}^{i-1}\frac{m_{j}}{ \gamma _{j}}\tilde{\omega }_{j}^{T}\omega _{j}+\sum _{j=1}^{i-1}\frac{n_{j}}{ a_{j}}\tilde{\varepsilon }_{j}^{T}\varepsilon _{j} \nonumber \\{} & {} +\sum _{j=1}^{i-1}\frac{h_{j}}{b_{j}}\tilde{\epsilon }_{j}^{T}\epsilon _{j}+\sum _{j=1}^{i-1}\rho _{j}(1+\epsilon _{j}^{*})+\sum _{j=1}^{i-1} \frac{3}{4}l_{j}^{2} \nonumber \\{} & {} +\varepsilon _{i}z_{i}^{3}+\frac{1}{b_{j}}\tilde{\epsilon } _{j}^{T}(b_{j}z_{j}^{3}\tau _{j}-\dot{\epsilon }_{j})+\eta _{i}\dot{\eta }_{i}+ \frac{1}{2}\eta _{i+1}^{2} \nonumber \\{} & {} +\frac{1}{\underline{g}_{i}}z_{i}^{3}g_{i}\alpha _{i}+\frac{1}{\gamma _{i}} \tilde{\omega }_{i}^{T}(\gamma _{i}z_{i}^{3}\xi _{i}(x_{i})-\dot{\omega }_{i}) \nonumber \\{} & {} +\frac{1}{a_{i}}\tilde{\varepsilon }_{i}^{T}(a_{i}z_{i}^{3}-\dot{\varepsilon }_{i})+\epsilon _{i}z_{i}^{3}\tau _{i}+\rho _{i}\epsilon _{i}^{*}+2(i-2)\iota \nonumber \\{} & {} +\sum _{j=2}^{i}\left( \frac{1}{2}-\frac{1}{\chi _{j}}+\frac{1}{2\iota } \bar{M}_{j}^{2}(\cdot )\right) \eta _{j}^{2}+\frac{1}{4}\frac{g_{i}}{ \underline{g}_{i}}z_{i+1}^{4} \nonumber \\{} & {} +z_{i}^{3}\bar{\alpha }_{i}-z_{i}^{3}\bar{\alpha }_{i}+\omega _{i}z_{i}^{3}\xi _{i}(x_{i}) \end{aligned}$$

(68)

where $\bar{\alpha }_{i}=-c_{i}z_{i}+\epsilon _{i}\tau _{i}+\omega _{i}\xi _{i}(x_{i})+\varepsilon _{i}$, $c_{i}>0,$ and $M_{i}(\cdot )$ is defined as a continuous function,

$$\begin{aligned} M_{i}(\cdot )= & {} \sum _{j=1}^{i}\frac{\partial \alpha _{i}}{\partial x_{j}} (g_{j}x_{j+1}+f_{j})+\sum _{j=0}^{i}\frac{\partial \alpha _{i}}{\partial y_{r}^{(i)}}y_{r}^{(j+1)} \nonumber \\{} & {} +\sum _{i=0}^{i}\frac{\partial \alpha _{i}}{\partial \hat{\omega }_{i}}\dot{ \omega }_{i}+\sum _{i=0}^{i}\frac{\partial \alpha _{i}}{\partial \hat{ \varepsilon }_{i}}\dot{\varepsilon }_{i} \nonumber \\{} & {} +\sum _{i=0}^{i}\frac{\partial \alpha _{i}}{\partial \hat{\epsilon }_{i}} \dot{\epsilon }_{i}+\frac{\partial \alpha _{i}}{\partial \hat{\theta }}\overset{.}{\hat{\theta }}+\frac{1}{2}\sum _{j=1}^{i}\frac{\partial ^{2}\alpha _{1}}{ \partial x_{p}\partial x_{q}}\varphi _{p}^{T}\varphi _{q}\nonumber \\ \end{aligned}$$

(69)

Applying Young’s inequality again, we get

$$\begin{aligned} \eta _{i}M_{i}(\cdot )\le \frac{1}{2\iota }\eta _{i}^{2}M_{i}^{2}(\cdot )+2\iota \end{aligned}$$

(70)

where $\iota$ is a positive constant.

Combining (68) with (70), we obtain

$$\begin{aligned} LV_{i}\le & {} -\sum _{i=1}^{i-1}c_{i}z_{i}^{4}+\sum _{j=1}^{i-1}\frac{m_{j}}{ \gamma _{j}}\tilde{\omega }_{j}^{T}\omega _{j}+\sum _{j=1}^{i-1}\frac{n_{j}}{ a_{j}}\tilde{\varepsilon }_{j}^{T}\varepsilon _{j} \nonumber \\{} & {} +\sum _{j=1}^{i-1}\frac{h_{j}}{b_{j}}\tilde{\epsilon }_{j}^{T}\epsilon _{j}+\sum _{j=1}^{i-1}\rho _{j}(1+\epsilon _{j}^{*})+z_{i}^{3}\bar{\alpha } _{i} \nonumber \\{} & {} +\sum _{j=1}^{i-1}\frac{3}{4}l_{j}^{2}+\frac{g_{i}}{\underline{g}_{i}} z_{i}^{3}\alpha _{i}+\frac{1}{\gamma _{i}}\tilde{\omega }_{i}^{T}(\gamma _{i}z_{i}^{3}\xi _{i}(x_{i})-\dot{\omega }_{i}) \nonumber \\{} & {} +\frac{1}{b_{j}}\tilde{\epsilon }_{j}^{T}(b_{j}z_{j}^{3}\tau _{j}-\dot{ \epsilon }_{j})+\frac{1}{a_{i}}\tilde{\varepsilon }_{i}^{T}(a_{i}z_{i}^{3}- \dot{\varepsilon }_{i}) \nonumber \\{} & {} +\omega _{i}z_{i}^{3}\xi _{i}(x_{i})+2(i-1)\iota +\frac{1}{4}\frac{g_{i}}{ \underline{g}_{i}}z_{i+1}^{4}+\frac{1}{2}\eta _{i+1}^{2} \nonumber \\{} & {} +\rho _{i}\epsilon _{i}^{*}+\sum _{j=2}^{i}\left( \frac{1}{2}-\frac{1}{ \chi _{j}}+\frac{1}{2\iota }M_{j}^{2}(\cdot )\right) \eta _{j}^{2} \end{aligned}$$

(71)

Virtual control law $\alpha _{i}$ and adaptation laws are devised as

$$\begin{aligned}{} & {} \alpha _{i}=-\frac{z_{i}^{3}\bar{\alpha }_{i}^{2}}{\sqrt{z_{i}^{6}\bar{\alpha } _{i}^{2}+\rho _{i}}} \end{aligned}$$

(72)

$$\begin{aligned}{} & {} \dot{\omega }_{i}=\gamma _{i}z_{i}^{3}\xi _{i}(x_{i})-m_{i}\omega _{i} \end{aligned}$$

(73)

$$\begin{aligned}{} & {} \dot{\varepsilon }_{i}=a_{i}z_{i}^{3}-n_{i}\varepsilon _{i} \end{aligned}$$

(74)

$$\begin{aligned}{} & {} \dot{\epsilon }_{i}=b_{i}z_{i}^{3}\tau _{i}-h_{i}\epsilon _{i} \end{aligned}$$

(75)

with parameters $m_{i}>0,n_{i}>0,h_{i}>0$.

From (72), we have the inequality

$$\begin{aligned} \frac{g_{i}}{\underline{g}_{i}}z_{i}^{3}\alpha _{i}+z_{i}^{3}\bar{\alpha } _{i}\le \rho _{i} \end{aligned}$$

(76)

Substituting (72)–(76) in (71) yields

$$\begin{aligned} LV_{i}\le & {} -\sum _{i=1}^{i}c_{i}z_{i}^{4}-\sum _{j=2}^{i}(\frac{1}{\chi _{j} }-\frac{1}{2\iota }M_{j}^{2}(\cdot )-\frac{1}{2})\eta _{j}^{2} \nonumber \\{} & {} +\sum _{j=1}^{i}\rho _{j}(1+\epsilon _{j}^{*})+\sum _{j=1}^{i}\frac{3}{4} l_{j}^{2}+2(i-1)\iota \nonumber \\{} & {} +\frac{1}{2}\eta _{i+1}^{2}+\frac{1}{4}\frac{g_{i}}{\underline{g}_{i}} z_{i+1}^{4}+\sum _{j=1}^{i}\frac{m_{j}}{2\gamma _{j}}\tilde{\omega } _{j}^{T}\omega _{j} \nonumber \\{} & {} +\sum _{j=1}^{i}\frac{n_{j}}{2a_{j}}\tilde{\varepsilon }_{j}^{T}\varepsilon _{j}+\sum _{j=1}^{i-1}\frac{h_{j}}{2b_{j}}\tilde{\epsilon }_{j}^{T}\epsilon _{j} \end{aligned}$$

(77)

To avoid continuously differentiating $\alpha _{i}$, we describe the first-order filter as

$$\begin{aligned} \chi _{i+1}\overset{.}{d}_{i+1}+d_{i+1}=\alpha _{i},d_{i+1}(0)=\alpha _{i}(0) \end{aligned}$$

(78)

where $\chi _{i+1}>0$ is a constant.

Because $\eta _{i+1}=d_{i+1}-\alpha _{i},$ we have

$$\begin{aligned} \dot{\eta }_{i+1}=\overset{.}{d}_{i+1}-\dot{\alpha }_{i}=-\frac{\eta _{i+1}}{ \chi _{i+1}}+M_{i+1}(\cdot ) \end{aligned}$$

(79)

where $M_{i+1}(\cdot )$ is defined as a continuous function,

$$\begin{aligned} M_{i+1}(\cdot )= & {} \sum _{j=1}^{i}\frac{\partial \alpha _{i}}{\partial x_{j}} (g_{j}x_{j+1}+f_{j})+\sum _{j=0}^{i}\frac{\partial \alpha _{i}}{\partial y_{r}^{(i)}}y_{r}^{(j+1)} \nonumber \\{} & {} +\sum _{i=1}^{i}\frac{\partial \alpha _{i}}{\partial \hat{\omega }_{i}}\dot{ \omega }_{i}+\sum _{i=1}^{i}\frac{\partial \alpha _{i}}{\partial \hat{ \varepsilon }_{i}}\dot{\varepsilon }_{i} \nonumber \\{} & {} +\sum _{i=1}^{i}\frac{\partial \alpha _{i}}{\partial \hat{\epsilon }_{i}} \dot{\epsilon }_{i}+\frac{1}{2}\sum _{j=1}^{i}\frac{\partial ^{2}\alpha _{1}}{ \partial x_{p}\partial x_{q}}\varphi _{p}^{T}\varphi _{q} \end{aligned}$$

(80)

Step n: According to (1) and (15), we obtain

$$\begin{aligned} V_{n}= & {} V_{n-1}+\frac{1}{4\underline{g}_{n}}z_{n}^{4}+\frac{1}{2\gamma _{n}} \tilde{\omega }_{n}^{T}\tilde{\omega }_{n}+\frac{1}{2a_{n}}\tilde{\varepsilon } _{n}^{T}\tilde{\varepsilon }_{n} \nonumber \\{} & {} +\frac{1}{2b_{n}}\tilde{\epsilon }_{n}^{T}\tilde{\epsilon }_{n}+\frac{1}{2} \eta _{n}^{2} \end{aligned}$$

(81)

with parameters $\gamma _{n}>0,a_{n}>0$, and $b_{n}>0$. $\tilde{\omega }_{n}$ is the estimation error and $\omega _{n}$ is the estimations of $\omega _{n}^{*}$ and $\tilde{\omega }_{n}=\omega _{n}^{*}-\omega _{n}$.

According to (14), (15), and (7)

$$\begin{aligned} LV_{n}= & {} LV_{n-1}+\frac{1}{\underline{g}_{n}}z_{n}^{3}\left( g_{n}u-Ld_{n}\right) \nonumber \\{} & {} +\frac{3}{2\underline{g}_{n}}z_{n}^{2}\left\| \varphi _{n}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{n-1}}{\partial x_{n-1}}\varphi _{n-1}\right\| ^{2} \nonumber \\{} & {} +\frac{1}{\gamma _{n}}\tilde{\omega }_{n}^{T}\overset{.}{\tilde{\omega }_{n}} +\frac{1}{a_{n}}\tilde{\varepsilon }_{n}^{T}\overset{.}{\tilde{\varepsilon }} _{n}+\frac{1}{b_{n}}\tilde{\epsilon }_{n}^{T}\overset{.}{\tilde{\epsilon }}_{n} \nonumber \\{} & {} +\eta _{n}\dot{\eta }_{n}+\varepsilon _{n}^{*}z_{n}^{3}+\omega _{n}^{*}z_{n}^{3}\xi _{n}(x_{n}) \end{aligned}$$

(82)

Similar to Step i, virtual control law can be designed as

$$\begin{aligned} u=\alpha _{n}=-\frac{z_{n}^{3}\bar{\alpha }_{n}^{2}}{\sqrt{z_{n}^{6}\bar{ \alpha }_{n}^{2}+\rho _{n}}} \end{aligned}$$

(83)

where $\bar{\alpha }_{n}=-c_{n}z_{n}+\epsilon _{n}\tau _{n}+\omega _{n}\xi _{n}(x_{n})+\varepsilon _{n}.$

An improved event-triggered strategy is obtained by noting the existence of the tracking error term. Therefore, the continuous controller can be devised as

$$\begin{aligned}{} & {} g(t)=\alpha _{n}-\bar{\upsilon }\tanh \left( \frac{z^{3}\bar{\upsilon }}{\mu } \right) \end{aligned}$$

(84)

$$\begin{aligned}{} & {} \dot{\omega }_{n}=\gamma _{n}z_{n}^{3}\xi _{n}(x_{n})-m_{n}\omega _{n} \end{aligned}$$

(85)

$$\begin{aligned}{} & {} \dot{\varepsilon }_{n}=a_{n}z_{n}^{3}-n_{n}\varepsilon _{n} \end{aligned}$$

(86)

$$\begin{aligned}{} & {} \dot{\epsilon }_{n}=b_{n}z_{n}^{3}\tau _{n}-h_{n}\epsilon _{n} \end{aligned}$$

(87)

where $m_{n}, n_{n}, h_{n}, \bar{\upsilon }$, and $\mu$ are positive parameters, and $\upsilon ^{*}$ is a threshold that can be expressed as

$$\begin{aligned} \upsilon ^{*}=\upsilon +\frac{p_{1}}{\left| z_{1}\right| +p_{2}}, \end{aligned}$$

(88)

where $\upsilon ^{*}<\upsilon \left( \bar{\upsilon }>\upsilon \right)$, $p_{1}>0$, and $p_{2}>0$. The triggered event is described as

$$\begin{aligned} u\left( t\right)= & {} g\left( t_{k}\right) ,\ \forall t\in \left[ t_{k},t_{k+1}\right) , \end{aligned}$$

(89)

$$\begin{aligned} t_{k+1}= & {} \inf \left\{ t\in R|\,\left| e(t)\right| \ge \upsilon ^{*}\right\} ,t_{1}=0, \end{aligned}$$

(90)

where $e\left( t\right) =g\left( t\right) -u\left( t\right)$ is measurement error between the continuous controller and the actual controller. $t_{k},\ k\in z^{+}$, is the controller update time, i.e., whenever (90) is triggered, the time is updated to $t_{k+1}$, and the input signal $u\left( t_{k+1}\right)$ is used in the system. Within time $t\in \left[ t_{k},t_{k+1}\right)$, the signal $g\left( t_{k}\right)$ remains constant.

Supposing that the t is over a time interval $\left[ t_{k},t_{k+1}\right)$, combined with (90), we get $\left| g\left( t\right) -u\left( t\right) \right| \le \upsilon$. Thus, it exists a continuous time-varying parameter $\varpi \left( t\right)$ also $\varpi \left( t_{k}\right) =0,\ \varpi \left( t_{k+1}\right) =\pm 1$ and $\left| \varpi \left( t\right) \right| \le 1$. In addition, $g\left( t\right) =u\left( t\right) +\varpi \left( t\right) \upsilon$. Based on the above, we can get

$$\begin{aligned} LV_{n}= & {} LV_{n-1}+\frac{3}{2\underline{g}_{n}}z_{n}^{2}\left\| \varphi _{n}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{n-1}}{\partial x_{n-1}}\varphi _{n-1}\right\| ^{2} \nonumber \\{} & {} +\frac{1}{\underline{g}_{n}}z_{n}^{3}g_{n}\left( \alpha _{n}-\bar{\upsilon } \tanh \left( \frac{z^{3}\bar{\upsilon }}{\mu }\right) -\varpi \left( t\right) \upsilon \right) \nonumber \\{} & {} +\omega _{n}^{*}z_{n}^{3}\xi _{n}(x_{n})+\frac{1}{\gamma _{n}}\tilde{ \omega }_{n}^{T}\overset{.}{\tilde{\omega }_{n}}-\frac{1}{\underline{g}_{n}} z_{n}^{3}Ld_{n} \nonumber \\{} & {} +\frac{1}{b_{n}}\tilde{\epsilon }_{n}\overset{.}{\tilde{\epsilon }}_{n}+\eta _{n}\dot{\eta }_{n}+\varepsilon _{n}^{*}z_{n}^{3}+\frac{1}{a_{n}}\tilde{ \varepsilon }_{n}^{T}\overset{.}{\tilde{\varepsilon }}_{n} \nonumber \\\le & {} LV_{n-1}+\frac{1}{\underline{g}_{n}}z_{n}^{3}\left( g_{n}\alpha _{n}- \dot{d}_{n}\right) +\omega _{n}^{*}z_{n}^{3}\xi _{n}(x_{n}) \nonumber \\{} & {} +\frac{3}{2\underline{g}_{n}}z_{n}^{2}\left\| \varphi _{n}-\sum _{j=1}^{i-1}\frac{\partial \alpha _{n-1}}{\partial x_{n-1}}\varphi _{n-1}\right\| ^{2} \nonumber \\{} & {} +\frac{g_{n}}{\underline{g}_{n}}0.2785\mu \end{aligned}$$

(91)

where

$$\begin{aligned}{} & {} z_{n}^{3}\left( -\bar{\upsilon }\tanh \left( \frac{z^{3}\bar{\upsilon }}{\mu }\right) -\varpi \left( t\right) \upsilon \right) \nonumber \\\le & {} \left( \left| z_{n}^{3}\upsilon \right| -z_{n}^{3}\bar{ \upsilon }\tanh \left( \frac{z^{3}\bar{\upsilon }}{\mu }\right) \right) \nonumber \\\le & {} 0.2785\mu \end{aligned}$$

(92)

where $v_{n}^{T}=[\left\| \varphi _{n}-\sum _{j=1}^{n-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\varphi _{j}\right\| ^{4}/g_{n}^{2}l_{n}^{2}$, $g_{n}/\underline{g}_{n}$,$-1/g_{n}$,$g_{n-1}g_{n}/\underline{g}_{n-1}]$ and $S_{n}^{T}=[3z_{n}/4$, $0.2785\mu$,$Ld_{n}$,$z_{n}/4]$.

Substituting (83)–(87) in (82) yields

$$\begin{aligned} LV_{n}\le & {} -\sum _{i=1}^{n}c_{i}z_{i}^{4}-\sum _{i=2}^{n}(\frac{1}{\chi _{i} }-\frac{1}{2\iota }\bar{M}_{i}^{2}(\cdot )-\frac{1}{2})\eta _{i}^{2} \nonumber \\{} & {} +\sum _{i=1}^{n}\rho _{i}(1+\epsilon _{i}^{*})+\frac{3}{4} \sum _{i=1}^{n}l_{i}^{2} \nonumber \\{} & {} +2(n-1)\iota +\sum _{j=1}^{n-1}\frac{h_{j}}{2b_{j}}\tilde{\epsilon } _{j}^{T}\epsilon _{j} \nonumber \\{} & {} +\sum _{j=1}^{n-1}\frac{m_{j}}{2\gamma _{j}}\tilde{\omega }_{j}^{T}\omega _{j}+\sum _{j=1}^{n-1}\frac{n_{j}}{2a_{j}}\tilde{\varepsilon } _{j}^{T}\varepsilon _{j} \end{aligned}$$

(93)

where $\left| M_{i}(\cdot )\right| \le \bar{M}_{i}, i=2,\ldots ,n,$ where $\bar{M}_{i}$ are positive constants.

Exploiting Young’s inequality, we obtain

$$\begin{aligned}{} & {} \frac{m_{n}}{\gamma _{n}}\tilde{\omega }_{n}^{T}\omega _{n}\le -\frac{m_{n}}{ 2\gamma _{n}}\tilde{\omega }_{n}^{T}\tilde{\omega }_{n}+\frac{m_{n}}{2\gamma _{n}}\tilde{\omega }_{n}^{*T}\omega _{n}^{*} \end{aligned}$$

(94)

$$\begin{aligned}{} & {} \frac{n_{n}}{a_{n}}\tilde{\varepsilon }_{n}^{T}\varepsilon _{n}\le -\frac{ n_{n}}{2a_{n}}\tilde{\varepsilon }_{n}^{T}\tilde{\varepsilon }_{n}+\frac{n_{n} }{2a_{n}}\tilde{\varepsilon }_{n}^{T}\varepsilon _{n}^{*} \end{aligned}$$

(95)

$$\begin{aligned}{} & {} \frac{h_{n}}{b_{n}}\tilde{\epsilon }_{n}^{T}\epsilon _{n}\le -\frac{h_{n}}{ 2b_{n}}\tilde{\epsilon }_{n}^{T}\tilde{\epsilon }_{n}+\frac{h_{n}}{2b_{n}} \epsilon _{n}^{*T}\epsilon _{n}^{*} \end{aligned}$$

(96)

Substituting (94)–(96) in (93) yields

$$\begin{aligned} LV_{n}\le & {} -\sum _{i=1}^{n}c_{i}z_{i}^{4}-\sum _{i=2}^{n}\varpi _{i}\eta _{i}^{2}+\sum _{i=1}^{n}\rho _{i}(1+\epsilon _{i}^{*}) \nonumber \\{} & {} +\frac{3}{4}\sum _{i=1}^{n}l_{i}^{2}\sum _{i=1}^{n}\frac{m_{i}}{2\gamma _{i}} \tilde{\omega }_{i}^{T}\tilde{\omega }_{i}+\sum _{i=1}^{n}\frac{m_{i}}{2\gamma _{i}}\omega _{i}^{*T}\omega _{i}^{*} \nonumber \\{} & {} -\sum _{i=1}^{n}\frac{n_{i}}{2a_{i}}\tilde{\varepsilon }_{i}^{T}\tilde{ \varepsilon }_{i}+\sum _{i=1}^{n}\frac{n_{i}}{2a_{i}}\varepsilon _{i}^{*T}\varepsilon _{i}^{*} \nonumber \\{} & {} -\sum _{i=1}^{n}\frac{h_{i}}{2b_{i}}\tilde{\epsilon }_{i}^{T}\tilde{\epsilon } _{i}+\sum _{i=1}^{n}\frac{h_{i}}{2b_{i}}\epsilon _{i}^{*T}\epsilon _{i}^{*}+2(n-1)\iota \nonumber \\ \end{aligned}$$

(97)

Select some appropriate parameters with $\varpi _{i}=\frac{1}{\chi _{i}}- \frac{1}{2\iota }\bar{M}_{i}^{2}(\cdot )-\frac{1}{2}>0,i=2,\ldots ,n.$

Selecting

$$\begin{aligned} c=\min \left\{ 4c_{i}\underline{g}_{i},2\varpi _{i},m_{i},n_{i},h_{i}\right\} \end{aligned}$$

(98)

Substituting (94)–(96) in (97) gives

$$\begin{aligned} LV_{n}\le & {} -c\sum _{i=1}^{n}z_{i}^{4}-c\sum _{i=2}^{n}\eta _{i}^{2}+-c\sum _{i=1}^{n}\frac{1}{2\gamma _{i}}\tilde{\omega }_{i}^{T}\tilde{ \omega }_{i} \nonumber \\{} & {} -c\sum _{i=1}^{n}\frac{1}{2a_{i}}\tilde{\varepsilon }_{i}^{T}\tilde{ \varepsilon }_{i}-c\sum _{i=1}^{n-1}\frac{1}{2b_{i}}\tilde{\epsilon }_{i}^{T} \tilde{\epsilon }_{i}+b \end{aligned}$$

(99)

where

$$\begin{aligned} b= & {} \sum _{i=1}^{n}\frac{m_{i}}{2\gamma _{i}}\omega _{i}^{*T}\omega _{i}^{*}+\sum _{i=1}^{n}\frac{n_{i}}{2a_{i}}\varepsilon _{i}^{*T}\varepsilon _{i}^{*}+\sum _{i=1}^{n}\frac{h_{i}}{2b_{i}}\epsilon _{i}^{*T}\epsilon _{i}^{*} \nonumber \\{} & {} +2(n-1)\iota +\sum _{i=1}^{n}\rho _{i}(1+\epsilon _{i}^{*})+\sum _{i=1}^{n}\frac{3}{4}l_{i}^{2} \end{aligned}$$

(100)