Event-based decentralized adaptive finite-time tracking control of interconnected nonlinear time-varying systems

Abstract

This paper studies the event-based decentralized adaptive finite-time tracking control problem of the interconnected nonlinear time-varying systems. A novel tracking control strategy associating event-triggered techniques, dynamic surface control, and finite-time control is presented. Correspondingly, the newly designed controller not only ensures finite-time convergence but also decreases the communication burden between the controller and the actuator. Moreover, the complexity explosion problem caused by the backstep** design procedure can be excluded. In addition, the difficulty caused by the system uncertainty is solved by utilizing bound estimation methods and constructing a suitable smooth function. Simulation results verify the effectiveness of our proposed control strategy.

Appendices

Appendix: Derivation of inequality (28)

Proof

Based on Assumption 2 and Lemma 1, one can obtain

$$\begin{aligned} z_{i,1}g_{i,1}(t)\alpha _{i,1}&=-g_{i,1}(t)\frac{z_{i,1}^2 \hat{\ell }_{i,1}^2\bar{\alpha }_{i,1}^2}{\sqrt{z_{i,1}^2 \hat{\ell }_{i,1}^2\bar{\alpha }_{i,1}^2+\epsilon _{i,1}^2}}\nonumber \\&\le -\underline{g}_{i,1}\frac{z_{i,1}^2\hat{\ell }_{i,1}^2 \bar{\alpha }_{i,1}^2}{\sqrt{z_{i,1}^2\hat{\ell }_{i,1}^2 \bar{\alpha }_{i,1}^2+\epsilon _{i,1}^2}}\nonumber \\&=\underline{g}_{i,1}\left[ -\frac{z_{i,1}^2\hat{\ell }_{i,1}^2 \bar{\alpha }_{i,1}^2}{\sqrt{z_{i,1}^2 \hat{\ell }_{i,1}^2\bar{\alpha }_{i,1}^2+\epsilon _{i,1}^2}}\right] \nonumber \\&\le \underline{g}_{i,1}(\epsilon _{i,1} -\hat{\ell }_{i,1} |z_{i,1}\bar{\alpha }_{i,1} |)\nonumber \\&\le \underline{g}_{i,1}(\epsilon _{i,1} -\hat{\ell }_{i,1}z_{i,1}\bar{\alpha }_{i,1}) \end{aligned}$$

(A.1)

Then, from (26), one has

$$\begin{aligned} \dot{V}_{i,1}&\le -c_{i,1}z_{i,1}^{2\beta } +z_{i,1}g_{i,1}(t)z_{i,2}+\epsilon _{i,1}(\underline{g}_{i,1}+\nu _i)\nonumber \\&\quad +(\underline{g}_{i,1}\tilde{\ell }_{i,1} -\underline{g}_{i,1}\hat{\ell }_{i,1})z_{i,1}\bar{\alpha }_{i,1}+z_{i,1}\bar{\alpha }_{i,1}\nonumber \\&\quad +\sum _{j=1}^{N}\varrho _{i,1,j}\phi _{i,1,j}(y_j)-\rho _iz_{i,1}\varphi _i\nonumber \\&\quad +\frac{1}{\gamma _{\nu _i}}\tilde{\nu }_i(\dot{\hat{\nu }}_i -\tau _{i,1})-\sigma _{\nu _i}\tilde{\nu }_i\hat{\nu }_i -\sigma _{\rho _i}\tilde{\rho }_i\hat{\rho }_i\nonumber \\&\quad -\sigma _{\ell _{i,1}}\tilde{\ell }_{i,1}\hat{\ell }_{i,1} +\phi _{i,2}\dot{{\phi }}_{i,2}+\frac{1}{2}\phi _{i,2}^2 \end{aligned}$$

(A.2)

Noting $\underline{g}_{i,1}\tilde{\ell }_{i,1}-\underline{g}_{i,1}\hat{\ell }_{i,1}=-\underline{{g}}_{i,1}\ell _{i,1}=-1$, which yields (28). This ends the proof of (28). $\square$

Appendix: Derivation of inequality (75)

Proof

Define the compact set as follows $\varPi _{i0}:=\{(y_{di},\dot{y}_{di}, \ddot{y}_{di}):y^2_{di}+\dot{y}_{di}^2+\ddot{y}_{di}^2\le K _{0}\}$, $\varPi _{i,k}:=\{V_i(t)\le p_i\}$ where $K_{0}$, $p_i>0$, $i=1,\ldots ,N$, $k=1,\ldots ,n_i-1$. Clearly, for every i and k, $\varPi _{i0}\times \varPi _{i,k}$ is also a compact set. Therefore, the continuous function $B_{i,k}$ has a maximum, say, $M_{i,k}$, on $\varPi _{i0}\times \varPi _{i,k}$. Hence, one has $|B_{i,k}|\le M_{i,k}$.

Thus, according to Lemma 3, and substituting (70)–(72) into (69), we get

$$\begin{aligned} \dot{V}&\le \sum _{i=1}^{N}\left[ -c_i\left( \sum _{q=1}^{n_i}\frac{1}{2}z_{i,q}^2\right) ^{\beta } -c_i\left( \sum _{q=2}^{n_i}\frac{1}{2}\phi _{i,q}^2\right) ^{\beta }\right. \nonumber \\&\quad +c_i\left( \sum _{q=2}^{n_i}\frac{1}{2}\phi _{i,q}^2\right) ^{\beta }-c_i\sum _{q=1}^{n_i-1} \frac{1}{2}\phi _{i,q+1}^2\nonumber \\&\quad -c_i\left( \frac{1}{2\gamma _{\nu _i}}\tilde{\nu }_i^2\right) ^{\beta }\nonumber \\&\quad +c_i\left( \frac{1}{2\gamma _{\nu _i}}\tilde{\nu }_i^2\right) ^{\beta }-c_i\left( \frac{1}{2\gamma _{\nu _i}} \tilde{\nu }_i^2\right) -c_i\left( \frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2\right) ^{\beta }\nonumber \\&\quad +c_i\left( \frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2\right) ^{\beta } -c_i\left( \frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2\right) \nonumber \\&\quad -c_i\left( \sum _{q=1}^{n_i}\frac{ \underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2\right) ^{\beta }\nonumber \\&\quad +c_i\left( \sum _{q=1}^{n_i}\frac{\underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2\right) ^{\beta } -c_i\left( \sum _{q=1}^{n_i}\frac{ \underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2\right) \nonumber \\&\quad +\sum _{q=1}^{n_i}\epsilon _{i,q}( \underline{g}_{i,q}+\nu _i) +\frac{\sigma _{\nu _i}{\nu }_i^2}{2}+\frac{\sigma _{\rho _i}{\rho }_i^2}{2}\nonumber \\&\quad +\sum _{q=1}^{n_i}\sigma _{\ell _{i,q}} \underline{g}_{i,q}\frac{\ell _{i,q}^2}{2} +\sum _{q=1}^{n_i-1}\frac{1}{2}M_{i,q+1}^2\nonumber \\&\quad +0.557\bar{g}_{i,n_i}\varepsilon _i+h_i\bigg ] \end{aligned}$$

(B.1)

with $c_{i}=\min _{1\le i\le N, 1\le q\le n_i}\{2^{\beta -1} \underline{k_i}, 2\left( \frac{1}{\lambda _{i,q}}-1\right) ,\sigma _{\nu _i}\gamma _{\nu _i},\sigma _{\rho _i}\gamma _{\rho _i},\sigma _{\ell _{i,q}}\gamma _{\ell _{i,q}}\}$, $\underline{k_i}=\min _{1\le q\le n_i}\{k_{i,q}\}$ and $h_i=\sum _{j=1}^{N}\sum _{q=1}^{n_j}\varrho _{j,q,i}\phi _{j,q,i}(y_i) -\rho _iz_{i,1}\varphi _i$ are the uncertain terms generated by interactions.

Due to $\phi _{j,q,i}\ge 0$ and the definitions of $\rho _i$ and $\varphi _i$, we have

$$\begin{aligned} h_i&\le \rho _i\sum _{j=1}^{N}\sum _{q=1}^{n_j}\phi _{j,q,i}(y_i)-\rho _iz_{i,1}\varphi _i\nonumber \\&=\rho _i\frac{\lambda _i-z_{i,1}^2}{z_{i,1}^2+\lambda _i}\sum _{j=1}^{N}\sum _{q=1}^{n_j}\phi _{j,q,i}(y_i) \end{aligned}$$

(B.2)

By (B.2), it can be deduced that, for each $i=1,\ldots ,N$, on the one hand, if $\mid z_{i,1}\mid > \sqrt{\lambda _i}$, $h_i<0$. And on the other hand, if $\mid z_{i,1}\mid \le \sqrt{\lambda _i}$, $y_i$ is bounded from (8). In summary then, $h_i$ has an upper bound $H_i \ge 0$.

Then, applying Lemma 4, let $\var** _1=\sum _{q=2}^{n_i}\frac{1}{2}\phi _{i,q}^2$, $\var** _2=\frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2$, $\var** _3=\frac{1}{2\gamma _{\nu _i}}\tilde{\nu }_i^2$, $\var** _4=\sum _{q=1}^{n_i}\frac{ \underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2$, $\varDelta =1$, $\zeta _1=\beta$, $\zeta _2=1-\beta$, $\zeta _3=\beta ^{-1}$, it follows that

$$\begin{aligned} \left( \sum _{q=2}^{n_i}\frac{1}{2}\phi _{i,q}^2\right) ^{\beta }&\le \sum _{q=2}^{n_i}\frac{1}{2}\phi _{i,q}^2+(1-\beta )\beta ^{\frac{\beta }{1-\beta }}\nonumber \\ \left( \frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2\right) ^{\beta }&\le \frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2+(1-\beta )\beta ^{\frac{\beta }{1-\beta }}\nonumber \\ \left( \frac{1}{2\gamma _{\nu _i}}\tilde{\nu }_i^2\right) ^{\beta }&\le \frac{1}{2\gamma _{\nu _i}}\tilde{\nu }_i^2+(1-\beta )\beta ^{\frac{\beta }{1-\beta }}\nonumber \\ \left( \sum _{q=1}^{n_i}\frac{ \underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2\right) ^{\beta }&\le \sum _{q=1}^{n_i}\frac{ \underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2+(1-\beta )\beta ^{\frac{\beta }{1-\beta }} \end{aligned}$$

(B.3)

Then, substituting (B.2) and (B.3) into (B.1), we have

$$\begin{aligned} \dot{V}&\le \sum _{i=1}^{N}\left[ -c_i\left( \sum _{q=1}^{n_i}\frac{1}{2}z_{i,q}^2\right) ^{\beta } -c_i\left( \sum _{q=2}^{n_i}\frac{1}{2}\phi _{i,q}^2\right) ^{\beta }\right. \nonumber \\&\quad -c_i\left( \frac{1}{2\gamma _{\nu _i}}\tilde{\nu }_i^2\right) ^{\beta } -c_i\left( \frac{1}{2\gamma _{\rho _i}}\tilde{\rho }_i^2\right) ^{\beta }\nonumber \\&\quad -c_i\left( \sum _{q=1}^{n_i}\frac{ \underline{g}_{i,q}}{2\gamma _{\ell _{i,q}}}\tilde{\ell }_{i,q}^2\right) ^{\beta }\nonumber \\&\quad +\sum _{q=1}^{n_i}\epsilon _{i,q}( \underline{g}_{i,q}+\nu _i) +\frac{\sigma _{\nu _i}{\nu }_i^2}{2}+\frac{\sigma _{\rho _i}{\rho }_i^2}{2}\nonumber \\&\quad +\sum _{q=1}^{n_i}\sigma _{\ell _{i,q}} \underline{g}_{i,q}\frac{\ell _{i,q}^2}{2} +\sum _{q=1}^{n_i-1}\frac{1}{2}M_{i,q}^2+4(1-\beta )\beta ^{\frac{\beta }{1-\beta }}\nonumber \\&\quad +0.557\bar{g}_{i,n_i}\varepsilon _i+H_i\bigg ] \end{aligned}$$

(B.4)

Inequality (75) is then obtained. $\square$