Event-triggered output feedback control for feedforward nonlinear systems with unknown measurement sensitivity

Abstract

This paper investigates the event-triggered output feedback control problem for feedforward nonlinear systems with unknown measurement sensitivity. The nonlinear terms are assumed to be bounded by an unknown constant multiplied by fractional power of states and controller. We design an auxiliary system independent of the output signal to construct controller and propose a novel time-varying triggering mechanism. It is the first time to introduce the logarithmic function into the auxiliary system and state transformation to deal with event-triggered control problem. Under the designed event-triggered controller, the convergence of all states is obtained and the Zeno behavior is effectively avoided. Finally, two numerical examples are presented to demonstrate the effectiveness of the designed event-triggered controller.

Appendix

In this section, the estimations for nonlinear terms of (6) and (12) are presented.

From Assumption 1 and state transformations (4) and (7), we get

$$\begin{aligned}&|(\ln (t+\tau ))^{i}f_{i}|\\&\quad \le (\ln (t+\tau ))^{i}\left( \gamma \sum _{j=i+2}^{n}|x_{j}|^{\frac{n-1+\Delta }{n}}+\gamma |(u-\omega )+\omega |^{\frac{n-1+\Delta }{n}}\right) \\&\quad \le \gamma \sum _{j=i+2}^{n}\frac{(\ln (t+\tau ))^{i}}{(\ln (t+\tau ))^{\frac{j(n-1+\Delta )}{n}}}|\varepsilon _{j}+L^{j-1}z_{j}|^{\frac{n-1+\Delta }{n}}\\&\qquad +\gamma (\ln (t+\tau ))^{i}\left( \frac{\delta _{1} L^{n-1}}{(\ln (t+3))^{n+1}}+\frac{L^{n}\sum _{j=1}^{n}{\hat{b}}_{j}|z_{j}|}{\ln (t+\tau )^{n+1}}\right) ^{\frac{n-1+\Delta }{n}} \end{aligned}$$

$$\begin{aligned}&\le \frac{\gamma }{(\ln (t+\tau ))^{1+\sigma _{2}}} \sum _{j=i+2}^{n}|\varepsilon _{j}+L^{j-1}z_{j}|^{\sigma _{1}}+\frac{\gamma (\ln (t+\tau ))^{i}}{(\ln (t+\tau ))^{(n+1)\sigma _{1}}}\\&\qquad \times \bigg (\frac{\delta _{1} L^{n-1}(\ln (t+\tau ))^{n+1}}{(\ln (t+3))^{n+1}}+L^{n}\sum _{j=1}^{n}{\hat{b}}_{j}|z_{j}|\bigg ) ^{\sigma _{1}}\\&\quad \le \frac{\gamma }{(\ln (t+\tau ))^{1+\sigma _{2}}}\bigg (\sum _{j=i+2}^{n}|\varepsilon _{j}+L^{j-1}z_{j}|^{\sigma _{1}}\\&\qquad +\bigg (\frac{\delta _{1} L^{n-1}(\ln (t+\tau ))^{n+1}}{(\ln (t+3))^{n+1}}+L^{n}\sum _{j=1}^{n}{\hat{b}}_{j}|z_{j}|\bigg ) ^{\sigma _{1}}\bigg ), \end{aligned}$$

where $\sigma _{1}=\frac{n-1+\Delta }{n}$, and $\sigma _{2}{=}\min \{\frac{3\Delta }{n}{,}\frac{(n{+}1)\Delta {-}1}{n}\}>0$, ${\hat{b}}_{1}=\max \limits _{t>0}|\theta (t)|b_{1}$, and ${\hat{b}}_{j}=b_{j}, j=2,\ldots ,n$.

By Lemmas 3 and 4, we obtain the following inequality

$$\begin{aligned} \begin{aligned}&\sum _{j=1}^{n}|\varepsilon _{j}||(\ln (t+\tau ))^{i}f_{i}|\\&\quad \le \frac{2^{n\sigma _{1}}\gamma }{(\ln (t+\tau ))^{1+\sigma _{2}}}\bigg ((n-1)\sum \limits _{j=1}^{n} |\varepsilon _{j}|^{1+\sigma _{1}}\\&\qquad +n\sum \limits _{j=i+2}^{n}(|\varepsilon _{j}|^{1+\sigma _{1}}+L^{2n}|z_{j}|^{1+\sigma _{1}})\\&\qquad +n\bigg (\frac{\delta _{1} L^{n-1}(\ln (t+\tau ))^{n+1}}{(\ln (t+3))^{n+1}}\bigg )^{1+\sigma _{1}}\\&\qquad +nL^{2n}\sum _{j=1}^{n}{\hat{b}}_{j}^{1+\sigma _{1}}|z_{j}|^{1+\sigma _{1}}\bigg ) \\&\quad \le \frac{2^{n\sigma _{1}}\gamma }{(\ln (t+\tau ))^{1+\sigma _{2}}}\bigg ((2n-1)\sum \limits _{j=1}^{n}|\varepsilon _{j}|^{1+\sigma _{1}}\\&\qquad +nL^{2n}\sum _{j=1}^{n}(1+{\hat{b}}_{j}^{1+\sigma _{1}})|z_{j}|^{1+\sigma _{1}}\\&\qquad +n\bigg (\frac{\delta _{1} L^{n-1}(\ln (t+\tau ))^{n+1}}{(\ln (t+3))^{n+1}}\bigg )^{1+\sigma _{1}}\bigg ), \end{aligned} \end{aligned}$$

and we further have

$$\begin{aligned}&|2\varepsilon ^{T}P_{\varepsilon }F|\\&\quad \le 2\sqrt{n}\left( \sum _{i=1}^{n}|{\tilde{p}}_{1i}||\varepsilon _{1}|+\ldots +\sum _{i=1}^{n}|{\tilde{p}}_{ni}||\varepsilon _{n}|\right) \Vert F\Vert \\&\quad \le 2n{\tilde{p}}\sum _{j=1}^{n}|\varepsilon _{j}|\sum _{i=1}^{n-1}|(\ln (t+\tau ))^{i}f_{i}|\\&\quad \le \frac{2^{1+n\sigma _{1}}n^{2}\gamma {\tilde{p}}}{(\ln (t+\tau ))^{1+\sigma _{2}}}\bigg ((2n-1)\sum \limits _{j=1}^{n}|\varepsilon _{j}|^{1+\sigma _{1}}\\&\qquad +nL^{2n}\sum _{j=1}^{n}(1+{\hat{b}}_{j}^{1+\sigma _{1}})|z_{j}|^{1+\sigma _{1}}\\&\qquad +n\bigg (\frac{\delta _{1} L^{n-1}(\ln (t+\tau ))^{n+1}}{(\ln (t+3))^{n+1}}\bigg )^{1+\sigma _{1}}\bigg ).\\ \end{aligned}$$

With the help of

$$\begin{aligned} \begin{aligned} \sum \limits _{j=1}^{n}|\varepsilon _{j}|^{1+\sigma _{1}}&\le \sqrt{n}\left( \sum \limits _{j=1}^{n}|\varepsilon _{j}|^{2(1+\sigma _{1})}\right) ^{\frac{1}{2}}\le \sqrt{n}\left( \sum \limits _{j=1}^{n}|\varepsilon _{j}|^{2}\right) ^{\frac{1+\sigma _{1}}{2}}\\&\le \sqrt{n}\Vert \varepsilon \Vert ^{1+\sigma _{1}}, \end{aligned} \end{aligned}$$

the following inequality can be obtained

$$\begin{aligned} \begin{aligned} |2\varepsilon ^{T}P_{\varepsilon }F|&\le \frac{k_{1}\Vert \varepsilon \Vert ^{1+\sigma _{1}}}{(\ln (t+\tau ))^{1+\sigma _{2}}} +\frac{k_{2}\Vert z\Vert ^{1+\sigma _{1}}}{(\ln (t+\tau ))^{1+\sigma _{2}}}\\&\quad +\frac{k_{3}}{(\ln (t+\tau ))^{1+\sigma _{2}}}\bigg (\frac{(\ln (t+\tau )}{\ln (t+3)}\bigg )^{(n+1)(1+\sigma _{1})}, \end{aligned} \end{aligned}$$

where $k_{1}=(1+\max \{{\hat{b}}_{i}\})(2n-1)n^{3}2^{1+n\sigma _{1}}\gamma {\tilde{p}}$, $k_{2}=k_{1}L^{2n}$, $k_{3}=k_{1}(\delta _{1} L^{n-1})^{1+\sigma _{1}}$, ${\tilde{p}}=\max \{ \sum _{i=1}^{n}|{\tilde{p}}_{1i}|, \ldots , \sum _{i=1}^{n}|{\tilde{p}}_{ni}|\}$ and ${\tilde{p}}_{ij}$ is the element of $P_{\varepsilon }$.

Using the same method, the estimation for the nonlinear terms of (12) can be obtained as follows

$$\begin{aligned} \begin{aligned}&|2\ln (t+\tau )z^{T}P_{z}C_{1}f_{1}|\\&\quad \le \frac{k_{4}}{(\ln (t+\tau ))^{1+\sigma _{2}}}\Vert \varepsilon \Vert ^{1+\sigma _{1}} +\frac{k_{5}}{(\ln (t+\tau ))^{1+\sigma _{2}}}\Vert z\Vert ^{1+\sigma _{1}}\\&\qquad +\frac{k_{6}}{(\ln (t+\tau ))^{1+\sigma _{2}}}\bigg (\frac{(\ln (t+\tau )}{\ln (t+3)}\bigg )^{(n+1)(1+\sigma _{1})}, \end{aligned} \end{aligned}$$

where $k_{4}=(1+\max \{{\hat{b}}_{i}\})(2n-1)n^{2}2^{1+n\sigma _{1}}\gamma p$, $k_{5}=k_{4}L^{2n}$, $k_{6}=k_{4}(\delta _{1} L^{n-1})^{1+\sigma _{1}}$, $p=\max \{|p_{11}|,\ldots , |p_{n1}|\}$ and $p_{i1}$ is the element of $P_{z}$.