Event-triggered control for p-normal uncertain nonlinear systems

Abstract

When dealing with microprocessors with limited resources and networks with limited bandwidth, the research of event triggering technology becomes more and more important. In order to save communication resources, this paper is concerned with the event-triggered control problem for a class of p-normal uncertain nonlinear systems with unknown growth rate. At first, an event-triggered controller is designed for the nominal system by adding a power integrator technique. Then, a novel dynamic gain is introduced into the proposed event-triggered controller to render lower-triangular uncertain nonlinear systems globally asymptotically stable. At the same time, Zeno phenomenon is excluded. Finally, the proposed method is extended to event-triggered control of p-normal upper-triangular uncertain nonlinear systems. A practical circuit system and a numerical simulation are given to illustrate the effectiveness of the proposed method.

Appendix

In this section, a generic constant c is used to represent some positive constant values. It may be different in different places.

Proof of Proposition 1:

From the construction of $W_k$, one has

$$\begin{aligned} \sum _{i=1}^{k-1}\frac{\partial W_k}{\partial {x}_i} x_{i+1}^{p_i}&\le c\sum _{i=1}^{k-1} |\xi _k|^{\frac{2\mu -r_k-\tau }{\alpha }-1}|x_k-x_k^{*}|\nonumber \\&\quad \big |\frac{\partial {x}_k^{*\alpha /r_k}}{\partial x_i}x_{i+1}^{p_i}\big | \nonumber \\&\le c\sum _{i=1}^{k-1}|\xi _k|^{\frac{2\mu -\tau -\alpha }{\alpha }}\big |\frac{\partial {x}_k^{*\alpha /r_k}}{\partial x_i}x_{i+1}^{p_i}\big |.\nonumber \\ \end{aligned}$$

(A1)

From (10) and using Young’s inequality, it can be seen that

$$\begin{aligned}&\big |\frac{\partial x_k^{*\alpha /r_k}}{\partial x_i}x_{i+1}^{p_i}\big | =\beta _{k-1}^{\alpha /r_k}\big |\frac{\partial \xi _{k-1}}{\partial x_i}\big |\big |x_{i+1}^{p_i}\big |\nonumber \\&\quad =\beta _{k-1}^{\alpha /r_k}\big |\frac{\partial x_{k-1}^{*\alpha /r_{k-1}}}{\partial x_i}\big |\big |x_{i+1}^{p_i}\big |=\cdots \nonumber \\ {}&\quad =\beta _{k-1}^{\alpha /r_k}\dots \beta _{i+1}^{\alpha /r_{i+2}}\big |\frac{\partial x_{i+1}^{*\alpha /r_{i+1}}}{\partial x_i}\big |\big |x_{i+1}^{p_i}\big |\nonumber \\&\quad =\frac{\alpha \beta _{k-1}^{\alpha /r_{k}}\dots \beta _i^{\alpha /r_{i+1}}}{r_i}\big |x_i^{\frac{\alpha }{r_i}-1}\big |\big |x_{i+1}^{p_i}\big |\nonumber \\&\quad =\frac{\alpha \beta _{k-1}^{\alpha /r_{k}}\dots \beta _i^{\alpha /r_{i+1}}}{r_i}\big |\xi _i-\beta _{i-1}^{\alpha /r_i}\xi _{i-1}\big |^{1-\frac{r_i}{\alpha }}\nonumber \\&\qquad \times \big |\xi _{i+1}-\beta _{i}^{\alpha /r_{i+1}}\xi _i\big |^{\frac{r_{i+1}p_i}{\alpha }}\nonumber \\&\quad \le c\big (|\xi _{i-1}|^\frac{\alpha +\tau }{\alpha }+|\xi _i|^\frac{\alpha +\tau }{\alpha }+|\xi _{i+1}|^\frac{\alpha +\tau }{\alpha }\big ). \end{aligned}$$

(A2)

Substituting (A2) into (A1) and using Young’s inequality, one has

$$\begin{aligned} \sum _{i=1}^{k-1}\frac{\partial W_k}{\partial x_i}x_{i+1}^{p_i}&\le \frac{1}{2}\sum _{i=1}^{k-1}\xi _i^\frac{2\mu }{\alpha }+\hat{c}_1\xi _k^\frac{2\mu }{\alpha } \end{aligned}$$

(A3)

with a constant ${\hat{c}}_1>0$.

Proof of Proposition 2:

By Lemma 3, one has

$$\begin{aligned} |x_{i-1}-{\hat{x}}_{i-1}|&=|(x_{i-1}^{p_{i-2}})^{\frac{1}{p_{i-2}}}-({\hat{x}}_{i-1}^{p_{i-2}})^{\frac{1}{p_{i-2}}}|\nonumber \\&\le 2^{1-\frac{1}{p_{i-2}}}|e_{i-1}|^{\frac{r_{i-1}}{\alpha }}. \end{aligned}$$

(A4)

From (10), one has

$$\begin{aligned}&|x_i|^{\frac{r_{i-1}}{r_i}}\le c(|\xi _i|^{\frac{r_{i-1}}{\alpha }}+|\xi _{i-1}|^{\frac{r_{i-1}}{\alpha }}),\nonumber \\&|x_{i+1}^{p_i}|\le c(|\xi _{i+1}|^{\frac{r_{i+1}p_i}{\alpha }}+|\xi _{i}|^{\frac{r_{i+1}p_i}{\alpha }}). \end{aligned}$$

(A5)

If $\tau >0$, then $\frac{r_{i-1}}{r_ip_{i-1}}=\frac{r_{i-1}}{r_{i-1}+\tau }<1$. Then, we have

$$\begin{aligned} |(x_i^{p_{i-1}})^{\frac{r_{i-1}}{r_i p_{i-1}}}-({\hat{x}}_i^{p_{i-1}})^{\frac{r_{i-1}}{r_i p_{i-1}}}|\le 2^{1-\frac{r_{i-1}}{r_ip_{i-1}}}|e_i|^{\frac{r_{i-1}}{\alpha }}. \end{aligned}$$

(A6)

If $\tau \le 0$, then $\frac{r_{i-1}}{r_ip_{i-1}}=\frac{r_{i-1}}{r_{i-1}+\tau }\ge 1$. By Lemma 3, one has

$$\begin{aligned} |(x_i^{p_{i-1}})^{\frac{r_{i-1}}{r_i p_{i-1}}}{-}({\hat{x}}_i^{p_{i-1}})^{\frac{r_{i-1}}{r_i p_{i-1}}}|{\le } c(|e_i|^{\frac{r_{i-1}}{\alpha }}{+}|x_i|^{\frac{r_{i{-}1}}{r_i}}). \end{aligned}$$

(A7)

From the above two cases, one has

$$\begin{aligned}&\rho _i x_i^{\rho _i-1}\big (x_i^{\frac{r_{i-1}}{r_i}}-\zeta _i\big )x_{i+1}^{p_i}\nonumber \\&\le c\big (|\xi _i|^\frac{(\rho _i-1)r_i}{\alpha }+|\xi _{i-1}|^\frac{(\rho _i-1)r_i}{\alpha }\big )\big (|\xi _{i+1}|^{\frac{r_{i+1}p_i}{\alpha }}+|\xi _{i}|^{\frac{r_{i+1}p_i}{\alpha }}\big )\nonumber \\&\quad \times \big (|e_i|^{\frac{r_{i-1}}{\alpha }}+|\xi _i|^{\frac{r_{i-1}}{\alpha }}+|\xi _{i-1}|^{\frac{r_{i-1}}{\alpha }}+l_{i-1}|e_{i-1}|^{\frac{r_{i-1}}{\alpha }}\big )\nonumber \\&\le \frac{1}{12}\sum _{j=i-1}^{i+1}\xi _j^\frac{2\mu }{\alpha }+\alpha _i e_i^\frac{2\mu }{\alpha }+g_i(l_{i-1})e_{i-1}^\frac{2\mu }{\alpha } \end{aligned}$$

(A8)

where $\alpha _i>0$ and $g_i$ is a continuous function of $l_{i-1}$ with $g_2(\cdot )=0$.

Proof of Proposition 3:

From the definition of $e_i$ and using Lemma 3, one has

$$\begin{aligned}&-l_{i-1}e_i^\frac{r_i p_{i-1}}{\alpha }\big (x_i^{\rho _i}-{\hat{x}}_i^{\rho _i}\big )\nonumber \\ {}&\quad {=}{-}l_{i-1}(x_i^{p_{i-1}}{-}{\hat{x}}_i^{p_{i-1}})\big ((x_i^{p_{i-1}})^{\frac{\rho _i}{p_{i-1}}}{-}({\hat{x}}_i^{p_{i{-}1}})^{\frac{\rho _i}{p_{i{-}1}}}\big )\nonumber \\&\quad \le -l_{i-1}2^{1-\frac{\rho _i}{p_{i-1}}}(x_i^{p_{i-1}}-{\hat{x}}_i^{p_{i-1}})^{\frac{\rho _i}{p_{i-1}}+1}\nonumber \\&\quad :=-l_{i-1}m_i e_i^\frac{2\mu }{\alpha } \end{aligned}$$

(A9)

where $m_i=2^{1-\rho _i/p_{i-1}}$.

Proof of Proposition 4:

From the definitions of $\rho _i$ and $r_i$, one has $r_i\rho _i/r_{i-1}>1$. By Lemma 3, from (21) one has

$$\begin{aligned}&-l_{i-1}e_i^\frac{r_i p_{i-1}}{\alpha }\big ({\hat{x}}_i^{\rho _i}-\zeta _i^{\frac{r_i \rho _i}{r_{i-1}}}\big )\nonumber \\ {}&\quad \le l_{i-1}|e_i|^\frac{r_ip_{i-1}}{\alpha }|(\eta _i+l_{i-1}{\hat{x}}_{i-1})^{\frac{r_i\rho _i}{r_{i-1}}}\nonumber \\&\qquad -(\eta _i+l_{i-1}x_{i-1})^{\frac{r_i\rho _i}{r_{i-1}}}|\nonumber \\&\quad \le c l_{i-1}^2|e_i|^\frac{r_ip_{i-1}}{\alpha }|e_{i-1}|^{\frac{r_{i-1}}{\alpha }}\nonumber \\&\qquad \times \big (l_{i-1}^{\frac{r_i\rho _i-r_{i-1}}{r_{i-1}}}|e_{i-1}|^{\frac{r_i\rho _i-r_{i-1}}{\alpha }}+|{\hat{x}}_i|^{\frac{r_i\rho _i-r_{i-1}}{r_i}}\big ). \end{aligned}$$

(A10)

From the definition of $e_i$ and (10), one has

$$\begin{aligned} |{\hat{x}}_i|^{p_{i-1}}&=|(\xi _i-\beta _{i-1}^{\alpha /r_i}\xi _{i-1})^{\frac{r_ip_{i-1}}{\alpha }}-e_i^{\frac{r_ip_{i-1}}{\alpha }}|\nonumber \\&\le c(|\xi _i|^{\frac{r_ip_{i-1}}{\alpha }}+|\xi _{i-1}|^{\frac{r_ip_{i-1}}{\alpha }}+|e_i|^{\frac{r_ip_{i-1}}{\alpha }}). \end{aligned}$$

(A11)

Substituting (A11) into (A10) and using Young’s inequality, one has

$$\begin{aligned}&-l_{i-1}e_i^\frac{r_i p_{i-1}}{\alpha }\big ({\hat{x}}_i^{\rho _i}-\zeta _i^{\frac{r_i \rho _i}{r_{i-1}}}\big )\nonumber \\&\quad \le e_i^\frac{2\mu }{\alpha }+\frac{1}{16}(\xi _{i-1}^\frac{2\mu }{\alpha }+\xi _i^\frac{2\mu }{\alpha })+h_i(l_{i-1})e_{i-1}^\frac{2\mu }{\alpha } \end{aligned}$$

(A12)

where $h_i(\cdot )$ is a continuous function of $l_{i-1}$. It should be pointed out that ${\hat{x}}_i^{\rho _i}-\zeta _i^{r_i\rho _i/r_{i-1}}=0$ for $i=2$.

Proof of Proposition 5:

From (22) and the definition of $e_i$, one has

$$\begin{aligned} |u({\hat{x}})|&\le c(|{\hat{x}}_n|^{\frac{\alpha }{r_n}}+\dots +|{\hat{x}}_2|^{\frac{\alpha }{r_2}}+|x_1|^{\frac{\alpha }{r_1}})^\frac{r_{n+1}}{\alpha }\nonumber \\&\le c(\big |x_n^{p_{n-1}}-e_n^{{(r_n p_{n-1})}/{\alpha }}\big |^{\frac{\alpha }{r_n p_{n-1}}}+\cdots \nonumber \\&\quad +\big |x_2^{p_1}-e_2^{{(r_2 p_1)}/{\alpha }}\big |^{\frac{\alpha }{r_2 p_1}}+|x_1|^{\frac{\alpha }{r_1}})^\frac{r_{n+1}}{\alpha }\nonumber \\&\le c \left( \sum _{i=1}^n|x_i|^{\frac{r_{n+1}}{r_i}}+\sum _{i=2}^n|e_i|^{\frac{r_{n+1}}{\alpha }} \right) \end{aligned}$$

(A13)

From (10) and (33), one has

$$\begin{aligned} |u|&\le c\left( \sum _{i=1}^n|x_i|^{\frac{r_{n+1}}{r_i}}+\sum _{i=2}^n|e_i|^{\frac{r_{n+1}}{\alpha }}+L^{-1}\hbox {e}^{-a_0t}\right) \nonumber \\&\le c\left( \sum _{i=1}^n|\xi _i|^{\frac{r_{n+1}}{\alpha }}+\sum _{i=2}^n|e_i|^{\frac{r_{n+1}}{\alpha }}+L^{-1}\hbox {e}^{-a_0t}\right) . \end{aligned}$$

(A14)

From the definition of $e_{n-1}$, one has

$$\begin{aligned} |x_{n-1}-{\hat{x}}_{n-1}|&=|(x_{n-1}^{p_{n-2}})^{\frac{1}{p_{n-2}}}-({\hat{x}}_{n-1}^{p_{n-2}})^{\frac{1}{p_{n-2}}}|\nonumber \\&\le 2^{1-\frac{1}{p_{n-2}}}|e_{n-1}|^{\frac{r_{n-1}}{\alpha }}. \end{aligned}$$

(A15)

If $\tau \ge 0$, then $\frac{r_{n-1}}{r_np_{n-1}}=\frac{r_{n-1}}{r_{n-1}+\tau }\le 1$. By Lemma 3, one has

$$\begin{aligned} |(x_n^{p_{n-1}})^{\frac{r_{n-1}}{r_np_{n-1}}}-({\hat{x}}_n^{p_{n-1}})^{\frac{r_{n-1}}{r_np_{n-1}}}|\le 2^{1-\frac{r_{n-1}}{r_n p_{n-1}}}|e_n|^{\frac{r_{n-1}}{\alpha }}. \end{aligned}$$

(A16)

If $\tau < 0$, then $\frac{r_{n-1}}{r_np_{n-1}}=\frac{r_{n-1}}{r_{n-1}+\tau }>1$. From (10) and Lemma 3, one has

$$\begin{aligned}&|(x_n^{p_{n-1}})^{\frac{r_{n-1}}{r_np_{n-1}}}-({\hat{x}}_n^{p_{n-1}})^{\frac{r_{n-1}}{r_n p_{n-1}}}|\nonumber \\ {}&\quad \le c(|e_n|^{\frac{r_{n-1}}{\alpha }}+|x_n|^{\frac{r_{n-1}}{r_n}})\nonumber \\&\quad \le c(|e_n|^{\frac{r_{n-1}}{\alpha }}+|\xi _n-\beta _{n-1}^\frac{\alpha }{r_n}\xi _{n-1}|^\frac{r_{n-1}}{\alpha })\nonumber \\&\quad \le c(|e_n|^{\frac{r_{n-1}}{\alpha }}+|\xi _n|^{\frac{r_{n-1}}{\alpha }}+|\xi _{n-1}|^{\frac{r_{n-1}}{\alpha }}). \end{aligned}$$

(A17)

Using Young’s inequality and the above inequalities, one has

$$\begin{aligned}&\rho _n x_n^{\rho _n-1}(x_n^{\frac{r_{n-1}}{r_n}}-\zeta _n)u\nonumber \\&\quad \le c|x_n|^{\rho _n-1}|(x_n^{p_{n-1}})^{\frac{r_{n-1}}{r_np_{n-1}}}-({\hat{x}}_n^{p_{n-1}})^{\frac{r_{n-1}}{r_np_{n-1}}}\nonumber \\&\qquad -l_{n-1}(x_{n-1}-{\hat{x}}_{n-1})| |u|\nonumber \\&\quad \le c(|\xi _n|^{\frac{(\rho _n-1)r_n}{\alpha }}+|\xi _{n-1}|^{\frac{(\rho _n-1)r_n}{\alpha }})\nonumber \\&\quad {\times }\big (|e_n|^{\frac{r_{n{-}1}}{\alpha }}{+}|\xi _n|^{\frac{r_{n-1}}{\alpha }}{+}|\xi _{n-1}|^{\frac{r_{n-1}}{\alpha }}{+}l_{n-1}|e_{n-1}|^{\frac{r_{n-1}}{\alpha }}\big )\nonumber \\&\qquad \times \left( \sum _{i=1}^n|\xi _i|^{\frac{r_{n+1}}{\alpha }}+\sum _{i=2}^n|e_i|^{\frac{r_{n+1}}{\alpha }}+L^{-1}\hbox {e}^{-a_0t}\right) \nonumber \\&\quad \le \frac{1}{8}\sum _{i=1}^{n}\xi _i^\frac{2\mu }{\alpha }+\bar{\alpha }\sum _{i=2}^n e_i^\frac{2\mu }{\alpha }+g_n(l_{n-1})e_{n-1}^\frac{2\mu }{\alpha }\nonumber \\&\qquad +c_1 L^{-\frac{2\mu }{r_{n+1}}}\hbox {e}^{-\frac{2\mu a_0}{r_{n+1}} t} \end{aligned}$$

(A18)

where ${\bar{\alpha }}, c_1$ are constants and $g_n$ is a continuous function of $l_{n-1}$.

Proof of Proposition 6:

From (33), one has

$$\begin{aligned}&\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }(u-x_{n+1}^{*})\nonumber \\&\quad =\frac{1+\delta _0 L^{-1}}{1+b_{1}(t)\delta _0 L^{-1}}\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }\big (u({\hat{x}})-x_{n+1}^{*}\big )\nonumber \\ {}&\quad +\frac{b_{2}(t)\delta _1 L^{-1} \hbox {e}^{-a_0t}}{1+b_{1}(t)\delta _0 L^{-1}}\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }\nonumber \\&\qquad +\left( \frac{1+\delta _0 L^{-1}}{1+b_1(t)\delta _0L^{-1}}-1\right) \xi _n^\frac{2\mu -\tau -r_{n}}{\alpha } x_{n+1}^{*}\nonumber \\&\quad \le \frac{1+\delta _0 L^{-1}}{1+b_{1}(t)\delta _0 L^{-1}}\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }\big (u({\hat{x}})-x_{n+1}^{*}\big )\nonumber \\&\qquad +\frac{b_{2}(t)\delta _1 L^{-1} \hbox {e}^{-a_0t}}{1+b_{1}(t)\delta _0 L^{-1}}\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha } \end{aligned}$$

(A19)

where the last relation is obtained by $\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }x_{n+1}^{*}\le 0$ and $\frac{1+\delta _0 L^{-1}}{1+b_1(t)\delta _0 L^{-1}}-1\ge 0$.

Now, we estimate the first term of (A19).

$$\begin{aligned}&\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }\big (u(\hat{x})-x_{n+1}^{*}\big )\nonumber \\&\le |\xi _n|^\frac{2\mu -\tau -r_{n}}{\alpha }\big |\beta _n \left( {\hat{x}}_n^{\frac{\alpha }{r_n}}+\sum _{i=1}^{n-1}(\prod _{j=i}^{n-1}\beta _j^{\frac{\alpha }{r_{j+1}}}){\hat{x}}_i^{\frac{\alpha }{r_i}}\right) ^\frac{r_{n+1}}{\alpha }\nonumber \\&\quad -\beta _n\left( x_n^{\frac{\alpha }{r_n}}+\sum _{i=1}^{n-1}\left( \prod _{j=i}^{n-1}\beta _j^{\frac{\alpha }{r_{j+1}}}\right) x_i^{\frac{\alpha }{r_i}}\right) ^\frac{r_{n+1}}{\alpha }\big |\nonumber \\&\le c|\xi _n|^\frac{2\mu -\tau -r_{n}}{\alpha }\left( \sum _{i=2}^{n} |\hat{x}_i^{\frac{\alpha }{r_i}}-x_i^{\frac{\alpha }{r_i}}|\right) ^\frac{r_{n+1}}{\alpha }. \end{aligned}$$

(A20)

From (10), one has $|x_i^{{\alpha }/{r_i}}|\le c(|\xi _i|+|\xi _{i-1}|)$. By Lemma 3, we have

$$\begin{aligned} |\hat{x}_i^{{\alpha }/{r_i}}-x_i^{{\alpha }/{r_i}}|&=|(\hat{x}_i^{p_{i-1}})^{\frac{\alpha }{r_ip_{i-1}}}-(x_i^{p_{i-1}})^{\frac{\alpha }{r_ip_{i-1}}}|\nonumber \\&\le c(|e_i|+|x_i|^{{\alpha }/{r_i}})\nonumber \\&\le c(|e_i|+|\xi _i|+|\xi _{i-1}|). \end{aligned}$$

(A21)

Substituting (A21) into (A20) yields

$$\begin{aligned}&\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }\big (u(\hat{x})-x_{n+1}^{*}\big )\nonumber \\ {}&\quad \le c|\xi _n|^\frac{2\mu -\tau -r_{n}}{\alpha }\Big (\sum _{i=2}^n(|e_i|+|\xi _i|+|\xi _{i-1}|)\Big )^{\frac{r_{n+1}}{\alpha }}\nonumber \\&\quad \le c|\xi _n|^\frac{2\mu -\tau -r_{n}}{\alpha }\sum _{i=2}^n \big (|e_i|^\frac{r_{n+1}}{\alpha } +|\xi _i|^\frac{r_{n+1}}{\alpha }+|\xi _{i-1}|^\frac{r_{n+1}}{\alpha }\big )\nonumber \\&\quad \le \frac{1}{6}\sum _{i=1}^{n}\xi _i^\frac{2\mu }{\alpha }+{\tilde{\alpha }} \sum _{i=2}^n e_i^\frac{2\mu }{\alpha } \end{aligned}$$

(A22)

where ${\tilde{\alpha }}>0$ is a constant.

By using Young’s inequality, one has

$$\begin{aligned} \frac{b_{2}(t)\delta _1 L^{-1} \hbox {e}^{-a_0t}}{1+b_{1}(t)\delta _0L^{-1}}\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }&\le c|L^{-1}\hbox {e}^{-a_0 t}||\xi _n|^{\frac{2\mu -\tau -r_{n}}{\alpha }}\nonumber \\&\le \frac{1}{12}\xi _n^{\frac{2\mu }{\alpha }}+c_2 L^{-\frac{2\mu }{r_{n+1}}}\hbox {e}^{-\frac{2\mu a_0}{r_{n+1}}t} \end{aligned}$$

(A23)

where $c_2>0$ is a constant.

Substituting (A22) and (A23) into (A19), one has

$$\begin{aligned}&\xi _n^\frac{2\mu -\tau -r_{n}}{\alpha }(u-x_{n+1}^{*})\nonumber \\ {}&\quad \le \frac{1}{4}\sum _{i=1}^{n}\xi _i^\frac{2\mu }{\alpha }+{\tilde{\alpha }} \sum _{i=2}^n e_i^\frac{2\mu }{\alpha }+c_2 L^{-\frac{2\mu }{r_{n+1}}}\hbox {e}^{-\frac{2\mu a_0}{r_{n+1}}t}. \end{aligned}$$

(A24)