Global dynamic output feedback for high-order nonlinear systems with uncertain output function

Abstract

In this article, a dynamic output feedback controller is proposed for a family of high-order nonlinear systems with uncertain output function. First, the nominal system is globally stabilized by designing a homogeneous output feedback controller. Subsequently, the nonlinear terms are tackled by introducing a well-designed gain into the output feedback controller. It can be proved that the system states can globally converge to origin, while the dynamic gain is bounded. Furthermore, the proposed control scheme can be applied to stabilize a family of high-order nonlinear upper-triangular systems. Finally, three examples illustrate the feasibility of the control method intuitively.

Appendix

Proof of Proposition 2

$$\begin{aligned}&\frac{2\sigma -r_{1}-\tau }{r_2} x_2^{\frac{2\sigma -r_{1}-\tau -r_2}{r_2}}\left( x_2^{\frac{r_{1}}{r_2}}-\lambda _2\right) x_{3}^{p_2}\nonumber \\&\quad =\frac{2\sigma -r_{1}-\tau }{r_2} x_2^{\frac{2\sigma -r_{1}-\tau -r_2}{r_2}}\left( x_2^{\frac{r_{1}}{r_2}}-{{\hat{x}}}_2^{\frac{r_{1}}{r_2}}\right) x_{3}^{p_2}. \end{aligned}$$

(102)

Noting that $\frac{r_{1}}{r_2p_{1}}=\frac{r_{1}}{r_{1}+\tau }\le 1$, by Lemma 2, one has

$$\begin{aligned} |x_2^{\frac{r_{1}}{r_2}}-{{\hat{x}}}_2^{\frac{r_{1}}{r_2}}|&=|\left( x_2^{p_{1}}\right) ^{\frac{r_{1}}{r_2p_{1}}}-\left( {{\hat{x}}}_2^{p_{1}}\right) ^{\frac{r_{1}}{r_2p_{1}}}|\nonumber \\&\le 2^{1-\frac{r_1}{r_2p_1}}|e_2|^\frac{r_1}{\sigma }\le {\tilde{d}}|e_2|^\frac{r_1}{\sigma }. \end{aligned}$$

(103)

From the definition of $x_2^*$ in (17), we can get

$$\begin{aligned} |x_2|^\frac{2\sigma -r_1-\tau -r_2}{r_2}&\le {\tilde{d}}\left( |\xi _2|^\frac{2\sigma -r_1-\tau -r_2}{\sigma }+|\xi _{1}|^\frac{2\sigma -r_1-\tau -r_2}{\sigma }\right) ,\nonumber \\ |x_{3}^{p_2}|&\le {\tilde{d}}\left( |\xi _{3}|^\frac{r_3p_2}{\sigma }+|\xi _2|^\frac{r_3p_2}{\sigma }\right) . \end{aligned}$$

(104)

By Lemma 4, it yields

$$\begin{aligned} |x_2^{\frac{2\sigma -r_1-\tau -r_2}{r_2}}x_{3}^{p_2} |&\le {\tilde{d}}\left( |\xi _2|^\frac{2\sigma -r_1-\tau -r_2}{\sigma } +|\xi _{1}|^\frac{2\sigma -r_1-\tau -r_2}{\sigma }\right) \nonumber \\&\quad \times \left( |\xi _{3}|^\frac{r_3p_2}{\sigma }+|\xi _2|^\frac{r_3p_2}{\sigma }\right) \nonumber \\&\le {\tilde{d}}\left( |\xi _{1}|^\frac{2\sigma -r_1}{\sigma }+|\xi _{2}|^\frac{2\sigma -r_1}{\sigma }+|\xi _{3}|^\frac{2\sigma -r_1}{\sigma }\right) . \end{aligned}$$

(105)

Substituting (103) and (105) into (102) and using Lemma 4, it yields

$$\begin{aligned}&\frac{2\sigma -r_1-\tau }{r_2} x_2^{\frac{2\sigma -r_1-\tau -r_2}{r_2}}(x_2^{\frac{r_1}{r_2}}-\lambda _2)x_{3}^{p_2}\nonumber \\&\quad \le {\tilde{d}}(|\xi _{1}|^\frac{2\sigma -r_1}{\sigma }+|\xi _{2}|^\frac{2\sigma -r_1}{\sigma }+|\xi _{3}|^\frac{2\sigma -r_1}{\sigma })|e_2|^\frac{r_1}{\sigma }\nonumber \\&\quad \le \frac{1}{12}\sum _{j=1}^{3}\xi _j^2+\alpha _2e_2^2 \end{aligned}$$

(106)

with a constant $\alpha _2>0$.

Proof of Proposition 3

$$\begin{aligned}&l_{1}\left( 1-\frac{\partial h}{\partial x_1}\right) {{\hat{x}}}_2^{p_{1}}\left( x_2^{\frac{2\sigma -r_{1}-\tau }{r_2}}-\lambda _2^{\frac{2\sigma -r_{1}-\tau }{r_{1}}}\right) \nonumber \\&=l_{1}\left( 1-\frac{\partial h}{\partial x_1}\right) {{\hat{x}}}_2^{p_{1}}\left( x_2^{\frac{2\sigma -r_{1}-\tau }{r_2}}-{{\hat{x}}}_2^{\frac{2\sigma -r_{1}-\tau }{r_2}}\right) \nonumber \\&\le {\tilde{d}}l_{1}|{{\hat{x}}}_2^{p_{1}}||x_2^{\frac{2\sigma -r_{1}-\tau }{r_2}}-{{\hat{x}}}_2^{\frac{2\sigma -r_{1}-\tau }{r_{1}}}|. \end{aligned}$$

(107)

Noting from the definition of $\xi _2$ and $e_2$, one has

$$\begin{aligned} |{{\hat{x}}}_2^{p_1}|&\le |x_2^{p_{1}}-e_2^{r_2p_{1}/\sigma }|\nonumber \\&\le {\tilde{d}}\left( |e_2|^{r_2p_{1}/\sigma }+|\xi _2|^{r_2p_{1}/\sigma }+|\xi _{1}|^{r_2p_{1}/\sigma }\right) . \end{aligned}$$

(108)

Noting that $\frac{2\sigma -r_1-\tau }{r_{1}}\ge 1$, by Lemma 2, one has

$$\begin{aligned} |x_2^{\frac{2\sigma -r_{1}-\tau }{r_2}}-{{\hat{x}}}_2^{\frac{2\sigma -r_{1}-\tau }{r_{1}}}|&=|\left( x_2^{p_{1}}\right) ^{\frac{2\sigma -r_{1}-\tau }{r_2p_{1}}}-\left( {{\hat{x}}}_2^{p_{1}}\right) ^{\frac{2\sigma -r_{1}-\tau }{r_2p_{1}}}|\nonumber \\&\le {\tilde{d}}|e_2|^\frac{2\sigma -r_{1}-\tau }{\sigma }+{{\bar{d}}}_1|x_2|^{\frac{2\sigma -r_{1}-\tau }{r_2}} \end{aligned}$$

(109)

where ${{\bar{d}}}_1$ is a small enough constant.

Similar to (104), we have

$$\begin{aligned} |x_2|^{\frac{2\sigma -r_{1}-\tau }{r_2}}\le {\tilde{d}}\left( |\xi _2|^\frac{2\sigma -r_1-\tau }{\sigma }+|\xi _{1}|^\frac{2\sigma -r_1-\tau }{\sigma }\right) . \end{aligned}$$

(110)

Substituting (108), (109) and (110) into (107) with using Lemma 4, it yields

$$\begin{aligned}&l_{1}\left( 1-\frac{\partial h}{\partial x_1}\right) {{\hat{x}}}_2^{p_{1}}\left( x_2^{\frac{2\sigma -r_{1}-\tau }{r_2}}-\lambda _2^{\frac{2\sigma -r_{1}-\tau }{r_{1}}}\right) \nonumber \\&\quad \le {\tilde{d}}l_1\left( |e_2|^{r_2p_{1}/\sigma }+|\xi _2|^{r_2p_{1}/\sigma }+|\xi _{1}|^{r_2p_{1}/\sigma }\right) \nonumber \\&\qquad \times \left( {\tilde{d}}|e_2|^\frac{2\sigma -r_{1}-\tau }{\sigma }+{{\bar{d}}}_1{\tilde{d}}\left( |\xi _2|^\frac{2\sigma -r_1-\tau }{\sigma }+|\xi _{1}|^\frac{2\sigma -r_1-\tau }{\sigma }\right) \right) \nonumber \\&\quad \le l_1{{\check{\alpha }}}\left( \xi _1^2+\xi _2^2\right) +l_1{{\hat{\alpha }}} e_2^2 \end{aligned}$$

(111)

where ${{\check{\alpha }}}>0$ and $0<{{\hat{\alpha }}}<m_2$ are positive constants.

Proof of Proposition 5:

$$\begin{aligned}&\frac{2\sigma -r_{n-1}-\tau }{r_n} x_n^{\frac{2\sigma -r_{n-1}-\tau -r_n}{r_n}}\left( x_n^{\frac{r_{n-1}}{r_n}}-\lambda _n\right) u^{p_n}\nonumber \\&\quad =\frac{2\sigma -r_{n-1}-\tau }{r_n} x_n^{\frac{2\sigma -r_{n-1}-\tau -r_n}{r_n}}u^{p_n} \nonumber \\&\quad \times \left( \left( x_n^{p_{n-1}}\right) ^{\frac{r_{n-1}}{r_np_{n-1}}}-\left( {{\hat{x}}}_n^{p_{n-1}}\right) ^{\frac{r_{n-1}}{r_np_{n-1}}}-l_{n-1}\left( x_{n-1}-{{\hat{x}}}_{n-1}\right) \right) \end{aligned}$$

(112)

By the homogeneous degree of u from (32), one has

$$\begin{aligned} |u^{p_n}|\le {\tilde{d}}\Vert {{\hat{x}}}\Vert _{\varDelta }^{r_n+\tau } \end{aligned}$$

(113)

where $\Vert {{\hat{x}}}\Vert _{\varDelta }=(|y|^{2/r_1}+\sum _{i=2}^{n}|{{\hat{x}}}_i|^{2/r_i})^{1/2}$, $\varDelta _{{{\hat{x}}}}=(r_1,r_2,\dots ,r_n)$. Thus, by the definition of $e_i$, one has

$$\begin{aligned} |u^{p_n}|&\le {\tilde{d}}\left( |y|^\frac{2}{r_1}+\sum _{i=2}^{n}|{{\hat{x}}}_i|^\frac{2}{r_i}\right) ^\frac{r_n+\tau }{2}\nonumber \\&\le {\tilde{d}}\left( |y|^\frac{2}{r_1}+\sum _{i=2}^{n}|x_i^{p_{i-1}}-e_i^{r_ip_{i-1}/\sigma }|^\frac{2}{r_ip_{i-1}}\right) ^\frac{r_n+\tau }{2}\nonumber \\&\le {\tilde{d}}\left( |y|^\frac{r_n+\tau }{r_1}+\sum _{i=2}^{n}|x_i|^\frac{r_n+\tau }{r_i}+\sum _{i=2}^{n}|e_i|^\frac{r_n+\tau }{\sigma }\right) \nonumber \\&\le {\tilde{d}}\left( \sum _{i=1}^{n}|\xi _i|^\frac{r_n+\tau }{\sigma }+\sum _{i=2}^{n}|e_i|^\frac{r_n+\tau }{\sigma }\right) . \end{aligned}$$

(114)

Noting that $\frac{r_{n-1}}{r_np_{n-1}}=\frac{r_{n-1}}{r_{n-1}+\tau }\le 1$ and by Lemma 2, it yields

$$\begin{aligned}&|x_{n-1}-{{\hat{x}}}_{n-1}|= |\left( x_{n-1}^{p_{n-2}}\right) ^\frac{1}{p_{n-2}}-\left( {{\hat{x}}}_{n-1}^{p_{n-2}}\right) ^\frac{1}{p_{n-2}}|\nonumber \\&\quad \le 2^{1-\frac{1}{p_{n-2}}}|e_{n-1}|^\frac{r_{n-1}}{\sigma }\le {\tilde{d}}|e_{n-1}|^\frac{r_{n-1}}{\sigma },\nonumber \\&\qquad |\left( x_n^{p_{n-1}}\right) ^\frac{r_{n-1}}{r_np_{n-1}}-\left( {{\hat{x}}}_n^{p_{n-1}}\right) ^\frac{r_{n-1}}{r_np_{n-1}}| \le 2^{1-\frac{r_{n-1}}{r_np_{n-1}}}|e_n|^\frac{r_{n-1}}{\sigma }\nonumber \\&\quad \le {\tilde{d}}|e_n|^\frac{r_{n-1}}{\sigma }. \end{aligned}$$

(115)

From the definition of $x_n^*$, we can get

$$\begin{aligned} |x_n|^\frac{2\sigma -r_{n-1}-\tau -r_n}{r_n}&\le {\tilde{d}}\left( |\xi _n|^\frac{2\sigma -r_{n-1}-\tau -r_n}{\sigma }\right. \nonumber \\&\left. +|\xi _{n-1}|^\frac{2\sigma -r_{n-1}-\tau -r_n}{\sigma }\right) . \end{aligned}$$

(116)

With (114), (115) and (116) and Lemma 4 in mind, one has

$$\begin{aligned}&\frac{2\sigma -r_{n-1}-\tau }{r_n} x_n^{\frac{2\sigma -r_{n-1}-\tau -r_n}{r_n}}(x_n^{\frac{r_{n-1}}{r_n}}-\lambda _n) u^{p_n}\nonumber \\&\quad \le {\tilde{d}}\left( |\xi _{n-1}|^{\frac{2\sigma -r_{n-1}-\tau -r_n}{\sigma }}+|\xi _{n}|^{\frac{2\sigma -r_{n-1}-\tau -r_n}{\sigma }}\right) \nonumber \\&\qquad \times \left( \sum _{i=1}^{n}|\xi _i|^\frac{r_n+\tau }{\sigma }+\sum _{i=2}^{n}|e_i|^\frac{r_n+\tau }{\sigma }\right) \nonumber \\&\qquad \times \left( {\tilde{d}}|e_n|^\frac{r_{n-1}}{\sigma }+l_{n-1}{\tilde{d}}|e_{n-1}|^\frac{r_{n-1}}{\sigma }\right) \nonumber \\&\quad \le \frac{1}{8}\sum _{i=1}^{n}\xi _i^2+\bar{\alpha }\sum _{i=2}^n e_i^2+g_n\left( l_{n-1}\right) e_{n-1}^2 \end{aligned}$$

(117)

with ${\bar{\alpha }}$ being a positive constant and $g_n(l_{n-1})$ being a continuous function of $l_{n-1}$.

Proof of Proposition 7:

In the light of the definitions of $x_{n+1}^*$ and u, it follows that

$$\begin{aligned}&\xi _n^\frac{2\sigma -r_n-\tau }{\sigma }\left( u^{p_n}-x_{n+1}^{*p_n}\right) \nonumber \\&\quad \le {\tilde{d}}|\xi _n|^\frac{2\sigma -r_n-\tau }{\sigma }|\beta _n^{p_n}\left( {{\hat{x}}}_n^{\frac{\sigma }{r_n}}+\beta _{n-1}^{\frac{\sigma }{r_n}}\right. \nonumber \\&\qquad \left. \left( {{\hat{x}}}_{n-1}^{\frac{\sigma }{r_{n-1}}}+\cdots +\beta _2^{\frac{\sigma }{r_3}}\left( {{\hat{x}}}_2^{\frac{\sigma }{r_2}} \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. +\beta _1^{\frac{\sigma }{r_2}}y^\frac{\sigma }{r_1}\right) \cdots \right) \right) ^\frac{r_{n+1}p_n}{\sigma }-\beta _n^{p_n}\left( x_n^{\frac{\sigma }{r_n}}+\beta _{n-1}^{\frac{\sigma }{r_n}}\right. \nonumber \\&\qquad \left. \left( x_{n-1}^{\frac{\sigma }{r_{n-1}}} +\cdots +\beta _2^{\frac{\sigma }{r_3}}\right. \right. \nonumber \\&\qquad \left. \left. \left( x_2^{\frac{\sigma }{r_2}}+\beta _1^{\frac{\sigma }{r_2}}y^\frac{\sigma }{r_1}\right) \cdots \right) \right) ^\frac{r_{n+1}p_n}{\sigma }|\nonumber \\&\le {\tilde{d}}|\xi _n|^\frac{2\sigma -r_n-\tau }{\sigma }\left( \sum _{i=2}^{n}|x_i^\frac{\sigma }{r_i}-{{\hat{x}}}_i^\frac{\sigma }{r_i}|\right) ^\frac{r_{n+1}p_n}{\sigma }. \end{aligned}$$

(118)

Since $\sigma /(r_ip_{i-1})\ge 1$, by the definition of $e_i$ and Lemma 2, one has

$$\begin{aligned} |x_i^\frac{\sigma }{r_i}-{{\hat{x}}}_i^\frac{\sigma }{r_i}|&\le |\left( x_i^{p_{i-1}}\right) ^\frac{\sigma }{r_ip_{i-1}}-\left( {{\hat{x}}}_i^{p_{i-1}}\right) ^\frac{\sigma }{r_ip_{i-1}}|\nonumber \\&\le {\tilde{d}}|e_i|+{{\bar{d}}}_2|x_i|^\frac{\sigma }{r_i} \end{aligned}$$

(119)

with ${{\bar{d}}}_2>0$ a small enough constant. In line with (116), one has

$$\begin{aligned} |x_i|^\frac{\sigma }{r_i} \le {\tilde{d}}\left( |\xi _i|+|\xi _{i-1}|\right) . \end{aligned}$$

(120)

Substituting (119) and (120) into (118), such that

$$\begin{aligned}&\xi _n^\frac{2\sigma -r_n-\tau }{\sigma }\left( u^{p_n}-x_{n+1}^{*p_n}\right) \nonumber \\&\quad \le {\tilde{d}}|\xi _n|^\frac{2\sigma -r_n-\tau }{\sigma }\sum _{i=2}^{n}\left( |e_i|^\frac{r_{n+1}p_n}{\sigma }+{{\bar{d}}}_2\right. \nonumber \\&\quad \left. \left( |\xi _i|^\frac{r_{n+1}p_n}{\sigma }+|\xi _{i-1}|^\frac{r_{n+1}p_n}{\sigma }\right) \right) \nonumber \\&\quad \le \frac{1}{4}\sum _{i=1}^{n}\xi _i^2+{\tilde{\alpha }}\sum _{j=2}^n e_j^2 \end{aligned}$$

(121)

with a positive constant ${\tilde{\alpha }}$.