Robust output feedback tracking control for a class of high-order time-delay nonlinear systems with input dead-zone and disturbances

Abstract

This paper is concerned with the tracking control problem for a class of high-order nonlinear systems. Different from the related studies, the considered systems allow the existence of input dead-zone, external disturbances and polynomial growing conditions with time-varying delays. A new Lyapunov–Krasovskii functional is skillfully constructed and a robust output feedback tracking controller is designed by using a modified homogeneous domination method. It is guaranteed that all signals of the closed-loop system are bounded and the tracking error can converge to a compact domain which can be tuned sufficiently small. A simulation example is provided to show the validity of our control strategy.

Appendix: Some proofs

Proof of Proposition 1

To prove Proposition 1, we consider the auxiliary system

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{x}_i = Sx_{i+1}^{p}, i=1,2,\ldots ,n-1,\\ \dot{x}_n = Su_1^p(v), \end{array} \right. \end{aligned}$$

(A.1)

and show that the derivative of $V_n$ along system (A.1) satisfies (8). The proof is given as follows.

Step 1: Taking the time derivative of $V_{1}$ along the solutions of system (A.1), we have $\dot{V}_1= Sz_1(x_{2}^{p}-\alpha ^{p}_1)+Sz_1\alpha ^{p}_1.$ Take the virtual controller $\alpha _1 =-(na_{n1})^{1/p}z_1=: -g_1z_1,$ with $a_{11}$ being a positive constant independent of S. Therefore, we have

$$\begin{aligned} \dot{V}_1 \le -Sna_{n1}z^{p+1}_1+S|z_1||x_{2}^{p}-\alpha ^{p}_1|. \end{aligned}$$

Step$k ( k=2,3,\ldots ,n)$: Suppose that $V_{k-1}$ satisfies $\dot{V}_{k-1} \le -S(n-k+2)\sum _{i=1}^{k-1}a_{n,i}z_i^{p+1} +S|z_{k-1}|\big |x_k^{p}-\alpha _{k-1}^{p}\big |.$ Then, taking the time derivative of the positive definite and radially unbounded function $V_k$, we have

$$\begin{aligned} \dot{V}_{k}\le & {} -S(n-k+2)\sum _{i=1}^{k-1}a_{n,i}z_i^{p+1}\\&+\,S|z_{k-1}||x_k^{p}-\alpha _{k-1}^{p}| \\&+\,Sz_kx^p_{k+1}-z_k\sum _{j=1}^{k-1} \frac{\partial \alpha _{k-1}}{\partial x_j}Sx_{j+1}^{p}. \end{aligned}$$

Using Lemmas 1 and 3, it can be deduced that $S|z_{k-1}||x_{k}^{p}-\alpha _{k-1}^p| \le \frac{a_{n,k-1}}{2} Sz^{p+1}_{k-1}+Sb_{kk}z^{p+1}_k,$ where $a_{n,k-1}$ and $b_{kk}$ are positive constants independent of S. Similarly, we have

$$\begin{aligned} -z_k\sum _{i=1}^{k-1} \frac{\partial \alpha _{k-1}}{\partial x_i}Sx_{i+1}^{p}\le & {} \frac{a_{n,k-1}}{2}Sz^{p+1}_{k-1}\\&+\, S\sum _{j=1}^{k-2}a_{nj}z^{p+1}_j \\&+\,S{\bar{b}}_{kk}z^{p+1}_k, \end{aligned}$$

where $a_{n,j},j=1,2,\ldots ,k-2$ and ${\bar{b}}_{kk}$ are positive constants independent of S. Now, we define ${\bar{a}}_{nk}=b_{kk}+{\bar{b}}_{kk}$ and choose the virtual controller $\alpha _k = -\big ((n-k+1)a_{nk}+{\bar{a}}_{nk}\big )^{1/p}z_k=: -g_kz_k$ with $a_{nk}$ being a positive constant independent of S. We get

$$\begin{aligned} \dot{V}_{k}\le & {} -S(n-k+1)\sum _{i=1}^{k}a_{ni}z_i^{p+1} +S|z_{k}|\big |x_{k+1}^{p}-\alpha _{k}^{p}\big |, \end{aligned}$$

This completes Step k.

When $k=n$, select $V_n=V_{n-1}+\frac{1}{2}z_n^2$ and choose $\alpha _n$$= -g_nz_n=-(\beta _1x_1+\beta _2x_2+\cdots +\beta _nx_n)$ with $\beta _{1},\beta _{2},\ldots ,$$\beta _{n}$ being positive constants independent of S. Similarly, we obtain $\dot{V}_{n} \le -S\sum _{i=1}^{n}a_{ni}z_i^{p+1} +S|z_{n}|\big |u_1^{p}(v)-\alpha _{n}^{p}\big |,$ which shows that Proposition 1 holds. $\square$

Proof of Proposition 2

Considering (7), it can be deduced that

$$\begin{aligned}&\left| \frac{\partial V_n}{\partial x_i}\right| \le C_i(|z_i|+|z_{i+1}|+\cdots +|z_{n}|),~i=1,2,\ldots ,n, \end{aligned}$$

where $C_i>0$ is a constant. Noting that $q_k\le p$, it is easy to obtain that

$$\begin{aligned}&\sum _{k=1}^m|\zeta _j|^{q_k} \le m(1+|\zeta _j|^{p}) \le m|\zeta _j|^{p}+m,\\&\quad \sum _{k=1}^m|\zeta _j(t-\tau _j(t))|^{q_k} \le m\big (1+|\zeta _j(t-\tau _j(t))|^{p}\big )\\&\quad \quad \le m|\zeta _j(t-\tau _j(t))|^{p}+m. \end{aligned}$$

From which and Assumption 1, we have

$$\begin{aligned}&|\psi _i(\cdot )| \le Cm\bigg (\sum _{j=1}^i|\zeta _j|^{p}+\sum _{j=1}^i|\zeta _j(t-\tau _j(t))|^{p}\bigg )\nonumber \\&\quad \quad \quad \quad \quad +\,2mi+d. \end{aligned}$$

Using the transformation (3), it follows that $|\psi _i|\le Cm\big (\sum _{j=1}^iS^{l_jp}|x_j|^p +\sum _{j=1}^iS^{l_jp}|x_j(t-\tau _j)|^p\big )+2mi+d$. Then, in view of $-l_i+l_jp=-\big (\frac{1}{p}+\frac{1}{p^2}+\cdots +\frac{1}{p^{i-1}}\big ) +\big (\frac{1}{p}+\frac{1}{p^2}+\cdots +\frac{1}{p^{j-2}}\big )+1\le 1-\frac{1}{p^{i-1}} ,1\le j\le i\le n$, (7) and $S>1$, it follows that

$$\begin{aligned} |f_1|\le & {} C\big (|x_1+y_r|^{p}+|x_{1}(t-\tau _1)+y_r(t-\tau _1)|^{p}\big ) + d-\dot{y}_r\nonumber \\\le & {} {\bar{C}}\big (|z_1|^{p}+|z_{1}(t-\tau _1)|^{p}\big ) +\frac{{\bar{d}}}{S^{l_1}}, \end{aligned}$$

(A.2)

$$\begin{aligned} |f_l|\le & {} CS^{1-1/p^{l-1}}\sum _{j=1}^{l}\big (|x_j|^{p}+|x_{j}(t-\tau _j)|^{p}\big ) +\frac{{\bar{d}}}{S^{l_l}} \nonumber \\\le & {} {\bar{C}}S^{1-1/p^{l-1}}\sum _{j=1}^{l}\big (|z_j|^{p}+|z_{j}(t-\tau _j)|^{p}\big )\nonumber \\&+\,{\bar{C}}S^{1-1/p^{l-1}}\sum _{j=1}^{l-1}|z_{j}(t-\tau _{j+1})|^{p} +\frac{{\bar{d}}}{S^{l_l}}, \end{aligned}$$

(A.3)

where $l=2,3,\ldots ,n$ and ${\bar{d}},{\bar{C}}$ are appropriate constants. For $i=1, 2,\ldots ,n$, there are constants $\lambda , B_{ij}(\lambda ), {\bar{B}}_{ij}$ and ${\hat{B}}_{ij}$ such that

$$\begin{aligned}&C_i\big (|z_i|+|z_{i+1}|+\cdots +|z_{n}|\big )|f_i| \nonumber \\&\quad \le C_i\big (|z_i|+|z_{i+1}|+\cdots +|z_{n}|\big )\bigg ({\bar{C}}S^{1-1/p^n} \sum _{j=1}^{i}\big (|z_j|^{p}\nonumber \\&\quad +\,|z_{j}(t-\tau _j)|^{p}\big ) +{\bar{C}}S^{1-1/p^n}\sum _{j=1}^{i-1}|z_{j}(t-\tau _{j+1})|^{p} +\frac{{\bar{d}}}{S^{l_i}}\bigg )\nonumber \\&\quad \le e^{-m\gamma _1}(1-\gamma _2)S^{1-1/p^{n}}\sum _{j=1}^{i-1}{\hat{B}}_{ij}z^{p+1}_j(t-\tau _{j+1})\nonumber \\&\quad +\,e^{-m\gamma _1}(1-\gamma _2)S^{1-1/p^{n}}\sum _{j=1}^{i}{\bar{B}}_{ij}z^{p+1}_j(t-\tau _j)\nonumber \\&\quad +\,S^{1-1/p^{n}}\sum _{j=1}^{n}B_{ij}(\lambda )z^{p+1}_j +\frac{1}{\lambda }\left( \frac{{\bar{d}}}{S^{l_i}}\right) ^{\frac{p+1}{p}}. \end{aligned}$$

(A.4)

Defining $b_j=\sum _{i=1}^{n}B_{ij}$, ${\bar{b}}_j=\sum _{i=1}^{n}{\bar{B}}_{ij}$, ${\hat{b}}_j=\sum _{i=1}^{n}{\hat{B}}_{ij}$, and using (A.4) and inequality $\left| \frac{\partial V_n}{\partial x}F\right| \le \sum _{i=1}^{n}\big |\frac{\partial V_n}{\partial x_i}f_i\big |,$ Proposition 2 holds.

Proof of Proposition 3

In view of Lemma 1, (A.2) and (A.3), one has

$$\begin{aligned} {\tilde{\varepsilon }}_{i}f_i\le & {} {\bar{C}}S^{1-1/p^{i-1}}|{\tilde{\varepsilon }}_{i}| \bigg (\sum _{j=1}^{i}|z_j|^p+\sum _{j=1}^{i}|z_j(t-\tau _j)|^p \nonumber \\&+\,\sum _{j=1}^{i-1}|z_j(t-\tau _{j+1})|^p \bigg ) +|{\tilde{\varepsilon }}_{i}|\frac{{\bar{d}}}{S^{l_i}}\nonumber \\\le & {} S^{1-1/p^{n}}\bigg (\sum _{j=1}^{i}C_{ij}z^{p+1}_j +\sum _{j=1}^{i}{\bar{C}}_{ij}z^{p+1}_j(t-\tau _j) \nonumber \\&+\,e^{-m\gamma _1}(1-\gamma _2){\sum _{j=1}^{i-1}}{\hat{C}}_{ij}z^{p+1}_j(t-\tau _{j+1})\bigg ) \nonumber \\&+\,S^{1-1/p^{i}}Q_i(\lambda ){\tilde{\varepsilon }}^{p+1}_{i} +\frac{1}{2\lambda }\left( \frac{{\bar{d}}}{S^{l_i}}\right) ^{\frac{p+1}{p}}, \end{aligned}$$

(A.5)

where $C_{ij},{\bar{C}}_{ij},{\hat{C}}_{ij},Q_i(\lambda )$ are positive constants independent of S. Similarly, it follows that

$$\begin{aligned}&M_i{\tilde{\varepsilon }}_{i}f_{i-1} \nonumber \\&\le {\bar{C}}M_iS^{1-1/p^{i-2}}|{\tilde{\varepsilon }}_{i}| \bigg (\sum _{j=1}^{i-2}|z_j(t-\tau _{j+1})|^p\nonumber \\&\quad +\sum _{j=1}^{i-1}|z_j|^p+\sum _{j=1}^{i-1}|z_j(t-\tau _j)|^p \bigg )+ M_i|{\tilde{\varepsilon }}_{i}|\frac{{\bar{d}}}{S^{l_{i-1}}} \nonumber \\&\le S^{1-1/p^{n}}\sum _{j=1}^{i-1}D_{ij}z^{p+1}_j +S^{1-1/p^{n}}\sum _{j=1}^{i-1}{\bar{D}}_{ij}z^{p+1}_j(t-\tau _j) \nonumber \\&\quad +\,e^{-m\gamma _1}(1-\gamma _2)S^{1-1/p^{n}}\sum _{j=1}^{i-2}{\hat{D}}_{ij}z^{p+1}_j(t-\tau _{j+1}) \nonumber \\&\quad +\,S^{1-1/p^{i}}{\bar{Q}}_i(M_i,\lambda ){\tilde{\varepsilon }}^{p+1}_{i} +\frac{1}{2\lambda }\left( \frac{{\bar{d}}}{S^{l_{i}}}\right) ^{\frac{p+1}{p}}, \end{aligned}$$

(A.6)

where $D_{ij},{\bar{D}}_{ij},{\hat{D}}_{ij},{\bar{Q}}_i(M_i,\lambda )$ are positive constants independent of S. Utilizing (A.5) and (A.6), and defining $c_j=\sum _{i=1}^{n}(C_{ij}+D_{ij})$, ${\bar{c}}_j=\sum _{i=1}^{n}({\bar{C}}_{ij}+{\bar{D}}_{ij})$, ${\hat{c}}_j=\sum _{i=1}^{n}({\hat{C}}_{ij}+{\hat{D}}_{ij})$, $A_i(M_i,\lambda )=Q_i(\lambda )+{\bar{Q}}_i(M_i,\lambda )$, we complete the proof. $\square$

Proof of Proposition 4

By Lemma 1, there exist constants ${\bar{H}}_{i-1}, {\bar{H}}_i$ and $B_0$ such that

$$\begin{aligned}&4^{p-1}pg^{p-1}_i|{\tilde{\varepsilon }}_{i-1}||\varepsilon _{i}|\big (z^{p-1}_i +z^{p-1}_{i-1}+\varepsilon ^{p-1}_{i}\big ) \nonumber \\&\quad \le {\bar{H}}_iz^{p+1}_i+{\bar{H}}_{i-1}z^{p+1}_{i-1}+B_0{\tilde{\varepsilon }}^{p+1}_{i-1}+\varepsilon ^{p+1}_{i}. \end{aligned}$$

(A.7)

Using (5), it yields that

$$\begin{aligned} \varepsilon ^{p+1}_{i}= & {} \big ({\tilde{\varepsilon }}_{i}+M_i{\tilde{\varepsilon }}_{i-1}+ M_iM_{i-1}{\tilde{\varepsilon }}_{i-2}+\cdots \nonumber \\&+\, M_iM_{i-1}\cdots M_3{\tilde{\varepsilon }}_{2}\big )^{p+1}\nonumber \\\le & {} \sum _{j=2}^{i-1} {\bar{B}}_j(M_{j+1},M_{j},\ldots ,M_i){\tilde{\varepsilon }}^{p+1}_{j} \nonumber \\&\quad +\bar{B}_i{\tilde{\varepsilon }}^{p+1}_{i}, \end{aligned}$$

(A.8)

where ${\bar{B}}_j,j=2,3,\ldots ,i-1$ are positive constants depended on the constants $M_{j+1},M_{j+2},\ldots ,M_i$, and ${\bar{B}}_i>0$ is a constant. By (5), Lemmas 1 and 3, we have $x^p_i-{\hat{x}}^p_i \le p|x_i-\hat{x}_i|\big (|x_i-{\hat{x}}_i|^{p-1}+|{\hat{x}}_i|^{p-1}\big ) \le p|\varepsilon _{i}|\big (\varepsilon _{i}^{p-1} +2^{p-2}\big (x^{p-1}_i+\varepsilon ^{p-1}_{i}\big )\big ) \le 2^{p-1}p|\varepsilon _{i}| \big (\big (z_i-g_{i-1}z_{i-1}\big )^{p-1}+\varepsilon ^{p-1}_{i}\big ) \le 4^{p-1}pg^{p-1}_{i-1}|\varepsilon _{i}| \big (z^{p-1}_i+z^{p-1}_{i-1}+\varepsilon ^{p-1}_{i}\big ).$ From which and (A.7), (A.8), we obtain

$$\begin{aligned} {\tilde{\varepsilon }}_{i-1}(x^p_i-{\hat{x}}^p_i)\le & {} \sum _{j=2}^{i-2} {\bar{B}}_j(M_{j+1},M_{j+2},\ldots ,M_i){\tilde{\varepsilon }}^{p+1}_{j} \nonumber \\&+\,\big (B_0+{\bar{B}}_{i-1}(M_i)\big ){\tilde{\varepsilon }}^{p+1}_{i-1} +\,{\bar{B}}_i{\tilde{\varepsilon }}^{p+1}_{i}\nonumber \\&+\,{\bar{H}}_iz^{p+1}_i+{\bar{H}}_{i-1}z^{p+1}_{i-1}. \end{aligned}$$

(A.9)

It follows from (A.9) that

$$\begin{aligned}&S\sum _{i=3}^{n}{\tilde{\varepsilon }}_{i-1}(x^p_i-{\hat{x}}^p_i)\nonumber \\&\quad \le \, S\sum _{i=2}^{n-1} {\hat{B}}_i(M_{i+1},M_{i+2},\ldots ,M_n){\tilde{\varepsilon }}^{p+1}_{i} \nonumber \\&\quad \quad +\,S\sum _{i=2}^{n}{\hat{H}}_iz^{p+1}_i+S{\hat{B}}_n{\tilde{\varepsilon }}^{p+1}_{n} , \end{aligned}$$

(A.10)

where ${\hat{B}}_i,i=2,3,\ldots ,n-1$ are positive constants depended on the constants $M_{i+1},M_{i+2},\ldots ,M_n$, and ${\hat{H}}_1,$${\hat{H}}_2,\ldots ,{\hat{H}}_n,{\hat{B}}_n$ are constants. Similarly, it can be deduced that

$$\begin{aligned}&-S\sum _{i=2}^{n}M_i{\tilde{\varepsilon }}_{i}\Big (x_i^p-\big (\hat{x}_i+{\tilde{\varepsilon }}_{i}\big )^p\Big ) \le S\sum _{i=1}^{n}{\tilde{H}}_iz^{p+1}_i\nonumber \\&\quad +\,S\sum _{i=2}^{n-1} {\tilde{B}}_i(M_{i+1},M_{i+2},\ldots ,M_n){\tilde{\varepsilon }}^{p+1}_{i}\nonumber \\&\quad +\,S{\tilde{B}}_n{\tilde{\varepsilon }}^{p+1}_{n}, \end{aligned}$$

(A.11)

where ${\tilde{H}}_1,{\tilde{H}}_2,\ldots ,{\tilde{H}}_n,{\tilde{B}}_n$ are constants and ${\tilde{B}}_i(M_{i+1},$$M_{i+2},\ldots ,M_n),i=2,3,\ldots ,n-1$ are positive constants independent of S. In view of Lemma 4, we have $-{\tilde{\varepsilon }}_{i}((\hat{x}_i+{\tilde{\varepsilon }}_{i})^p-{\hat{x}}^p_i) \le -\frac{1}{2^{p-1}}{\tilde{\varepsilon }}^{p+1}_{i},$ which indicates

$$\begin{aligned}&\quad -S\sum _{i=2}^{n}M_i{\tilde{\varepsilon }}_{i}\Big (\big (\hat{x}_i+{\tilde{\varepsilon }}_{i}\big )^p-{\hat{x}}^p_i\Big )\nonumber \\&\quad \le -S\sum _{i=2}^{n}\frac{M_i}{2^{p-1}}{\tilde{\varepsilon }}^{p+1}_{i}. \end{aligned}$$

(A.12)

Using (A.11) and (A.12), it is clear that

$$\begin{aligned}&-S\sum _{i=2}^{n}M_i{\tilde{\varepsilon }}_{i} \big (x_i^p-{\hat{x}}^p_i\big ) \nonumber \\&\quad = -S\sum _{i=2}^{n}M_i{\tilde{\varepsilon }}_{i}\Big (x_i^p-\big (\hat{x}_i+{\tilde{\varepsilon }}_{i}\big )^p\Big )\nonumber \\&\quad \quad -\,S\sum _{i=2}^{n}M_i{\tilde{\varepsilon }}_{i}\Big (\big (\hat{x}_i+{\tilde{\varepsilon }}_{i}\big )^p-{\hat{x}}^p_i\Big )\nonumber \\&\quad \le -S\sum _{i=2}^{n}\frac{M_i}{2^{p-1}}{\tilde{\varepsilon }}^{p+1}_{i} +S\sum _{i=1}^{n}{\tilde{H}}_iz^{p+1}_i +S{\tilde{B}}_n{\tilde{\varepsilon }}^{p+1}_{n}\nonumber \\&\quad \quad +\,S\sum _{i=2}^{n-1} {\tilde{B}}_i(M_{i+1},M_{i+2},\ldots ,M_n){\tilde{\varepsilon }}^{p+1}_{i}. \end{aligned}$$

(A.13)

Considering (A.10) and (A.13), and defining $h_1={\tilde{H}}_1$, $h_i={\hat{H}}_i+{\tilde{H}}_i$ and $B_i={\hat{B}}_i+{\tilde{B}}_i, i=2,3,\ldots ,n$, Proposition 4 is proved. $\square$

Proof of the inequality (12)

It is easy to obtain from (2), (3) and (9) that

$$\begin{aligned} u^p_1(v)-{\bar{u}}^p= \left\{ \begin{array}{ll} m^p_{r}\Big (\frac{{\bar{u}}}{m_r}+\frac{{\bar{b}}_r-b_{r}(t)}{S^{l_{n+1}}}\Big )^p-{\bar{u}}^p, ~~~{\bar{u}}>0,\\ 0,\qquad \qquad \qquad \qquad \qquad \quad ~~ {\bar{u}}=0,\\ m^p_{l}\Big (\frac{{\bar{u}}}{m_l}+ \frac{-{\bar{b}}_l+b_{l}(t)}{S^{l_{n+1}}}\Big )^p-{\bar{u}}^p, \,~~{\bar{u}}<0. \end{array} \right. \end{aligned}$$

When ${\bar{u}}>0$, we obtain

$$\begin{aligned}&u^p_1(v)-{\bar{u}}^p \nonumber \\&\quad = m^p_{r}\left( \frac{{\bar{u}}}{m_r}+\frac{{\bar{b}}_r-b_{r}(t)}{S^{l_{n+1}}}\right) ^p -{\bar{u}}^p \nonumber \\&\quad =m^p_{r}\sum _{k=0}^{p}\frac{p!}{k!(n-k)!}\left( \frac{{\bar{u}}}{m_r}\right) ^k \left( \frac{{\bar{b}}_r-b_{r}(t)}{S^{l_{n+1}}}\right) ^{p-k} -{\bar{u}}^p \nonumber \\&\quad =\sum _{k=0}^{p-1}\frac{p!}{k!(n-k)!} {\bar{u}}^k\bigg (\frac{m_r\big ({\bar{b}}_r-b_{r}(t)\big )}{S^{l_{n+1}}}\bigg )^{p-k}. \end{aligned}$$

(A.14)

Noticing $x_1=z_1,{\hat{x}}_i=x_i-\varepsilon _i=z_i-g_{i-1}z_{i-1}-\varepsilon _i, i=2,3,\ldots ,n$, there exists a constant $\beta _0$ satisfying

$$\begin{aligned} {\bar{u}}= & {} -\big (\beta _1x_1+\beta _2{\hat{x}}_2+\cdots +\beta _n{\hat{x}}_n\big )\nonumber \\\le & {} \beta _0\bigg (\sum _{i=1}^{n}|z_i|+\sum _{j=2}^{n}|\varepsilon _j|\bigg ). \end{aligned}$$

(A.15)

It follows from (A.14) and (A.15) that

$$\begin{aligned}&|z_{n}||u_1^p(v)-{\bar{u}}^p| \le \sum _{i=1}^{n}\varGamma _iz^{p+1}_i +\sum _{j=2}^{n} \gamma _{1j}\varepsilon ^{p+1}_{j}\nonumber \\&\quad +\,\frac{\delta }{S^{l_{n+1}(p+1)}}, \end{aligned}$$

(A.16)

where $\varGamma _i,i=1,2,\ldots ,n$ and $\gamma _{1j},j=2,3,\ldots ,n$ are constants, and $\delta =\max \{{\bar{b}}_r^{p+1},{\bar{b}}_l^{p+1}\}$ is a constant. With the similar method, one can show that (A.16) still holds for ${\bar{u}}\le 0$. Next, applying Lemma 3, $\varepsilon _i=x_i-{\hat{x}}_i,i=2,3,\ldots ,n$, (A.15), and considering the definition of ${\bar{u}}$ and $\alpha _n$, there exists a constant $\mu >0$ such that

$$\begin{aligned} |{\bar{u}}^p-\alpha ^p_n|= & {} \big |-\big (\beta _1x_1+\beta _2{\hat{x}}_2+\cdots +\beta _n{\hat{x}}_n\big )^p \\&+\big (\beta _1x_1+\beta _2x_2+\cdots +\beta _n x_n\big )^p\big | \\\le & {} \Big ( \big |\beta _2\varepsilon _2 +\beta _3\varepsilon _3+\cdots +\beta _n\varepsilon _n\big |^{p-1} +|{\bar{u}}|^{p-1} \Big ) \\&+p\big |\beta _2\varepsilon _2+\beta _3\varepsilon _3+\cdots +\beta _n\varepsilon _n\big | \\\le & {} \mu \sum _{j=2}^{n}|\varepsilon _j| \bigg (\sum _{i=1}^{n}|z_i|^{p-1}+\sum _{i=2}^{n}|\varepsilon _i|^{p-1}\bigg ). \end{aligned}$$

Therefore, in view of Lemma 1, there exist constants $\varLambda _i$ and $\gamma _{2j}$ satisfying

$$\begin{aligned} |z_{n}||{\bar{u}}^p-\alpha ^p_n|\le & {} \mu |z_{n}|\sum _{j=2}^{n}|\varepsilon _j| \bigg (\sum _{i=1}^{n}|z_i|^{p-1}+\sum _{i=2}^{n}|\varepsilon _i|^{p-1}\bigg ) \nonumber \\\le & {} \sum _{i=1}^{n}\varLambda _iz^{p+1}_i +\sum _{j=2}^{n} \gamma _{2j}\varepsilon ^{p+1}_{j}. \end{aligned}$$

(A.17)

Using (5), we have $\varepsilon _i = {\tilde{\varepsilon }}_i+\sum _{j=2}^{i-1}M_iM_{i-1}\cdots M_{j+1}{\tilde{\varepsilon }}_j,$ which further indicates that

$$\begin{aligned}&\sum _{j=2}^{n} (\gamma _{1j}+\gamma _{2j})\varepsilon ^{p+1}_{j}\nonumber \\&\quad \le \sum _{j=2}^{n-1} D_j(M_{j+1},M_{j+2},\ldots ,M_n){\tilde{\varepsilon }}^{p+1}_{j} \nonumber \\&\quad +D_n{\tilde{\varepsilon }}^{p+1}_{n}, \end{aligned}$$

(A.18)

where $D_j$ is a constant depended on $M_{j+1},M_{j+2},\ldots ,$$M_n$. Substituting (A.16), (A.17) and (A.18) into $|z_n|\cdot |u_1^p(v)-\alpha _n^p| \le |z_n|\cdot |u_1^p(v)-{\bar{u}}^p|+|z_n|\cdot |{\bar{u}}^p-\alpha _n^p|$ and defining ${\hat{d}}_i=\varGamma _i+\varLambda _i$, we show that the conclusion holds. $\square$