logo

SCIENCE CHINA Information Sciences, Volume 60, Issue 12: 122201(2017) https://doi.org/10.1007/s11432-016-0448-2

Global practical tracking for stochastic time-delay nonlinearsystems with SISS-like inverse dynamics

More info
  • ReceivedSep 15, 2016
  • AcceptedOct 31, 2016
  • PublishedApr 25, 2017

Abstract

This paper investigates the practical tracking problem ofstochastic delayed nonlinear systems. The powers of the nonlinearterms are relaxed to a certain interval rather than a preciselyknown point. Based on the Lyapunov-Krasovskii (L-K) functionalmethod and the modified adding a power integrator technique, a newcontroller is constructed to render the solutions of the consideredsystem to be bounded in probability, and furthermore, thetracking error in sense of the mean square can be made small enoughby adjusting some designed parameters. A simulation example isprovided to demonstrate the validity of the method in this paper.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573227, 61633014, 61673242, 61603231), State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant No. LAPS16011), Research Fund for the Taishan Scholar Project of Shandong Province of China, Postgraduate Innovation Funds of SDUST (No. SDKDYC170229), SDUST Research Fund (Grant No. 2015TDJH105), and Shandong Provincial Natural Science Foundation of China (Grant No. 2016ZRB01076).


Supplement

Appendix

Proof of Proposition pro2

By transformation (a1), it can be deduced that \begin{align}\alpha_i^{p_{i-1}} &= -\beta_{i-1}^{\frac{r_ip_{i-1}}{\mu}}z_{i-1}^{\frac{r_ip_{i-1}}{\mu}} =-\beta_{i-1}^{\frac{r_ip_{i-1}}{\mu}} \bigg(x_{i-1}^{\frac{\mu}{r_{i-1}}}-\alpha_{i-1}^{\frac{\mu}{r_{i-1}}}\bigg)^{\frac{r_ip_{i-1}}{\mu}} \\ &= -\beta_{i-1}^{\frac{r_ip_{i-1}}{\mu}} \bigg(x_{i-1}^{\frac{\mu}{r_{i-1}}}+\beta_{i-2}z_{i-2}\bigg)^{\frac{r_ip_{i-1}}{\mu}} =-\beta_{i-1}^{\frac{r_ip_{i-1}}{\mu}} \bigg(x_{i-1}^{\frac{\mu}{r_{i-1}}} +\beta_{i-2}\bigg(x_{i-2}^{\frac{\mu}{r_{i-2}}}-\alpha_{i-2}^{\frac{\mu}{r_{i-2}}}\bigg) \bigg)^{\frac{r_ip_{i-1}}{\mu}} \\ &= -\beta_{i-1}^{\frac{r_ip_{i-1}}{\mu}} \bigg(x_{i-1}^{\frac{\mu}{r_{i-1}}} +\beta_{i-2}\bigg(x_{i-2}^{\frac{\mu}{r_{i-2}}}+\cdots +\beta_{2}\bigg(x_{2}^{\frac{\mu}{r_{2}}}+\beta_1z_1\bigg)\cdots \bigg)\bigg)^{\frac{r_ip_{i-1}}{\mu}}. \end{align}

Proof of Proposition pro6

By defining $\delta_{i0}=\max\{1,2^{\frac{r_i+\varpi-r_1}{r_1}}\}$, $i=1,\ldots,n$ and using Assumption ass1, Lemma lem2 and (a1), it yields that \begin{equation}|x_1|^{\frac{r_i+\varpi}{r_1}} \leq \delta_{i0} \left(|z_1|^{\frac{r_i+\varpi}{\mu}}+M^{\frac{r_i+\varpi}{r_1}}\right), |x_{k}|^{\frac{r_i+\varpi}{r_{k}}} \leq |z_{k}|^{\frac{r_i+\varpi}{\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{r_i+\varpi}{\mu}}, \tag{47}\end{equation} from which and Assumptions ass1ass2, one gets \begin{equation} |f_i| &\leq& C|\zeta_0|^{r_i+\varpi} +\delta_{i0}C \left( |z_1|^{\frac{r_i+\varpi}{\mu}}+|z_1(t-\eta)|^{\frac{r_i+\varpi}{\mu}} \right) +2\delta_{i0}CM^{\frac{r_i+\varpi}{r_1}} +l_{i1} \cr & & + \sum_{k=2}^{i} C\bigg( |z_k|^{\frac{r_i+\varpi}{\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{r_i+\varpi}{\mu}} +|z_k(t-\eta )|^{\frac{r_i+\varpi}{\mu}} +|\beta_{k-1}z_{k-1}(t-\eta)|^{\frac{r_i+\varpi}{\mu}} \bigg)\cr &\leq& \widehat C_{i1}\bigg( 1+|\zeta_0|^{r_i+\varpi} +\sum_{k=1}^{i} \bigg( |z_k|^{\frac{r_i+\varpi}{\mu}} +|z_k(t-\eta )|^{\frac{r_i+\varpi}{\mu}} \bigg)\bigg), \tag{48}\end{equation} where $\widehat~C_{i1}\triangleq~\max_{2\leq~k\leq i-1}\{C,C(\delta_{i0}+\beta_1^{\frac{r_i+\varpi}{\mu}}), C\beta_k^{\frac{r_i+\varpi}{\mu}}, 2\delta_{i0}CM^{\frac{r_i+\varpi}{r_1}}+l_{i1}\}$. For simplicity, let $\sum_{k=2}^1x_k=0$ for all $x_k$.

Similarly, define $\delta_{i2}\triangleq\max\{1,2^{\frac{2r_i+\varpi-2r_1}{2r_1}}\}$. By $0<\frac{\mu-2r_k}{\mu}<1$, one has \begin{equation}|x_1|^{\frac{2r_i+\varpi}{2r_1}} = |z_1^{\frac{r_1}{\mu}}+y_{\rm r}|^{\frac{2r_i+\varpi}{2r_1}} \leq \delta_{i2} \left(|z_1|^{\frac{2r_i+\varpi}{2\mu}}+M^{\frac{2r_i+\varpi}{2r_1}}\right), |x_k|^{\frac{2r_i+\varpi}{2r_k}} \leq |z_k|^{\frac{2r_i+\varpi}{2\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{2r_i+\varpi}{2\mu}}. \tag{49}\end{equation} Then one can find two positive constants $\widehat{C}_{i2}$ and $\widehat{C}_{i3}$, such that \begin{equation} |g_i| &\leq& C|\zeta_0|^{\frac{2r_i+\varpi}{2}} +\delta_{i2}C \left( |z_1|^{\frac{2r_i+\varpi}{2\mu}}+|z_1(t-\eta)|^{\frac{2r_i+\varpi}{2\mu}} \right) +2\delta_{i2}C M^{\frac{2r_i+\varpi}{2r_1}} +l_{i2} \cr & &+ \sum_{k=2}^{i} C\left( |z_k|^{\frac{2r_i+\varpi}{2\mu}} +|\beta_{k-1}z_{k-1}|^{\frac{2r_i+\varpi}{2\mu}} +|z_k(t-\eta )|^{\frac{2r_i+\varpi}{2\mu}} +|\beta_{k-1}z_{k-1}(t-\eta )|^{\frac{2r_i+\varpi}{2\mu}} \right)\cr &\leq& \widehat{C}_{i2} \bigg(1+|\zeta_0|^{\frac{2r_i+\varpi}{2}} +\sum_{k=1}^{i} \bigg( |z_k|^{\frac{2r_i+\varpi}{2\mu}} +|z_k(t-\eta )|^{\frac{2r_i+\varpi}{2\mu}} \bigg)\bigg), \tag{50}\end{equation} \begin{equation} |g_i|^2 \leq \widehat{C}_{i3} \bigg(1+|\zeta_0|^{2r_i+\varpi} +\sum_{k=1}^{i} \bigg( |z_k|^{\frac{2r_i+\varpi}{\mu}} +|z_k(t-\eta )|^{\frac{2r_i+\varpi}{\mu}} \bigg)\bigg). \tag{51}\end{equation} From Assumption ass1, Lemma lem1 and (48), there exist a small enough parameter $C_{11}>0$ and a constant $r_{11}>0$ such that \begin{equation} z_1^{\frac{4\lambda-\varpi-r_1}{\mu}}\big(f_1-\dot y_{\rm r}\big) &\leq& |z_1|^{\frac{4\lambda-\varpi-r_1}{\mu}} \bigg( \widehat C_{11}\bigg( 1+|\zeta_0 |^{r_1+\varpi} + |z_1|^{\frac{r_1+\varpi}{\mu}} +|z_1(t-\eta )|^{\frac{r_1+\varpi}{\mu}}\bigg) +M \bigg) \cr &\leq& \frac{ c_0l_3}{2(n+1)}|\zeta_0|^{4\lambda} +\frac{1-\bar \tau}{3}{\rm e}^{-\tau}z_1^{\frac{4\lambda}{\mu}}(t-\eta) +r_{11}z_1^{\frac{4\lambda}{\mu}}+C_{11}. \tag{52}\end{equation} Similar to the proof of (52), one obtains \begin{equation}& &c_0l_4\left(x_1^{\frac{4\lambda}{r_1}} +x_1^{\frac{4\lambda}{r_1}}(t-\eta)\right) \leq c_0l_4\bigg(\bigg(z_1^{\frac{r_1}{\mu}}+y_{\rm r}\bigg)^{\frac{4\lambda}{r_1}} +\bigg(z_1^{\frac{r_1}{\mu}}(t-\eta)+y_{\rm r}(t-\eta)\bigg)^{\frac{4\lambda}{r_1}}\bigg)\cr & & \leq c_0l_42^{\frac{4\lambda-r_1}{r_1}}\bigg( z_1^{\frac{4\lambda}{\mu}} +z_1^{\frac{4\lambda}{\mu}}(t-\eta) \bigg) +c_0l_4(2M)^{\frac{4\lambda}{r_1}} \leq \frac{1-\bar \tau}{3}{\rm e}^{-\tau}z_1^{\frac{4\lambda}{\mu}}(t-\eta) +r_{12}z_1^{\frac{4\lambda}{\mu}}+C_{12}, \tag{53}\end{equation} where $C_{12}$ is a small enough design parameter and $r_{12}>0$. By Lemma lem1, (a1) and (51), one can find a constant $r_{13}>0$ and a small enough design parameter $C_{13}>0$, such that \begin{align}&\displaystyle\frac{4\lambda-\varpi -\mu}{2r_1} \mbox{Tr} \left\{g_1^\top z_1^{\frac{4\lambda-\varpi-2r_1}{\mu}}g_1\right\} \leq \frac{4\lambda-\varpi -\mu}{2r_1}|z_1|^{\frac{4\lambda-\varpi-2r_1}{\mu}}|g_1|^2 \\ & \leq\displaystyle \frac{4\lambda-\varpi -\mu}{2r_1}|z_1|^{\frac{4\lambda-\varpi-2r_1}{\mu}} \widehat{C}_{13} \bigg(1+|\zeta_0|^{2r_1+\varpi} +|z_1|^{\frac{2r_1+\varpi}{\mu}} +|z_1(t-\eta )|^{\frac{2r_1+\varpi}{\mu}} \bigg) \\ & \leq\displaystyle \frac{ c_0l_3}{2(n+1)}|\zeta_0|^{4\lambda} +\frac{1-\bar \tau}{3}{\rm e}^{-\tau}z_1^{\frac{4\lambda}{\mu}}(t-\eta) +r_{13}z_1^{\frac{4\lambda}{\mu}}+C_{13}. \end{align}

Proof of Proposition pro3

For simplicity, we define $A_i=4\lambda-\varpi-r_i,B_i=4\lambda-\varpi-\mu-r_i$ and $D_i=4\lambda-\varpi-2\mu-r_i$. Then the function $U_i$ can be rewritten as $U_i=\int_{\alpha_{i}}^{x_i}(~s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}}) ^{\frac{A_i}{\mu}}\mbox~ds,$ $i=2,\ldots,n$. By Definition def1 and (sect. 3), one has \begin{equation}\mathcal{L}U_i &=& \frac{\partial U_i}{\partial z_1}\frac{\mu}{r_1}z_1^{\frac{\mu-r_1}{\mu}} (a_1x_2^{p_1}+f_1-\dot y_{\rm r}) +\sum_{j=2}^{i-1}\frac{\partial U_i}{\partial x_j}(a_jx_{j+1}^{p_j}+f_j) +\frac{\partial U_i}{\partial x_i}(a_ix_{i+1}^{p_i}+f_i)\cr & &+\frac{1}{2} \frac{\partial^2U_i}{\partial z_1^2} \bigg|\frac{\mu}{r_1}z_1^{\frac{\mu-r_1}{\mu}}g_1\bigg|^2 +\frac{1}{2}\sum_{j=2}^{i-1}\frac{\partial^2U_i}{\partial x_j^2}|g_j|^2 +\frac{1}{2}\frac{\partial^2U_i}{\partial x_i^2}|g_i|^2 +\sum_{j=2}^i\frac{\partial^2U_i}{\partial z_1\partial x_j} \bigg|\frac{\mu}{r_1}z_1^{\frac{\mu-r_1}{\mu}}g_1\bigg||g_j| \cr & & +\frac{1}{2}\sum_{k,j=2,k\neq j}^{i-1}\frac{\partial^2U_i}{\partial x_k\partial x_j}|g_k||g_j| +\sum_{j=2}^{i-1}\frac{\partial^2U_i}{\partial x_i\partial x_j}|g_i||g_j|. \tag{54}\end{equation} Next, we estimate the terms in (54).

Term 1. By Lemmas lem5, lem8 and (a1), \begin{equation} \bigg| \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{B_i}{\mu}}\mbox ds \bigg| \leq |z_i|^{\frac{B_i}{\mu}}|x_i-\alpha_i| \leq 2^{\frac{\mu-r_i}{\mu}}|z_i|^{\frac{4\lambda-\varpi-\mu}{\mu}}. \tag{55}\end{equation} By the definitions of $U_i,A_i,B_i$ and Proposition pro2, the partial derivative $\frac{\partial~U_i}{\partial~z_1}$ is given by \begin{equation} \frac{\partial U_i}{\partial z_1} = -\frac{A_i}{\mu} \frac{\partial \alpha_i^{\frac{\mu}{r_i}}}{\partial z_1} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{B_i}{\mu}}\mbox ds = \frac{A_i}{\mu} \beta_{i-1}\cdots\beta_1 \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{B_i}{\mu}}\mbox ds. \tag{56}\end{equation} Taking the absolute value on both sides of (56), by (55), we have \begin{equation} \bigg|\frac{\partial U_i}{\partial z_1}\bigg| \leq \frac{A_i}{\mu}2^{\frac{\mu-r_i}{\mu}} \beta_{i-1}\cdots\beta_1 |z_i|^{\frac{4\lambda-\varpi-\mu}{\mu}}. \tag{57}\end{equation} From $0<a_1\leq~\bar~\epsilon$, $|x_2|^{p_1}~\leq |z_2|^{\frac{r_1+\varpi}{\mu}}+|\beta_1z_1|^{\frac{r_1+\varpi}{\mu}}$ and (48), it follows that \begin{equation} |a_1x_2^{p_1}+f_1-\dot y_{\rm r}| &\leq& \bar \epsilon \left( |z_2|^{\frac{r_1+\varpi}{\mu}} +|\beta_1z_1|^{\frac{r_1+\varpi}{\mu}} \right) +\widehat C_{11}\bigg( 1+|\zeta_0|^{r_1+\varpi} + |z_1|^{\frac{r_1+\varpi}{\mu}} +|z_1(t-\eta)|^{\frac{r_1+\varpi}{\mu}} \bigg) +M\cr &\leq& \widehat{C}_{14}\bigg(1+|\zeta_0|^{r_1+\varpi} +|z_1|^{\frac{r_1+\varpi}{\mu}}+|z_1(t-\eta)|^{\frac{r_1+\varpi}{\mu}} +|z_2|^{\frac{r_1+\varpi}{\mu}} \bigg) \tag{58}\end{equation} with $\widehat{C}_{14}=\max\{\bar~\epsilon,\bar \epsilon\beta_1^{\frac{r_1+\varpi}{\mu}}+\widehat~C_{11},~\widehat C_{11}+M\}$. By using Lemma lem1, (57) and (58), the first term of (54) is estimated \begin{equation} \bigg|\frac{\partial U_i}{\partial z_1}\frac{\mu}{r_1}z_1^{\frac{\mu-r_1}{\mu}} (a_1x_2^{p_1}+f_1-\dot y_{\rm r})\bigg| &\leq& \frac{A_i}{r_1}2^{\frac{\mu-r_i}{\mu}} \beta_{i-1}\cdots\beta_1 |z_i|^{\frac{4\lambda-\varpi-\mu}{\mu}} |z_1|^{\frac{\mu-r_1}{\mu}} |a_1x_2^{p_1}+f_1-\dot y_{\rm r}| \cr &\leq& \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1}{9}\left(z_1^{\frac{4\lambda}{\mu}}+z_2^{\frac{4\lambda}{\mu}}\right) +\frac{1-\bar \tau}{9}{\rm e}^{-\tau}z_1^{\frac{4\lambda}{\mu}}(t-\eta) +r_{i1}z_i^{\frac{4\lambda}{\mu}}+C_{i1}. \tag{59}\end{equation}

Term 2. Similar to the proofs of (56) and (57), it can be verified that \begin{equation} \frac{\partial U_i}{\partial x_j} = -\frac{A_i}{\mu} \frac{\partial \alpha_i^{\frac{\mu}{r_i}}}{\partial x_j} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^\frac{B_i}{\mu}\mbox ds = \frac{A_i}{r_j} \beta_{i-1}\cdots\beta_jx_j^{\frac{\mu-r_j}{r_j}} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{B_i}{\mu}}\mbox ds, \tag{60}\end{equation} \begin{equation}\bigg|\frac{\partial U_i}{\partial x_j}\bigg| \leq \frac{A_i}{r_j} 2^{\frac{\mu-r_i}{\mu}}\beta_{i-1}\cdots\beta_j \left( |z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}} \right) |z_i|^{\frac{4\lambda-\varpi-\mu}{\mu}}, j=2,\ldots,i-1,\end{equation} from which and using Lemma lem1, (48) and $0<a_j\leq~\bar~\epsilon$, the second term of (54) is estimated as \begin{equation}& &\left|\sum_{j=2}^{i-1}\frac{\partial U_i}{\partial x_j}\left(a_jx_{j+1}^{p_j}+f_j\right)\right|\cr & & \leq \sum_{j=2}^{i-1} \frac{A_i}{r_j} 2^{\frac{\mu-r_i}{\mu}}\beta_{i-1}\cdots\beta_j \left( |z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}} \right) |z_i|^{\frac{4\lambda-\varpi-\mu}{\mu}} \bigg( \bar \epsilon \bigg( |z_{j+1}|^{\frac{r_j+\varpi}{\mu}}+|\beta_jz_j|^{\frac{r_j+\varpi}{\mu}} \bigg)+|f_j|\bigg) \cr & & \leq \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1}{9}\sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +r_{i2}z_i^{\frac{4\lambda}{\mu}}+C_{i2}. \tag{61}\end{equation}

Term 3. By Lemma lem1, (48) and $ \frac{\partial~U_i}{\partial~x_i} =z_i^{\frac{A_i}{\mu}}, $ there holds \begin{equation}& &\frac{\partial U_i}{\partial x_i}\left(a_ix_{i+1}^{p_i}+f_i\right) \leq a_iz_i^{\frac{A_i}{\mu}}x_{i+1}^{p_i} + |z_i|^{\frac{A_i}{\mu}} |f_i| \cr & & \leq \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1}{9}\sum_{j=1}^{i-1}z_i^{\frac{4\lambda}{\mu}} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +a_iz_i^{\frac{A_i}{\mu}}x_{i+1}^{p_i} +r_{i3}z_i^{\frac{4\lambda}{\mu}}+C_{i3}. \tag{62}\end{equation}

Term 4. By (56) and Proposition pro2, it follows that \begin{equation} \frac{\partial^2 U_i}{\partial z_1^2} = \frac{A_iB_i}{\mu^2} \bigg(\frac{\partial \alpha_i^{\frac{\mu}{r_i}}}{\partial z_1}\bigg)^2 \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{D_i}{\mu}}\mbox ds = \frac{A_iB_i}{\mu^2} (\beta_{i-1}\cdots\beta_1)^2 \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{D_i}{\mu}}\mbox ds. \tag{63}\end{equation} Furthermore, using the inequality \begin{equation} \bigg| \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{D_i}{\mu}}\mbox ds \bigg| \leq 2^{\frac{\mu-r_i}{\mu}}|z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}}, \tag{64}\end{equation} we deduce that \begin{equation} \bigg|\frac{\partial^2 U_i}{\partial z_1^2}\bigg| \leq \frac{A_iB_i}{\mu^2}2^{\frac{\mu-r_i}{\mu}} (\beta_{i-1}\cdots\beta_1)^2 |z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}}. \tag{65}\end{equation} Finally, using Lemma lem1, (51) and (65), \begin{equation}\frac{1}{2} \bigg|\frac{\partial^2U_i}{\partial z_1^2}\bigg| \bigg|\frac{\mu}{r_1}z_1^{\frac{\mu-r_1}{\mu}}g_1\bigg|^2 &\leq& \frac{A_iB_i}{2 r_1^2}2^{\frac{\mu-r_i}{\mu}} (\beta_{i-1}\cdots\beta_1)^2 |z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}} |z_1|^{\frac{2\mu-2r_1}{\mu}} |g_1|^2 \cr &\leq& \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1}{9}z_1^{\frac{4\lambda}{\mu}} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} z_1^{\frac{4\lambda}{\mu}}(t-\eta) +r_{i4}z_i^{\frac{4\lambda}{\mu}}+C_{i4}. \tag{66}\end{equation}

Term 5. Taking the partial derivative of the equation (60), we have \begin{equation}\frac{\partial^2 U_i}{\partial x_j^2} &=& \frac{A_i(\mu-r_j)}{r_j^2} \beta_{i-1}\cdots\beta_j x_j^{\frac{\mu-2r_j}{r_j}} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{B_i}{\mu}}\mbox ds +\frac{A_iB_i}{r_j^2} (\beta_{i-1}\cdots\beta_j)^2 x_j^{\frac{2\mu-2r_j}{r_j}} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{D_i}{\mu}}\mbox ds. \tag{67}\end{equation} In addition, considering $|x_k|^{\frac{2\mu-2r_k}{r_k}} \leq 2(|z_k|^{\frac{2\mu-2r_k}{\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{2\mu-2r_k}{\mu}})$, $|x_k|^{\frac{\mu-2r_k}{r_k}} \leq |z_k|^{\frac{\mu-2r_k}{\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{\mu-2r_k}{\mu}},$ and using (55) and (64), we obtain \begin{equation} \bigg|\frac{\partial^2 U_i}{\partial x_j^2}\bigg| &\leq& \frac{A_i(\mu-r_j)}{r_j^2}2^{\frac{\mu-r_i}{\mu}} \beta_{i-1}\cdots\beta_j \left(|z_j|^{\frac{\mu-2r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-2r_j}{\mu}}\right) |z_i|^{\frac{4\lambda-\varpi-\mu}{\mu}} \cr & & + \frac{A_iB_i}{r_j^2}2^{\frac{2\mu-r_i}{\mu}} (\beta_{i-1}\cdots\beta_j)^2 \left(|z_j|^{\frac{2\mu-2r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{2\mu-2r_j}{\mu}}\right) |z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}}. \tag{68}\end{equation} Then, by Lemma lem1, (51) and (68), the fifth term of (54) is estimated as follows: \begin{equation}\frac{1}{2}\sum_{j=2}^{i-1}\bigg|\frac{\partial^2U_i}{\partial x_j^2}\bigg||g_j|^2 \leq \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +\frac{1}{9}\sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}} +r_{i5}z_i^{\frac{4\lambda}{\mu}}+C_{i5}. \tag{69}\end{equation}

Term 6. By utilizing $~|\frac{\partial^2~U_i}{\partial x_i^2}|~= |\frac{A_i}{r_i}x_i^{\frac{\mu-r_i}{r_i}}z_i^{\frac{B_i}{\mu}}| \leq \frac{A_i}{r_i}(|z_i|^{\frac{\mu-r_i}{\mu}}+|\beta_{i-1}z_{i-1}|^{\frac{\mu-r_i}{\mu}}) |z_i|^{\frac{B_i}{\mu}},~$ Lemma lem1 and (51), it yields \begin{equation}\frac{1}{2}\bigg|\frac{\partial^2U_i}{\partial x_i^2}\bigg||g_i|^2 &\leq& \frac{A_i}{2r_i}\left(|z_i|^{\frac{\mu-r_i}{\mu}}+|\beta_{i-1}z_{i-1}|^{\frac{\mu-r_i}{\mu}}\right) |z_i|^{\frac{B_i}{\mu}}|g_i|^2\cr &\leq& \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +\frac{1}{9}\sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}} +r_{i6}z_i^{\frac{4\lambda}{\mu}}+C_{i6}. \tag{70}\end{equation}

Term 7. Similar to the proof of Term 5, we can deduce \begin{equation}\frac{\partial^2 U_i}{\partial z_1\partial x_j} = \frac{A_iB_i}{r_j\mu} (\beta_{i-1}\cdots\beta_j)^2\beta_{j-1}\cdots\beta_1 x_j^{\frac{\mu-r_j}{r_j}} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{D_i}{\mu}}\mbox ds, \tag{71}\end{equation} \begin{equation}\bigg|\frac{\partial^2 U_i}{\partial z_1\partial x_j}\bigg| \leq \frac{A_iB_i}{r_j\mu}2^{\frac{\mu-r_i}{\mu}} (\beta_{i-1}\cdots\beta_j)^2\beta_{j-1}\cdots\beta_1 \left(|z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}}\right) |z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}}. \tag{72}\end{equation} Utilizing (50) and Lemma lem1, one has \begin{equation} & &\sum_{j=2}^i\bigg|\frac{\partial^2U_i}{\partial z_1\partial x_j}\bigg| \bigg|\frac{\mu}{r_1}z_1^{\frac{\mu-r_1}{\mu}}g_1\bigg||g_j|\cr & & \leq \sum_{j=2}^i \frac{A_iB_i}{r_1 r_j}2^{\frac{\mu-r_i}{\mu}} (\beta_{i-1}\cdots\beta_j)^2\beta_{j-1}\cdots\beta_1 \left(|z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}}\right) |z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}} |z_1|^{\frac{\mu-r_1}{\mu}}|g_1||g_j| \cr & & \leq \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +\frac{1}{9}\sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}} +r_{i7}z_i^{\frac{4\lambda}{\mu}}+C_{i7}. \tag{73}\end{equation}

Term 8. For $k\neq~j,2\leq~k,j\leq~i-1,$ we have \begin{equation}\frac{\partial^2 U_i}{\partial x_k\partial x_j} = \frac{A_iB_i}{r_kr_j} \beta_{i-1}\cdots\beta_k\cdot\beta_{i-1}\cdots\beta_j x_k^{\frac{\mu-r_k}{r_k}}x_j^{\frac{\mu-r_j}{r_j}} \int_{\alpha_i}^{x_i} \bigg( s^{\frac{\mu}{r_i}}-\alpha_i^{\frac{\mu}{r_i}} \bigg)^{\frac{D_i}{\mu}}\mbox ds. \tag{74}\end{equation} By $0<\frac{\mu-r_k}{\mu}<1$, $|x_k|^{\frac{\mu-r_k}{r_k}} \leq |z_k|^{\frac{\mu-r_k}{\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{\mu-r_k}{\mu}}$ and (64), it yields that \begin{equation} \bigg|\frac{\partial^2 U_i}{\partial x_k\partial x_j}\bigg| &\leq& \frac{A_iB_i}{r_kr_j}2^{\frac{\mu-r_i}{\mu}} \beta_{i-1}\cdots\beta_k\cdot\beta_{i-1}\cdots\beta_j \left(|z_k|^{\frac{\mu-r_k}{\mu}}+|\beta_{k-1}z_{k-1}|^{\frac{\mu-r_k}{\mu}}\right)\cr & &\cdot \left(|z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}}\right) |z_i|^{\frac{4\lambda-\varpi-2\mu}{\mu}}. \tag{75}\end{equation} Using Lemma lem1, (50), (75) and taking the similar manipulations of (73), we obtain \begin{equation}\frac{1}{2}\sum_{k,j=2,k\neq j}^{i-1}\bigg|\frac{\partial^2U_i}{\partial x_k\partial x_j}\bigg||g_k||g_j| \leq \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1}{9}\sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +r_{i8}z_i^{\frac{4\lambda}{\mu}}+C_{i8}. \tag{76}\end{equation}

Term 9. For $j<i$, notice \begin{equation} \bigg|\frac{\partial^2 U_i}{\partial x_i\partial x_j}\bigg| = \bigg|\frac{A_i}{r_j} \beta_{i-1}\cdots\beta_j x_j^{\frac{\mu-r_j}{r_j}} z_i^{\frac{B_i}{\mu}}\bigg| \leq \frac{A_i}{r_j} \beta_{i-1}\cdots\beta_j \left(|z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}}\right) |z_i|^{\frac{B_i}{\mu}}, \tag{77}\end{equation} and use Lemma lem1 and (50), the last term of (54) is estimated as \begin{equation} \sum_{j=2}^{i-1}\bigg|\frac{\partial^2U_i}{\partial x_i\partial x_j}\bigg ||g_i||g_j| &\leq& \sum_{j=2}^{i-1} \frac{A_i}{r_j} \beta_{i-1}\cdots\beta_j \left(|z_j|^{\frac{\mu-r_j}{\mu}}+|\beta_{j-1}z_{j-1}|^{\frac{\mu-r_j}{\mu}}\right) |z_i|^{\frac{B_i}{\mu}}|g_i||g_j|\cr &\leq& \frac{ c_0l_3}{9(n+1)}|\zeta_0|^{4\lambda} +\frac{1}{9}\sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}} +\frac{1-\bar \tau}{9}{\rm e}^{-\tau} \sum_{j=1}^{i-1}z_j^{\frac{4\lambda}{\mu}}(t-\eta) +r_{i9}z_i^{\frac{4\lambda}{\mu}}+C_{i9}. \tag{78}\end{equation} Substituting Terms 1–9 into (54) leads to Proposition pro3.


References

[1] Zhang W, Chen B S. On stabilizability and exact observability of stochastic systems with their applications. Automatica, 2004, 40: 87-94 CrossRef Google Scholar

[2] Zhang W, Zhang H, Chen B S. Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion. IEEE Trans Automat Contr, 2008, 53: 1630-1642 CrossRef Google Scholar

[3] Xue-Jun Xie , Na Duan , Xin Yu . State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics. IEEE Trans Automat Contr, 2011, 56: 1921-1926 CrossRef Google Scholar

[4] Zhao X, Deng F. Divided State Feedback Control of Stochastic Systems. IEEE Trans Automat Contr, 2015, 60: 1870-1885 CrossRef Google Scholar

[5] Xin Yu , Xue-Jun Xie . Output feedback regulation of stochastic nonlinear systems with stochastic iISS inverse dynamics. IEEE Trans Automat Contr, 2010, 55: 304-320 CrossRef Google Scholar

[6] Wu M, He Y, She J H. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 2004, 40: 1435-1439 CrossRef Google Scholar

[7] Zhou G P, Huang J H, Tian F X, et al. Sufficient and necessary conditions for global stability of genetic regulator networks with time delays. Sci China Inf Sci, 2016, 59: 012202. Google Scholar

[8] Liu Z W, Zhang H G, Sun Q Y. Static output feedback stabilization for systems with time-varying delay based on a matrix transformation method. Sci China Inf Sci, 2015, 58: 012201. Google Scholar

[9] Yan X H, Liu Y G. Global practical tracking by output-feedback for nonlinear systems with unknown growth rate. Sci China Inf Sci, 2011, 54: 2079-2090 CrossRef Google Scholar

[10] Li W, Wu Z. Output tracking of stochastic high-order nonlinear systems with Markovian switching. IEEE Trans Automat Contr, 2013, 58: 1585-1590 CrossRef Google Scholar

[11] Li W, Zhang J F. Distributed practical output tracking of high-order stochastic multi-agent systems with inherent nonlinear drift and diffusion terms. Automatica, 2014, 50: 3231-3238 CrossRef Google Scholar

[12] Li W, Liu L, Feng G. Distributed containment tracking of multiple stochastic nonlinear systems. Automatica, 2016, 69: 214-221 CrossRef Google Scholar

[13] Wu Z J, Yang J, Shi P. Adaptive tracking for stochastic nonlinear systems with Markovian switching. IEEE Trans Automat Contr, 2010, 55: 2135-2141 CrossRef Google Scholar

[14] Wu Z, Cui M, Shi P. Backstepping Control in Vector Form for Stochastic Hamiltonian Systems. SIAM J Control Optim, 2012, 50: 925-942 CrossRef Google Scholar

[15] Xie X J, Zhao C R, Duan N. Further results on state feedback stabilization of stochastic high-order nonlinear systems. Sci China Inf Sci, 2014, 57: 072202. Google Scholar

[16] Zhang K, Zhao C R, Xie X J. Global output feedback stabilisation of stochastic high-order feedforward nonlinear systems with time-delay. Int J Control, 2015, 88: 2477-2487 CrossRef Google Scholar

[17] Zhang W, Chen B S. State Feedback $H_\infty$ Control for a Class of Nonlinear Stochastic Systems. SIAM J Control Optim, 2006, 44: 1973-1991 CrossRef Google Scholar

[18] Hu Z, Deng F. Robust $H^\infty$ control for networked systems with transmission delays and successive packet dropouts under stochastic sampling. Int J Robust NOnlinear Control, 2017, 27: 84-107 CrossRef Google Scholar

[19] Xue-Jun Xie , Na Duan . Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system. IEEE Trans Automat Contr, 2010, 55: 1197-1202 CrossRef Google Scholar

[20] Jin S L, Liu Y G. Global practical tracking via adaptive output-feedback for uncertain nonlinear systems with generalized control coefficients. Sci China Inf Sci, 2016, 59: 012203. Google Scholar

[21] Xue L, Zhang W, Lin Y. Global output tracking control for high-order stochastic nonlinear systems with SISS inverse dynamics and time-varying delays. J Franklin Institute, 2016, 353: 3249-3270 CrossRef Google Scholar

[22] Sun Z Y, Liu Z G, Zhang X H. New results on global stabilization for time-delay nonlinear systems with low-order and high-order growth conditions. Int J Robust NOnlinear Control, 2015, 25: 878-899 CrossRef Google Scholar

[23] Liu S J, Zhang J F, Jiang Z P. Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica, 2007, 43: 238-251 CrossRef Google Scholar

[24] Liu S J, Ge S S, Zhang J F. Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays. Int J Control, 2008, 81: 1210-1220 CrossRef Google Scholar

[25] Sun Z Y, Xue L R, Zhang K. A new approach to finite-time adaptive stabilization of high-order uncertain nonlinear system. Automatica, 2015, 58: 60-66 CrossRef Google Scholar

[26] Mao X R. Stochastic Differential Equations and Applications. North-Holland: Elsevier, 2007. Google Scholar

[27] Mao X R. A note on the LaSalle-type theorems for stochastic differential delay equations. Appl Math Comput, 2002, 268: 125--142. Google Scholar

[28] Khalil H K, Grizzle J W. Nonlinear Systems. 3rd ed. New Jersey: Prentice Hall, 1996. 144--145. Google Scholar

[29] Wu Y Q, Liu Z G. Output feedback stabilization for time-delay nonholonomic systems with polynomial conditions.. ISA Trans, 2015, 58: 1-10 CrossRef PubMed Google Scholar

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1