Abstract
For a class of more general stochastic high-order feedforward nonlinear systems, this paper deals with the problem of state feedback stabilization. By introducing an appropriate coordinate transformation, the original system is transformed into an equivalent one with tunable gain. After that, by reasonably extending the homogeneous domination approach and skillfully choosing the low gain scale, a state feedback controller is explicitly constructed to render the closed-loop system globally asymptotically stable in probability. Two numerical examples are provided to demonstrate the effectiveness of the proposed design method.
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Acknowledgments
The authors would like to express sincere gratitude to the editor and reviewers for their helpful suggestions in improving the quality of this paper. This work was partially supported by the National Natural Science Foundation of China under Grants 61403041 and 61503036, the Program for Liaoning Excellent Talents in University under Grant LJQ2015001 and the Program for Liaoning Innovative Research Team in University under Grant LT2013023.
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Appendix
Appendix
Proof of Lemma 7
(i) We first prove \(r_{n}+\tau \ge \max _{1\le i\le n}\{2r_{i}\}\). By \(r_{1}=1, r_{i+1}=\frac{r_{i}+\tau }{p}\), one has
when \(i=1\), choosing \(d_1=\frac{(2p^{n-1}-1)(p-1)}{p^n-1}\) and \(\tau \ge d_1\), with (29) and \(p\in R_\mathrm{odd}^{>2}\), one can obtain
For \(i=2,\ldots ,n\), choosing \(d_i=\frac{(\frac{2}{p^{i-1}}-\frac{1}{p^{n-1}})(p-1)}{p-2+\frac{2}{p^{i-1}}-\frac{1}{p^{n-1}}}\) and \(\tau \ge d_i\), by (29) and \(p\in R_\mathrm{odd}^{>2}\), one gets
It is easy to conclude that \(d_1>1, 0<d_i<1, i=2,\ldots ,n\). By (30) and (31), \(r_{n}+\tau \ge \max _{1\le i\le n}\{2r_{i}\}\) holds for any \(p\in R_\mathrm{odd}^{>2}\) and \(\tau \in [d_1,+\infty )\).
(ii) For any \(p\in R_\mathrm{odd}^{>2}\) and \(\tau \in [d_1,+\infty )\), one can choose \(l_1\) to satisfy \(r_{n}+\tau \ge \max _{1\le i\le n}\{\frac{r_{i}+\tau }{l_1}\}\).
From (i) and (ii), it follows that \(r_{n}+\tau \ge \max _{1\le i\le n}\{2r_{i}, \frac{r_{i}+\tau }{l_1}\}\). According to the denseness of real number, there exists \(\mu \in R_\mathrm{odd}^{+}\) such that \(r_{n}+\tau \ge \mu \ge \max _{1\le i\le n}\{2r_{i}, \frac{r_{i}+\tau }{l_1}\}\) holds for any \(p\in R_\mathrm{odd}^{>2}\) and \(\tau \in [d_1,+\infty )\). \(\square \)
Proof of Lemma 8
We first prove that \(V_{i}(\eta _1,\ldots ,\eta _{i})\) is \({\mathcal {C}}^2\). By (10) and (12), it is easy to obtain
where \(j,k=1,\ldots ,i-1\) and the last equality is obtained by using \(\frac{\partial ^2{\eta _{i}^{*}}^{\frac{\mu }{r_{i}}}}{\partial \eta _{j}\partial \eta _{k}}=0, j\ne k\). By \(\mu \ge \max _{1\le i\le n}\{2r_{i}, \frac{r_{i}+\tau }{l_1}\}\), \(\frac{\partial {\eta _{i}^{*}}^{\frac{\mu }{r_{i}}}}{\partial \eta _{j}}=-\beta _{i-1}\cdots \beta _j \frac{\mu }{r_j}\eta _j^{\frac{\mu -r_j}{r_j}}\) and \(\frac{\partial ^2{\eta _{i}^{*}}^{\frac{\mu }{r_{i}}}}{\partial \eta _{j}^2}=-\beta _{i-1}\cdots \beta _j \frac{\mu (\mu -r_j)}{r_j^2}\eta _j^{\frac{\mu -2r_j}{r_j}}\), one has \(\frac{q_{i}}{\mu }-2\ge 0, \frac{\mu -r_j}{r_j}\ge 1\) and \(\frac{\mu -2r_j}{r_j}\ge 0\), from which and (32), we know that \(V_i\) is \({\mathcal {C}}^2\).
Next, we divide into two cases to prove that \(V_{i}(\eta _1,\ldots ,\eta _i)\) is positive definite and proper.
Case I When \(\eta _{i}^{*}\le \eta _{i}\), using Lemma 5, one leads to
Case II When \(\eta _{i}^{*}\ge \eta _{i}\), (33) can be proved similarly.
Therefore, \(V_i=V_{i-1}(\eta _1,\ldots ,\eta _{i-1})+U_{i}(\eta _1,\ldots ,\eta _{i})\ge V_{i-1}(\eta _1,\ldots ,\eta _{i-1})+m_{i}(\eta _{i}-\eta _{i}^{*})^{\frac{4l_1\mu -\tau }{r_{i}}}\), which implies that \(V_i(\bar{\eta }_i)\) is positive definite and proper, where \(m_{i}\) is a positive constant.
At last, we prove inequality (13). From (3), (10)–(12) and (32), it follows that
We concentrate on the last two terms on the right-hand side of (34).
When \(\frac{r_{i}p}{\mu }\le 1\), using (10) and Lemma 5, one obtains
when \(\frac{r_{i}p}{\mu }\ge 1\), by (10) and Lemmas 4, 6, there exist positive constants c, \(b_{i1}\) and \(\bar{b}_{i1}\) such that
Combining (35) and (36), by Lemma 4, one has
where \(\tilde{b}_{i1}=\max \{2^{1-\frac{r_{i}p}{\mu }},\bar{b}_{i1}\}\), \(l_{i,i-1,1}\) and \(\rho _{i1}\) are positive constants.
With the help of (10) and Lemmas 4 and 5, one obtains
where \(\bar{d}, \tilde{d}, l_{ij2} (j=1,\ldots ,i-1)\) and \(\rho _{i2}\) are positive constants.
Choosing
and substituting (37)–(39) into (34), the inequality (13) holds. \(\square \)
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Liu, L., Xing, X. & Gao, M. Global Stabilization for a Class of Stochastic High-Order Feedforward Nonlinear Systems Via Homogeneous Domination Approach. Circuits Syst Signal Process 35, 2723–2740 (2016). https://doi.org/10.1007/s00034-015-0170-x
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DOI: https://doi.org/10.1007/s00034-015-0170-x