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Adaptive tracking control for a class of non-affine switched stochastic nonlinear systems with unmodeled dynamics

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Abstract

This paper proposes an adaptive tracking control scheme for a class of non-affine switched stochastic nonlinear systems with unmeasured states and stochastic inverse dynamics. K-filters are used to estimate unmeasured states, and a changing supply function is introduced to deal with stochastic inverse dynamics. By using neural networks, dynamic surface technique and common Lyapunov function method, a controller and adaptive laws are designed for the considered system. The proposed control scheme guarantees that all signals of the closed-loop system are bounded in probability. Finally, a simulation example is presented to show the effectiveness of the proposed method.

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Acknowledgments

This work was supported by the National Science Foundation of China under Grant No. 61273120.

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Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Jun Mao, Shipei Huang & Zhengrong Xiang

Authors
  1. Jun Mao
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  2. Shipei Huang
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  3. Zhengrong Xiang
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Corresponding author

Correspondence to Zhengrong Xiang.

Appendix

Appendix

Proof

Consider the following two cases separately:

Case 1 If \(2\bar{r}(\left| y \right| ) \le {\alpha _0}(\left\| z \right\| )\), then we have

$$\begin{aligned} \varrho ({V_z}(z))(\bar{r}(\left| y \right| ) - {\alpha _0}(\left\| z \right\| ))= & {} \varrho ({V_z}(z))(\bar{r}(\left| y \right| ) - \frac{1}{2}{\alpha _0}(\left\| z \right\| )) - \frac{1}{2}\varrho ({V_z}(z)){\alpha _0}(\left\| z \right\| )\\&\le \varrho (\bar\eta (\left| y \right| ))\bar{r}(\left| y \right| ) - \frac{1}{2}\varrho ({V_z}(z)){\alpha _0}(\left\| z \right\| ). \end{aligned}$$

Case 2 If \(2\bar{r}(\left| y \right| ) > {\alpha _0}(\left\| z \right\| )\), then we get \({V_z}(z) \le \bar{\alpha } (\alpha _0^{ - 1}(2\bar{r}(\left| y \right| ))) = \bar\eta (\left| y \right| )\), and

$$\begin{aligned} \varrho ({V_z}(z))(\bar{r}(\left| y \right| )- {\alpha _0}(\left\| z \right\| ))&\le \varrho ({V_z}(z))\bar{r}(\left| y \right| ) - \frac{1}{2}\varrho ({V_z}(z)){\alpha _0}(\left\| z \right\| )\\&\le \varrho (\bar\eta (\left| y \right| ))\bar{r}(\left| y \right| ) - \frac{1}{2}\varrho ({V_z}(z)){\alpha _0}(\left\| z \right\| ). \end{aligned}$$

The proof is completed. \(\square\)

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Mao, J., Huang, S. & Xiang, Z. Adaptive tracking control for a class of non-affine switched stochastic nonlinear systems with unmodeled dynamics. Neural Comput & Applic 28 (Suppl 1), 1069–1081 (2017). https://doi.org/10.1007/s00521-016-2381-x

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