SCIENTIA SINICA Informationis, Volume 51 , Issue 3 : 430(2021) https://doi.org/10.1360/SSI-2020-0162

Constrained sliding mode control of MIMO nonlinear systems

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  • ReceivedJun 5, 2020
  • AcceptedJul 27, 2020
  • PublishedFeb 23, 2021


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命题 rho~reduction 的证明

参考定理 3.1 的证明, 约束 (23) 表示状态空间 $\xi_{i}(t)\in\mathbb{R}^{3}$ 的一个子集, 其可以重新定义为 $\Omega_{\xi_{i}}=\{\xi_{i}(t)\in\mathbb{R}^{3}|~|A\xi_{i}(t)|\le~c_{r_{i}}\chi_{i}+\beta_{i}\}$, 其中 $A_i~\triangleq[ \begin{array}{ccc} -\eta_{1}~&~~-\frac{1}{c_{r_{i}}}~&~~1 \end{array} ]$. 而后可以通过引理 2.8 得到 Lyapunov 泛函 (12) 的最大正不变集 $\Omega_{p_{i}}=\left\{\left.\xi_{i}(t)\in\Omega_{\xi_{i}}(t)\right|~V_{i}(\xi_{i}(t))\le~V_{p_{i}}\right\}\subset\Omega_{\xi_{i}}$, 其中 $V_{p_{i}}\triangleq\left(c_{r_{i}}\chi_{i}+\beta_{i}\right)^{2}\delta_{i}$, $\delta_{i}\triangleq\frac{|P^{i}|}{|AP^{i}A^{\rm{T}}|}$.

同时, 将关于 $s_{i}(t)\in\mathbb{R}$ 的不变集表示为 $\Omega_{s_{i}}=\left\{\left.s_{i}(t)\in\Omega_{\xi_{i}}(t)\right|~V_{i}(s_{i}(t))\le~V_{p_{i}}\right\}$, 其也是 $\Omega_{\xi_{i}}$ 的子集, 即 $\Omega_{s_{i}}\subset\Omega_{\xi_{i}}$. 因此得到 $ V_{i}(s_{i}(t))\le~p_{22}^{i}|s_{i}(t)|^{2}~\le~V_{p_{i}}$, 其中 $p_{22}^{i}\triangleq\eta_{3}+\frac{1}{2~c_{r_{i}}^{2}}$ 是矩阵 $P^{i}$ 的第 2 行和第 2 列. 由此可得 \begin{eqnarray*}|s_{i}(t)|=\left|C_{i}\hat{x}^{i}(t)\right|\le \sqrt{\frac{2c_{r_{i}}^{2}V_{p_{i}}}{2\eta_{3}c_{r_{i}}^{2}+1}} =c_{r_{i}}\left(c_{r_{i}}\chi_{i}+\beta_{i}\right)\sqrt{\frac{2\delta_{i}}{2\eta_{3}c_{r_{i}}^{2}+1}}. \end{eqnarray*} 至此我们得到了关于 $s_{i}(t)$ 的水平集. 接下来我们将分析 $|\Pi_{i}(t)|$ 的水平集的边界.

状态变量 $\hat{x}_{j}^{i}(t)$ ($i=1,2,\ldots,m$, $j=1,2,\ldots,r_{i}$) 可以渐近收敛到零, 这在定理 3.1 中得到了证明. 由于 $s_{i}(t)$ 和 $\Pi_{i}(\hat{x},t)$ 都是 $\hat{x}_{j}^{i}(t)$ 的线性正组合, 它们在有限时间内有界并收敛到零. 因此, \begin{eqnarray}|\Pi_{i}(\hat{x},t)|&=&|C_{i}C_{d}\hat{x}^{i}(t)-c_{i1}\hat{x}_{1}^{i}(t)| \le\|C_{d}\||C_{i}\hat{x}^{i}(t)|+|c_{i1}\hat{x}_{1}^{i}(t)| \\ &\le&\|C_{d}\||s_{i}(t)|+c_{i1}\alpha_{1}^{i} \le\kappa_{i}\left(c_{r_{i}}\chi_{i}+\beta_{i}\right)+c_{i1}\alpha_{1}^{i}=\rho_{i}, \tag{36} \end{eqnarray} 其中 $C_{d}~\triangleq~{\rm{diag}}\{1,\frac{c_{i1}}{c_{i2}},\ldots,\frac{c_{i2}}{c_{i3}},\frac{c_{ir_{i}-1}}{c_{r_{i}}}\}\in\mathbb{R}^{r_{i}\times~r_{i}}$, $\kappa_{i}\triangleq~c_{r_{i}}\|C_{d}\|\sqrt{\frac{2\delta_{i}}{2\eta_{3}c_{r_{i}}^{2}+1}}$. 此外, $\beta_{i}$ 可以用 $\rho_{i}$ 代替, 以此进一步逼近 $|\Pi_{i}(\hat{x},t)|$ 的实际值. 可以将式 (36) 重写为 $|\Pi_{i}(\hat{x},t)|\le~\kappa_{i}(c_{r_{i}}\chi_{i}+\rho_{i})+c_{i1}\alpha_{1}^{i}$. 因此, 在条件 (23) 下, 总是存在参数 $\rho_{i}<\beta_{i}$ 满足式 (36).


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