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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

Abstract


Funded by

科技部重点研发计划项目(2019YFB1312001)

国家自然科学基金(62033005,62022030,41772377)

国网黑龙江省电力有限公司科技项目(522417190057)

黑龙江省科学基金(F2018012)

先进焊接与连接国家重点实验室开放课题


Supplement

Appendix

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).


References

[1] Pittet C, Tarbouriech S, Burgat C. Stability regions for linear systems with saturating controls via circle and popov criteria. In: Proceedings of the 36th IEEE Conference on Decision & Control, 1997. 4518--4523. Google Scholar

[2] Hindi H, Stephen B. Analysis of linear systems with saturation using convex optimization. In: Proceedings of the 37th IEEE Conference on Decision & Control, 1998. 903--908. Google Scholar

[3] Hu T, Lin Z, Chen B M. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 2002, 38: 351-359 CrossRef Google Scholar

[4] Hu T, Lin Z, Chen B M. Analysis and design for discrete-time linear systems subject to actuator saturation. Syst Control Lett, 2002, 45: 97-112 CrossRef Google Scholar

[5] Valmorbida G, Tarbouriech S, Garcia G. Design of Polynomial Control Laws for Polynomial Systems Subject to Actuator Saturation. IEEE Trans Automat Contr, 2013, 58: 1758-1770 CrossRef Google Scholar

[6] Chisci L, Rossiter J A, Zappa G. Systems with persistent disturbances: predictive control with restricted constraints. Automatica, 2001, 37: 1019-1028 CrossRef Google Scholar

[7] Dinuzzo F, Ferrara A. Higher Order Sliding Mode Controllers With Optimal Reaching. IEEE Trans Automat Contr, 2009, 54: 2126-2136 CrossRef Google Scholar

[8] Edwards C, Herrmann G, Hredzak B, et al. A discrete-time sliding mode scheme with constrained input. In: Proceedings of the 46th IEEE Conference on Decision & Control, 2007. 3203--3208. Google Scholar

[9] Chen M, Ge S S, How B V E. Robust Adaptive Position Mooring Control for Marine Vessels. IEEE Trans Contr Syst Technol, 2013, 21: 395-409 CrossRef Google Scholar

[10] Sun L, Wang Y, Feng G. Control Design for a Class of Affine Nonlinear Descriptor Systems With Actuator Saturation. IEEE Trans Automat Contr, 2015, 60: 2195-2200 CrossRef Google Scholar

[11] Battilotti S. Robust observer design under measurement noise with gain adaptation and saturated estimates. Automatica, 2017, 81: 75-86 CrossRef Google Scholar

[12] Utkin V. Variable structure systems with sliding modes. IEEE Trans Automat Contr, 1977, 22: 212-222 CrossRef Google Scholar

[13] Utkin V. Sliding Modes in Control and Optimization. Berlin: Springer-Verlag, 1992. Google Scholar

[14] Fulwani D, Bandyopadhyay B. Design of sliding mode controller with actuator saturation. In: Advances in Sliding Mode Control. Berlin: Springer-Verlag, 2013. Google Scholar

[15] Castillo I, Steinberger M, Fridman L. Saturated super-twisting algorithm: Lyapunov based approach. In: Proceedings of the 14th International Workshop on Variable Structure Systems (VSS), 2016. 269--273. Google Scholar

[16] Behera A K, Chalanga A, Bandyopadhyay B. A new geometric proof of super-twisting control with actuator saturation. Automatica, 2018, 87: 437-441 CrossRef Google Scholar

[17] Ferrara A, Rubagotti M. A Sub-Optimal Second Order Sliding Mode Controller for Systems With Saturating Actuators. IEEE Trans Automat Contr, 2009, 54: 1082-1087 CrossRef Google Scholar

[18] Tanizawa H, Ohta Y. Sliding mode control under state and control constraints. In: Proceedings of the 16th IEEE International Conference on Control Aplications, Amsterdam, 2007, 1173--1178. Google Scholar

[19] Incremona G P, Rubagotti M, Ferrara A. Sliding Mode Control of Constrained Nonlinear Systems. IEEE Trans Automat Contr, 2017, 62: 2965-2972 CrossRef Google Scholar

[20] Davila J, Fridman L, Levant A. Second-order sliding-mode observer for mechanical systems. IEEE Trans Automat Contr, 2005, 50: 1785-1789 CrossRef Google Scholar

[21] Kalsi K, Lian J, Hui S. Sliding-mode observers for systems with unknown inputs: A high-gain approach. Automatica, 2010, 46: 347-353 CrossRef Google Scholar

[22] Wang X, Tan C P, Zhou D. A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions. Automatica, 2017, 79: 290-295 CrossRef Google Scholar

[23] Soh Y C, Rath J J, Defoort M. Higher-order sliding mode observer for estimation of tyre friction in ground vehicles. IET Control Theor Appl, 2014, 40: 399-408 CrossRef Google Scholar

[24] Chalanga A, Kamal S, Fridman L M. Implementation of Super-Twisting Control: Super-Twisting and Higher Order Sliding-Mode Observer-Based Approaches. IEEE Trans Ind Electron, 2016, 63: 3677-3685 CrossRef Google Scholar

[25] Levant A. Higher-order sliding modes, differentiation and output-feedback control. Int J Control, 2003, 76: 924-941 CrossRef Google Scholar

[26] Pisano A, Usai E. Output-feedback control of an underwater vehicle prototype by higher-order sliding modes. Automatica, 2004, 40: 1525-1531 CrossRef Google Scholar

[27] Fridman L, Shtessel Y, Edwards C. Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems. Int J Robust NOnlinear Control, 2008, 18: 399-412 CrossRef Google Scholar

[28] Belkhatir Z, Laleg-Kirati T M. High-order sliding mode observer for fractional commensurate linear systems with unknown input. Automatica, 2017, 82: 209-217 CrossRef Google Scholar

[29] Yang J, Su J, Li S. High-Order Mismatched Disturbance Compensation for Motion Control Systems Via a Continuous Dynamic Sliding-Mode Approach. IEEE Trans Ind Inf, 2014, 10: 604-614 CrossRef Google Scholar

[30] Efimov D, Edwards C, Zolghadri A. Enhancement of adaptive observer robustness applying sliding mode techniques. Automatica, 2016, 72: 53-56 CrossRef Google Scholar

[31] Isidori A. Nonlinear Control Systems. Berlin: Springer Science & Business Media, 2013. Google Scholar

[32] Liu X, Han Y. Finite time control for MIMO nonlinear system based on higher-order sliding mode. ISA Trans, 2014, 53: 1838-1846 CrossRef Google Scholar

[33] Yu S, Yu X, Shirinzadeh B. Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica, 2005, 41: 1957-1964 CrossRef Google Scholar

[34] Incremona G P, Cucuzzella M, Ferrara A. Second order sliding mode control for nonlinear affine systems with quantized uncertainty. Automatica, 2017, 86: 46-52 CrossRef Google Scholar

[35] Moreno J A, Osorio M. Strict Lyapunov Functions for the Super-Twisting Algorithm. IEEE Trans Automat Contr, 2012, 57: 1035-1040 CrossRef Google Scholar

[36] Filippov A F. Differential Equations With Discontinuous Righthand Sides: Control Systems. Berlin: Springer Science & Business Media, 2013. Google Scholar

[37] Stankovic S S, Stanojevic M J, Siljak D D. Decentralized overlapping control of a platoon of vehicles. IEEE Trans Contr Syst Technol, 2000, 8: 816-832 CrossRef Google Scholar

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