常微分–波方程系统滑模与自抗扰控制及分析
Sliding Mode and Active Disturbance Rejection Control of Ordinary Differential-Wave Equations with Perturbations, and Their Analysis
摘要: 本文研究一类具有应变耦合特征的常微分(ODE)–波方程的级联系统。针对波方程控制端受限于纽曼(Neumann)边界且存在未知有界扰动，而非受控端受限于狄利克雷(Dirichlet)边界的复杂工况，分别构建了滑模与自抗扰控制器。首先，利用可逆变换简化原级联耦合系统。对于滑模控制，通过构造滑模面，证明了闭环系统的指数稳定性。对于自抗扰控制，构建高增益观测器，对扰动进行实时补偿，并证明了闭环系统的渐近稳定性。数值仿真结果验证了两种控制策略的有效性。
Abstract: In this paper, we consider an ODE-wave equation cascaded system characterized by strain coupling. To address the complex boundary configuration, where the controlled end of the wave equation is subject to a Neumann condition under unknown bounded disturbances, while the uncontrolled end follows a Dirichlet condition-both sliding mode control (SMC) and active disturbance rejection control (ADRC) are formulated. Initially, an invertible transformation is applied to simplify the original coupled dynamics. Under the SMC framework, a sliding surface is designed to establish the exponential stability of the closed-loop system. Alternatively, the ADRC approach utilizes a high-gain observer for real-time compensation of the disturbance, subsequently ensuring the asymptotic stability of the overall system. Finally, numerical simulations are provided to demonstrate the effectiveness of the two proposed control strategies.
文章引用：王子雄, 杨坤一. 常微分–波方程系统滑模与自抗扰控制及分析[J]. 应用数学进展, 2026, 15(10): 66-82. https://doi.org/10.12677/aam.2026.1510397

参考文献

[1] 田强, 刘铖, 李培, 胡海岩. 多柔体系统动力学研究进展与挑战[J]. 动力学与控制学报, 2017, 15(5): 385-405.
[2] 徐建亮, 毛海军, 徐友洪, 毕少平. 基于自适应动态规划的柔性机械臂跨尺度运动控制研究[J]. 动力系统与控制, 2025, 14(2): 116-129.
[3] 洪嘉振, 尤超蓝. 刚柔耦合系统动力学研究进展[J]. 动力学与控制学报, 2004, 2(2): 1-6.
[4] Han, Z.J. and Zuazua, E. (2016) Decay Rates for 1-d Heat-Wave Planar Networks. Networks and Heterogeneous Media, 11, 655-692.
https://doi.org/10.3934/nhm.2016013
[5] Valein, J. and Zuazua, E. (2009) Stabilization of the Wave Equation on 1-D Networks. SIAM Journal on Control and Optimization, 48, 2771-2797.
https://doi.org/10.1137/080733590
[6] Than, A.A. and Wang, J. (2021) Output Feedback Stabilization of Cascaded ODE-Wave Equations with Time Delay in Observation. Asian Journal of Control, 23, 449-462.
https://doi.org/10.1002/asjc.2217
[7] Guo, B.Z. and Jin, F.F. (2013) The Active Disturbance Rejection and Sliding Mode Control Approach to the Stabilization of the Euler-Bernoulli Beam Equation with Boundary Input Disturbance. Automatica, 49, 2911-2918.
https://doi.org/10.1016/j.automatica.2013.06.018
[8] Guo, B.Z. and Jin, F.F. (2012) Sliding Mode and Active Disturbance Rejection Control to Stabilization of One-Dimensional Anti-Stable Wave Equations Subject to Disturbance in Boundary Input. IEEE Transactions on Automatic Control, 58, 1269-1274.
https://doi.org/10.1109/tac.2012.2218669
[9] Guo, B.Z. and Liu, J.J. (2014) Sliding Mode Control and Active Disturbance Rejection Control to the Stabilization of One-Dimensional Schrödinger Equation Subject to Boundary Control Matched Disturbance. International Journal of Robust and Nonlinear Control, 24, 2194-2212.
https://doi.org/10.1002/rnc.2977
[10] Liu, J.J. and Wang, J.M. (2015) Active Disturbance Rejection Control and Sliding Mode Control of One-Dimensional Unstable Heat Equation with Boundary Uncertainties. IMA Journal of Mathematical Control and Information, 32, 97-117.
https://doi.org/10.1093/imamci/dnt034
[11] Kang, W. and Fridman, E. (2016) Sliding Mode Control of Schrödinger Equation-ODE in the Presence of Unmatched Disturbances. Systems & Control Letters, 98, 65-73.
https://doi.org/10.1016/j.sysconle.2016.10.009
[12] Liu, J.J. (2018) Sliding Mode Control to Stabilization of an ODE-Schrödinger Cascade Systems Subject to Boundary Control Matched Disturbance. Journal of Systems Science and Complexity, 31, 1146-1163.
https://doi.org/10.1007/s11424-018-6286-x
[13] Liu, J.J. and Wang, J.M. (2017) Boundary Stabilization of a Cascade of ODE‐Wave Systems Subject to Boundary Control Matched Disturbance. International Journal of Robust and Nonlinear Control, 27, 252-280.
https://doi.org/10.1002/rnc.3572
[14] Wang, J.M., Liu, J.J., Ren, B., et al. (2015) Sliding Mode Control to Stabilization of Cascaded Heat PDE-ODE Systems Subject to Boundary Control Matched Disturbance. Automatica, 52, 23-34.
https://doi.org/10.1016/j.automatica.2014.10.117
[15] Zhou, H.C. and Guo, B.Z. (2019) Stabilization of ODE with Hyperbolic Equation Actuator Subject to Boundary Control Matched Disturbance. International Journal of Control, 92, 12-26.
https://doi.org/10.1080/00207179.2016.1235286
[16] Guo, B.Z., Zhou, H.C., Al-Fhaid, A.S., et al. (2014) Stabilization of Euler-Bernoulli Beam Equation with Boundary Moment Control and Disturbance by Active Disturbance Rejection Control and Sliding Mode Control Approaches. Journal of Dynamical and Control Systems, 20, 539-558.
https://doi.org/10.1007/s10883-014-9241-8
[17] Guo, B.Z., Liu, J.J., Al-Fhaid, A.S., et al. (2015) The Active Disturbance Rejection Control Approach to Stabilisation of Coupled Heat and ODE System Subject to Boundary Control Matched Disturbance. International Journal of Control, 88, 1554-1564.
https://doi.org/10.1080/00207179.2015.1010179
[18] Krstic, M. and Smyshlyaev, A. (2008) Boundary Control of PDEs: A Course on Backstepping Designs. Society for Industrial and Applied Mathematics.
https://doi.org/10.1137/1.9780898718607
[19] Susto, G.A. and Krstic, M. (2010) Control of PDE-ODE Cascades with Neumann Interconnections. Journal of the Franklin Institute, 347, 284-314.
https://doi.org/10.1016/j.jfranklin.2009.09.005
[20] Adams, R.A. and Fournier, J.J.F. (2003) Sobolev Spaces. Elsevier.
[21] Müller, V. (2007) Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Birkhäuser Basel.
[22] Yang, K., Ren, X. and Zhang, J. (2016) Output Feedback Stabilization of an Unstable Wave Equation with Observations Subject to Time Delay. Journal of Systems Science and Complexity, 29, 99-118.
https://doi.org/10.1007/s11424-016-5169-2
[23] Pazy, A. (2012) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Science & Business Media.
[24] Tucsnak, M. and Weiss, G. (2009) Observation and Control for Operator Semigroups. Birkhäuser Basel.
[25] Weiss, G. (1989) Admissibility of Unbounded Control Operators. SIAM Journal on Control and Optimization, 27, 527-545.
https://doi.org/10.1137/0327028
[26] Guo, B.Z. and Zhao, Z. (2011) On the Convergence of an Extended State Observer for Nonlinear Systems with Uncertainty. Systems & Control Letters, 60, 420-430.
https://doi.org/10.1016/j.sysconle.2011.03.008
[27] Komornik, V. and Loreti, P. (2005) Fourier Series in Control Theory. Springer.

为你推荐