Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Feedback control for a class of second order hyperbolic distributed parameter systems

一类二阶双曲型分布参数系统的反馈控制

Abstract

This paper deals with the problem of state feedback control for a class of the distributed parameter systems with the disturbance term. And the considered distributed parameter systems are composed of the second order hyperbolic partial differential equations. Two different classes of restrictions on the disturbance term are given, one is that the disturbance term satisfies the linear growth constraint condition to the state variables of the system, and the other is that the disturbance term obeys the bound constraint under the significance of L 2. Based on a variable structure method, the state feedback controllers are obtained by means of constructing appropriate Lyapunov functional. The closed-loop systems are globally asymptotically stable on W 1,2(0, 1) × L 2(0, 1) space under the effect of the state feedback control laws. Simulation results illustrate the effectiveness of the proposed method.

摘要

创新点

研究一类带扰动项的分布参数系统的状态反馈控制问题, 该类分布参数系统由二阶双曲型偏微分方程构成。 通过构建一种新的 Lyapunov 泛函, 基于变结构方法, 设计得到一种新的变结构状态反馈控制律。 当该类控制律作用于系统时, 若扰动项满足线性增长性条件, 则闭环系统的强解于W1,2(0,1)*L2(0,1)空间内全局渐近稳定; 若扰动项满足有界性约束条件, 则闭环系统的广义解于W1,2(0,1)*L2(0,1)空间内全局渐近稳定。 当扰动项为零时, 已有的变结构控制方法, 获得的结论为: 广义解意义下, 全局渐近稳定。 由于本文采用了新的 Lyapunov 泛函及反馈控制器, 当扰动项为零时, 得到的结论为: 强解意义下, 全局渐近稳定, 由此改进了已有的相应结论。

This is a preview of subscription content, log in to check access.

References

  1. 1

    Smyshlyaev A, Krstic M. Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary. Syst Control Lett, 2009, 58: 617–623

  2. 2

    Smyshlyaev A, Krstic M. Backstepping observers for a class of parabolic PDEs. Syst Control Lett, 2005, 54: 613–625

  3. 3

    Krstic M, Smyshlyaev A. Adaptive control of PDEs. Ann Rev Control, 2008, 32: 149–160

  4. 4

    Cheng M B, Radisavljevic V, Su W C. Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica, 2011, 47: 381–387

  5. 5

    Orlov Y. Discontinuous unit feedback control of uncertain infinite-dimensional systems. IEEE Trans Autom Control, 2000, 45: 834–843

  6. 6

    Fu Q. Iterative learning control for second order nonlinear hyperbolic distributed parameter systems (in Chinese). J Syst Sci Math Sci, 2014, 34: 284–293

  7. 7

    Orlov Y, Pisano A, Usai E. Continuous state-feedback tracking of the uncertain heat diffusion process. Syst Control Lett, 2010, 59: 754–759

  8. 8

    Orlov Y, Pisano A, Usai E. Exponential stabilization of the uncertain wave equation via distributed dynamic input extension. IEEE Trans Autom Control, 2011, 56: 212–217

  9. 9

    Pisano A, Orlov Y, Usai E. Tracking control of the uncertain heat and wave equation via power-fractional and slidingmode techniques. SIAM J Control Optim, 2011, 49: 363–382

  10. 10

    Orlov Y. Discontinuous Systems Lyapunov Analysis and Robust Synthesis Under Uncertainty Conditions. Berlin: Springer-Verlag, 2009

Download references

Author information

Correspondence to Qin Fu.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fu, Q., Gu, W., Gu, P. et al. Feedback control for a class of second order hyperbolic distributed parameter systems. Sci. China Inf. Sci. 59, 92206 (2016). https://doi.org/10.1007/s11432-016-5554-4

Download citation

Keywords

  • state feedback
  • globally asymptotically stable
  • second order hyperbolic distributed parameter systems
  • variable structure method
  • Lyapunov functional

关键词

  • 状态反馈
  • 全局渐近稳定
  • 二阶双曲型分布参数系统
  • 变结构方法
  • Lyapunov 泛函