Adaptive Boundary Control for a Class of Uncertain Heat Equations
-
摘要: 研究了一类含有不确定控制系数和边界扰动的热方程自适应状态反馈边界控制设计问题. 通过Lyapunov方法, 显式地得到了仅需系统边界状态信息的自适应控制器. 证明了闭环系统状态是L2[0,1]稳定的, 特别是当边界扰动消逝时, 该状态收敛到0. 此外, 通过灵活选取参数调节律的初始条件, 适当放宽了相关文献中相容性条件对系统初始条件的限制. 仿真算例验证了本文方法的有效性.Abstract: This paper investigates the adaptive state-feedback boundary control design for a class of heat equations with uncertain control coefficient and boundary disturbance. By Lyapunov method, the desirable controller is explicitly constructed, which only needs the boundary state of the system. It is shown that the closed-loop system state is L2[0,1] stable, and particularly, the state converges to zero when the boundary disturbance vanishes. Moreover, by skillfully choosing the initial condition of parameter updating law, the restriction on the initial condition of the system is moderately relaxed, which is usually described by the so-called compatible condition in the related literature. A simulation example is presented to illustrate the effectiveness of the proposed method.
-
[1] Triggiani R. Boundary feedback stabilizability of parabolic equations. Applied Mathematics and Optimization, 1980, 6(1): 201-220[2] Kim J U, Renardy Y. Boundary control of the Timoshenko beam. SIAM Journal on Control and Optimization, 1987, 25(6): 1417-1429[3] Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge: Cambridge University Press, 2000[4] Boskovic D M, Krstic M, Liu W J. Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE Transactions on Automatic Control, 2001, 46(12): 2022-2028[5] Liu W J. Boundary feedback stabilization of an unstable heat equation. SIAM Journal on Control and Optimization, 2004, 42(3): 1033-1043[6] Smyshlyaev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Transactions on Automatic Control, 2004, 49(12): 2185-2202[7] Krstic M, Smyshlyaev A. Adaptive boundary control for unstable parabolic PDEs-Part I: Lyapunov design. IEEE Transactions on Automatic Control, 2008, 53(7): 1575-1591[8] Smyshlyaev A, Krstic M. Adaptive boundary control for unstable parabolic PDEs-Part II: estimation-based designs. Automatica, 2007, 43(9): 1543-1556[9] Liu W J, Krstic M. Adaptive control of Burgers' equation with unknown viscosity. International Journal of Adaptive Control and Signal Processing, 2001, 15(7): 745-766[10] Nguyen Q C, Hong K S. Asymptotic stabilization of a nonlinear axially moving string by adaptive boundary control. Journal of Sound and Vibration, 2010, 329(22): 4588-4603[11] Fard M P, Sagatun S I. Exponential stabilization of a transversely vibrating beam by boundary control via Lyapunov's direct method. Journal of Dynamic Systems, Measurement, and Control, 2001, 123(2): 195-200[12] Queiroz M S, Dawson D M, Nagarkatti S P, Zhang F M. Lyapunov-Based Control of Mechanical Systems. Boston: Birkhauser, 2000[13] Krstic M. On global stabilization of Burgers' equation by boundary control. Systems and Control Letters, 1999, 37(3): 123-141[14] Yang W Y, Cao W, Chung T S, Morris J. Applied Numerical Methods Using MATLAB. New Jersey: John Wiley and Sons, 2005
点击查看大图
计量
- 文章访问数: 2264
- HTML全文浏览量: 68
- PDF下载量: 925
- 被引次数: 0