带有时滞控制的一维热传导方程的参数化控制器设计与稳定性研究
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摘要
研究具有Dirichlet边界条件和内部时滞控制的热方程的镇定问题.文章的目标是设计一个状态反馈控制器,使得闭环系统以指定的衰减率λ指数衰减.与早期的控制器设计方法不同,文章探索一种新的控制器设计方法——对偏微分方程的参数化控制器设计.首先,将带有时滞的控制系统转换成由传输方程和热方程构成的串联系统.然后,构建一个具有指数稳定性并且与文章所研究的系统具有类似结构的目标系统.最后,选择合适的核函数,使其构成的有界线性变换可以将闭环系统映到目标系统.通过选择不同的核函数,可以得到由目标系统到闭环系统的逆变换.
In this paper, we study the stabilization problem of a heat equation with Dirichlet boundary conditions and internal delayed control. Our goal is to design a state feedback control such that the closed-loop system decays exponentially at designated decay rate λ. Different from earlier approach of controller design, in the present paper, we explore a new approach of controller design — The paramerization controller for Partial Differential Equations(PDEs). Firstly, we formulate the system with delayed control into a cascaded system of a transport equation and heat equation. Then, we construct a target system that has designated stability and a similar structure as the system under consideration. Finally, we select the suitable kernel functions of parameterization controller that forms a bounded linear transformation and maps the closed-loop system to the target system. By selecting different kernel function of controller we obtain the inverse transformation from the target system and the closed-loop system.
In this paper, we study the stabilization problem of a heat equation with Dirichlet boundary conditions and internal delayed control. Our goal is to design a state feedback control such that the closed-loop system decays exponentially at designated decay rate λ. Different from earlier approach of controller design, in the present paper, we explore a new approach of controller design — The paramerization controller for Partial Differential Equations(PDEs). Firstly, we formulate the system with delayed control into a cascaded system of a transport equation and heat equation. Then, we construct a target system that has designated stability and a similar structure as the system under consideration. Finally, we select the suitable kernel functions of parameterization controller that forms a bounded linear transformation and maps the closed-loop system to the target system. By selecting different kernel function of controller we obtain the inverse transformation from the target system and the closed-loop system.
引文
[1] Yang D G, Liao X F, Chen Y, et al. New delay-dependent global asymptotic stability criteria of delayed Hopfield neural networks. Nonlinear Analysis:Real World Applications, 2009, 9(5):1894-1904.
[2] Wang Q, Liu X Z. Impulsive stabilization of delay differential systems via the LyapunovRazumikhin method. Applied Mathematics Letters, 2007, 20(8):839-845.
[3] Stamova I M, Stamov G T. Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics. Journal of Computational&Applied Mathematics, 2001, 130(1-2):163-171.
[4] Dumitru Baleanu A N, Ranjbar S J, Sadati H D, et al. Lyapunov-Krasovskii stability theorem for fractional systems with delay. Romanian Journal of Physics, 2011, 56(5-6):636-643.
[5] Fridman E. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Systems&Control Letters, 2001, 43(4):309-319.
[6] Kharitonov V L, Zhabko A P. Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems. Automatica, 2003, 39(1):15-20.
[7] Zeng Z G, Fu C J, Liao X X. Stability analysis of neural networks with infinite time-varying delay. Journal of Math, 2002, 22(4):391-396.
[8] Xu G Q, Yung S P, Li L K. Stabilization of wave system with input delay in the boundary control.ESAIM:Control, Optimization and Calculus of Variations, 2006, 12(2):770-785.
[9] Guo Y N, Chen Y L, Xu G Q, et al. Exponential stabilization of variable coefficient wave equation in a generic tree with small time-delays in the nodal feedback. Journal of Mathematical Analysis and Applications, 2012, 395(2):727-746.
[10] Shang Y F, Xu G Q. Stabilization of an Euler-Bernoulli beam with input delay in the boundary control. System Control Letters, 2012, 61(11):1069-1078.
[11] Xu G Q,Wang H X. Stabilization of Tmoshenko beam system with delay in the boundary control.International Journal of Control, 2013, 86(6):1165-1178.
[12] Shang Y F,Xu G Q. Dynamic feedback control and exponential stabilization of a compound system. Journal of Mathematical Analysis and Applications, 2015, 422(2):858-879.
[13] Liu X F, Xu G Q. Exponential stabilization for Timoshenko beam with distributed delays in the boundary control. Abstract and Applied Anallysis, 2013, 2013(193):1-15.
[14] Zheng F, Zhang C L, Guo B Z. Exponential stability of the mono-tubular heat exchanger equation with time delay in boundary observation. Mathematical Problems in Engineering, 2017, 2017(6),https://doi.org/10.1155/2017/8952647.
[15] Aassila M. Stability and asymptotic behaviour of solutions of the heat equation. IMA Journal of Applied Mathematics, 2017, 69(1):93-109.
[16] Altm(u|¨)ller N, Gr(u|¨)ne L. A comparative stability analysis of Neumann and Dirichlet boundary MPC for the Heat equation. IFAC Proceedings Volumes, 2013, 46(26):133-138.
[17] Feng X X, Xu G Q, Chen Y L. Rapid stabilization of an Euler-Bernoulli beam with the internal delay control. International Journal of Control, 2017, 00(20):1-21.
[2] Wang Q, Liu X Z. Impulsive stabilization of delay differential systems via the LyapunovRazumikhin method. Applied Mathematics Letters, 2007, 20(8):839-845.
[3] Stamova I M, Stamov G T. Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics. Journal of Computational&Applied Mathematics, 2001, 130(1-2):163-171.
[4] Dumitru Baleanu A N, Ranjbar S J, Sadati H D, et al. Lyapunov-Krasovskii stability theorem for fractional systems with delay. Romanian Journal of Physics, 2011, 56(5-6):636-643.
[5] Fridman E. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Systems&Control Letters, 2001, 43(4):309-319.
[6] Kharitonov V L, Zhabko A P. Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems. Automatica, 2003, 39(1):15-20.
[7] Zeng Z G, Fu C J, Liao X X. Stability analysis of neural networks with infinite time-varying delay. Journal of Math, 2002, 22(4):391-396.
[8] Xu G Q, Yung S P, Li L K. Stabilization of wave system with input delay in the boundary control.ESAIM:Control, Optimization and Calculus of Variations, 2006, 12(2):770-785.
[9] Guo Y N, Chen Y L, Xu G Q, et al. Exponential stabilization of variable coefficient wave equation in a generic tree with small time-delays in the nodal feedback. Journal of Mathematical Analysis and Applications, 2012, 395(2):727-746.
[10] Shang Y F, Xu G Q. Stabilization of an Euler-Bernoulli beam with input delay in the boundary control. System Control Letters, 2012, 61(11):1069-1078.
[11] Xu G Q,Wang H X. Stabilization of Tmoshenko beam system with delay in the boundary control.International Journal of Control, 2013, 86(6):1165-1178.
[12] Shang Y F,Xu G Q. Dynamic feedback control and exponential stabilization of a compound system. Journal of Mathematical Analysis and Applications, 2015, 422(2):858-879.
[13] Liu X F, Xu G Q. Exponential stabilization for Timoshenko beam with distributed delays in the boundary control. Abstract and Applied Anallysis, 2013, 2013(193):1-15.
[14] Zheng F, Zhang C L, Guo B Z. Exponential stability of the mono-tubular heat exchanger equation with time delay in boundary observation. Mathematical Problems in Engineering, 2017, 2017(6),https://doi.org/10.1155/2017/8952647.
[15] Aassila M. Stability and asymptotic behaviour of solutions of the heat equation. IMA Journal of Applied Mathematics, 2017, 69(1):93-109.
[16] Altm(u|¨)ller N, Gr(u|¨)ne L. A comparative stability analysis of Neumann and Dirichlet boundary MPC for the Heat equation. IFAC Proceedings Volumes, 2013, 46(26):133-138.
[17] Feng X X, Xu G Q, Chen Y L. Rapid stabilization of an Euler-Bernoulli beam with the internal delay control. International Journal of Control, 2017, 00(20):1-21.