具有内部控制和边界观测具有时滞的单管热交换方程的指数稳定性
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摘要
本文研究了具有时滞边界观测和内部控制的单管热交换方程的指数稳定性.首先,将闭环系统转换为合适状态空间上的抽象柯西问题.通过验证,闭环系统生成一个一致有界的C_0半群,意味着系统存在唯一解.其次,分析了系统的谱分布,通过某些预解集上的预解式估计得到生成半群的最终可微性和最终紧性,这意味着系统的谱确定增长假设成立.最后,给出了系统指数稳定性的一个充分条件,此充分条件与物理参数有关而与时滞无关.
In this note, the exponential stability of the mono-tubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated.Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded Co-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum determined growth assumption hold. Finally, a sufficient condition, which is related to the physical parameters in the system and independent of the time delay,of the exponential stability of the closed-loop system was given.
In this note, the exponential stability of the mono-tubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated.Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded Co-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum determined growth assumption hold. Finally, a sufficient condition, which is related to the physical parameters in the system and independent of the time delay,of the exponential stability of the closed-loop system was given.
引文
[1]Xu C Z, Gauthier J P, Kupka I. Exponential stability of the heat exchanger equation. In:Proceedings of the Second European Control Conference, Vol.l, Groningen, 1993, 303-307
[2]Kanoh H. Control of heat exchangers by placement of closed-loop poles. In:Proceedings of the IFAC Symposium on Design Methods of Control Systems, Zurich, 1991, 662-667
[3]Sano H. Exponential stability of a mono-tubular heat exchanger equation with output feedback.Systems&Control Letters, 2003, 50):363-369
[4]Guo B Z, Liang X Y. Differentiability of the Co-semigroup and Failure of Riesz Basis for a Monotubular Heat Exchanger Equation with Output Feedback:a Case Study. Semigroup Forum, 2004, 69:462-471
[5]Yu Xu, Liu K S. Eventual regularity of the semigroup associated with the mono-tubular heat exchanger equation with output feedback. Systems&Control Letters, 2006, 55:859-862
[6]Gumowski I, Mira C. Optimization in Control Theory and Practice. Cambridge:Cambridge University Press, 1968
[7]Datko R. Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks. SIAM J. Control Optim., 1988, 26:697-713
[8]Suh I H, Bien Z. Use of time delay action in the controller design. IEEE Trans. Automat. Control,1980, 25:600-603
[9]Kwon W H, Lee G W, Kim S W. Performance improvement, using time delays in multi-variable controller design. INT J. Control, 1990, 52:1455-1473
[10]Logemann H, Rebarber R, Weiss G. Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM J. Control Optim., 1996,34:572-600
[11]Guo B Z, Yang K Y. Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica, 2009, 45:1468-1475
[12]Guo B Z, Xu C Z, Hammouri H. Output Feedback Stabilization of a One-dimensional Wave Equation with an Arbitrary Time Delay in Boundary Observation,ESAIM:COCV, 2012, 18:22-35
[13]Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim., 2006, 45:1561-1585
[14]Zheng F, Fu L L. Spectral method for the reliability analysis of a complex redundant system. ICIC Express Letters, Part B:Applications, 2013, 4(6):1535-1541
[15]Xu G Q, Yung S P, Li L K. Stabilization of Wave Systems with Input Delay in the Boundary Control.ESAIM:COCV, 2006, 12:770-785
[16]Zheng F, Fu L L, Teng M Y. Exponential Stability of a Linear Distributed Parameter Bioprocess with Input Delay in Boundary Control. J. Fun. Spac. App., DOI:10.1155/2013/274857, 2013
[17]Curtain R F, Zwart H J. An Introduction to Infinitedimensional Linear Systems Theory. In:Textsin Applied Mathematics, Vol. 21, Springer, New York, 1995
[18]Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin:Springer, 1983
[19]Ruan S G, Wei J J. On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. IMA J. Math. Appl. Medic. Biol., 2001, 18:41-52
[20]Li X, Wei J J. Stability and bifurcation analysis in a system of four coupled neurons with multiple delays. Acta Math. Appl. Sin.(English Series), 2013, 29:425-448
[2]Kanoh H. Control of heat exchangers by placement of closed-loop poles. In:Proceedings of the IFAC Symposium on Design Methods of Control Systems, Zurich, 1991, 662-667
[3]Sano H. Exponential stability of a mono-tubular heat exchanger equation with output feedback.Systems&Control Letters, 2003, 50):363-369
[4]Guo B Z, Liang X Y. Differentiability of the Co-semigroup and Failure of Riesz Basis for a Monotubular Heat Exchanger Equation with Output Feedback:a Case Study. Semigroup Forum, 2004, 69:462-471
[5]Yu Xu, Liu K S. Eventual regularity of the semigroup associated with the mono-tubular heat exchanger equation with output feedback. Systems&Control Letters, 2006, 55:859-862
[6]Gumowski I, Mira C. Optimization in Control Theory and Practice. Cambridge:Cambridge University Press, 1968
[7]Datko R. Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks. SIAM J. Control Optim., 1988, 26:697-713
[8]Suh I H, Bien Z. Use of time delay action in the controller design. IEEE Trans. Automat. Control,1980, 25:600-603
[9]Kwon W H, Lee G W, Kim S W. Performance improvement, using time delays in multi-variable controller design. INT J. Control, 1990, 52:1455-1473
[10]Logemann H, Rebarber R, Weiss G. Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM J. Control Optim., 1996,34:572-600
[11]Guo B Z, Yang K Y. Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica, 2009, 45:1468-1475
[12]Guo B Z, Xu C Z, Hammouri H. Output Feedback Stabilization of a One-dimensional Wave Equation with an Arbitrary Time Delay in Boundary Observation,ESAIM:COCV, 2012, 18:22-35
[13]Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim., 2006, 45:1561-1585
[14]Zheng F, Fu L L. Spectral method for the reliability analysis of a complex redundant system. ICIC Express Letters, Part B:Applications, 2013, 4(6):1535-1541
[15]Xu G Q, Yung S P, Li L K. Stabilization of Wave Systems with Input Delay in the Boundary Control.ESAIM:COCV, 2006, 12:770-785
[16]Zheng F, Fu L L, Teng M Y. Exponential Stability of a Linear Distributed Parameter Bioprocess with Input Delay in Boundary Control. J. Fun. Spac. App., DOI:10.1155/2013/274857, 2013
[17]Curtain R F, Zwart H J. An Introduction to Infinitedimensional Linear Systems Theory. In:Textsin Applied Mathematics, Vol. 21, Springer, New York, 1995
[18]Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin:Springer, 1983
[19]Ruan S G, Wei J J. On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. IMA J. Math. Appl. Medic. Biol., 2001, 18:41-52
[20]Li X, Wei J J. Stability and bifurcation analysis in a system of four coupled neurons with multiple delays. Acta Math. Appl. Sin.(English Series), 2013, 29:425-448