分布参数控制系统的小时滞鲁棒稳定性
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摘要
在任何反馈控制系统中，时滞总是存在。因此研究反馈闭环系统中的小时滞对控制系统的敏感性是极其重要的，并已引起广泛关注。本文研究分布参数控制系统的指数稳定对小时滞的鲁棒性(简称小时滞鲁棒稳定性)，对含无界算子的微分系统用基本算子刻画小时滞的鲁棒稳定性为本文首次采用。首先，对Banach空间X中的时滞抽象微分方程引入基本算子，并用基本算子和预解式刻画时滞系统在状态空间X×L~P[-r,0]；X)上的小时滞鲁棒稳定性。特别是在Hilbert空间中，可分别用预解式在虚轴上的一致平方可积性和一致有界性刻画系统的小时滞鲁棒稳定性，并很容易得到已有的一些结论。我们进一步研究了时滞项含无界算子的抽象微分方程的小时滞鲁棒稳定性，获得了一些充分必要条件，并应用所得结论讨论了具解析半群生成元的时滞系统。其次，给出了非自治线性系统具时变小时滞鲁棒稳定性的一个充分条件，此结论被应用到非自治抛物系统。最后，对两端自由的变系数无阻尼Euler-Bernoulli梁采用旋转角、角速度和位移、速度的线性边界反馈指数稳定化问题进行了讨论。我们引入动态状态空间，采用算子半群技巧、乘子方法和频域反证法巧妙结合的方法，获得了此系统能量按指数律衰减的最佳条件。
In the implementation of any feedback control system, it is very likely that time delays will occur. It is therefore of vital importance to understand the sensitivity of control system to introduction of small delays in the feedback loop and problems of this type have attracted a lot of attention. This paper is concerned with the robustness with respect to small delays for exponential stability of distributed parameter control systems, which is first characterized by fundamental operators.
We first introduce fundamental operators for abstract differential equations with delays in Banach spaces, and characterize the robustness with respect to small delays via the associated fundamental operators. Specially, in Hilbert spaces, some necessary and sufficient conditions are given in terms of the uniformly square integrability and uniform boundness of the resolvent on the imaginary axis respectively. Furthermore, we consider this robustness for abstract differential equations with unbounded operator in the delay term. Next, sufficient conditions are given for the nonautonomous systems to be robust with respect to small time-varing delays, and these results are applied to the nonautonomous parabolic system.
Finally, the boundary feedback control of the undamped Euler-Bemoulli beam with both ends free is concerned. We introduce the dynamical state space, combine the operator semigroup technique, the multiplier technique and the contradiction argument of frequency domain to obtain the best possible conditions for the energy of this system to be uniformly exponentially decay.
In the implementation of any feedback control system, it is very likely that time delays will occur. It is therefore of vital importance to understand the sensitivity of control system to introduction of small delays in the feedback loop and problems of this type have attracted a lot of attention. This paper is concerned with the robustness with respect to small delays for exponential stability of distributed parameter control systems, which is first characterized by fundamental operators.
We first introduce fundamental operators for abstract differential equations with delays in Banach spaces, and characterize the robustness with respect to small delays via the associated fundamental operators. Specially, in Hilbert spaces, some necessary and sufficient conditions are given in terms of the uniformly square integrability and uniform boundness of the resolvent on the imaginary axis respectively. Furthermore, we consider this robustness for abstract differential equations with unbounded operator in the delay term. Next, sufficient conditions are given for the nonautonomous systems to be robust with respect to small time-varing delays, and these results are applied to the nonautonomous parabolic system.
Finally, the boundary feedback control of the undamped Euler-Bemoulli beam with both ends free is concerned. We introduce the dynamical state space, combine the operator semigroup technique, the multiplier technique and the contradiction argument of frequency domain to obtain the best possible conditions for the energy of this system to be uniformly exponentially decay.
引文
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