随机镇定与反镇定概述

黄礼荣　邓飞其　宋明辉

摘要本文概述了随机镇定与反镇定理论的研究现状.主要回顾了一个微分方程的随机镇定与反镇定普遍理论及其发展，并围绕该理论的应用和扩展从四个方面阐述连续时间系统噪声镇定理论的当前发展概况.此外，本文还概述了離散时间系统随机镇定方面的最新进展.关键词几乎必然稳定性；连续时间系统；离散时间系统；噪声镇定；随机微分方程；随机镇定

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实际系统中通常存在噪声干扰，而在很多案例中这些噪声干扰不利于系统的运作，在较大程度上损害系统的良好性态，甚至破坏了系统的稳定性.长期以来，怎么处理系统运作中的噪声干扰是工程理论研究中的一个重要课题（参见文献[16]）.

不但有如何克服噪声干扰保持稳定性的工作，而且不少研究发现利用噪声既可使系统失去稳定性却亦可能使不稳定的系统镇定或稳定的系统更加稳定.后者引起了人们广泛的研究兴趣，由此得到了许多重要的研究成果（见文献[5，78，1012，1524，26，29，33，3637，42，68，7275]等）.利用噪声使（不稳定）系统镇定是有重要意义的研究课题，非常有助于工程系统的分析和设计.

在这些重要结果中，本文主要概述文献[20]提出的微分方程的随机镇定与反镇定普遍理论及其发展[15，21]，并以这一理论的应用和扩展从以下四个方面阐述当前连续时间系统噪声镇定理论的发展概况（每一方面均列举例子并配以一详细范例说明）：1）随机镇定理论的应用：如文献[26]应用文献[20]（Theorem 31，亦见文献[5]）中的随机镇定理论结合矩阵不等式提出利用噪声镇定的状态反馈控制器设计方法；2）控制策略及方式扩展至随机镇定理论：如文献[36]将采样数据控制策略（参见文献[6163]）扩展至随机镇定理论（参见文献[20]，Theorem 31）；3）随机镇定理论推广至多类系统：如文献[37]将对微分方程的随机镇定理论[20]推广至具有马氏切换的混合微分方程；4）利用其他类型噪声：如文献[73]以Lévy噪声替换文献[20]结果中的以布朗运动描述的噪声，得到了文献[20]（Theorem 31）的一个推广结果.

相对于连续时间系统随机镇定理论及应用，离散时间系统的随机镇定理论与应用仍然处于初级阶段.在离散时间系统分析和设计中，噪声通常作干扰处理.在离散时间系统随机镇定方面，本文主要概述了文献[28]提出的新理论及其应用在状态反馈控制器设计上的新方法.

随机镇定是一个有重要意义而且内容丰富的研究领域，其中很多问题有待研究：例如上述噪声镇定理论的发展和应用主要针对具有线性增长条件的微分方程，对（高阶）非线性的微分方程也可作相应发展；控制策略和方式拓展在随机镇定上的问题值得进一步探究（文献[31，36]）；离散时间系统噪声镇定理论与应用有许多需要研究的问题等.

The proposed control design method exploits the stabilizing role of noise in discretetime systems and applies to some cases where the other results in the literature do not work，which have been verified with examples in [28].

4Concluding remarks

This paper has given an overview of some recent advances of theory on stabilization and destabilization by noise.Particularly，this paper has reviewed a general theory proposed in [20] and developed in [15，21] as well as its applications and generalizations.It is observed，as reviewed in Section 3，that the results in [20] for systems with the linear growth condition have been used and/or generalized in many works （see，e.g.，[26，3637，73]） and now a few begin to apply the results developed in [15，21] for （highly） nonlinear systems （see，e.g.，[27]）.There is much work to do for （highly） nonlinear systems as well as developments of techniques such as input delay or sampled data control （in diffusion）.This paper has also reviewed the theory on stochastic stabilization of discretetime systems developed in [28].It appears that，compared with that for continuoustime systems，the theory of stabilization by noise for discretetime systems and its applications are in an early stage.In summary，the study of stochastic stabilization and destabilization is a very rich research field.

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