隨機鎮(zhèn)定與反鎮(zhèn)定概述

黃禮榮　鄧飛其　宋明輝

摘要本文概述了隨機鎮(zhèn)定與反鎮(zhèn)定理論的研究現(xiàn)狀.主要回顧了一個微分方程的隨機鎮(zhèn)定與反鎮(zhèn)定普遍理論及其發(fā)展，并圍繞該理論的應(yīng)用和擴展從四個方面闡述連續(xù)時間系統(tǒng)噪聲鎮(zhèn)定理論的當前發(fā)展概況.此外，本文還概述了離散時間系統(tǒng)隨機鎮(zhèn)定方面的最新進展.關(guān)鍵詞幾乎必然穩(wěn)定性；連續(xù)時間系統(tǒng)；離散時間系統(tǒng)；噪聲鎮(zhèn)定；隨機微分方程；隨機鎮(zhèn)定

中圖分類號TP13

文獻標志碼A

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本文原文為英文，希望感興趣的讀者進一步關(guān)注原文.

實際系統(tǒng)中通常存在噪聲干擾，而在很多案例中這些噪聲干擾不利于系統(tǒng)的運作，在較大程度上損害系統(tǒng)的良好性態(tài)，甚至破壞了系統(tǒng)的穩(wěn)定性.長期以來，怎么處理系統(tǒng)運作中的噪聲干擾是工程理論研究中的一個重要課題（參見文獻[16]）.

不但有如何克服噪聲干擾保持穩(wěn)定性的工作，而且不少研究發(fā)現(xiàn)利用噪聲既可使系統(tǒng)失去穩(wěn)定性卻亦可能使不穩(wěn)定的系統(tǒng)鎮(zhèn)定或穩(wěn)定的系統(tǒng)更加穩(wěn)定.后者引起了人們廣泛的研究興趣，由此得到了許多重要的研究成果（見文獻[5，78，1012，1524，26，29，33，3637，42，68，7275]等）.利用噪聲使（不穩(wěn)定）系統(tǒng)鎮(zhèn)定是有重要意義的研究課題，非常有助于工程系統(tǒng)的分析和設(shè)計.

在這些重要結(jié)果中，本文主要概述文獻[20]提出的微分方程的隨機鎮(zhèn)定與反鎮(zhèn)定普遍理論及其發(fā)展[15，21]，并以這一理論的應(yīng)用和擴展從以下四個方面闡述當前連續(xù)時間系統(tǒng)噪聲鎮(zhèn)定理論的發(fā)展概況（每一方面均列舉例子并配以一詳細范例說明）：1）隨機鎮(zhèn)定理論的應(yīng)用：如文獻[26]應(yīng)用文獻[20]（Theorem 31，亦見文獻[5]）中的隨機鎮(zhèn)定理論結(jié)合矩陣不等式提出利用噪聲鎮(zhèn)定的狀態(tài)反饋控制器設(shè)計方法；2）控制策略及方式擴展至隨機鎮(zhèn)定理論：如文獻[36]將采樣數(shù)據(jù)控制策略（參見文獻[6163]）擴展至隨機鎮(zhèn)定理論（參見文獻[20]，Theorem 31）；3）隨機鎮(zhèn)定理論推廣至多類系統(tǒng)：如文獻[37]將對微分方程的隨機鎮(zhèn)定理論[20]推廣至具有馬氏切換的混合微分方程；4）利用其他類型噪聲：如文獻[73]以Lévy噪聲替換文獻[20]結(jié)果中的以布朗運動描述的噪聲，得到了文獻[20]（Theorem 31）的一個推廣結(jié)果.

相對于連續(xù)時間系統(tǒng)隨機鎮(zhèn)定理論及應(yīng)用，離散時間系統(tǒng)的隨機鎮(zhèn)定理論與應(yīng)用仍然處于初級階段.在離散時間系統(tǒng)分析和設(shè)計中，噪聲通常作干擾處理.在離散時間系統(tǒng)隨機鎮(zhèn)定方面，本文主要概述了文獻[28]提出的新理論及其應(yīng)用在狀態(tài)反饋控制器設(shè)計上的新方法.

隨機鎮(zhèn)定是一個有重要意義而且內(nèi)容豐富的研究領(lǐng)域，其中很多問題有待研究：例如上述噪聲鎮(zhèn)定理論的發(fā)展和應(yīng)用主要針對具有線性增長條件的微分方程，對（高階）非線性的微分方程也可作相應(yīng)發(fā)展；控制策略和方式拓展在隨機鎮(zhèn)定上的問題值得進一步探究（文獻[31，36]）；離散時間系統(tǒng)噪聲鎮(zhèn)定理論與應(yīng)用有許多需要研究的問題等.

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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