线性随机系统精确可观测性新准则与应用分析

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本文《亚洲控制杂志》(AsianJournalofControl)第15卷第4期，2013年7月发表，作者为Ting Hou、Weihai Zhang和Hongji Ma，论文以"线性随机系统精确可观测性的一些性质及其应用"为题。该研究关注于线性随机系统中精确可观测性的新准则，并探讨了这一概念在实际控制问题中的重要性。首先，作者提出了一个关于线性随机系统精确可观测性的新颖判断标准，这有助于确定系统的状态或输出是否能被完全且无遗漏地测量。精确可观测性是现代控制理论中的关键概念，它确保了通过系统输出可以准确地追踪到初始状态，这对于系统的分析和设计至关重要。论文进一步深入研究了精确可观测性对状态/输出反馈的影响。通过理论分析，作者揭示了当系统稳定且具备精确可观测性时，状态跟踪和控制性能的显著提升。在这样的条件下，系统能够有效地处理噪声和不确定性，提高控制的精度和鲁棒性。此外，文章还讨论了由广义Lyapunov方程（GLEs）定义的一个集合的性质。广义Lyapunov方程在稳定性分析中扮演着核心角色，而与精确可观测性相结合，它们提供了一种更深入理解系统动态和行为的工具。论文指出，这些集合的特性对于确定系统是否具有良好的渐近均方稳定性（ASMS）有着重要意义，这是一种衡量系统长期行为的重要指标。本研究不仅深化了对线性随机系统精确可观测性的理解，而且展示了其在实际应用中的实用价值，如在设计稳定的控制器以及优化系统性能方面的潜力。通过将精确可观测性与广义Lyapunov方程的理论相结合，作者为控制工程师提供了宝贵的设计指导和理论支持。读者可以从这篇论文中了解到如何利用这些性质来改善控制系统的性能，尤其是在存在不确定性和噪声的环境中。

Asian Journal of Control, Vol. 15, No. 4, pp. 1 6, July 2013

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/asjc.391

–Brief Paper–

SOME PROPERTIES OF EXACT OBSERVABILITY OF LINEAR

STOCHASTIC SYSTEMS AND THEIR APPLICATIONS

Ting Hou, Weihai Zhang, and Hongji Ma

ABSTRACT

A new criterion is proposed for exact observability of linear stochastic

systems. As applications of this criterion, the effects of exact observability due

to state/output feedback are investigated. In addition, under the conditions of

stability and exact observability, we discuss the properties of a set deﬁned by

the generalized Lyapunov equations (GLEs).

Key Words: Exact observability, state/output feedback, asymptotic mean

square stability, GLEs, convex cone.

I. INTRODUCTION

It is well-known that stability and observability

are essential concepts in modern control theory,

especially in system analysis and design. For linear

stochastic systems, several different interesting notions

of observability have appeared and a rich source of

references can be found in [1–6]. References [2] and

[4] have extended complete observability of determin-

istic linear systems to exact observability of Itˆo-type

stochastic systems. Exact observability has come into

current use in discussing the inﬁnite horizon stochastic

linear quadratic (LQ) problems as well as the properties

Manuscript received March 29, 2010; revised October 29,

2010; accepted January 28, 2011.

T. Hou is with the College of Information and Electrical

Engineering and College of Science, Shandong University

of Science and Technology, Qingdao 266510, China (e-mail:

math@sina.com).

W. Zhang (corresponding author) is with the College of

Information and Electrical Engineering, Shandong University

of Science and Technology, Qingdao 266510, China (e-mail:

hzhang@163.com).

H. Ma is with the College of Science, Shandong University

of Science and Technology, Qingdao 266510, China (e-mail:

mhj3027@sina.com).

This work was supported by the Key Project of Natural

Science Foundation of Shandong Province (Grant No.

ZR2009GZ001), the Specialized Research Fund for the

Doctoral Program of Higher Education of China (Grant No.

20103718110006) and the National Natural Science Founda-

tion of China (Grant No. 60874032).

of the related coupled algebraic Riccati equations

as done in [7]. A further step in the development

of this aspect is the establishment of its sufﬁcient

and necessary condition by the spectrum technique,

which may be viewed as the generalized version of the

Popov-Belevitch-Hautus (PBH) criterion for complete

observability [2]. Our ﬁrst objective in the present

paper is to contribute to presenting a new criterion for

the exact observability of linear stochastic systems. As

is known to all, for deterministic linear systems, the

observability of an open-loop system is not affected

by the output feedback, however, the observability of

open-loop and closed-loop state feedback systems is

unrelated. As applications of our obtained criterion,

we will generalize the previous classical results to

stochastic systems.

In the domain of stochastic stability and stabiliza-

tion of Itˆo differential systems, several authors have

been interested by these aspects for decades and some

extensive and elaborate results concerned with these

aspects have been derived since the fundamental works

[8, 9] appeared; See, e.g. [10–14] and the references

therein. We note that most of these works are concen-

trated on the investigation of mean square stabilization,

which has been playing a central role in many prob-

lems, such as inﬁnite horizon stochastic optimal control

problems [10], robust and stochastic H

∞

problems

[15], etc. Similar to the deterministic system case,

some necessary and sufﬁcient conditions on stabi-

lization of stochastic systems have been obtained in

terms of stochastic algebraic Riccati equations and



2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Asian Journal of Control, Vol. 14, No. 3, pp. 868–873, May 2012

Published online 28 April 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.391

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