随机时滞下Markov跳跃线性系统鲁棒稳定与延迟反馈控制

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本文探讨了具有随机时滞的不确定马尔可夫跳跃线性系统在鲁棒随机稳定性与延迟状态反馈镇定方面的关键问题。作者们利用李亚普诺夫稳定性理论和稳健分析方法，针对这类系统中的不确定性以及随机时延特性进行深入研究。在马尔可夫跳跃过程中，系统的行为可能会因时间的随机变化而变得复杂，这就对系统设计提出了更高的要求。随机时滞的存在意味着系统状态更新的时间并非固定，而是服从一定的概率分布，这会显著影响系统的动态行为。因此，研究具有随机时滞的系统不仅涉及到传统的稳定性分析，还需要考虑时变性和随机性的双重挑战。文章的主要目标是寻找一种方法，即使在面对不确定性和随机时延的情况下，也能保证系统的稳定性并实现有效的控制。通过建立合适的Lyapunov函数，作者试图找到一个稳健的稳定条件，这个条件应该能够包容系统可能遇到的所有不确定性，并确保在随机时滞的范围内系统仍能保持稳定。延迟状态反馈是一种常用的控制策略，它通过引入适当的反馈信号来补偿系统的时滞效应。在本文中，作者探讨了如何设计一个延迟状态反馈控制器，既能克服时滞带来的问题，又能抵抗系统参数的不确定性，从而实现系统的稳定和可控。为了达到这些目标，文中可能涉及以下几个关键步骤： 1. 建立系统的数学模型，包括随机时滞、不确定性和马尔可夫跳跃的数学表达。 2. 设计Lyapunov函数的形式，通过其导数的负值来评估系统的稳定性。 3. 利用矩阵不等式技术进行稳健稳定性分析，以确保在所有可能的参数组合下，系统都能保持稳定性。 4. 提出一种具体的反馈律设计，可能是基于线性矩阵 inequalities (LMI) 或其他优化技术。 5. 通过仿真或理论证明，展示所提出的策略对于实际应用的有效性和可行性。这篇研究论文提供了一个重要的理论框架，有助于工程师们理解和控制具有随机时滞和不确定性特征的马尔可夫跳跃线性系统，这对于现代工业控制系统的安全性和可靠性至关重要。通过将稳定性理论与实际工程问题相结合，本文的成果可以促进未来此类复杂系统的实时控制设计和实践应用。

IET Control Theory & Applications

Research Article

Robust stochastic stability and

delayed-state-feedback stabilisation of

uncertain Markovian jump linear systems

with random time delays

ISSN 1751-8644

Received on 24th October 2014

Revised on 30th January 2015

Accepted on 23rd March 2015

doi: 10.1049/iet-cta.2014.1138

www.ietdl.org

Li Qiu

,Yang Shi

, Bugong Xu

, FengqiYao

Shenzhen Key Laboratory of Urban RailTransit, College of Mechatronics and Control Engineering, Shenzhen University,

Shenzhen 518060, People’s Republic of China

Department of Mechanical Engineering, University of Victoria, Victoria BC V8W 3P6, Canada

College of Automation, South China University ofTechnology, Guangzhou 510641, People’s Republic of China

School of Electrical Engineering and Information, Anhui University ofTechnology, Ma’anshan 243000, People’s Republic of China

E-mail: yshi@uvic.ca

Abstract: The problem of robust stochastic stability and delayed-state-feedback stabilisation of uncertain Markovian jump

linear systems with random Markov delays is investigated. Based on the Lyapunov stability theory and robust analysis

techniques, some robust stochastic stability criteria are derived in terms of linear matrix inequalities. Robust delayed-

state-feedback controllers that stochastically stabilise the uncertain Markovian jump linear systems is also proposed. The

state variable on the controller is assumed to be dependent on the Markov delay that has uncertain transition probabilities.

Finally, numerical examples are provided to illustrate the feasibility and effectiveness of the proposed methods.

1 Introduction

Markovian jump linear systems (MJLSs) represent an important

class of stochastic hybrid dynamic systems subject to abrupt

variations in their structure. This condition is partly caused by

phenomena such as changing subsystem interconnections, failures

or repairs, and abrupt environmental disturbances. Recently, MJLSs

have been an active research topic because of their extensive appli-

cations in networked control systems, electrical power systems,

communication systems, circuit systems, robot manipulator system

and economic systems. Many control topics of Markovian jump

systems or related systems with Markov process have been widely

investigated; notable developments have been achieved to ensure

stochastic Lyapunov stability, for example, robust control [1–4],

∞

control [5–8], fault-tolerant control [9–11], quantised control

[12], predictive control [13], H

∞

control [14–16], synchroni-

sation [17–21], and stability and stabilisation [22–27]. The ﬁltering

problem, such as ﬁlter design [28–30], Kalman ﬁltering [31, 32],

∞

ﬁltering [33–35] and exponential H

∞

ﬁltering [36], have also

gained signiﬁcant attention. Most of the reported research ﬁndings

on MJSs concern the Markovian transition probabilities that are

known a priori.

In practical processes, obtaining complete knowledge on the

Markovian transition probabilities is unlikely, or costly [37]. As

errors are usually generated through tests or experience, it is dif-

ﬁcult to derive the exact transition probabilities of a real physical

system. Therefore investigating general Markovian jump processes

with uncertain transition probabilities further from robust control

perspectives is signiﬁcant and challenging, especially when ran-

dom Markov time delays are included by communication network.

In the case of networked control systems with forward random time

delay, or random forward and feedback channel packet dropouts,

the random time delay in the forward and feedback channel or

the data packet dropouts in the forward and feedback channels are

modelled using a homogeneous Markov chain. However, ﬁnding

a probabilistic relation between the dropout phenomena that occur

in two channels is difﬁcult. The same problems may arise in other

practical systems with Markovian jump processes. Thus, in-depth

study of more general jump systems with uncertain transition

probabilities from control perspectives, especially with parameter

uncertainties and random Markov time delays included, is signif-

icant. Recently, some well-established results have been obtained

for MJSs with unknown or partially known transition probabilities.

For example, a class of continuous-time MJSs with bounded or

unknown transition probabilities have been studied in [38–40]. The

fault detection problem and stabilisation problem for the discrete-

time MJLs with partly known transition probabilities have been

addressed in [41] and in [37], respectively. However, parameter

uncertainties are inherent features of many physical processes and

are often encountered in engineering systems. Also, the transi-

tion probabilities of random Markov time delays are difﬁcult to

obtain but are bounded. The topic of delayed-state-feedback stabil-

isation problem for uncertain MJLSs with Markov time delays and

with uncertain transition probabilities has received limited atten-

tion. This observation motivates us to investigate the robust control

and delayed-state-feedback controller design problem.

This study focuses on the robust stochastic stability analysis

and delayed-state-feedback stabilisation synthesis problems for a

class of discrete-time uncertain MJLSs with uncertain transition

probabilities and random Markov time delays. The contribution of

this study is twofold. First, by using Jensen’s inequality approach,

a robust stochastic stability criteria are obtained for the MJLSs

with uncertain parameters and random time delays modelled as

a Markov process with uncertain transition probability. Second,

mode-dependent delayed-state-feedback control scheme is devel-

oped, and less conservative robust stability and stabilisation condi-

tions for the uncertain MJLSs are derived. The rest of this paper is

organised as follows. Section 2 provides the problem formulation

and Section 3 presents robust stochastic stability analysis. Section 4

establishes the robust delayed-state-feedback stabilisation condi-

tions for uncertain MJLSs with uncertain transition probabilities.

Illustrative examples are presented in Section 5 and the conclusion

is provided in Section 6 and two appendices containing formal

proofs of the results in the main body.

Notations: Throughout this paper, we let R

and R

n×m

denote the

n-dimensional Euclidean space and the set of all n × m real matri-

ces, respectively. The notation A > 0(A < 0) indicates that A is

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1878–1886

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