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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 839648, 17 pages
http://dx.doi.org/10.1155/2011/839648
Research Article

Robust 𝐿𝟐-𝐿∞ Filtering of Time-Delay Jump Systems with Respect to the Finite-Time Interval

Shuping He1,2,3Β and Fei Liu1,2

1Key Laboratory of Advanced Process Control for Light Industry, Jiangnan University, Ministry of Education, Wuxi 214122, China
2Institute of Automation, Jiangnan University, Wuxi 214122, China
3Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 1QD, UK

Received 10 September 2010; Accepted 22 September 2010

Academic Editor: MingΒ Li

Copyright Β© 2011 Shuping He and Fei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studied the problem of stochastic finite-time boundedness and disturbance attenuation for a class of linear time-delayed systems with Markov jumping parameters. Sufficient conditions are provided to solve this problem. The 𝐿2-𝐿∞ filters are, respectively, designed for time-delayed Markov jump linear systems with/without uncertain parameters such that the resulting filtering error dynamic system is stochastically finite-time bounded and has the finite-time interval disturbance attenuation 𝛾 for all admissible uncertainties, time delays, and unknown disturbances. By using stochastic Lyapunov-Krasovskii functional approach, it is shown that the filter designing problem is in terms of the solutions of a set of coupled linear matrix inequalities. Simulation examples are included to demonstrate the potential of the proposed results.

1. Introduction

Since the introduction of the framework of the class of Markov jump linear systems (MJLSs) by Krasovskii and Lidskii [1], we have seen increasing interest for this class of stochastic systems. It was used to model a variety of physical systems, which may experience abrupt changes in structures and parameters due to, for instance, sudden environment changes, subsystem switching, system noises, and failures occurring in components or interconnections and executor faults. For more information regarding the use of this class of systems, we refer the reader to Sworder and Rogers [2], Athans [3], Arrifano and Oliveira [4], and the references therein. It has been recognized that the time-delays and parameter uncertainties, which are inherent features of many physical processes, are very often the cause for poor performance of systems. In the past few years, considerable attention has been given to the robust control and filtering for linear uncertain time-delayed systems. A great amount of progress has been made on this general topic; see, for example, [5–8], and the references therein. As for MJLSs with time-delays and uncertain parameters, the results of stochastic stability, robust controllability, observability, filtering, and fault detection have been well investigated, and recent results can be found in [9–15].

It is now worth pointing out that the control performances mentioned above concern the desired behavior of control dynamics over an infinite-time interval and it always deals with the asymptotic property of system trajectories. But in some practical processes, a Lyapunov asymptotically stable system over an infinite-time interval does not mean that it has good transient characteristics, for instance, biochemistry reaction system, robot control system, and communication network system. Moreover, the main attention in these dynamics may be related to the behavior over a fixed finite-time interval. Therefore, we need to check the unacceptable values to see whether the system states remain within the prescribed bound in a fixed finite-time interval or not. To discuss this transient performance of control systems, Dorato [16] gave the concept of finite-time stability (or short-time stability [17, 18]) in the early 1960s. Then, some attempts on finite-time stability can be found in [19–24]. More recently, the concept of finite-time stability has been revisited in the light of recent results coming from linear matrix inequalities (LMIs) techniques, which relate the computationally appealing conditions guaranteeing finite-time stabilization [25, 26] of dynamic systems. Towards each case above, more details are related to linear control dynamics [27–29]. However, very few results in the literature consider the related control and filtering problems [30] of stochastic MJLSs in the finite-time interval. These motivate us to research this topic.

As we all know, since the Kalman filtering theory has been introduced in the early 1960s, the filtering problem has been extensively investigated, whose objective is to estimate the unavailable state variables (or a linear combination of the states) of a given system. During the past decades, the filtering technique regains increasing interest, and many filtering schemes have been developed. Among these filtering approaches, the 𝐿2-𝐿∞ filtering problem [31, 32] has received less attention. But in practical engineering applications, the peak values of filtering error should be considered. And compared with 𝐻∞ filtering, the stochastic noise disturbances are both assumed energy bounded in these two filtering techniques, but 𝐿2-𝐿∞ filtering setting requires the 𝐿2-𝐿∞ performance prescribed bounded from unknown noise disturbances to filtering error.

This paper is concerned with the robust 𝐿2-𝐿∞ filtering problem for a class of continuous time-delay uncertain systems with Markov jumping parameters. We aim at designing a robust 𝐿2-𝐿∞ filter such that, for all admissible uncertainties, time delays, and unknown disturbances, the filtering error dynamic system is stochastically finite-time bounded (FTB) and satisfies the given finite-time interval induced 𝐿2-𝐿∞ norm of the operator from the unknown disturbance to the output error. By using stochastic Lyapunov-Krasovskii functional approach, we show that the filter designing problem can be dealt with by solving a set of coupled LMIs. In order to illustrate the proposed results, two simulation examples are given at last.

In the sequel, unless otherwise specified, matrices are assumed to have compatible dimensions. The notations used throughout this paper are quite standard. β„œπ‘› and β„œπ‘›Γ—π‘š denote, respectively, the 𝑛 dimensional Euclidean space and the set of all π‘›Γ—π‘š real matrices. 𝐴T and π΄βˆ’1 denote the matrix transpose and matrix inverse. diag{𝐴𝐡} represents the block-diagonal matrix of 𝐴 and 𝐡. If 𝐴 is a symmetric matrix, denote by 𝜎min(𝐴) and 𝜎max(𝐴) its smallest and largest eigenvalues, respectively. βˆ‘π‘π‘–<𝑗denotes, for example, for 𝑁=3, βˆ‘π‘π‘–<π‘—π‘Žπ‘–π‘—β‡”π‘Ž12+π‘Ž13+π‘Ž23. 𝐄{βˆ—} stands for the mathematics statistical expectation of the stochastic process or vector and β€–βˆ—β€– is the Euclidean vector norm. 𝐿𝑛2[0𝑇] is the space of 𝑛 dimensional square integrable function vector over [0𝑇]. 𝑃>0 stands for a positive-definite matrix. 𝐼 is the unit matrix with appropriate dimensions. 0 is the zero matrix with appropriate dimensions. In symmetric block matrices, we use β€œβˆ—β€ as an ellipsis for the terms that are introduced by symmetry.

The paper is organized as follows. In Section 2, we derive the new definitions about stochastic finite-time filtering of MJLSs. In Section 3, we give the main results of 𝐿2-𝐿∞ filtering problem of MJLSs and extend this to uncertain dynamic MJLSs in Section 4. In Section 5, we demonstrate two simulation examples to show the validity of the developed methods.

2. Problem Formulation

Given a probability space (Ξ©,𝐹,π‘ƒπ‘Ÿ) where Ξ© is the sample space, 𝐹 is the algebra of events and π‘ƒπ‘Ÿ is the probability measure defined on 𝐹. Let the random form process {π‘Ÿπ‘‘,𝑑β‰₯0} be the continuous-time discrete-state Markov stochastic process taking values in a finite set 𝑀={1,2,…,𝑁} with transition probability matrix π‘ƒπ‘Ÿ={𝑃𝑖𝑗(𝑑),𝑖,π‘—βˆˆπ‘€} given byπ‘ƒπ‘Ÿ=π‘ƒπ‘–π‘—ξ€½π‘Ÿ(𝑑)=𝑃𝑑+Δ𝑑=π‘—βˆ£π‘Ÿπ‘‘ξ€Ύ=ξƒ―πœ‹=𝑖𝑖𝑗Δ𝑑+π‘œ(Δ𝑑),𝑖≠𝑗,1+πœ‹π‘–π‘–Ξ”π‘‘+π‘œ(Δ𝑑),𝑖=𝑗,(2.1) where Δ𝑑>0 and limΔ𝑑→0π‘œ(Δ𝑑)/Δ𝑑→0. πœ‹π‘–π‘—β‰₯0 is the transition probability rates from mode 𝑖 at time 𝑑 to mode 𝑗(𝑖≠𝑗) at time 𝑑+Δ𝑑, and βˆ‘π‘π‘—=1,π‘—β‰ π‘–πœ‹π‘–π‘—=βˆ’πœ‹π‘–π‘–.

Consider the following time-delay dynamic MJLSs over the probability space (Ξ©,𝐹,π‘ƒπ‘Ÿ):ξ€·π‘ŸΜ‡π‘₯(𝑑)=𝐴𝑑π‘₯(𝑑)+π΄π‘‘ξ€·π‘Ÿπ‘‘ξ€Έπ‘₯ξ€·π‘Ÿ(π‘‘βˆ’π‘‘)+π΅π‘‘ξ€Έπ‘€ξ€·π‘Ÿ(𝑑),𝑦(𝑑)=πΆπ‘‘ξ€Έξ€·π‘Ÿπ‘₯(𝑑)+π·π‘‘ξ€Έπ‘§ξ€·π‘Ÿπ‘€(𝑑),(𝑑)=𝐿𝑑π‘₯(𝑑),π‘₯(𝑑)=πœ†(𝑑),π‘Ÿπ‘‘ξ€Ίπ‘‘=πœ‰(𝑑),π‘‘βˆˆ0βˆ’π‘‘π‘‘0ξ€»,(2.2) where π‘₯(𝑑)βˆˆβ„œπ‘› is the state, 𝑦(𝑑)βˆˆβ„œπ‘™ is the measured output, 𝑀(𝑑)βˆˆβ„œπ‘ is the unknown input, 𝑧(𝑑)βˆˆβ„œπ‘ž is the controlled output, 𝑑>0 is the constant time-delay, 𝜎(𝑑) is a vector-valued initial continuous function defined on the interval [𝑑0βˆ’π‘‘π‘‘0], and πœ‰(𝑑) is the initial mode. 𝐴(π‘Ÿπ‘‘), 𝐴𝑑(π‘Ÿπ‘‘), 𝐡(π‘Ÿπ‘‘), 𝐢(π‘Ÿπ‘‘), 𝐷(π‘Ÿπ‘‘), and 𝐿(π‘Ÿπ‘‘) are known mode-dependent constant matrices with appropriate dimensions, and π‘Ÿπ‘‘ represents a continuous-time discrete state Markov stochastic process with values in the finite set 𝑀={1,2,…,𝑁}.

Remark 2.1. For convenience, when π‘Ÿπ‘‘=𝑖, we denote 𝐴(π‘Ÿπ‘‘), 𝐴𝑑(π‘Ÿπ‘‘), 𝐡(π‘Ÿπ‘‘), 𝐢(π‘Ÿπ‘‘), 𝐷(π‘Ÿπ‘‘), and 𝐿(π‘Ÿπ‘‘) as 𝐴𝑖, 𝐴𝑑𝑖, 𝐡𝑖, 𝐢𝑖, 𝐷𝑖, and 𝐿𝑖. Notice that the time-delays in (2.2) are constant and only dependent of the system structure, and they are not dependent on the defined stochastic process. To simplify the study, we take the initial time 𝑑0=0 and let the initial values {πœ†(𝑑)}π‘‘βˆˆ[βˆ’π‘‘0] and {πœ‰(𝑑)=π‘Ÿπ‘‘}π‘‘βˆˆ[βˆ’π‘‘0] be fixed. At each mode, we assume that the time-delay MJLSs have the same dimension.

We now construct the following full-order linear filter for MJLSs (2.2) asΜ‡Μ‚π‘₯(𝑑)=𝐴𝑓𝑖̂π‘₯(𝑑)+𝐴𝑑𝑖̂π‘₯(π‘‘βˆ’π‘‘)+𝐡𝑓𝑖𝑦(𝑑),̂𝑧(𝑑)=𝐢𝑓𝑖̂π‘₯(𝑑),Μ‚π‘₯(𝑑)=πœ‘(𝑑),π‘Ÿπ‘‘ξ€Ίξ€»,=πœ‰(𝑑),π‘‘βˆˆβˆ’π‘‘0(2.3) where Μ‚π‘₯(𝑑)βˆˆβ„œπ‘› is the filter state, ̂𝑧(𝑑)βˆˆβ„œπ‘ž is the filter output, πœ‘(𝑑) is a continuous vector-valued initial function, and the mode-dependent matrices 𝐴𝑓𝑖, 𝐡𝑓𝑖, and 𝐢𝑓𝑖 are unknown filter parameters to be designed for each value π‘–βˆˆπ‘€.

Define 𝑒(𝑑)=π‘₯(𝑑)βˆ’Μ‚π‘₯(𝑑) and π‘Ÿ(𝑑)=𝑧(𝑑)βˆ’Μ‚π‘§(𝑑), then we can get the following filtering error system:𝐴̇𝑒(𝑑)=π‘–βˆ’π΅π‘“π‘–πΆπ‘–ξ€Έπ‘₯(𝑑)βˆ’π΄π‘“π‘–Μ‚π‘₯(𝑑)+𝐴𝑑𝑖[π‘₯]+𝐡(π‘‘βˆ’π‘‘)βˆ’Μ‚π‘₯(π‘‘βˆ’π‘‘)π‘–βˆ’π΅π‘“π‘–π·π‘–ξ€Έπ‘‘π‘Ÿ(𝑑),(𝑑)=𝐿𝑖π‘₯(𝑑)βˆ’πΆπ‘“π‘–Μ‚π‘₯(𝑑).(2.4) Let π‘₯𝑇(𝑑)=[π‘₯𝑇(𝑑)𝑒𝑇(𝑑)], the filtering error system (2.4) can be rewritten asΜ‡β€Œπ‘₯(𝑑)=𝐴𝑖π‘₯(𝑑)+𝐴𝑑𝑖π‘₯(π‘‘βˆ’π‘‘)+𝐡𝑖𝑀(𝑑),π‘Ÿ(𝑑)=𝐢𝑖π‘₯(𝑑),π‘₯𝑇(𝑑)=πœ†(𝑑)πœ†(𝑑)βˆ’πœ‘(𝑑),π‘Ÿπ‘‘[],=πœ‰(𝑑),π‘‘βˆˆβˆ’π‘‘0(2.5) where𝐴𝑖=𝐴𝑖0π΄π‘–βˆ’π΄π‘“π‘–βˆ’π΅π‘“π‘–πΆπ‘–π΄π‘“π‘–ξ‚Ή,𝐴𝑑𝑖=𝐴𝑑𝑖00𝐴𝑑𝑖,𝐡𝑖=ξ‚Έπ΅π‘–π΅π‘–βˆ’π΅π‘“π‘–π·π‘–ξ‚Ή,𝐢𝑖=ξ€ΊπΏπ‘–βˆ’πΆπ‘“π‘–πΆπ‘“π‘–ξ€».(2.6)

The objective of this paper consists of designing the finite-time filter of time-delay MJLSs in (2.1) and obtaining an estimate Μ‚β€Œz(𝑑) of the signal 𝑧(𝑑) such that the defined guaranteed 𝐿2-𝐿∞ performance criteria are minimized. For some given initial conditions [24–27], the general idea of finite-time filtering can be formalized through the following definitions over a finite-time interval.

Assumption. The external disturbance 𝑀(𝑑) is time-varying and satisfies ξ€œπ‘‡0𝑀𝑇(𝑑)𝑀(𝑑)π‘‘π‘‘β‰€π‘Š,π‘Šβ‰₯0.(2.7)

Definition 2.2. For a given time-constant 𝑇>0, the filtering error MJLSs system (2.5) with 𝑀(𝑑)=0 is stochastically finite-time stable (FTS) if there exist positive matrix π‘…π‘–βˆˆβ„œ2𝑛×2𝑛>0 and scalars 𝑐1>0 and 𝑐2>0, such that 𝐄π‘₯𝑇𝑑1𝑅𝑖π‘₯𝑑1≀𝑐1ξ‚†βŸΉπ„π‘₯𝑇𝑑2𝑅𝑖π‘₯𝑑2<𝑐2,𝑑1βˆˆξ€Ίξ€»βˆ’π‘‘0,𝑑2].∈(0𝑇(2.8)

Definition 2.3 (FTB). For a given time-constant 𝑇>0, the filtering error (2.5) is stochastically finite-time bounded (FTB) with respect to (𝑐1𝑐2π‘‡π‘…π‘–π‘Š) if condition (2.8) holds.

Remark 2.4. Notice that FTB and FTS are open-loop concepts. FTS can be recovered as a particular case of FTB with π‘Š=0 and FTS leads to the concept of FTB in the presence of external inputs. FTB implies finite-time stability, but the converse is not true. It is necessary to point out that Lyapunov stability and FTS are independent concepts. Different with the concept of Lyapunov stability [33–35] which is largely known to the control community, a stochastic MJLSs is FTS if, once we fix a finite time-interval [36, 37], its state remain within prescribed bounds during this time-interval. Moreover, an MJLS which is FTS may not be Lyapunov stochastic stability; conversely, a Lyapunov stochastically stable MJLS could be not FTS if its states exceed the prescribed bounds during the transients.

Definition 2.5 (Feng et al. [34], Mao [35]). Let 𝑉(π‘₯(𝑑),π‘Ÿπ‘‘,𝑑>0) be the stochastic positive functional; define its weak infinitesimal operator as ℑ𝑉π‘₯(𝑑),π‘Ÿπ‘‘ξ€Έ=𝑖,𝑑=limΞ”π‘‘β†’πŸŽ1𝐄𝑉Δ𝑑π‘₯(𝑑+Δ𝑑),π‘Ÿπ‘‘+Δ𝑑,𝑑+Ξ”π‘‘βˆ£π‘₯(𝑑),π‘Ÿπ‘‘ξ€Ύξ€·=π‘–βˆ’π‘‰π‘₯(𝑑),π‘Ÿπ‘‘.=𝑖,𝑑(2.9)

Definition 2.6. For the filtering error MJLSs (2.5), if there exist filter parameters 𝐴𝑓𝑖, 𝐡𝑓𝑖, and 𝐢𝑓𝑖, and a positive scalar 𝛾, such that (2.5) is FTB and under the zero-valued initial condition, the system output error satisfies the following cost function inequality for 𝑇>0 with attenuation 𝛾>0 and for all admissible 𝑀(𝑑) with the constraint condition (2.7), ‖𝐽=π„β€–π‘Ÿ(𝑑)2βˆžξ€Ύβˆ’π›Ύ2‖‖𝑑(𝑑)22<0,(2.10) where 𝐄{β€–π‘Ÿ(𝑑)β€–2∞}=𝐄{supπ‘‘βˆˆ[0𝑇][π‘Ÿπ‘‡(𝑑)π‘Ÿ(𝑑)]}, ‖𝑀(𝑑)β€–22=βˆ«π‘‡0𝑀T(𝑑)𝑀(𝑑)𝑑𝑑.
Then, the filter (2.3) is called the stochastic finite-time 𝐿2-𝐿∞ filter of time-delay dynamic MJLSs (2.2) with 𝛾-disturbance attenuation.

Remark 2.7. In stochastic finite-time 𝐿2-𝐿∞ filtering process, the unknown noises 𝑀(𝑑) are assumed to be arbitrary deterministic signals of bounded energy and the problem of this paper is to design a filter that guarantees a prescribed bounded for the finite-time interval induced 𝐿2-𝐿∞ norm of the operator from the unknown noise inputs 𝑀(𝑑) to the output error π‘Ÿ(𝑑), that is, the designed stochastic finite-time 𝐿2-𝐿∞ filter is supposed to satisfy inequality (2.10) with attenuation 𝛾.

3. Finite-Time 𝐿2-𝐿∞ Filtering for MJLSs

In this section, we will study the stochastic finite-time 𝐿2-𝐿∞ filtering problem for time-delay dynamic MJLSs (2.2).

Theorem 3.1. For a given time-constant 𝑇>0, the filtering error MJLSs (2.5) are stochastically FTB with respect to (𝑐1𝑐2π‘‡π‘…π‘–π‘Š) and has a prescribed 𝐿2-𝐿∞ performance level 𝛾>0 if there exist a set of mode-dependent symmetric positive-definite matrices π‘ƒπ‘–βˆˆβ„œ2𝑛×2𝑛 and symmetric positive-definite matrix π‘„βˆˆβ„œ2𝑛×2𝑛, satisfying the following matrix inequalities for all π‘–βˆˆπ‘€, βŽ‘βŽ’βŽ’βŽ’βŽ£Ξ›π‘–βˆ’π›Όπ‘ƒπ‘–π‘ƒπ‘–π΄π‘‘π‘–π‘ƒπ‘–π΅π‘–βˆ—βˆ’βˆ—π‘„0π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ¦βˆ—βˆ’πΌ<0,(3.1)𝐢𝑇𝑖𝐢𝑖<𝛾2𝑃𝑖,(3.2)𝑐1ξ€·πœŽπ‘ƒ+π‘‘πœŽπ‘„ξ€Έ+π‘Šπ›Όξ€·1βˆ’π‘’βˆ’π›Όπ‘‡ξ€Έ<π‘’βˆ’π›Όπ‘‡π‘2πœŽπ‘,(3.3) where Λ𝑖=𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+βˆ‘π‘„+𝑁𝑗=1πœ‹π‘–π‘—π‘ƒπ‘—, πœŽπ‘ƒ=minπ‘–βˆˆπ‘€πœŽmin(𝑃𝑖), πœŽπ‘ƒ=maxπ‘–βˆˆπ‘€πœŽmax(𝑃𝑖), πœŽπ‘„=maxπ‘–βˆˆπ‘€πœŽmax(𝑄𝑖), 𝑄𝑖=π‘…π‘–βˆ’1/2Qπ‘…π‘–βˆ’1/2, and 𝑃𝑖=π‘…π‘–βˆ’1/2π‘ƒπ‘–π‘…π‘–βˆ’1/2.

Proof. For the given symmetric positive-definite matrices π‘ƒπ‘–βˆˆβ„œ2𝑛×2𝑛 and Qβˆˆβ„œ2𝑛×2𝑛, we define the following stochastic Lyapunov-Krasovskii functional as 𝑉=π‘₯(𝑑),𝑖π‘₯𝑇(𝑑)π‘ƒπ‘–ξ€œπ‘₯(𝑑)+π‘‘π‘‘βˆ’π‘‘π‘₯𝑇(𝜍)𝑄π‘₯(𝜍)π‘‘πœ.(3.4)
Then referring to Definition 2.5 and along the trajectories of the resulting closed-loop MJLSs (2.7), we can derive the corresponding time derivative of 𝑉(π‘₯(𝑑),𝑖) as ℑ𝑉π‘₯ξ€Έ=(𝑑),𝑖π‘₯𝑇(𝑑)Λ𝑖π‘₯(𝑑)+2π‘₯𝑇(𝑑)𝑃𝑖𝐴𝑑𝑖π‘₯(π‘‘βˆ’π‘‘)+2π‘₯𝑇(𝑑)𝑃𝑖𝐡𝑖𝑀(𝑑)βˆ’π‘₯𝑇(π‘‘βˆ’π‘‘)𝑄π‘₯(π‘‘βˆ’π‘‘).(3.5)
Considering the 𝐿2-𝐿∞ filtering performance for the dynamic filtering error system (2.5), we introduce the following cost function by Definition 2.6 with 𝑑β‰₯0, 𝐽1ξ€½ξ€·(𝑑)=𝐄ℑ𝑉π‘₯𝑉(𝑑),π‘–ξ€Έξ€Ύβˆ’π›Όπ„π‘₯(𝑑),π‘–ξ€Έξ€Ύβˆ’π‘€π‘‡(𝑑)𝑀(𝑑).(3.6)
According to relation (3.1), it follows that 𝐽1(𝑑)<0, that is, 𝐄ℑ𝑉π‘₯𝑉(𝑑),𝑖<𝛼𝐄π‘₯(𝑑),𝑖+𝑀𝑇(𝑑)𝑀(𝑑).(3.7)
Then, multiplying the above inequality by π‘’βˆ’π›Όπ‘‘, we have β„‘ξ€½π‘’βˆ’π›Όπ‘‘π„ξ€Ίπ‘‰ξ€·π‘₯(𝑑),𝑖<π‘’βˆ’π›Όπ‘‘π‘€π‘‡(𝑑)𝑀(𝑑).(3.8)
In the following, we assume zero initial condition, that is, π‘₯(𝑑)=0, for π‘‘βˆˆ[βˆ’π‘‘0], and integrate the above inequality from 0 to 𝑇; then π‘’βˆ’π›Όπ‘‡π„ξ€Ίπ‘‰ξ€·<ξ€œπ‘₯(𝑇),𝑖𝑇0π‘’βˆ’π›Όπ‘‘π‘€π‘‡(𝑑)𝑀(𝑑)𝑑𝑑.(3.9) Recalling to the defined Lyapunov-Krasovskii functional, it can be verified that, 𝐄π‘₯𝑇(𝑇)𝑃𝑖𝑉π‘₯(𝑇)<𝐄<ξ€œπ‘₯(𝑇),𝑖𝑇0π‘’βˆ’π›Όπ‘‘π‘€π‘‡(𝑑)𝑀(𝑑)𝑑𝑑.(3.10) By (3.2) and within the finite-time interval [0𝑇], we can also get π„ξ€½π‘Ÿπ‘‡ξ€Ύξ‚†(𝑇)π‘Ÿ(𝑇)=𝐄π‘₯𝑇(𝑇)𝐢𝑇𝑖𝐢𝑖π‘₯(𝑇)<𝛾2𝐄π‘₯𝑇(𝑇)𝑃𝑖π‘₯(𝑇)=𝛾2𝑉π‘₯(𝑇),𝑖<𝛾2π‘’π›Όπ‘‡ξ€œπ‘‡0π‘’βˆ’π›Όπ‘‘π‘€π‘‡(𝑑)𝑀(𝑑)𝑑𝑑<𝛾2π‘’π›Όπ‘‡ξ€œπ‘‡0𝑀𝑇(𝑑)𝑀(𝑑)𝑑𝑑.(3.11) Therefore, the cost function inequality (2.10) can be guaranteed by setting βˆšπ›Ύ=𝑒𝛼𝑇𝛾, which implies 𝐽=E{β€–π‘Ÿ(𝑑)β€–2∞}βˆ’π›Ύ2‖𝑀(𝑑)β€–22<0.
On the other hand, by integrating the above inequality (3.8) between 0 to π‘‘βˆˆ[0𝑇], it yields π‘’βˆ’π›Όπ‘‘π„ξ€½π‘‰ξ€·ξ€½π‘‰ξ€½π‘₯(𝑑),π‘–ξ€Έξ€Ύβˆ’π„π‘₯(0),π‘Ÿπ‘‘<ξ€œ=πœ‰(0)𝑑0π‘’βˆ’π›Όπ‘ π‘€π‘‡(𝑑)𝑀(𝑑)𝑑𝑠.(3.12) Denote 𝑃𝑖=π‘…π‘–βˆ’1/2π‘ƒπ‘–π‘…π‘–βˆ’1/2, 𝑄𝑖=π‘…π‘–βˆ’1/2Qπ‘…π‘–βˆ’1/2, πœŽπ‘ƒ=minπ‘–βˆˆπ‘€πœŽmin(𝑃𝑖), πœŽπ‘ƒ=maxπ‘–βˆˆπ‘€πœŽmax(𝑃𝑖), and πœŽπ‘„=maxπ‘–βˆˆπ‘€πœŽmax(𝑄𝑖). Note that 𝛼>0, 0≀𝑑≀𝑇; then 𝐄π‘₯𝑇(𝑇)𝑃𝑖𝑉π‘₯(𝑇)≀𝐄π‘₯(𝑑),𝑖<𝑒𝛼𝑑𝐄𝑉π‘₯(0),π‘Ÿπ‘‘=πœ‰(0)ξ€Έξ€Ύ+π‘’π›Όπ‘‘ξ€œπ‘‘0π‘’βˆ’π›Όπ‘ π‘€π‘‡(𝑠)𝑀(𝑠)𝑑𝑠<𝑒𝛼𝑑𝐄𝑉π‘₯(𝑑),π‘Ÿπ‘‘||ξ€Έξ€Ύπ‘‘βˆˆ[βˆ’π‘‘0]+π‘’π›Όπ‘‘π‘Šξ€œπ‘‘0π‘’βˆ’π›Όπ‘ π‘‘π‘ <𝑒𝛼𝑇𝑐1ξ€·πœŽπ‘ƒ+π‘‘πœŽπ‘„ξ€Έ+π‘Šπ›Όξ€·1βˆ’π‘’βˆ’π›Όπ‘‡ξ€Έξ‚„.(3.13) From the selected stochastic Lyapunov-Krasovskii function, we can obtain 𝐄π‘₯𝑇(𝑑)𝑃𝑖π‘₯(𝑑)β‰₯πœŽπ‘π„ξ€Ίπ‘₯𝑇(𝑑)𝑅𝑖.π‘₯(𝑑)(3.14)
Then we can get 𝐄π‘₯𝑇(𝑑)𝑅𝑖<𝑒π‘₯(𝑑)𝛼𝑇𝑐1ξ€·πœŽπ‘ƒ+π‘‘πœŽπ‘„ξ€Έ+ξ€·(π‘Š/𝛼)1βˆ’π‘’βˆ’π›Όπ‘‡ξ€Έξ€»πœŽπ‘(3.15) which implies 𝐄[π‘₯𝑇(𝑑)𝑅𝑖π‘₯(𝑑)]<𝑐2 for βˆ€π‘‘βˆˆ[0𝑇]. This completes the proof.

Theorem 3.2. For a given time-constant 𝑇>0, the filtering error dynamic MJLSs (2.5) are FTB with respect to (𝑐1𝑐2π‘‡π‘…π‘–π‘Š) with 𝑅𝑖=diag{𝑉𝑖𝑉𝑖} and has a prescribed 𝐿2-𝐿∞performance level 𝛾>0 if there exist a set of mode-dependent symmetric positive-definite matrices π‘ƒπ‘–βˆˆβ„œπ‘›Γ—π‘›, symmetric positive-definite matrix π‘„βˆˆβ„œπ‘›Γ—π‘›, a set of mode-dependent matrices 𝑋𝑖, π‘Œπ‘–, and 𝐢𝑓𝑖 and positive scalars 𝜎1, 𝜎2 satisfying the following matrix inequalities for all π‘–βˆˆπ‘€, βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£Ξ›1π‘–βˆ—π‘ƒπ‘–π΄π‘‘π‘–0𝑃𝑖𝐡𝑖Λ2𝑖Λ3𝑖0π‘ƒπ‘–π΄π‘‘π‘–π‘ƒπ‘–π΅π‘–βˆ’π‘Œπ‘–π·π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ—βˆ—βˆ’π‘„00βˆ—βˆ—βˆ—βˆ’π‘„0βˆ—βˆ—βˆ—βˆ—βˆ’πΌ<0,(3.16)βŽ‘βŽ’βŽ’βŽ’βŽ£π‘ƒπ‘–0πΏπ‘‡π‘–βˆ’πΆπ‘‡π‘“π‘–βˆ—π‘ƒπ‘–πΆπ‘‡π‘“π‘–βˆ—βˆ—π›Ύ2𝐼⎀βŽ₯βŽ₯βŽ₯⎦>0,(3.17)𝑉𝑖<𝑃𝑖<𝜎1𝑉𝑖,(3.18)0<𝑄<𝜎2𝑉𝑖,(3.19)𝑐1𝜎1+𝑐1π‘‘πœŽ2+π‘Šπ›Όξ€·1βˆ’π‘’βˆ’π›Όπ‘‡ξ€Έ<π‘’βˆ’π›Όπ‘‡π‘2,(3.20) where Ξ›1𝑖=𝐴𝑇𝑖𝑃𝑖+π‘ƒπ‘–π΄π‘–βˆ‘+𝑄+𝑁𝑗=1πœ‹π‘–π‘—π‘ƒπ‘—βˆ’π›Όπ‘ƒπ‘–, Ξ›2𝑖=π‘ƒπ‘–π΄π‘–βˆ’π‘‹π‘–βˆ’π‘Œπ‘–πΆπ‘–, and Ξ›3𝑖=𝑋𝑇𝑖+π‘‹π‘–βˆ‘+𝑄+𝑁𝑗=1πœ‹π‘–π‘—π‘ƒπ‘—βˆ’π›Όπ‘ƒπ‘–.
Moreover, the suitable filter parameters can be given as 𝐴𝑓𝑖=π‘ƒπ‘–βˆ’1𝑋𝑖,𝐡𝑓𝑖=π‘ƒπ‘–βˆ’1π‘Œπ‘–,𝐢𝑓𝑖=𝐢𝑓𝑖.(3.21)

Proof. For convenience, we set 𝑃𝑖=diag{𝑃𝑖,𝑃𝑖}, 𝑄=diag{𝑄,𝑄}. Then inequalities (3.1) and (3.2) are equivalent to LMIs (3.16) and (3.17) by letting 𝑋𝑖=𝑃𝑖𝐴𝑓𝑖, π‘Œπ‘–=𝑃𝑖𝐡𝑓𝑖.On the other hand, by setting 𝑅𝑖=diag{𝑉𝑖𝑉𝑖}, LMIs (3.18) and (3.19) imply that 1<πœŽπ‘ƒ=minπ‘–βˆˆπ‘€πœŽmin𝑃𝑖,πœŽπ‘ƒ=maxπ‘–βˆˆπ‘€πœŽmax𝑃𝑖<𝜎1,πœŽπ‘„=maxπ‘–βˆˆπ‘€πœŽmaxξ€·π‘„π‘–ξ€Έβ‰€πœŽ2.(3.22) Then recalling condition (3.3), we can get LMI (3.20). This completes the proof.

Remark 3.3. It can be seen that if we choose the infinite time-interval, that is, π‘‡β†’βˆž, the main results in Theorems 3.1 and 3.2 can reduce to conclusions of regular 𝐿2-𝐿∞ filtering. And other filtering schemes, such as Kalman, 𝐻∞, and 𝐻2 filtering of stochastic jump systems can be also handled, referring to [9–13, 29, 32]. When the delays in MJLSs (2.2) satisfy 𝑑=0, it reduces to a delay-free system. We can immediately get the corresponding results implied in Theorems 3.1 and 3.2 by choosing the stochastic Lyapunov-Krasovskii functional as 𝑉(π‘₯(𝑑),𝑖)=π‘₯𝑇(𝑑)𝑃𝑖π‘₯(𝑑) and following the similar proofs.

4. Extension to Uncertain MJLSs

It has been recognized that the unknown disturbances and parameter uncertainties are inherent features of many physical process and often encountered in engineering systems, their presences must be considered in realistic filter design. For these, we consider the following stochastic time-delay MJLSs with uncertain parameters, ξ€Ίπ΄ξ€·π‘ŸΜ‡π‘₯(𝑑)=π‘‘ξ€Έξ€·π‘Ÿ+Δ𝐴𝑑π‘₯𝐴,𝑑(𝑑)+π‘‘ξ€·π‘Ÿπ‘‘ξ€Έ+Ξ”π΄π‘‘ξ€·π‘Ÿπ‘‘π‘₯ξ€·π‘Ÿ,𝑑(π‘‘βˆ’π‘‘)+π΅π‘‘ξ€Έπ‘€ξ€·π‘Ÿ(𝑑),𝑦(𝑑)=πΆπ‘‘ξ€Έξ€·π‘Ÿπ‘₯(𝑑)+π·π‘‘ξ€Έπ‘§ξ€·π‘Ÿπ‘€(𝑑),(𝑑)=𝐿𝑑π‘₯(𝑑),π‘₯(𝑑)=πœ†(𝑑),π‘Ÿπ‘‘ξ€Ίπ‘‘=πœ‰(𝑑),π‘‘βˆˆ0βˆ’π‘‘π‘‘0ξ€».(4.1)

Assumption. The time-varying but norm-bounded uncertainties Δ𝐴(π‘Ÿπ‘‘,𝑑) and Δ𝐴𝑑(π‘Ÿπ‘‘,𝑑) satisfy ξ€Ίξ€·π‘ŸΞ”π΄π‘‘ξ€Έ,π‘‘Ξ”π΄π‘‘ξ€·π‘Ÿπ‘‘,𝑑=𝑀𝑖Γ𝑖𝑁(𝑑)1i𝑁2iξ€»,(4.2) where 𝑀𝑖, 𝑁1𝑖, and 𝑁2𝑖 are known real constant matrices of appropriate dimensions, Γ𝑖(𝑑) is unknown, time-varying matrix function satisfying ‖Γ𝑖(𝑑)β€–2≀1, and the elements of Γ𝑖(𝑑) are Lebesgue measurable for any π‘–βˆˆπ‘€.

Remark 4.1. In condition (4.2), 𝑀𝑖 is chosen as a full row rank matrix. And the parameter uncertainty structure is an extension of the admissible condition. In fact, it is always impossible to obtain the exact mathematical model of the practical dynamics due to environmental noises, complexity process, time-varying parameters, and many measuring difficulties, and so forth. These motivate us to consider system (4.1) containing uncertainties Δ𝐴(π‘Ÿπ‘‘,𝑑) and Δ𝐴𝑑(π‘Ÿπ‘‘,𝑑). Moreover, the uncertainties Δ𝐴(π‘Ÿπ‘‘,𝑑) and Δ𝐴𝑑(π‘Ÿπ‘‘,𝑑)within (4.2) reflect the inexactness in mathematical modeling of jump dynamical systems. To simplify the study, Δ𝐴(π‘Ÿπ‘‘,𝑑) and Δ𝐴𝑑(π‘Ÿπ‘‘,𝑑) can be abbreviated as Δ𝐴𝑖 and Δ𝐴𝑑𝑖. It is necessary to point out that the unknown mode-dependent matrix Γ𝑖(𝑑) in (4.2) can also be allowed to be state-dependent, that is, Γ𝑖(𝑑)=Γ𝑖(𝑑,π‘₯(𝑑)), as long as ‖Γ𝑖(𝑑,π‘₯(𝑑))‖≀1 is satisfied.
For this case, we can get the following filtering error system by letting π‘₯𝑇π‘₯(𝑑)=[𝑇(𝑑)𝑒𝑇](𝑑): Μ‡β€Œπ‘₯(𝑑)=𝐴𝑖π‘₯(𝑑)+𝐴𝑑𝑖π‘₯(π‘‘βˆ’π‘‘)+𝐡𝑖𝑀(𝑑),π‘Ÿ(𝑑)=𝐢𝑖π‘₯(𝑑),(4.3) where 𝐴𝑖=𝐴𝑖+Δ𝐴𝑖0𝐴𝑖+Ξ”π΄π‘–βˆ’π΄π‘“π‘–βˆ’π΅π‘“π‘–πΆπ‘–π΄π‘“π‘–ξ‚Ή,𝐴𝑑𝑖=𝐴𝑑𝑖+Δ𝐴𝑑𝑖0Δ𝐴𝑑𝑖𝐴𝑑𝑖,𝐡𝑖=ξ‚Έπ΅π‘–π΅π‘–βˆ’π΅π‘“π‘–π·π‘–ξ‚Ή,𝐢𝑖=ξ€ΊπΏπ‘–βˆ’πΆπ‘“π‘–πΆπ‘“π‘–ξ€».(4.4)

Lemma 4.2 (Wang et al. [38]). Let 𝑇, 𝑀, and 𝑁 be real matrices with appropriate dimensions. Then for all time-varying unknown matrix function 𝐹(𝑑) satisfying ‖𝐹(𝑑)‖≀1, the following relation []𝑇+𝑀𝐹(𝑑)𝑁+𝑀𝐹(𝑑)𝑁𝑇<0(4.5) holds if and only if there exists a positive scalar 𝛼>0, such that 𝑇+π›Όβˆ’1𝑀𝑀𝑇+𝛼𝑁𝑇𝑁<0.(4.6)
By following the similar lines and the main proofs of Theorems 3.1 and 3.2 and using the above Lemma 4.2, one can get the results stated as follows.

Theorem 4.3. For a given time-constant 𝑇>0, the filtering error MJLSs (4.3) with uncertainties are stochastically FTB with respect to (𝑐1𝑐2π‘‡π‘‰π‘–π‘Š) and has a prescribed 𝐿2-𝐿∞ performance level 𝛾>0 if there exist a set of mode-dependent symmetric positive-definite matrices π‘ƒπ‘–βˆˆβ„œπ‘›Γ—π‘›, symmetric positive-definite matrix π‘„βˆˆβ„œπ‘›Γ—π‘›, a set of mode-dependent matrices𝑋𝑖, π‘Œπ‘–, and 𝐢𝑓𝑖 and positive scalars 𝜎1, 𝜎1and πœ€π‘– satisfying LMIs (3.17)–(3.20), and the following matrix inequalities for all π‘–βˆˆπ‘€, βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£Ξ›1𝑖+πœ€π‘–π‘π‘‡1𝑖𝑁1π‘–βˆ—π‘ƒπ‘–π΄π‘‘π‘–+πœ€π‘–π‘π‘‡1𝑖𝑁2𝑖0𝑃𝑖𝐡𝑖𝑃𝑖𝑀𝑖Λ2𝑖Λ3𝑖0π‘ƒπ‘–π΄π‘‘π‘–π‘ƒπ‘–π΅π‘–βˆ’π‘Œπ‘–π·π‘–π‘ƒπ‘–π‘€π‘–βˆ—βˆ—βˆ’π‘„+πœ€π‘–π‘π‘‡2𝑖𝑁2𝑖000βˆ—βˆ—βˆ—βˆ’π‘„00βˆ—βˆ—βˆ—βˆ—βˆ’πΌ0βˆ—βˆ—βˆ—βˆ—βˆ—βˆ’πœ€π‘–πΌβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦<0.(4.7) And the suitable stochastic finite-time 𝐿2-𝐿∞ filter can be derived by (3.21).

Remark 4.4. Theorems 3.2 and 4.3 have presented the sufficient condition of designing the stochastic finite-time 𝐿2-𝐿∞filter of time-delay MJLSs. Notice that the coupled LMIs (3.16)–(3.20) (or LMIs (4.7), (3.17)–(3.20)) are with respect to 𝑃𝑖, 𝑄, 𝑉𝑖, 𝑋𝑖, π‘Œπ‘–, 𝐢𝑓𝑖, 𝑐1, 𝑐2, 𝜎1, 𝜎2, 𝑇, π‘Š, 𝛾2, and πœ€π‘–. Therefore, for given 𝑉𝑖, 𝑐1, 𝑇, and π‘Š, we can take 𝛾2 as optimal variable, that is, to obtain an optimal stochastic finite-time 𝐿2-𝐿∞filter, the attenuation lever 𝛾2 can be reduced to the minimum possible value such that LMIs (3.16)–(3.20) (or LMIs (4.7), (3.17)–(3.20)) are satisfied. The optimization problem [39] can be described as follows: min𝑃𝑖,𝑋𝑖,π‘Œπ‘–,𝐢𝑓𝑖,𝜎1,𝜎2,𝑐2,πœ€π‘–,𝜌𝜌s.t.LMIs(3.16)-(3.20)orLMIs(4.7),(3.17)-(3.20)with𝜌=𝛾2.(4.8)

Remark 4.5. As we did in previous Remark 4.4, we can also fix 𝛾 and look for the optimal admissible 𝑐1 or 𝑐2 guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties.

5. Numeral Examples

Example 5.1. Consider a class of constant time-delay MJLSs with parameters described as follows:Mode 1
𝐴1=ξ‚Έξ‚Ήβˆ’12βˆ’3βˆ’2,𝐴𝑑1=ξ‚Έξ‚Ήβˆ’0.20βˆ’0.1βˆ’0.2,𝐡1=ξ‚Έξ‚Ή,πΆβˆ’0.20.11=ξ€Ίξ€»10.5,𝐷1=[]0.1,𝐿1=ξ€Ίξ€»;0.61(5.1)
Mode 2
𝐴2=ξ‚Έξ‚Ή03βˆ’1βˆ’2,𝐴𝑑2=ξ‚Έξ‚Ή0βˆ’0.10.20.3,𝐡2=ξ‚Έξ‚Ή,𝐢0.1βˆ’0.22=ξ€Ίξ€»11,𝐷2=[]βˆ’0.2,𝐿2=ξ€Ίξ€».0.2βˆ’1(5.2)

Let the transition rate matrix be Ξ =[βˆ’334βˆ’4]. With the initial value for 𝛼=0.5, π‘Š=2, 𝑇=4, and 𝑉𝑖=𝐼2, we fix 𝛾=0.8 and look for the optimal admissible 𝑐2 of different 𝑐1 guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties. Table 1 and Figure 1, respectively, give the optimal minimal admissible 𝑐2 with different initial upper bound 𝑐1.
For 𝑐1=1, we solve LMIs (3.16)–(3.20) by Theorem 3.2 and optimization algorithm (4.8) and get the following optimal 𝐿2-𝐿∞ filters as 𝐴𝑓1=ξ‚Έξ‚Ήβˆ’1.06950.3912βˆ’1.3394βˆ’4.2562,𝐡𝑓1=ξ‚Έξ‚Ήβˆ’1.37053.2483,𝐢𝑓1=ξ€Ίξ€»,π΄βˆ’0.30.4𝑓2=ξ‚Έξ‚Ήβˆ’3.21698.70421.5785βˆ’8.0938,𝐡𝑓2=ξ‚Έξ‚Ή1.9832βˆ’2.6324,𝐢𝑓2=ξ€Ίξ€».0.1βˆ’0.5(5.3) And then, we can also get the attenuation lever as 𝛾=0.0755.

tab1
Table 1: The optimal minimal admissible 𝑐2 with different initial upper bound 𝑐1.
839648.fig.001
Figure 1: The optimal minimal upper bound 𝑐2 with different initial 𝑐1.

Example 5.2. Consider a class of constant time-delay MJLSs with uncertain parameters described as follows: 𝑀1=ξ‚Έξ‚Ή0.10.2,𝑁11=ξ€Ίξ€»0.10,𝑁12=ξ€Ίξ€»,π‘€βˆ’0.10.122=ξ‚Έξ‚Ήβˆ’0.10.1,𝑁11=ξ€Ίξ€»0.1βˆ’0.2,𝑁12=ξ€Ίξ€».0.10.3(5.4) The modes, transition rate matrix, the matrices parameters and initial conditions are defined similarly as Example 5.1.
By solving LMIs (4.7), (3.17)–(3.20) by Theorem 4.3 and Remark 4.4, we can get the optimal value 𝛾min=0.0759, and the mode-dependent optimized 𝐿2-𝐿∞ filtering performance can be easily obtained as follows: 𝐴𝑓1=ξ‚Έξ‚Ήβˆ’1.0520βˆ’0.3571βˆ’1.3063βˆ’4.1243,𝐡𝑓1=ξ‚Έξ‚Ήβˆ’1.34183.2853,𝐢𝑓1=ξ€Ίξ€»,π΄βˆ’0.30.4𝑓2=ξ‚Έξ‚Ήβˆ’3.16648.60581.5604βˆ’8.0587,𝐡𝑓2=ξ‚Έξ‚Ή1.9745βˆ’2.6445,𝐢𝑓2=ξ€Ίξ€».0.1βˆ’0.5(5.5)
In this work, the simulation time is selected as ]π‘‘βˆˆ[010. Assume the initial conditions are π‘₯1(0)=π‘₯𝑓1(0)=1.0, π‘₯2(0)=π‘₯𝑓2(0)=0.8 and π‘Ÿ0=1. The unknown inputs are selected as 𝑀(𝑑)=1.2π‘’βˆ’0.2𝑑sin(100𝑑),𝑑β‰₯0,0,𝑑<0.(5.6)
The simulation results of jump mode (the estimation of changing between modes during the simulation with the initial mode 1), the response of system states (real states and estimated states) and filtering output error are shown in Figures 2–5, which show the effective of the proposed approaches.
It is clear from Figures 3–5 that the estimated states can track the real states smoothly. Furthermore, the presented 𝐿2-𝐿∞ filter guarantees a prescribed bounded for the induced finite-time 𝐿2-𝐿∞ norm of the operator from the unknown disturbance to the filtering output error with attenuation 𝛾=0.0759, which illustrates the effectiveness of the proposed techniques.

839648.fig.002
Figure 2: The estimation of changing between modes during the simulation with the initial mode 1.
839648.fig.003
Figure 3: The response of the system state π‘₯1(𝑑).
839648.fig.004
Figure 4: The response of the system state π‘₯2(𝑑).
839648.fig.005
Figure 5: The response of the output error π‘Ÿ(𝑑).

6. Conclusions

In the paper, we have studied the design of stochastic finite-time 𝐿2-𝐿∞ filter for uncertain time-delayed MJLSs. It ensures the finite-time stability and finite-time boundedness for the filtering error dynamic MJLSs. By selecting the appropriate Lyapunov-Krasovskii function and applying matrix transformation and variable substitution, the main results are provided in terms of LMIs form. Simulation examples demonstrate the effectiveness of the developed techniques.

Acknowledgments

This work was partially supported by National Natural Science Foundation of P. R. China (no. 60974001, 60873264, and 61070214), National Natural Science Foundation of Jiangsu Province (no. BK2009068), Six Projects Sponsoring Talent Summits of Jiangsu Province, and Program for Postgraduate Scientific Research and Innovation of Jiangsu Province.

References

  1. N. M. Krasovskii and E. A. Lidskii, β€œAnalytical design of controllers in systems with random attributes,” Automation and Remote Control Remote Control, vol. 22, pp. 1021–1025, 1141–1146, 1289–1294, 1961. View at Google Scholar
  2. D. D. Sworder and R. O. Rogers, β€œAn LQG solution to a control problem with solar thermal receiver,” IEEE Transactions on Automatic Control, vol. 28, no. 10, pp. 971–978, 1983. View at Publisher Β· View at Google Scholar Β· View at Scopus
  3. M. Athans, β€œCommand and control theory: a challenge to control science,” IEEE Transactions on Automatic Control, vol. 32, no. 4, pp. 286–293, 1987. View at Publisher Β· View at Google Scholar Β· View at Scopus
  4. N. S. D. Arrifano and V. A. Oliveira, β€œRobust H∞ fuzzy control approach for a class of Markovian jump nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 6, pp. 738–754, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus
  5. D. Yue, S. Won, and O. Kwon, β€œDelay dependent stability of neutral systems with time delay: an LMI approach,” IEE Proceedings: Control Theory and Applications, vol. 150, no. 1, pp. 23–27, 2003. View at Publisher Β· View at Google Scholar Β· View at Scopus
  6. S. He and F. Liu, β€œExponential stability for uncertain neutral systems with Markov jumps,” Journal of Control Theory and Applications, vol. 7, no. 1, pp. 35–40, 2009. View at Publisher Β· View at Google Scholar
  7. D. Liu, X. Liu, and S. Zhong, β€œDelay-dependent robust stability and control synthesis for uncertain switched neutral systems with mixed delays,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 828–839, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. X.-M. Zhang and Q.-L. Han, β€œRobust H∞ filtering for a class of uncertain linear systems with time-varying delay,” Automatica, vol. 44, no. 1, pp. 157–166, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. H. Zhao, S. Xu, and Y. Zou, β€œRobust H∞ filtering for uncertain Markovian jump systems with mode-dependent distributed delays,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 1, pp. 83–94, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  10. E. K. Boukas and Z. K. Liu, β€œRobust H∞ filtering for polytopic uncertain time-delay systems with Markov jumps,” Computers and Electrical Engineering, vol. 28, no. 3, pp. 171–193, 2002. View at Publisher Β· View at Google Scholar Β· View at Scopus
  11. M. S. Mahmoud, β€œDelay-dependent H∞ filtering of a class of switched discrete-time state delay systems,” Signal Processing, vol. 88, no. 11, pp. 2709–2719, 2008. View at Publisher Β· View at Google Scholar Β· View at Scopus
  12. S. He and F. Liu, β€œUnbiased H∞ filtering for neutral Markov jump systems,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 175–185, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. S. He and F. Liu, β€œRobust peak-to-peak filtering for Markov jump systems,” Signal Processing, vol. 90, no. 2, pp. 513–522, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  14. S. He and F. Liu, β€œControlling uncertain fuzzy neutral dynamic systems with Markov jumps,” Journal of Systems Engineering and Electronics, vol. 21, no. 3, pp. 476–484, 2010. View at Publisher Β· View at Google Scholar
  15. S. He and F. Liu, β€œFuzzy model-based fault detection for Markov jump systems,” International Journal of Robust and Nonlinear Control, vol. 19, no. 11, pp. 1248–1266, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. P. Dorato, β€œShort-time stability,” IRE Transactions on Automatic Control, vol. 6, pp. 86–86, 1961. View at Google Scholar
  17. E. G. Bakhoum and C. Toma, β€œRelativistic short range phenomena and space-time aspects of pulse measurements,” Mathematical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008. View at Google Scholar Β· View at Zentralblatt MATH
  18. G. Mattioli, M. Scalia, and C. Cattani, β€œAnalysis of large-amplitude pulses in short time intervals: application to neuron interactions,” Mathematical Problems in Engineering, vol. 2010, Article ID 895785, 15 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. L. Weiss and E. F. Infante, β€œFinite time stability under perturbing forces and on product spaces,” IEEE Transactions on Automatic Control, vol. 12, pp. 54–59, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. L. Weiss, β€œConverse theorems for finite time stability,” SIAM Journal on Applied Mathematics, vol. 16, pp. 1319–1324, 1968. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  21. W. L. Garrard, β€œFurther results on the synthesis of finite-time stable systems,” IEEE Transactions on Automatic Control, vol. 17, pp. 142–144, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  22. F. A. San Filippo and P. Dorato, β€œShort-time parameter optimization with flight control application,” Automatica, vol. 10, no. 4, pp. 425–430, 1974. View at Publisher Β· View at Google Scholar Β· View at Scopus
  23. F. Amato, M. Ariola, and C. Cosentino, β€œFinite-time stabilization via dynamic output feedback,” Automatica, vol. 42, no. 2, pp. 337–342, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  24. F. Amato, M. Ariola, and P. Dorato, β€œFinite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459–1463, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  25. F. Amato and M. Ariola, β€œFinite-time control of discrete-time linear systems,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 724–729, 2005. View at Publisher Β· View at Google Scholar
  26. F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, β€œFinite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354–1358, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  27. S. He and F. Liu, β€œRobust finite-time stabilization of uncertain fuzzy jump systems,” International Journal of Innovative Computing, Information and Control, vol. 6, pp. 3853–3862, 2010. View at Google Scholar
  28. S. He and F. Liu, β€œStochastic finite-time stabilization for uncertain jump systems via state feedback,” Journal of Dynamic Systems, Measurement and Control, vol. 132, no. 3, Article ID 034504, 4 pages, 2010. View at Publisher Β· View at Google Scholar
  29. S. He and F. Liu, β€œFinite-time H∞ filtering of time-delay stochastic jump systems with unbiased estimation,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering. In press.
  30. S. He and F. Liu, β€œObserver-based finite-time control of time-delayed jump systems,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2327–2338, 2010. View at Google Scholar
  31. H. Zhang, A. S. Mehr, and Y. Shi, β€œImproved robust energy-to-peak filtering for uncertain linear systems,” Signal Processing, vol. 90, no. 9, pp. 2667–2675, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  32. S. He and F. Liu, β€œL2-L∞ fuzzy filtering for nonlinear time-delay jump systems with uncertain parameters,” in Proceedings of the International Conference on Intelligent Systems and Knowledge Engineering, pp. 589–596, Chengdu, China, October 2007.
  33. M. Li, β€œExperimental stability analysis of a test system for doing fatigue tests under random loading,” Journal of Testing and Evaluation, vol. 34, no. 4, pp. 364–367, 2006. View at Google Scholar Β· View at Scopus
  34. X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, β€œStochastic stability properties of jump linear systems,” IEEE Transactions on Automatic Control, vol. 37, no. 1, pp. 38–53, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  35. X. Mao, β€œStability of stochastic differential equations with Markovian switching,” Stochastic Processes and Their Applications, vol. 79, no. 1, pp. 45–67, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  36. S. Y. Chen, Y. F. Li, and J. Zhang, β€œVision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167–176, 2008. View at Publisher Β· View at Google Scholar
  37. S. Y. Chen and Y. F. Li, β€œDetermination of stripe edge blurring for depth sensing,” IEEE Sensors Journal. In press. View at Publisher Β· View at Google Scholar
  38. Y. Y. Wang, L. Xie, and C. E. de Souza, β€œRobust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  39. M. Li, β€œAn optimal controller of an irregular wave maker,” Applied Mathematical Modelling, vol. 29, no. 1, pp. 55–63, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus