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. ℜ𝑛 and ℜ𝑛×𝑚 denote, respectively, the 𝑛 dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. 𝐴T and 𝐴−1 denote the matrix transpose and matrix inverse. diag{𝐴𝐵} represents the block-diagonal matrix of 𝐴 and 𝐵. If 𝐴 is a symmetric matrix, denote by 𝜎min(𝐴) and 𝜎max(𝐴) its smallest and largest eigenvalues, respectively. ∑𝑁𝑖<𝑗denotes, for example, for 𝑁=3, ∑𝑁𝑖<𝑗𝑎𝑖𝑗⇔𝑎12+𝑎13+𝑎23. 𝐄{∗} stands for the mathematics statistical expectation of the stochastic process or vector and ‖∗‖ 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 “∗” 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 (Ω,𝐹,𝑃𝑟) where Ω is the sample space, 𝐹 is the algebra of events and 𝑃𝑟 is the probability measure defined on 𝐹. Let the random form process {𝑟𝑡,𝑡≥0} be the continuous-time discrete-state Markov stochastic process taking values in a finite set 𝑀={1,2,…,𝑁} with transition probability matrix 𝑃𝑟={𝑃𝑖𝑗(𝑡),𝑖,𝑗∈𝑀} given by𝑃𝑟=𝑃𝑖𝑗𝑟(𝑡)=𝑃𝑡+Δ𝑡=𝑗∣𝑟𝑡=𝜋=𝑖𝑖𝑗Δ𝑡+𝑜(Δ𝑡),𝑖≠𝑗,1+𝜋𝑖𝑖Δ𝑡+𝑜(Δ𝑡),𝑖=𝑗,(2.1) where Δ𝑡>0 and limΔ𝑡→0𝑜(Δ𝑡)/Δ𝑡→0. 𝜋𝑖𝑗≥0 is the transition probability rates from mode 𝑖 at time 𝑡 to mode 𝑗(𝑖≠𝑗) at time 𝑡+Δ𝑡, and ∑𝑁𝑗=1,𝑗≠𝑖𝜋𝑖𝑗=−𝜋𝑖𝑖.

Consider the following time-delay dynamic MJLSs over the probability space (Ω,𝐹,𝑃𝑟):𝑟̇𝑥(𝑡)=𝐴𝑡𝑥(𝑡)+𝐴𝑑𝑟𝑡𝑥𝑟(𝑡−𝑑)+𝐵𝑡𝑤𝑟(𝑡),𝑦(𝑡)=𝐶𝑡𝑟𝑥(𝑡)+𝐷𝑡𝑧𝑟𝑤(𝑡),(𝑡)=𝐿𝑡𝑥(𝑡),𝑥(𝑡)=𝜆(𝑡),𝑟𝑡𝑡=𝜉(𝑡),𝑡∈0−𝑑𝑡0,(2.2) where 𝑥(𝑡)∈ℜ𝑛 is the state, 𝑦(𝑡)∈ℜ𝑙 is the measured output, 𝑤(𝑡)∈ℜ𝑝 is the unknown input, 𝑧(𝑡)∈ℜ𝑞 is the controlled output, 𝑑>0 is the constant time-delay, 𝜎(𝑡) is a vector-valued initial continuous function defined on the interval [𝑡0−𝑑𝑡0], and 𝜉(𝑡) is the initial mode. 𝐴(𝑟𝑡), 𝐴𝑑(𝑟𝑡), 𝐵(𝑟𝑡), 𝐶(𝑟𝑡), 𝐷(𝑟𝑡), and 𝐿(𝑟𝑡) are known mode-dependent constant matrices with appropriate dimensions, and 𝑟𝑡 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 𝑟𝑡=𝑖, we denote 𝐴(𝑟𝑡), 𝐴𝑑(𝑟𝑡), 𝐵(𝑟𝑡), 𝐶(𝑟𝑡), 𝐷(𝑟𝑡), and 𝐿(𝑟𝑡) as 𝐴𝑖, 𝐴𝑑𝑖, 𝐵𝑖, 𝐶𝑖, 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 𝑡0=0 and let the initial values {𝜆(𝑡)}𝑡∈[−𝑑0] and {𝜉(𝑡)=𝑟𝑡}𝑡∈[−𝑑0] 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) aṡ̂𝑥(𝑡)=𝐴𝑓𝑖̂𝑥(𝑡)+𝐴𝑑𝑖̂𝑥(𝑡−𝑑)+𝐵𝑓𝑖𝑦(𝑡),̂𝑧(𝑡)=𝐶𝑓𝑖̂𝑥(𝑡),̂𝑥(𝑡)=𝜑(𝑡),𝑟𝑡,=𝜉(𝑡),𝑡∈−𝑑0(2.3) where ̂𝑥(𝑡)∈ℜ𝑛 is the filter state, ̂𝑧(𝑡)∈ℜ𝑞 is the filter output, 𝜑(𝑡) is a continuous vector-valued initial function, and the mode-dependent matrices 𝐴𝑓𝑖, 𝐵𝑓𝑖, and 𝐶𝑓𝑖 are unknown filter parameters to be designed for each value 𝑖∈𝑀.

Define 𝑒(𝑡)=𝑥(𝑡)−̂𝑥(𝑡) and 𝑟(𝑡)=𝑧(𝑡)−̂𝑧(𝑡), then we can get the following filtering error system:𝐴̇𝑒(𝑡)=𝑖−𝐵𝑓𝑖𝐶𝑖𝑥(𝑡)−𝐴𝑓𝑖̂𝑥(𝑡)+𝐴𝑑𝑖[𝑥]+𝐵(𝑡−𝑑)−̂𝑥(𝑡−𝑑)𝑖−𝐵𝑓𝑖𝐷𝑖𝑑𝑟(𝑡),(𝑡)=𝐿𝑖𝑥(𝑡)−𝐶𝑓𝑖̂𝑥(𝑡).(2.4) Let 𝑥𝑇(𝑡)=[𝑥𝑇(𝑡)𝑒𝑇(𝑡)], the filtering error system (2.4) can be rewritten aṡ‌𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐴𝑑𝑖𝑥(𝑡−𝑑)+𝐵𝑖𝑤(𝑡),𝑟(𝑡)=𝐶𝑖𝑥(𝑡),𝑥𝑇(𝑡)=𝜆(𝑡)𝜆(𝑡)−𝜑(𝑡),𝑟𝑡[],=𝜉(𝑡),𝑡∈−𝑑0(2.5) where𝐴𝑖=𝐴𝑖0𝐴𝑖−𝐴𝑓𝑖−𝐵𝑓𝑖𝐶𝑖𝐴𝑓𝑖,𝐴𝑑𝑖=𝐴𝑑𝑖00𝐴𝑑𝑖,𝐵𝑖=𝐵𝑖𝐵𝑖−𝐵𝑓𝑖𝐷𝑖,𝐶𝑖=𝐿𝑖−𝐶𝑓𝑖𝐶𝑓𝑖.(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 ̂‌z(𝑡) of the signal 𝑧(𝑡) 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 𝑤(𝑡) is time-varying and satisfies 𝑇0𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑡≤𝑊,𝑊≥0.(2.7)

Definition 2.2. For a given time-constant 𝑇>0, the filtering error MJLSs system (2.5) with 𝑤(𝑡)=0 is stochastically finite-time stable (FTS) if there exist positive matrix 𝑅𝑖∈ℜ2𝑛×2𝑛>0 and scalars 𝑐1>0 and 𝑐2>0, such that 𝐄𝑥𝑇𝑡1𝑅𝑖𝑥𝑡1≤𝑐1⟹𝐄𝑥𝑇𝑡2𝑅𝑖𝑥𝑡2<𝑐2,𝑡1∈−𝑑0,𝑡2].∈(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 (𝑐1𝑐2𝑇𝑅𝑖𝑊) 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 𝑊=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(𝑥(𝑡),𝑟𝑡,𝑡>0) be the stochastic positive functional; define its weak infinitesimal operator as ℑ𝑉𝑥(𝑡),𝑟𝑡=𝑖,𝑡=limΔ𝑡→𝟎1𝐄𝑉Δ𝑡𝑥(𝑡+Δ𝑡),𝑟𝑡+Δ𝑡,𝑡+Δ𝑡∣𝑥(𝑡),𝑟𝑡=𝑖−𝑉𝑥(𝑡),𝑟𝑡.=𝑖,𝑡(2.9)

Definition 2.6. For the filtering error MJLSs (2.5), if there exist filter parameters 𝐴𝑓𝑖, 𝐵𝑓𝑖, and 𝐶𝑓𝑖, 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 𝑤(𝑡) with the constraint condition (2.7), ‖𝐽=𝐄‖𝑟(𝑡)2∞−𝛾2‖‖𝑑(𝑡)22<0,(2.10) where 𝐄{‖𝑟(𝑡)‖2∞}=𝐄{sup𝑡∈[0𝑇][𝑟𝑇(𝑡)𝑟(𝑡)]}, ‖𝑤(𝑡)‖22=∫𝑇0𝑤T(𝑡)𝑤(𝑡)𝑑𝑡.
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 𝑤(𝑡) 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 𝑤(𝑡) to the output error 𝑟(𝑡), 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 (𝑐1𝑐2𝑇𝑅𝑖𝑊) and has a prescribed 𝐿2-𝐿∞ performance level 𝛾>0 if there exist a set of mode-dependent symmetric positive-definite matrices 𝑃𝑖∈ℜ2𝑛×2𝑛 and symmetric positive-definite matrix 𝑄∈ℜ2𝑛×2𝑛, satisfying the following matrix inequalities for all 𝑖∈𝑀, ⎡⎢⎢⎢⎣Λ𝑖−𝛼𝑃𝑖𝑃𝑖𝐴𝑑𝑖𝑃𝑖𝐵𝑖∗−∗𝑄0𝑖⎤⎥⎥⎥⎦∗−𝐼<0,(3.1)𝐶𝑇𝑖𝐶𝑖<𝛾2𝑃𝑖,(3.2)𝑐1𝜎𝑃+𝑑𝜎𝑄+𝑊𝛼1−𝑒−𝛼𝑇<𝑒−𝛼𝑇𝑐2𝜎𝑝,(3.3) where Λ𝑖=𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+∑𝑄+𝑁𝑗=1𝜋𝑖𝑗𝑃𝑗, 𝜎𝑃=min𝑖∈𝑀𝜎min(𝑃𝑖), 𝜎𝑃=max𝑖∈𝑀𝜎max(𝑃𝑖), 𝜎𝑄=max𝑖∈𝑀𝜎max(𝑄𝑖), 𝑄𝑖=𝑅𝑖−1/2Q𝑅𝑖−1/2, and 𝑃𝑖=𝑅𝑖−1/2𝑃𝑖𝑅𝑖−1/2.

Proof. For the given symmetric positive-definite matrices 𝑃𝑖∈ℜ2𝑛×2𝑛 and Q∈ℜ2𝑛×2𝑛, we define the following stochastic Lyapunov-Krasovskii functional as 𝑉=𝑥(𝑡),𝑖𝑥𝑇(𝑡)𝑃𝑖𝑥(𝑡)+𝑡𝑡−𝑑𝑥𝑇(𝜍)𝑄𝑥(𝜍)𝑑𝜍.(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(𝑥(𝑡),𝑖) as ℑ𝑉𝑥=(𝑡),𝑖𝑥𝑇(𝑡)Λ𝑖𝑥(𝑡)+2𝑥𝑇(𝑡)𝑃𝑖𝐴𝑑𝑖𝑥(𝑡−𝑑)+2𝑥𝑇(𝑡)𝑃𝑖𝐵𝑖𝑤(𝑡)−𝑥𝑇(𝑡−𝑑)𝑄𝑥(𝑡−𝑑).(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 𝑡≥0, 𝐽1(𝑡)=𝐄ℑ𝑉𝑥𝑉(𝑡),𝑖−𝛼𝐄𝑥(𝑡),𝑖−𝑤𝑇(𝑡)𝑤(𝑡).(3.6)
According to relation (3.1), it follows that 𝐽1(𝑡)<0, that is, 𝐄ℑ𝑉𝑥𝑉(𝑡),𝑖<𝛼𝐄𝑥(𝑡),𝑖+𝑤𝑇(𝑡)𝑤(𝑡).(3.7)
Then, multiplying the above inequality by 𝑒−𝛼𝑡, we have ℑ𝑒−𝛼𝑡𝐄𝑉𝑥(𝑡),𝑖<𝑒−𝛼𝑡𝑤𝑇(𝑡)𝑤(𝑡).(3.8)
In the following, we assume zero initial condition, that is, 𝑥(𝑡)=0, for 𝑡∈[−𝑑0], and integrate the above inequality from 0 to 𝑇; then 𝑒−𝛼𝑇𝐄𝑉<𝑥(𝑇),𝑖𝑇0𝑒−𝛼𝑡𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑡.(3.9) Recalling to the defined Lyapunov-Krasovskii functional, it can be verified that, 𝐄𝑥𝑇(𝑇)𝑃𝑖𝑉𝑥(𝑇)<𝐄<𝑥(𝑇),𝑖𝑇0𝑒−𝛼𝑡𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑡.(3.10) By (3.2) and within the finite-time interval [0𝑇], we can also get 𝐄𝑟𝑇(𝑇)𝑟(𝑇)=𝐄𝑥𝑇(𝑇)𝐶𝑇𝑖𝐶𝑖𝑥(𝑇)<𝛾2𝐄𝑥𝑇(𝑇)𝑃𝑖𝑥(𝑇)=𝛾2𝑉𝑥(𝑇),𝑖<𝛾2𝑒𝛼𝑇𝑇0𝑒−𝛼𝑡𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑡<𝛾2𝑒𝛼𝑇𝑇0𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑡.(3.11) Therefore, the cost function inequality (2.10) can be guaranteed by setting √𝛾=𝑒𝛼𝑇𝛾, which implies 𝐽=E{‖𝑟(𝑡)‖2∞}−𝛾2‖𝑤(𝑡)‖22<0.
On the other hand, by integrating the above inequality (3.8) between 0 to 𝑡∈[0𝑇], it yields 𝑒−𝛼𝑡𝐄𝑉𝑉𝑥(𝑡),𝑖−𝐄𝑥(0),𝑟𝑡<=𝜉(0)𝑡0𝑒−𝛼𝑠𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑠.(3.12) Denote 𝑃𝑖=𝑅𝑖−1/2𝑃𝑖𝑅𝑖−1/2, 𝑄𝑖=𝑅𝑖−1/2Q𝑅𝑖−1/2, 𝜎𝑃=min𝑖∈𝑀𝜎min(𝑃𝑖), 𝜎𝑃=max𝑖∈𝑀𝜎max(𝑃𝑖), and 𝜎𝑄=max𝑖∈𝑀𝜎max(𝑄𝑖). Note that 𝛼>0, 0≤𝑡≤𝑇; then 𝐄𝑥𝑇(𝑇)𝑃𝑖𝑉𝑥(𝑇)≤𝐄𝑥(𝑡),𝑖<𝑒𝛼𝑡𝐄𝑉𝑥(0),𝑟𝑡=𝜉(0)+𝑒𝛼𝑡𝑡0𝑒−𝛼𝑠𝑤𝑇(𝑠)𝑤(𝑠)𝑑𝑠<𝑒𝛼𝑡𝐄𝑉𝑥(𝑡),𝑟𝑡||𝑡∈[−𝑑0]+𝑒𝛼𝑡𝑊𝑡0𝑒−𝛼𝑠𝑑𝑠<𝑒𝛼𝑇𝑐1𝜎𝑃+𝑑𝜎𝑄+𝑊𝛼1−𝑒−𝛼𝑇.(3.13) From the selected stochastic Lyapunov-Krasovskii function, we can obtain 𝐄𝑥𝑇(𝑡)𝑃𝑖𝑥(𝑡)≥𝜎𝑝𝐄𝑥𝑇(𝑡)𝑅𝑖.𝑥(𝑡)(3.14)
Then we can get 𝐄𝑥𝑇(𝑡)𝑅𝑖<𝑒𝑥(𝑡)𝛼𝑇𝑐1𝜎𝑃+𝑑𝜎𝑄+(𝑊/𝛼)1−𝑒−𝛼𝑇𝜎𝑝(3.15) which implies 𝐄[𝑥𝑇(𝑡)𝑅𝑖𝑥(𝑡)]<𝑐2 for ∀𝑡∈[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 (𝑐1𝑐2𝑇𝑅𝑖𝑊) 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 𝑃𝑖∈ℜ𝑛×𝑛, symmetric positive-definite matrix 𝑄∈ℜ𝑛×𝑛, a set of mode-dependent matrices 𝑋𝑖, 𝑌𝑖, and 𝐶𝑓𝑖 and positive scalars 𝜎1, 𝜎2 satisfying the following matrix inequalities for all 𝑖∈𝑀, ⎡⎢⎢⎢⎢⎢⎣Λ1𝑖∗𝑃𝑖𝐴𝑑𝑖0𝑃𝑖𝐵𝑖Λ2𝑖Λ3𝑖0𝑃𝑖𝐴𝑑𝑖𝑃𝑖𝐵𝑖−𝑌𝑖𝐷𝑖⎤⎥⎥⎥⎥⎥⎦∗∗−𝑄00∗∗∗−𝑄0∗∗∗∗−𝐼<0,(3.16)⎡⎢⎢⎢⎣𝑃𝑖0𝐿𝑇𝑖−𝐶𝑇𝑓𝑖∗𝑃𝑖𝐶𝑇𝑓𝑖∗∗𝛾2𝐼⎤⎥⎥⎥⎦>0,(3.17)𝑉𝑖<𝑃𝑖<𝜎1𝑉𝑖,(3.18)0<𝑄<𝜎2𝑉𝑖,(3.19)𝑐1𝜎1+𝑐1𝑑𝜎2+𝑊𝛼1−𝑒−𝛼𝑇<𝑒−𝛼𝑇𝑐2,(3.20) where Λ1𝑖=𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖∑+𝑄+𝑁𝑗=1𝜋𝑖𝑗𝑃𝑗−𝛼𝑃𝑖, Λ2𝑖=𝑃𝑖𝐴𝑖−𝑋𝑖−𝑌𝑖𝐶𝑖, and Λ3𝑖=𝑋𝑇𝑖+𝑋𝑖∑+𝑄+𝑁𝑗=1𝜋𝑖𝑗𝑃𝑗−𝛼𝑃𝑖.
Moreover, the suitable filter parameters can be given as 𝐴𝑓𝑖=𝑃𝑖−1𝑋𝑖,𝐵𝑓𝑖=𝑃𝑖−1𝑌𝑖,𝐶𝑓𝑖=𝐶𝑓𝑖.(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, 𝑌𝑖=𝑃𝑖𝐵𝑓𝑖.On the other hand, by setting 𝑅𝑖=diag{𝑉𝑖𝑉𝑖}, LMIs (3.18) and (3.19) imply that 1<𝜎𝑃=min𝑖∈𝑀𝜎min𝑃𝑖,𝜎𝑃=max𝑖∈𝑀𝜎max𝑃𝑖<𝜎1,𝜎𝑄=max𝑖∈𝑀𝜎max𝑄𝑖≤𝜎2.(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, 𝑇→∞, 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(𝑥(𝑡),𝑖)=𝑥𝑇(𝑡)𝑃𝑖𝑥(𝑡) 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, 𝐴𝑟̇𝑥(𝑡)=𝑡𝑟+Δ𝐴𝑡𝑥𝐴,𝑡(𝑡)+𝑑𝑟𝑡+Δ𝐴𝑑𝑟𝑡𝑥𝑟,𝑡(𝑡−𝑑)+𝐵𝑡𝑤𝑟(𝑡),𝑦(𝑡)=𝐶𝑡𝑟𝑥(𝑡)+𝐷𝑡𝑧𝑟𝑤(𝑡),(𝑡)=𝐿𝑡𝑥(𝑡),𝑥(𝑡)=𝜆(𝑡),𝑟𝑡𝑡=𝜉(𝑡),𝑡∈0−𝑑𝑡0.(4.1)

Assumption. The time-varying but norm-bounded uncertainties Δ𝐴(𝑟𝑡,𝑡) and Δ𝐴𝑑(𝑟𝑡,𝑡) satisfy 𝑟Δ𝐴𝑡,𝑡Δ𝐴𝑑𝑟𝑡,𝑡=𝑀𝑖Γ𝑖𝑁(𝑡)1i𝑁2i,(4.2) where 𝑀𝑖, 𝑁1𝑖, and 𝑁2𝑖 are known real constant matrices of appropriate dimensions, Γ𝑖(𝑡) is unknown, time-varying matrix function satisfying ‖Γ𝑖(𝑡)‖2≤1, and the elements of Γ𝑖(𝑡) are Lebesgue measurable for any 𝑖∈𝑀.

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(𝑟𝑡,𝑡) and Δ𝐴𝑑(𝑟𝑡,𝑡). Moreover, the uncertainties Δ𝐴(𝑟𝑡,𝑡) and Δ𝐴𝑑(𝑟𝑡,𝑡)within (4.2) reflect the inexactness in mathematical modeling of jump dynamical systems. To simplify the study, Δ𝐴(𝑟𝑡,𝑡) and Δ𝐴𝑑(𝑟𝑡,𝑡) can be abbreviated as Δ𝐴𝑖 and Δ𝐴𝑑𝑖. It is necessary to point out that the unknown mode-dependent matrix Γ𝑖(𝑡) in (4.2) can also be allowed to be state-dependent, that is, Γ𝑖(𝑡)=Γ𝑖(𝑡,𝑥(𝑡)), as long as ‖Γ𝑖(𝑡,𝑥(𝑡))‖≤1 is satisfied.
For this case, we can get the following filtering error system by letting 𝑥𝑇𝑥(𝑡)=[𝑇(𝑡)𝑒𝑇](𝑡): ̇‌𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐴𝑑𝑖𝑥(𝑡−𝑑)+𝐵𝑖𝑤(𝑡),𝑟(𝑡)=𝐶𝑖𝑥(𝑡),(4.3) where 𝐴𝑖=𝐴𝑖+Δ𝐴𝑖0𝐴𝑖+Δ𝐴𝑖−𝐴𝑓𝑖−𝐵𝑓𝑖𝐶𝑖𝐴𝑓𝑖,𝐴𝑑𝑖=𝐴𝑑𝑖+Δ𝐴𝑑𝑖0Δ𝐴𝑑𝑖𝐴𝑑𝑖,𝐵𝑖=𝐵𝑖𝐵𝑖−𝐵𝑓𝑖𝐷𝑖,𝐶𝑖=𝐿𝑖−𝐶𝑓𝑖𝐶𝑓𝑖.(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(𝑡) satisfying ‖𝐹(𝑡)‖≤1, the following relation []𝑇+𝑀𝐹(𝑡)𝑁+𝑀𝐹(𝑡)𝑁𝑇<0(4.5) holds if and only if there exists a positive scalar 𝛼>0, such that 𝑇+𝛼−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 (𝑐1𝑐2𝑇𝑉𝑖𝑊) and has a prescribed 𝐿2-𝐿∞ performance level 𝛾>0 if there exist a set of mode-dependent symmetric positive-definite matrices 𝑃𝑖∈ℜ𝑛×𝑛, symmetric positive-definite matrix 𝑄∈ℜ𝑛×𝑛, a set of mode-dependent matrices𝑋𝑖, 𝑌𝑖, and 𝐶𝑓𝑖 and positive scalars 𝜎1, 𝜎1and 𝜀𝑖 satisfying LMIs (3.17)–(3.20), and the following matrix inequalities for all 𝑖∈𝑀, ⎡⎢⎢⎢⎢⎢⎢⎢⎣Λ1𝑖+𝜀𝑖𝑁𝑇1𝑖𝑁1𝑖∗𝑃𝑖𝐴𝑑𝑖+𝜀𝑖𝑁𝑇1𝑖𝑁2𝑖0𝑃𝑖𝐵𝑖𝑃𝑖𝑀𝑖Λ2𝑖Λ3𝑖0𝑃𝑖𝐴𝑑𝑖𝑃𝑖𝐵𝑖−𝑌𝑖𝐷𝑖𝑃𝑖𝑀𝑖∗∗−𝑄+𝜀𝑖𝑁𝑇2𝑖𝑁2𝑖000∗∗∗−𝑄00∗∗∗∗−𝐼0∗∗∗∗∗−𝜀𝑖𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎦<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, 𝑌𝑖, 𝐶𝑓𝑖, 𝑐1, 𝑐2, 𝜎1, 𝜎2, 𝑇, 𝑊, 𝛾2, and 𝜀𝑖. Therefore, for given 𝑉𝑖, 𝑐1, 𝑇, and 𝑊, 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,𝑌𝑖,𝐶𝑓𝑖,𝜎1,𝜎2,𝑐2,𝜀𝑖,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=−12−3−2,𝐴𝑑1=−0.20−0.1−0.2,𝐵1=,𝐶−0.20.11=10.5,𝐷1=[]0.1,𝐿1=;0.61(5.1)Mode 2
𝐴2=03−1−2,𝐴𝑑2=0−0.10.20.3,𝐵2=,𝐶0.1−0.22=11,𝐷2=[]−0.2,𝐿2=.0.2−1(5.2)
Let the transition rate matrix be Π=[−334−4]. With the initial value for 𝛼=0.5, 𝑊=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=−1.06950.3912−1.3394−4.2562,𝐵𝑓1=−1.37053.2483,𝐶𝑓1=,𝐴−0.30.4𝑓2=−3.21698.70421.5785−8.0938,𝐵𝑓2=1.9832−2.6324,𝐶𝑓2=.0.1−0.5(5.3) And then, we can also get the attenuation lever as 𝛾=0.0755.

Table 1 
The optimal minimal admissible 𝑐2 with different initial upper bound 𝑐1.

Figure 1 
The optimal minimal upper bound 𝑐 2 with different initial 𝑐 1 .

Example 5.2. Consider a class of constant time-delay MJLSs with uncertain parameters described as follows: 𝑀1=0.10.2,𝑁11=0.10,𝑁12=,𝑀−0.10.122=−0.10.1,𝑁11=0.1−0.2,𝑁12=.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=−1.0520−0.3571−1.3063−4.1243,𝐵𝑓1=−1.34183.2853,𝐶𝑓1=,𝐴−0.30.4𝑓2=−3.16648.60581.5604−8.0587,𝐵𝑓2=1.9745−2.6445,𝐶𝑓2=.0.1−0.5(5.5)
In this work, the simulation time is selected as ]𝑡∈[010. Assume the initial conditions are 𝑥1(0)=𝑥𝑓1(0)=1.0, 𝑥2(0)=𝑥𝑓2(0)=0.8 and 𝑟0=1. The unknown inputs are selected as 𝑤(𝑡)=1.2𝑒−0.2𝑡sin(100𝑡),𝑡≥0,0,𝑡<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.


Figure 2 
The estimation of changing between modes during the simulation with the initial mode 1.

Figure 3 
The response of the system state 𝑥 1 ( 𝑡 ) .

Figure 4 
The response of the system state 𝑥 2 ( 𝑡 ) .

Figure 5 
The response of the output error 𝑟 ( 𝑡 ) .

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.