Finite-Frequency Filter Design for Networked Control Systems with Missing Measurements

Abstract

This paper is concerned with the problem of robust filter design for networked control systems (NCSs) with random missing measurements. Different from existing robust filters, the proposed one is designed in finite-frequency domain. With consideration of possible missing data, the NCSs are first modeled to Markov jump systems (MJSs). A finite-frequency stochastic 𝐻_∞ performance is subsequently given that extends the standard 𝐻_∞ performance, and then a sufficient condition guaranteeing the system to be with such a performance is derived in terms of linear matrix inequality (LMI). With the aid of this condition, a procedure of filter synthesis is proposed to deal with noises in the low-, middle-, and high-frequency domains, respectively. Finally, an example about the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) is carried out to illustrate the effectiveness of the proposed method.

1. Introduction

Due to the rapid development in communication network and computer technology, networked control systems (NCSs) have received much more research attentions. Networked control systems (NCSs) are control systems in which controller and plant are connected via a communication channel. The defining feature of an NCS is that information (reference input, plant output, control input, etc.) is exchanged using a network among control system components (sensors, controller, actuators, etc.). NCSs have many advantages such as easy diagnosis, low cost, and high mobility. And thus NCSs have been applied in many industrial systems such as automobiles, manufacturing plants, and aircrafts; see, for example, [1, 2]. Motivated by this wide spectrum of applications, the new problems arising from the limit resources of the communication channel are gradually taken into account when designing the NCSs. For example, the issues of network-induced delay, packet dropout, and quantization are considered in [3–12], respectively.

On the other hand, state estimation has been widely studied and has found many practical applications over the past decades [13]. When a priori statistical information on the external noise signals is unknown, the celebrated Kalman filtering cannot be employed. To address this issue, 𝐻_∞ filtering is introduced, which aims to make the worst case 𝐻_∞ norm from the process noise to the estimation error minimized. More recently, there have appeared a few results on 𝐻_∞ filter design [14–18]. However, it should be noticed that all the aforementioned methods are proposed in full-frequency domain. Nevertheless, practical industry systems often employ large, complex, or lightweight structures, which include finite-frequency fundamental vibration modes [19]. In these situations, it is more reasonable and precise to design filters in finite-frequency domain. Fortunately, the Kalman-Yakubovich-Popov (KYP) lemma is generalized in [20] to characterize frequency domain inequalities with (semi)finite-frequency ranges in terms of linear matrix inequalities (LMIs). The generalized KYP lemma is an effective tool to deal with the finite-frequency problem of linear time-invariant systems [20–26]. There are some results concerned with this meaningful problem, for example, [27] proposed an effective filter for fuzzy nonlinear systems. However, to the best of the author’s knowledge, for system engaging random packet dropout, few results have been published for such class of NCSs. This instance motivates our present investigation.

This paper studies the robust filter design problem with finite-frequency specifications for networked control systems (NCSs) subject to random missing measurements. First, the considered systems are modeled in the framework of Markov jump systems (MJSs). Motivated by Iwasaki et al. [22], a definition of finite-frequency stochastic 𝐻_∞ norm is subsequently given to measure the robustness, which extends the standard 𝐻_∞ norm and contains the frequency information of noises. Then based on Projection Lemma, an analysis condition is presented to guarantee the MJS being with such a performance in the framework of linear matrix inequalities (LMIs). Further, a procedure of filter synthesis is designed to deal with noises in low-, middle-, and high-frequency domains, respectively. Finally, an example about the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) is given to illustrate the effectiveness of the proposed method.

The rest of the paper is organized as follows. The problem statement for NCSs with random packet dropout is formulated in Section 2. Section 3 provides sufficient condition to meet the performance request and a design procedure of the robust filter. In Section 4, an example is given to illustrate the effectiveness of the proposed method. Finally, some conclusions end the paper in Section 5.

Notations. Throughout the paper, the superscripts 𝑇 and −1 stand for, respectively, the transposition and the inverse of a matrix; 𝑀>0 means that 𝑀 is real symmetric and positive definite; ‖⋅‖ denotes the Euclidean norm; 𝑙₂ denotes the Hilbert space of square integrable functions. In block symmetric matrices or long matrix expressions, we use ∗ to represent a term that is induced by symmetry. The sum of a square matrix 𝐴 and its transposition 𝐴^𝑇 is denoted by 𝐻𝑒(𝐴)∶=𝐴+𝐴^𝑇.

2. Problem Formulation

Considering the NCS depicted in Figure 1, the continuous-time plant model is 𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡)(𝑡),𝑧(𝑡)(𝑡),=𝐴𝑥+𝐵𝑤=𝐶𝑥+𝐷𝑤=𝐸𝑥 (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state, 𝑦(𝑡)∈ℝ^𝑚 is the measured output, 𝑧(𝑡)∈ℝ^𝑝 is the controlled output, and 𝑤(𝑡)∈ℝ^𝑑 is the exogenous disturbance which belongs to 𝑙₂[0,∞). 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 are known real constant matrices with appropriate dimensions.

Figure 1 

The structure of the NCSs.

It is assumed that, as shown in Figure 1, the measurement signals will be transmitted via the networks wherein missing data may occur. Further, assume the interval between the 𝑘th and the (𝑘+1)th successfully received measurements at the filter is 𝛼_𝑘ℎ, where ℎ is the sampling period of the sensor. It is obvious that the number of the missed packets at time instant (𝑘+1)ℎ is 𝛼_𝑘−1, which can be modeled by a time-homogeneous Markov chain 𝛼_𝑘 with the range set 𝕊={1,2,…,𝑁} and the transition probability matrix Λ=𝜆_{𝑖𝑗𝑖,𝑗∈𝕊}=𝐏𝛼_𝑘+1=𝑗∣𝛼_𝑘=𝑖_{𝑖,𝑗∈𝕊}=𝜆₁₁𝜆₁₂…𝜆_1𝑁𝜆₂₁𝜆₂₂…𝜆_2𝑁𝜆⋮⋮⋱⋮_𝑁1𝜆_𝑁2…𝜆_𝑁𝑁. (2) In this situation, the dynamics of plant (1) together with the missing measurements at time instant 𝑡_𝑘 can be approximated by 𝑥(𝑘+1)=𝐴_𝛼𝑘𝑥(𝑘)+𝐵_𝛼𝑘𝑤(𝑘),𝑦_𝑟(𝑘)(𝑘)(𝑘),𝑧(𝑘)(𝑘),=𝐶𝑥+𝐷𝑤=𝐸𝑥 (3) where 𝑡₀=0, 𝑡_𝑘=∑^𝑘−1_𝑖=0𝛼_𝑖ℎ(𝑘≥1), 𝑥(𝑘)=𝑥(𝑡_𝑘), 𝐴_𝛼𝑘=𝑒^{𝐴𝑐(𝛼𝑘ℎ)}, and 𝐵_𝛼𝑘=∫^𝛼𝑘ℎ₀𝑒^𝐴𝑐𝑡𝐵_𝑐𝑑𝑡. It can be seen that, after the above treatment, the possible missing measurements can be converted to the jumping parameter of the MJS (3) with the transition probability Λ.

In this paper, the filter is chosen as the following form: 𝑥(𝑘+1)=𝐴_𝑓𝛼𝑘𝑥(𝑘)+𝐵_𝑓𝛼𝑘𝑦_𝑟(𝑘),𝑧(𝑘)=𝐶_𝑓𝛼𝑘𝑥(𝑘)+𝐷_𝑓𝛼𝑘𝑦_𝑟(𝑘), (4) where 𝑥(𝑘)∈ℝ^𝑛 is the filter's state, 𝑧(𝑘) is the estimated output, and 𝐴_𝑓𝛼𝑘, 𝐵_𝑓𝛼𝑘, 𝐶_𝑓𝛼𝑘, and 𝐷_𝑓𝛼𝑘 are filter gains to be designed.

To ensure the achievement of filter design objective, a basic assumption, that is, 𝐴_𝑐 is stable, is also assumed to be valid.

Remark 1. This assumption is required to get a stable filtering error dynamics. If this assumption is not satisfied, a stabilizing output feedback controller is required.

For convenience, 𝐴_𝛼𝑘, 𝐵_𝛼𝑘, 𝐴_𝑓𝛼𝑘, 𝐵_𝑓𝛼𝑘, 𝐶_𝑓𝛼𝑘, and 𝐷_𝑓𝛼𝑘 are notated as 𝐴_𝑖, 𝐵_𝑖, 𝐴_𝑓𝑖, 𝐵_𝑓𝑖, 𝐶_𝑓𝑖, and 𝐷_𝑓𝑖 when 𝛼_𝑘=𝑖, respectively. Denoting 𝜂(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘)]^𝑇 and 𝑒(𝑘)=𝑧(𝑘)−𝑧(𝑘), the filtering error system can be described by the following system: 𝜂(𝑘+1)=𝒜_𝑖𝜂(𝑘)+ℬ_𝑖𝑤(𝑘),𝑒_𝑟(𝑘)=𝒞_𝑖𝜂(𝑘)+𝒟_𝑖𝑤(𝑘), (5) where 𝒜_𝑖ℬ_𝑖𝒞_𝑖𝒟_𝑖=𝐴_𝑖0𝐵_𝑖𝐵_𝑓𝑖𝐶𝐴_𝑓𝑖𝐵_𝑓𝑖𝐷𝐸−𝐷_𝑓𝑖𝐶−𝐶_𝑓𝑖−𝐷_𝑓𝑖𝐷. (6)

In order to present the objective of this paper clearly, the following definition is first given.

Definition 2 (MSS). The filter error system (5) is said to be mean-square stable (MSS) with 𝑤(𝑘)=0, if lim_𝑘→∞𝔼𝜂(𝑘)²=0 (7) holds for all 𝛼_𝑘=𝑖∈𝕊.

Definition 3 (finite-frequency stochastic 𝐻_∞ norm). For all the solutions of (5) which satisfied the following inequalities under zero initial condition for nonzero disturbance:(i)for the low-frequency range |𝜃|≤𝜗_𝑙^∞_𝑘=0𝜂(𝑘+1)−𝜂(𝑘)𝜂(𝑘+1)−𝜂(𝑘)^𝑇≤𝜗2sin^𝑙2^2∞_𝑘=0𝜂(𝑘)𝜂(𝑘)^𝑇, (8)(ii)for the middle-frequency range 𝜗₁≤𝜃≤𝜗₂𝑒^{𝑗𝜗𝑤∞}_𝑘=0𝜂(𝑘+1)−𝑒^𝑗𝜗1𝜂(𝑘×𝜂())𝑘+1−𝑒^−𝑗𝜗2𝜂(𝑘)^𝑇≤0, (9) where 𝜗_𝑤=(𝜗₂−𝜗₁)/2,(iii)for the high-frequency range |𝜃|≥𝜗_ℎ^∞_𝑘=0𝜂(𝑘+1)−𝜂(𝑘)𝜂(𝑘+1)−𝜂(𝑘)^𝑇≥𝜗2sin^ℎ2^2∞_𝑘=0𝜂(𝑘)𝜂(𝑘)^𝑇. (10)the given constant 𝛾>0 is said to be the finite-frequency stochastic 𝐻_∞ norm of (5) if the following inequality 𝔼^∞_𝑘=0𝑒_𝑟(𝑘)²≤𝛾²𝔼^∞_𝑘=0‖𝑤(𝑘)‖² (11) holds.

Remark 4. Definition 3 is motivated by the work of [20, 27], which can be regarded as an extension in finite-frequency domain of standard 𝐻_∞ norm. Noticeably, it expresses the robustness from 𝑤(𝑘) to 𝑒(𝑘) in finite-frequency, that is, the smaller it is, the more robust to 𝑤(𝑘) the error 𝑒(𝑘) becomes.

Now, the problem to be addressed in this paper can be formulated as follows: design a stable filter (4) such that the filter error system (5) is mean-square stable, and with prescribed finite-frequency stochastic 𝐻_∞ norm 𝛾 for the external disturbance 𝑤(𝑘).

3. Main Results

The filter design problem proposed in the above section will be discussed in this section.

3.1. Conditions for Robustness

Before proceeding further, the following lemma will be recalled to help us derive our main results.

Lemma 5 (Projection Lemma [27]). For arbitrary Γ,Λ,Θ, there exists matrix 𝐹 satisfying Γ𝐹Λ+(Γ𝐹Λ)^𝑇+Θ<0 if and only if the following two conditions hold: Γ^⟂ΘΓ^⟂𝑇<0,Λ^𝑇⟂ΘΛ^𝑇⟂𝑇<0. (12)

The following lemma will be given which provides a sufficient condition for the desired performance (11) of system (5).

Lemma 6. Assume the MJLS (5) is mean-square stable; let 𝛾>0 be a given constant, then system (5) has a finite-frequency stochastic 𝐻_∞ norm 𝛾 if there exist mode-dependent matrices 𝑃_𝑖=𝑃^𝑇_𝑖, 𝑄_𝑖=𝑄^𝑇_𝑖>0, 𝑖∈𝕊 such that the following inequalities hold: 𝒜_𝑖ℬ_𝑖𝐼0^𝑇Ξ_𝑖𝒜_𝑖ℬ_𝑖+𝒞𝐼0_𝑖𝒟_𝑖0𝐼^𝑇Π𝒞_𝑖𝒟_𝑖0𝐼<0, (13) where Π=^𝐼0_0−𝛾²_𝐼 and(i)for the low-frequency range |𝜃|≤𝜗_𝑙Ξ_𝑖=−𝑃_𝑖𝑄_𝑖𝑄_𝑖𝑃_𝑖−2cos𝜗_𝑙𝑄_𝑖, (14)(ii)for the middle-frequency range 𝜗₁≤𝜃≤𝜗₂Ξ_𝑖=−𝑃_𝑖𝑒^𝑗𝜗𝑐𝑄_𝑖𝑒^{−𝑗𝜗𝑐}𝑄_𝑖𝑃_𝑖−2cos𝜗_𝑤𝑄_𝑖, (15) where 𝜗_𝑐=(𝜗₂+𝜗₁)/2, 𝜗_𝑤=(𝜗₂−𝜗₁)/2,(iii)for the high-frequency range |𝜃|≥𝜗_ℎΞ_𝑖=−𝑃_𝑖−𝑄_𝑖−𝑄_𝑖𝑃_𝑖+2cos𝜗_ℎ𝑄_𝑖, (16) where 𝑃_𝑖=∑_𝑗∈𝕊𝜆_𝑖𝑗𝑃_𝑗.

Proof. We first consider the middle-frequency case for the system (5). Assume (13) holds, before and after multiplying it by 𝜂^𝑇(𝑘)𝑤^𝑇(𝑘) and its transpose, then we can derive 𝜂^𝑇(𝑘)𝑃_𝑖𝜂(𝑘)−𝜂^𝑇(𝑘+1)𝑃_𝑖𝜂(𝑘+1)+𝑒^𝑇_𝑟(𝑘)𝑒_𝑟(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)+tr𝑄𝑒^𝑗𝜗𝑐𝜂(𝑘)𝜂^𝑇(𝑘+1)+𝑒^{−𝑗𝜗𝑐}𝜂(𝑘+1)𝜂^𝑇(𝑘)−2cos𝜗_𝑤𝜂(𝑘)𝜂^𝑇(𝑘)≤0. (17) Since 𝜂(0)=0 and system (5) is mean-square stable, summing up (17) from 0 to ∞ with respect to 𝑘, it is straightforward to see that (17) is equal to 𝔼^∞_𝑘=0𝑒^𝑇_𝑟(𝑘)𝑒_𝑟(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)+tr𝑄_𝑖𝑆≤0, (18) where 𝑆∶=^∞_𝑘=0𝑒^𝑗𝜗𝑐𝜂(𝑘)𝜂^𝑇(𝑘+1)+𝑒^{−𝑗𝜗𝑐}𝜂(𝑘+1)𝜂^𝑇(𝑘)−2cos𝜗_𝑤𝜂(𝑘)𝜂^𝑇(𝑘). (19) It is easy to prove that −𝑆 is equal to the left-hand side of (9), so the 𝑆 is semipositive definite. Also, since 𝑄_𝑖>0, the term tr{𝑄_𝑖𝑆} is nonnegative when (9) is satisfied. Hence, we have 𝔼(∑^∞_𝑘=0(𝑒^𝑇_𝑟(𝑘)𝑒_𝑟(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘))≤0, which is equivalent to condition (11) for middle frequency in Definition 3.
Similarly, the results follow by choosing 𝜗₁∶=−𝜗_𝑙 and 𝜗₂∶=𝜗_𝑙 for low-frequency case and 𝜗₁∶=𝜗_ℎ and 𝜗₂∶=2𝜋−𝜗_ℎ for high-frequency case, respectively. The proof is completed.

Remark 7. If all the matrices in Lemma 6 are independent on 𝑖, the MJS will be reduced to a determinate linear system. In this case, Lemma 6 is equivalent to the GKYP in [20], which has been proved to be an effective tool to deal with the finite-frequency problem of linear time-invariant systems.

Theorem 8. Consider system (5) for all 𝛼_𝑘=𝑖∈𝕊; assume it is mean-square stable; for a given scalar 𝛾>0, the performance of (11) is guaranteed if there exist matrices 𝑈_𝑖, 𝑊_𝑖, 𝑉_𝑖, 𝓐_𝑓𝑖, 𝓑_𝑓𝑖, 𝐶_𝑓𝑖, 𝐷_𝑓𝑖, and 𝑃^𝑇_𝑖=𝑃_𝑖=𝑃_𝑖1∗𝑃_𝑖2𝑃_𝑖3,𝑄^𝑇_𝑖=𝑄_𝑖=𝑄_𝑖1∗𝑄_𝑖2𝑄_𝑖3, (20) such that the following LMIs hold: −𝑃_𝑖1−∗∗∗∗∗𝑃_𝑖2−𝑃_𝑖3𝑄∗∗∗∗_1𝑖−𝑈_𝑖𝑄^𝑇_2𝑖−𝑉_𝑖Ψ³₃𝑄∗∗∗_2𝑖−𝑊_𝑖𝑄_3𝑖−𝑉_𝑖Ψ⁴₃Ψ⁴₄∗∗00Ψ⁵₃Ψ⁵₄−𝛾²𝐼∗00𝐷_𝑓𝐶𝐶_𝑓𝐷_𝑓𝐷−𝐼<0, (21) where 𝑃_𝑖=∑_𝑗∈𝕊𝜆_𝑖𝑗𝑃_𝑗 and Ψ³₃=𝑃_𝑖1−2cos𝜗_𝑙𝑄_𝑖1+𝐻𝑒𝑈_𝑖𝐴_𝑖+𝓑_𝑓𝑖𝐶+𝐸^𝑇𝐸−𝐻𝑒𝐶^𝑇𝐷^𝑇_𝑓𝑖𝐸,Ψ⁴₃=𝑃_𝑖2−2cos𝜗_𝑙𝑄_𝑖2+𝑊_𝑖𝐴_𝑖+𝓑_𝑓𝑖𝐶+𝓐^𝑇_𝑓𝑖−𝐶^𝑇_𝑓𝑖Ψ𝐸,₄₄=𝑃_𝑖3−2cos𝜗_𝑙𝑄_𝑖3+𝐻𝑒𝓐_𝑓𝑖,Ψ⁵₃=𝐵^𝑇_𝑖𝑈^𝑇_𝑖+𝐷^𝑇𝓑^𝑇_𝑓𝑖−𝐷^𝑇𝐷^𝑇_𝑓Ψ𝐸,₅₄=𝐵^𝑇_𝑖𝑊^𝑇_𝑖+𝐷^𝑇𝓑^𝑇_𝑓𝑖. (22)

Proof. It can be concluded from Lemma 6 that if inequality (13) holds for all 𝛼_𝑘=𝑖∈𝕊, the performance of (11) can be reached. Further, (13) is equivalent to 𝒜_𝑖ℬ_𝑖𝐼00𝐼^𝑇Θ_𝑖𝒜_𝑖ℬ_𝑖𝐼00𝐼<0, (23) where Θ_𝑖=𝐽Ξ_𝑖𝐽^𝑇+𝐻_𝑖Π𝐻^𝑇_𝑖, (24) with 𝐽=^𝐼00𝐼₀₀ and 𝐻_𝑖=^𝒞00𝑖0_𝒟𝑖𝐼.
On the other hand, (13) implies that 𝐼0000𝐼^𝑇Θ_𝑖=𝐼00−00𝐼𝑃_𝑖00𝒟^𝑇_𝑖𝒟_𝑖−𝛾²𝐼<0. (25)
Combining (23) and (25), from Lemma 5, one can easily derive that (13) holds if and only if Θ_𝑖+𝐻𝑒𝒜−𝐼_𝑇𝑖ℬ^𝑇_𝑖𝑋^𝑇_𝑖0𝐼0<0, (26) where matrix 𝑋^𝑇_𝑖 is the slack variable with appropriate dimensions which is introduced by Lemma 5.
Rewrite 𝑋_𝑖 as the form of 𝑋_𝑖=𝑈_𝑖𝑉_𝑖𝑊_𝑖𝑉_𝑖. (27) One can conclude that the following inequality provides a sufficient condition for (26): −𝑃_𝑖𝑄∗∗_𝑖−𝑋_𝑖𝜓∗0ℬ^𝑇_𝑖𝑋^𝑇_𝑖+𝒟^𝑇_𝑖𝒞_𝑖𝒟^𝑇_𝑖𝒟_𝑖−𝛾²𝐼<0, (28) where 𝜓=𝑃_𝑖−2cos𝜗_𝑙𝑄_𝑖+𝐻𝑒(𝑋_𝑖𝒜_𝑖)+𝒞^𝑇_𝑖𝒞_𝑖.
After partitioning the matrices 𝑃_𝑖 and 𝑄_𝑖 as the following form 𝑃_𝑖=𝑃_𝑖1∗𝑃_𝑖2𝑃_𝑖3,𝑄_𝑖=𝑄_𝑖1∗𝑄_𝑖2𝑄_𝑖3 (29) and defining the following new variables 𝓐_𝑓𝑖=𝑉_𝑖𝐴_𝑓𝑖,𝓑_𝑓𝑖=𝑉_𝑖𝐵_𝑓𝑖, (30) inequality (21) can be derived. The proof is completed.

Remark 9. In Theorem 8, by introducing a variable 𝑋_𝑖, the coupling between the variable 𝑃_𝑖 and the filter gains will be eliminated. Such a matrix does not present any structure constraint; on the contrary, it may lead to potentially less conservative results.

3.2. Conditions for Stability

Theorem 8 can guarantee the filtering error system to be with a specific robust performance in a certain frequency range of relevance. However, the stability has not been captured, and hence, one may wish to include a stability constraint as an additional design specification. The following theorem will give a result for stability.

Theorem 10. The system (5) is mean-square stable with 𝑤(𝑘)=0 if there exist matrices 𝑈_𝑖, 𝑊_𝑖, 𝑉_𝑖, 𝓐_𝑓𝑖, 𝓑_𝑓𝑖, 𝐶_𝑓𝑖, 𝐷_𝑓𝑖, and 𝑃^𝑇_𝑠𝑖=𝑃_𝑠𝑖=𝑃_𝑠𝑖1∗𝑃_𝑠𝑖2𝑃_𝑠𝑖3>0, (31) such that the following inequality holds: 𝑃_𝑠𝑖1−𝐻𝑒𝑈_𝑖∗∗∗𝑃_𝑠𝑖2−𝑉^𝑇_𝑖−𝑊_𝑖𝑃_𝑠𝑖3−𝐻𝑒𝑉_𝑖𝐴∗∗_𝑇𝑖𝑈^𝑇_𝑖+𝐶^𝑇𝓑^𝑇_𝑓𝑖𝐴^𝑇_𝑖𝑊^𝑇_𝑖+𝐶^𝑇𝓑^𝑇_𝑓𝑖−𝑃_𝑠𝑖1∗𝓐^𝑇_𝑓𝑖𝓐^𝑇_𝑓𝑖−𝑃_𝑠𝑖2−𝑃_𝑠𝑖3<0, (32) where 𝑃_𝑠𝑖=∑_𝑗∈𝕊𝜆_𝑖𝑗𝑃_𝑠𝑗.

Proof. Combing the knowledge of the existing stability criteria for MJSs and the lemma, following the line of the proof for Theorem 8, the conclusion can be derived easily. We do not explain it specifically here.

Remark 11. It should be pointed out that inequalities (28)–(33) are all linear matrix inequalities which can be solved through LMI toolbox of MATLAB.

Based on the above analysis, a set of optimal solutions 𝓐_𝑓𝑖, 𝓑_𝑓𝑖, 𝐶_𝑓𝑖, and 𝐷_𝑓𝑖 can be obtained by solving the following optimization problem: (),().min𝛾s.t.2132 (33)

Then the filter gains can be computed by the following equalities: 𝐴_𝑓𝑖=𝑉⁻¹𝓐_𝑓𝑖,𝐵_𝑓𝑖=𝑉⁻¹𝓑_𝑓𝑖,𝐶_𝑓𝑖=𝐶_𝑓𝑖,𝐷_𝑓𝑖=𝐷_𝑓𝑖. (34)

4. An Illustrative Example

In this section, an example is given to illustrate the effectiveness of the proposed method.

According to some previous researches of the aircraft dynamic model such as [28, 29], it is easily concluded that the nonlinear aircraft dynamic model can be established according to Newton’s Second Law of motion. Furthermore, in order to decouple the nonlinear dynamics, the model is decomposed into two models along with longitudinal- and lateral-directional motions, respectively.

The model used in this example is the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) utilized in [30], that is, 𝑥̇(𝑡)=𝑥(𝑡)+𝑤(𝑡),𝑦(𝑡)=𝑥(𝑡)+𝑤(𝑡),𝑧(𝑡)=𝑥(𝑡),−0.1660.629−0.9971−12.97−1.7610.50833.191−0.1417−0.15291.80.7−1.460100010.96−0.42.8−0.521.31.70.9−0.4 (35) where the system state and output are, respectively, state=sideslipangle^∘rollrate^∘/𝑠yawrate^∘,/𝑠output=rollrate^∘/𝑠yawrate^∘./𝑠 (36)

Assume that the sampling period is ℎ=1s, and the sampled data are transmitted through a network, where the data packets may be lost. Further, the quantity of the lost packet (𝛼_𝑘−1), 𝛼_𝑘∈𝕊={1,2,3} at each sampling period is shown in Figure 2, which is subject to the following transition probability matrix: Λ=0.90.100.80.10.10.80.20. (37)

Figure 2 

The quantity of the dropped packet.

For prescribed 𝜗_𝑙=0.8, solving the optimization problem (33), we can obtain the optimal value for finite-frequency stochastic 𝐻_∞ norm, that is, the robust performance is 𝛾=0.001 with the corresponding filter gains as follows: 𝐴_𝑓1=,𝐵−0.55520.20721.84021.0360−1.5484−10.58040.43970.40412.0462_𝑓1=,𝐶0.23671.8933−1.3950−11.76540.20541.8380_𝑓1=,𝐷−2.8000−1.1960−5.4184−1.70000.09442.7866_𝑓1=,𝐴−1.7160−4.11840.99442.3866_𝑓2=,𝐵0.25430.0869−0.3916−0.2131−0.27983.04820.05330.0319−0.4729_𝑓2=,𝐶0.0765−0.3750−0.50823.03830.0340−0.7283_𝑓2=,𝐷−2.8000−0.2844−3.2306−1.7000−0.36771.6775_𝑓2=,𝐴−0.8044−1.93060.53231.2775_𝑓3=,𝐵−0.1377−0.07390.07530.35930.4907−0.66570.0806−0.14140.0216_𝑓3=,𝐶−0.06150.08000.5174−0.9498−0.1909−0.0263_𝑓3=,𝐷−2.80000.7070−0.8511−1.7000−0.74670.7680_𝑓3=.0.18700.44890.15330.3680 (38)

In order to show the advantage of the proposed method, we compare it with standard 𝐻_∞ filtering method for MJSs which can be found in many researches. In the following, the system will be simulated under zero initial condition, and the disturbance input 𝑤(𝑡) is 𝑤(𝑡)=sin(0.5𝑡),0s≤𝑡≤40s,0,otherwise, (39) which is shown in Figure 3. It is easy to see that the disturbance considered in the this paper is an instantaneous sinusoidal disturbance and its frequency is considered as zero, which belongs to the low frequency.

Figure 3 

The disturbance input 𝑤(𝑡).

The simulation results are shown in Figures 4 and 5, which confirm that all the expected system performance requirement are well achieved. Compared with the standard 𝐻_∞ filtering method, our method performs better even in the case of possible missing measurements.

Figure 4 

The estimation error 𝑒1(𝑘).

Figure 5 

The estimation error 𝑒2(𝑘).

5. Conclusions

In this paper, we have studied the robust filtering with finite-frequency specifications for NCSs subject to random missing measurements. Here, the NCSs are first modeled into MJLs. Then, a new robust filtering method has been proposed that makes full use of the frequency information of noises to reduce design conservatism by the introduction of finite-frequency stochastic 𝐻_∞ index for MJLSs. The design problem is formulated into solving a set of linear matrix inequalities, which can be computed by the LMI Control Toolbox. An example is included to show the effectiveness of the obtained theoretical results.

Acknowledgments

This work is supported by National Natural Science Foundation of China (nos. 61273155, 61273148), New Century Excellent Talents in University (no. NCET-11-0083), a Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China (no. 201157), the Fundamental Research Funds for the Central Universities (Grant no. N120504003), and the Foundation of State Key Laboratory of Robotics (no. 2012-001).