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

Authors: Dan Ye; Yue Long; Guang-Hong Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## 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 stochasticH∞ 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.


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## Body

## 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 Section2. 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 AT is denoted by He(A)∶=A+AT.


## 2. Problem Formulation

Considering the NCS depicted in Figure1, the continuous-time plant model is
(1)x˙(t)=Ax(t)+Bw(t),y(t)=Cx(t)+Dw(t),z(t)=Ex(t),
where x(t)∈ℝn is the state, y(t)∈ℝm is the measured output, z(t)∈ℝp is the controlled output, and w(t)∈ℝd 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 Figure1, 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 αkh, 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 αk-1, which can be modeled by a time-homogeneous Markov chain αk with the range set 𝕊={1,2,…,N} and the transition probability matrix
(2)Λ=(λij)i,j∈𝕊=(P(αk+1=j∣αk=i))i,j∈𝕊=[λ11λ12…λ1Nλ21λ22…λ2N⋮⋮⋱⋮λN1λN2…λNN].
In this situation, the dynamics of plant (1) together with the missing measurements at time instant tk can be approximated by
(3)x(k+1)=Aαkx(k)+Bαkw(k),yr(k)=Cx(k)+Dw(k),z(k)=Ex(k),
where t0=0, tk=∑i=0k-1‍αih(k≥1), x(k)=x(tk), Aαk=eAc(αkh), and Bαk=∫0αkh‍eActBcdt. 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:(4)x^(k+1)=Afαkx^(k)+Bfαkyr(k),z^(k)=Cfαkx(k)+Dfαkyr(k),
where x^(k)∈ℝn is the filter's state, z^(k) is the estimated output, and Afαk, Bfαk, Cfαk, and Dfαk are filter gains to be designed.To ensure the achievement of filter design objective, a basic assumption, that is,Ac 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,Aαk, Bαk, Afαk, Bfαk, Cfαk, and Dfαk are notated as Ai, Bi, Afi, Bfi, Cfi, and Dfi when αk=i, respectively. Denoting η(k)=[xT(k)x^T(k)]T and e(k)=z(k)-z^(k), the filtering error system can be described by the following system:
(5)η(k+1)=𝒜iη(k)+ℬiw(k),er(k)=𝒞iη(k)+𝒟iw(k),
where
(6)[𝒜iℬi𝒞i𝒟i]=[Ai0BiBfiCAfiBfiDE-DfiC-Cfi-DfiD].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
(7)limk→∞𝔼{∥η(k)∥2}=0
holds for all αk=i∈𝕊.Definition 3 (finite-frequency stochasticH∞ 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|≤ϑl(8)∑k=0∞‍(η(k+1)-η(k))(η(k+1)-η(k))T≤(2sinϑl2)2∑k=0∞‍η(k)η(k)T,(ii)
for the middle-frequency rangeϑ1≤θ≤ϑ2(9)ejϑw∑k=0∞‍(η(k+1)-ejϑ1η(k))×(η(k+1)-e-jϑ2η(k))T≤0,
where ϑw=(ϑ2-ϑ1)/2,(iii)
for the high-frequency range|θ|≥ϑh(10)∑k=0∞(η(k+1)-η(k))(η(k+1)-η(k))T‍≥(2sinϑh2)2∑k=0∞‍η(k)η(k)T.the given constant γ>0 is said to be the finite-frequency stochastic H∞ norm of (5) if the following inequality
(11)𝔼(∑k=0∞‍∥er(k)∥2)≤γ2𝔼(∑k=0∞‍∥w(k)∥2)
holds.Remark 4.
Definition3 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 arbitraryΓ,Λ,Θ, there exists matrix F satisfying ΓFΛ+(ΓFΛ)T+Θ<0 if and only if the following two conditions hold:
(12)Γ⊥ΘΓ⊥T<0,ΛT⊥ΘΛT⊥T<0.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 Pi=PiT, Qi=QiT>0, i∈𝕊 such that the following inequalities hold:
(13)[𝒜iℬiI0]TΞi[𝒜iℬiI0]+[𝒞i𝒟i0I]TΠ[𝒞i𝒟i0I]<0,
where Π=[I00-γ2I] and(i)
for the low-frequency range|θ|≤ϑl(14)Ξi=[-P-iQiQiPi-2cosϑlQi],(ii)
for the middle-frequency rangeϑ1≤θ≤ϑ2(15)Ξi=[-P-iejϑcQie-jϑcQiPi-2cosϑwQi],
where ϑc=(ϑ2+ϑ1)/2, ϑw=(ϑ2-ϑ1)/2,(iii)
for the high-frequency range|θ|≥ϑh(16)Ξi=[-P-i-Qi-QiPi+2cosϑhQi],
where P-i=∑j∈𝕊‍λijPj.Proof.
We first consider the middle-frequency case for the system (5). Assume (13) holds, before and after multiplying it by [ηT(k)wT(k)] and its transpose, then we can derive
(17)ηT(k)Piη(k)-ηT(k+1)P-iη(k+1)+erT(k)er(k)-γ2wT(k)w(k)+tr{Q(ejϑcη(k)ηT(k+1)+e-jϑcη(k+1)ηT(k)-2cosϑwη(k)ηT(k))}≤0.
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
(18)𝔼(∑k=0∞‍(erT(k)er(k)-γ2wT(k)w(k))+tr{QiS})≤0,
where
(19)S∶=∑k=0∞‍(ejϑcη(k)ηT(k+1)+e-jϑcη(k+1)ηT(k)-2cosϑwη(k)ηT(k)).
It is easy to prove that -S is equal to the left-hand side of (9), so the S is semipositive definite. Also, since Qi>0, the term tr{QiS} is nonnegative when (9) is satisfied. Hence, we have 𝔼(∑k=0∞‍(erT(k)er(k)-γ2wT(k)w(k))≤0, which is equivalent to condition (11) for middle frequency in Definition 3.
Similarly, the results follow by choosingϑ1∶=-ϑl and ϑ2∶=ϑl for low-frequency case and ϑ1∶=ϑh and ϑ2∶=2π-ϑh for high-frequency case, respectively. The proof is completed.Remark 7.
If all the matrices in Lemma6 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 αk=i∈𝕊; assume it is mean-square stable; for a given scalar γ>0, the performance of (11) is guaranteed if there exist matrices Ui, Wi, Vi, Afi, Bfi, Cfi, Dfi, and
(20)PiT=Pi=[Pi1*Pi2Pi3],QiT=Qi=[Qi1*Qi2Qi3],
such that the following LMIs hold:
(21)[-P-i1*****-P-i2-P-i3****Q1i-UiQ2iT-ViΨ33***Q2i-WiQ3i-ViΨ34Ψ44**00Ψ35Ψ45-γ2I*00DfCCfDfD-I]<0,
where P-i=∑j∈𝕊‍λijPj and
(22)Ψ33=Pi1-2cosϑlQi1+He(UiAi+BfiC)+ETE-He(CTDfiTE),Ψ34=Pi2-2cosϑlQi2+WiAi+BfiC+AfiT-CfiTE,Ψ44=Pi3-2cosϑlQi3+He(Afi),Ψ35=BiTUiT+DTBfiT-DTDfTE,Ψ45=BiTWiT+DTBfiT.Proof.
It can be concluded from Lemma6 that if inequality (13) holds for all αk=i∈𝕊, the performance of (11) can be reached. Further, (13) is equivalent to
(23)[𝒜iℬiI00I]TΘi[𝒜iℬiI00I]<0,
where
(24)Θi=JΞiJT+HiΠHiT,
with J=[I00I00] and Hi=[00𝒞i0𝒟iI].
On the other hand, (13) implies that
(25)[I0000I]TΘi[I0000I]=[-P-i00𝒟iT𝒟i-γ2I]<0.
Combining (23) and (25), from Lemma 5, one can easily derive that (13) holds if and only if
(26)Θi+He([-I𝒜iTℬiT]XiT[0I0])<0,
where matrix XiT is the slack variable with appropriate dimensions which is introduced by Lemma 5.
RewriteXi as the form of
(27)Xi=[UiViWiVi].
One can conclude that the following inequality provides a sufficient condition for (26):
(28)[-P-i**Qi-Xiψ*0ℬiTXiT+𝒟iT𝒞i𝒟iT𝒟i-γ2I]<0,
where ψ=Pi-2cosϑlQi+He(Xi𝒜i)+𝒞iT𝒞i.
After partitioning the matricesPi and Qi as the following form
(29)Pi=[Pi1*Pi2Pi3],Qi=[Qi1*Qi2Qi3]
and defining the following new variables
(30)Afi=ViAfi,Bfi=ViBfi,
inequality (21) can be derived. The proof is completed.Remark 9.
In Theorem8, by introducing a variable Xi, the coupling between the variable Pi 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

Theorem8 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 Ui, Wi, Vi, Afi, Bfi, Cfi, Dfi, and
(31)PsiT=Psi=[Psi1*Psi2Psi3]>0,
such that the following inequality holds:
(32)[P-si1-He(Ui)***P-si2-ViT-WiP-si3-He(Vi)**AiTUiT+CTBfiTAiTWiT+CTBfiT-Psi1*AfiTAfiT-Psi2-Psi3]<0,
where P-si=∑j∈𝕊‍λijPsj.Proof.
Combing the knowledge of the existing stability criteria for MJSs and the lemma, following the line of the proof for Theorem8, 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 solutionsAfi, Bfi, Cfi, and Dfi can be obtained by solving the following optimization problem:
(33)minγs.t.(21),(32).Then the filter gains can be computed by the following equalities:(34)Afi=V-1Afi,Bfi=V-1Bfi,Cfi=Cfi,Dfi=Dfi.


## 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 arbitraryΓ,Λ,Θ, there exists matrix F satisfying ΓFΛ+(ΓFΛ)T+Θ<0 if and only if the following two conditions hold:
(12)Γ⊥ΘΓ⊥T<0,ΛT⊥ΘΛT⊥T<0.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 Pi=PiT, Qi=QiT>0, i∈𝕊 such that the following inequalities hold:
(13)[𝒜iℬiI0]TΞi[𝒜iℬiI0]+[𝒞i𝒟i0I]TΠ[𝒞i𝒟i0I]<0,
where Π=[I00-γ2I] and(i)
for the low-frequency range|θ|≤ϑl(14)Ξi=[-P-iQiQiPi-2cosϑlQi],(ii)
for the middle-frequency rangeϑ1≤θ≤ϑ2(15)Ξi=[-P-iejϑcQie-jϑcQiPi-2cosϑwQi],
where ϑc=(ϑ2+ϑ1)/2, ϑw=(ϑ2-ϑ1)/2,(iii)
for the high-frequency range|θ|≥ϑh(16)Ξi=[-P-i-Qi-QiPi+2cosϑhQi],
where P-i=∑j∈𝕊‍λijPj.Proof.
We first consider the middle-frequency case for the system (5). Assume (13) holds, before and after multiplying it by [ηT(k)wT(k)] and its transpose, then we can derive
(17)ηT(k)Piη(k)-ηT(k+1)P-iη(k+1)+erT(k)er(k)-γ2wT(k)w(k)+tr{Q(ejϑcη(k)ηT(k+1)+e-jϑcη(k+1)ηT(k)-2cosϑwη(k)ηT(k))}≤0.
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
(18)𝔼(∑k=0∞‍(erT(k)er(k)-γ2wT(k)w(k))+tr{QiS})≤0,
where
(19)S∶=∑k=0∞‍(ejϑcη(k)ηT(k+1)+e-jϑcη(k+1)ηT(k)-2cosϑwη(k)ηT(k)).
It is easy to prove that -S is equal to the left-hand side of (9), so the S is semipositive definite. Also, since Qi>0, the term tr{QiS} is nonnegative when (9) is satisfied. Hence, we have 𝔼(∑k=0∞‍(erT(k)er(k)-γ2wT(k)w(k))≤0, which is equivalent to condition (11) for middle frequency in Definition 3.
Similarly, the results follow by choosingϑ1∶=-ϑl and ϑ2∶=ϑl for low-frequency case and ϑ1∶=ϑh and ϑ2∶=2π-ϑh for high-frequency case, respectively. The proof is completed.Remark 7.
If all the matrices in Lemma6 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 αk=i∈𝕊; assume it is mean-square stable; for a given scalar γ>0, the performance of (11) is guaranteed if there exist matrices Ui, Wi, Vi, Afi, Bfi, Cfi, Dfi, and
(20)PiT=Pi=[Pi1*Pi2Pi3],QiT=Qi=[Qi1*Qi2Qi3],
such that the following LMIs hold:
(21)[-P-i1*****-P-i2-P-i3****Q1i-UiQ2iT-ViΨ33***Q2i-WiQ3i-ViΨ34Ψ44**00Ψ35Ψ45-γ2I*00DfCCfDfD-I]<0,
where P-i=∑j∈𝕊‍λijPj and
(22)Ψ33=Pi1-2cosϑlQi1+He(UiAi+BfiC)+ETE-He(CTDfiTE),Ψ34=Pi2-2cosϑlQi2+WiAi+BfiC+AfiT-CfiTE,Ψ44=Pi3-2cosϑlQi3+He(Afi),Ψ35=BiTUiT+DTBfiT-DTDfTE,Ψ45=BiTWiT+DTBfiT.Proof.
It can be concluded from Lemma6 that if inequality (13) holds for all αk=i∈𝕊, the performance of (11) can be reached. Further, (13) is equivalent to
(23)[𝒜iℬiI00I]TΘi[𝒜iℬiI00I]<0,
where
(24)Θi=JΞiJT+HiΠHiT,
with J=[I00I00] and Hi=[00𝒞i0𝒟iI].
On the other hand, (13) implies that
(25)[I0000I]TΘi[I0000I]=[-P-i00𝒟iT𝒟i-γ2I]<0.
Combining (23) and (25), from Lemma 5, one can easily derive that (13) holds if and only if
(26)Θi+He([-I𝒜iTℬiT]XiT[0I0])<0,
where matrix XiT is the slack variable with appropriate dimensions which is introduced by Lemma 5.
RewriteXi as the form of
(27)Xi=[UiViWiVi].
One can conclude that the following inequality provides a sufficient condition for (26):
(28)[-P-i**Qi-Xiψ*0ℬiTXiT+𝒟iT𝒞i𝒟iT𝒟i-γ2I]<0,
where ψ=Pi-2cosϑlQi+He(Xi𝒜i)+𝒞iT𝒞i.
After partitioning the matricesPi and Qi as the following form
(29)Pi=[Pi1*Pi2Pi3],Qi=[Qi1*Qi2Qi3]
and defining the following new variables
(30)Afi=ViAfi,Bfi=ViBfi,
inequality (21) can be derived. The proof is completed.Remark 9.
In Theorem8, by introducing a variable Xi, the coupling between the variable Pi 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

Theorem8 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 Ui, Wi, Vi, Afi, Bfi, Cfi, Dfi, and
(31)PsiT=Psi=[Psi1*Psi2Psi3]>0,
such that the following inequality holds:
(32)[P-si1-He(Ui)***P-si2-ViT-WiP-si3-He(Vi)**AiTUiT+CTBfiTAiTWiT+CTBfiT-Psi1*AfiTAfiT-Psi2-Psi3]<0,
where P-si=∑j∈𝕊‍λijPsj.Proof.
Combing the knowledge of the existing stability criteria for MJSs and the lemma, following the line of the proof for Theorem8, 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 solutionsAfi, Bfi, Cfi, and Dfi can be obtained by solving the following optimization problem:
(33)minγs.t.(21),(32).Then the filter gains can be computed by the following equalities:(34)Afi=V-1Afi,Bfi=V-1Bfi,Cfi=Cfi,Dfi=Dfi.


## 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,
(35)x˙(t)=[-0.1660.629-0.9971-12.97-1.7610.50833.191-0.1417-0.1529]x(t)+[1.80.7-1.46]w(t),y(t)=[010001]x(t)+[0.96-0.4]w(t),z(t)=[2.8-0.521.31.70.9-0.4]x(t),
where the system state and output are, respectively,
(36)state=[sideslipangle(°)rollrate(°/s)yawrate(°/s)],output=[rollrate(°/s)yawrate(°/s)].Assume that the sampling period ish=1s, and the sampled data are transmitted through a network, where the data packets may be lost. Further, the quantity of the lost packet (αk-1), αk∈𝕊={1,2,3} at each sampling period is shown in Figure 2, which is subject to the following transition probability matrix:
(37)Λ=[0.90.100.80.10.10.80.20].Figure 2
The quantity of the dropped packet.For prescribedϑl=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:
(38)Af1=[-0.55520.20721.84021.0360-1.5484-10.58040.43970.40412.0462],Bf1=[0.23671.8933-1.3950-11.76540.20541.8380],Cf1=[-2.8000-1.1960-5.4184-1.70000.09442.7866],Df1=[-1.7160-4.11840.99442.3866],Af2=[0.25430.0869-0.3916-0.2131-0.27983.04820.05330.0319-0.4729],Bf2=[0.0765-0.3750-0.50823.03830.0340-0.7283],Cf2=[-2.8000-0.2844-3.2306-1.7000-0.36771.6775],Df2=[-0.8044-1.93060.53231.2775],Af3=[-0.1377-0.07390.07530.35930.4907-0.66570.0806-0.14140.0216],Bf3=[-0.06150.08000.5174-0.9498-0.1909-0.0263],Cf3=[-2.80000.7070-0.8511-1.7000-0.74670.7680],Df3=[0.18700.44890.15330.3680].In order to show the advantage of the proposed method, we compare it with standardH∞ 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
(39)w(t)={sin(0.5t),0s≤t≤40s,0,otherwise,
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 inputw(t).The simulation results are shown in Figures4 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 errore1(k).Figure 5
The estimation errore2(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 stochasticH∞ 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.



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*Source: 825143-2013-06-24.xml*