# Robust DistributedH∞ Filtering for Nonlinear Systems with Sensor Saturations and Fractional Uncertainties with Digital Simulation

Authors: Dong Liu; Guangfu Tang; Zhiyuan He; Yan Zhao; Hui Pang
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper is concerned with the robust distributedH∞ filtering problem for nonlinear systems subject to sensor saturations and fractional parameter uncertainties. A sufficient condition is derived for the filtering error system to reach the required H∞ performance in terms of recursive linear matrix inequality method. An iterative algorithm is then proposed to obtain the filter parameters recursively by solving the corresponding linear matrix inequality. A numerical example is presented to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

The sensor networks have gradually received more and more research interest in the past decades. A typical example is the power grids which have evolved over the past century from a series of small independent community based systems to large-scale and complex systems involving many kinds of intelligent electronic devices [1]. As is well known, the information collected by the sensor networks need to be further processed before used for diagnosis, control, optimization, and other transactions for the smart grids, which requires much more advanced methodologies that are more sensitive, reliable, and economic than those applied in the traditional centralized electricity networks. Due to its clear engineering insights, recently, sensor networks have gained increasing research interests during the past decades in various branches of theoretical research and industrial applications. In particular, the distributed filtering over sensor networks has been ongoing research issue that attracts special attention from researchers in the area. Different from traditional filtering techniques based on single or centralized structured/located sensors, the information available for the filter design on an individual node of the sensor network is not only from its own measurement but also from its neighboring sensors’ measurements according to the given topology [2]. As such, the objective of filtering based on a sensor network can be achieved in a distributed yet collaborative way. Such a problem is usually referred to as the distributed filtering problem. It is noticed that one of the main challenges in designing distributed filters lies in how to cope with the complicated couplings between one sensor and its neighboring sensors by reflecting such couplings in the filter structure specification.It is worth mentioning that in real-world applications, due to a variety of reasons, such as abrupt changes of working conditions, internal/external disturbances, erosions, equipment aging, the sensor outputs are usually corrupted by these different disadvantages. Such phenomena are always referred to as incomplete information. Up to now, filtering problems with incomplete information have been widely studied and a lot of results have been reported in the literature; see [3–14] for some latest publications. As far as the distributed filtering problems subject to incomplete information are concerned, some recent representative work can be summarized as follows. In [15], the distributed filtering problem was solved when randomly occurring saturations and successive packet dropouts appear over sensor networks. In [16], distributed H∞ filtering problem was studied for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks. In [17], distributed filtering for a class of time-varying systems was discussed over sensor networks with quantization errors and successive packet dropouts. The distributed H∞ state estimation with stochastic parameters and nonlinearities through sensor networks was solved in [18] over the finite-horizon. With randomly varying nonlinearities and missing measurements, [19] developed a distributed filtering method to achieve the performance requirements. In [20], distributed state estimation problem for uncertain Markov-type sensor networks with mode-dependent distributed delays was solved. Using linear matrix inequality approach, the distributed filters were given in [21, 22], respectively, for different types of stochastic systems. However, it is worth mentioning that, up to now, the distributed filtering problem has not been studied for nonlinear time-varying systems subject to randomly occurring sensor failures.Motivated by the above discussion, in this paper, it is the objective to design a robust filter for a class of discrete time-varying nonlinear stochastic systems subject to sensor saturations and fractional uncertainties. The contribution of this paper is twofold: (i) for the first time, the filtering problem is discussed while taking both sensor saturations and fractional uncertainties into consideration. (ii) An iterative algorithm is developed to solve the robustH∞ filtering problem and then seek the desired filter parameters step by step. The rest of this paper is organized as follows: Section 2 formulates the distributed event-based filter design problem for nonlinear discrete time-varying stochastic system. The main results are presented in Section 3 where sufficient conditions for the existence of the desired filter are given in terms of recursive linear matrix inequalities. Section 4 gives a numerical example. Section 5 is the conclusion.


## 2. Problem of Formulation

In this paper, it is assumed that the sensor network hasN sensor nodes which are distributed in the space according to a specific interconnection topology characterized by a directed graph G=(V,E,L), where V={1,2,…,N} denotes the set of sensor nodes, E⊆V×V is the set of edges, and L=[rij]N×N is the nonnegative adjacency matrix associated with the edges of the graph, that is, rij>0⇔edge(i,j)∈E, which means that there is information transmission from sensor j to sensor i. If (i,j)∈E, then node j is called one of the neighbors of node i. For all i∈V, denote Ni≜{j∈V∣(i,j)∈E}, which means that, in the sensor network, sensor node i can receive the information from its neighboring nodes j∈Ni according to the given network topology.Let us consider the discrete-time nonlinear stochastic system withN sensors defined on k∈[0,T]:(1)xk+1=Ak+ΔAkxk+gkxk+Dkωk,yi,k=σCi,kxk+Ei,kωk,zk=Mkxk,where xk, yi,k, zk, and ωk represent the state, the measured output of the ith sensor, estimated output, and disturbance belonging to l2, respectively. Ak, Ci,k, Dk, Ei,k, and Mk are known real time-varying matrices with appropriate dimensions.Assumption 1.
The nonlinear functiongk(xk) is assumed to obey the following constraint:(2)gkTxkSk-1gkxk≤δxkTxk,where Sk are known positive definite matrix sequence with appropriate dimensions describing the shape of the ellipsoids with 0 being the center of the ellipsoids and δ>0 is a known positive scalar.The nonlinear functiongk(xk) stands for the nonlinearity that is unknown, bounded, and deterministic but reside within an ellipsoidal set. Such a type of nonlinearity is usually occurring in practical engineering practice and is always a main origin for the degradation of the system performance.Assumption 2 (see [23]).
Δ
A
k has the linear fractional form as follows:(3)ΔAk=UΣkI-JΣk-1Vwith JTJ<I and ΣkTΣk<I, where U, V, and J are known constant matrices and Σk denotes the unknown matrix functions with Lebesgue measurable elements.Definition 3.
A nonlinear functionψ(·) is said to satisfy the sector-bounded condition if(4)ψθ-Θ1θTψθ-Θ2θ≤0for some real matrices Θ1 and Θ2, where Θ=Θ2-Θ1 is a symmetric positive definite matrix. In this case, we say that ψ(·) belongs to [Θ1,Θ2].As discussed in [23], there exist matrices K1 and K2 such that 0≤K1<I≤K2; the sensor saturation function σ(Ci,kxk) can be written as(5)σCi,kxk=K1Ci,kxk+φyCi,kxk,where φy(Ci,kxk) is a nonlinear vector-valued function satisfying the sector-bounded condition with Θ1=0 and Θ2=K. In this case, φy(Ci,kxk) can be represented as follows:(6)φyTCi,kxkφyCi,kxk-KCi,kxk≤0,where K=K2-K1.The objective of this paper is to design a filter with the following form for system (1):(7)x^i,k+1=Gi,kx^i,k+∑j∈NirijHij,kyj,k-Cj,kx^j,k,z^i,k=Mkx^i,k,where x^i,k is the state estimate, z^k is the output to be estimated, and Gi,k and Hij,k are the filter parameters to be determined.From system (1) and filter structure (7), one can obtain(8)ei,k+1=xk+1-x^i,k+1=Ak+ΔAkxk+gxk+Dkωk-Gi,kx^i,k+∑j∈NirijHij,kyj,k-Cj,kx^j,k=Ak+ΔAkxk+gxk+Dkωk-Gi,kx^i,k-∑j∈NirijHij,kσCj,kxk+Ej,kωk-Cj,kx^j,k=Ak+ΔAkxk+gxk+Dkωk-Gi,kx^i,k-∑j∈NirijHij,kK1Cj,kxk+φyCj,kxk+Ej,kωk-Cj,kx^j,k,which can be rewritten in a more compact form as follows:(9)ek+1=Akξk+g~k+Dkω~k-Gkx^k-∑i=1NΦiHkRiKkξk-∑i=1NΦiHkRiφ~y-∑i=1NΦiHkRiEkω~k+∑i=1NΦiHkRiCkx^k=Ak-Gk-∑i=1NΦiHkRiKk+∑i=1NΦiHkRiCkξk+Gk-∑i=1NΦiHkRiCkek+g~k+Dk-∑i=1NΦiHkRiEkω~k-∑i=1NΦiHkRiφ~y,where(10)ek=e1e2⋮eN,ξk=xkxk⋮xk,x^k=x^1x^2⋮x^N,g~k=gxkgxk⋮gxk,ω~k=ωkωk⋮ωk,φ~y=φC1,kxkφC2,kxk⋮φCN,kxk,Ak=diag⁡Ak+ΔAk,Ak+ΔAk,…,Ak+ΔAk,Dk=diag⁡Dk,Dk,…,Dk,Gk=diag⁡G1,k,G2,k,…,GN,k,Φi=diag⁡0,…,0︸i-1,I,0,…,0︸N-i,Hk=Hij,k=H11,kH12,k⋯H1N,kH21,kH22,k⋯H2N,k⋮⋮⋱⋮HN1,kHN2,k⋯HNN,k,Kk=diag⁡K1C1,k,K1C2,k,…,K1CN,k,Ck=diag⁡C1,k,C2,k,…,CN,k,Ek=diag⁡E1,k,E2,k,…,EN,k.Definez~i,k=Mkei,k; it is the objective of this paper to design the filter parameters Gk and Hk to satisfy(11)∑i=1Tz~k2≤γ2∑i=1Tω~k2+e0TΞe0for some given disturbance attenuation level γ>0 and given positive matrix Ξ=ΞT.


## 3. Main Results

Before giving the main results, some useful lemmas are firstly introduced.Lemma 4.
LetW0·,W1·,…,Wp· be quadratic functions of the variable η: Wi(η)≜ηTTiη with TiT=Ti(i=0,1,…,p). If there exist scalars τ1≥0,τ2≥0,…,τp≥0 such that(12)T0-∑i=1pτiTi≤0then the following is true(13)W1η≤0,W2η≤0,…,Wpη≤0⟹W0η≤0.Lemma 5 (Schur Complement Equivalence).
Given constant matricesQ1, Q2, and Q3, where Q1=Q1T and Q2=Q2T>0, then(14)Q1+Q3TQ2-1Q3<0if and only if(15)Q1Q3TQ3-Q2<0or, equivalently,(16)-Q2Q3Q3TQ1<0.Lemma 6 (see [23]).
LetU, V, and Γ be real matrices of appropriate dimensions with Γ satisfying ΓTΓ≤λI. Then, for any scalar ϵ>0, we have(17)UΓV+UΓVT≤ϵ-1UUT+ϵλVTV.


### 3.1.H∞ Performance Analysis

The following theorem gives a sufficient condition under which the requiredH∞ performance can be satisfied.Theorem 7.
Given aH∞ performance γ>0 and the initial matrix Ξ>0. Let the filter parameters Gk and Hk be given. If there exist three sequences of positive scalars τ1k, τ2k(0≤k≤T), and a sequence of positive definite matrices Pk(0≤k≤T+1) satisfying η0TP0η0≤γ2e0TΞe0 such that the following matrix inequality(18)Λ1100τ2kC~kTKT2A~kT∗-τ1kSk-100F~kT∗∗-γ2I0D~kT∗∗∗-τ2kIY~kT∗∗∗∗-Pk+1-1≤0,where (19)Λ11=-Pk+M~kTM~k+τ1kδI,A~k=Ak0A¯kG¯k,A¯k=Ak-Gk-∑i=1NΦiHkRiKk-Ck,G¯k=Gk-∑i=1NΦiHkRiCk,Y~k=0Y¯k,Y¯k=-∑i=1NΦiHkRiD~k=DkD¯k,D¯k=Dk-∑i=1NΦiHkRiEk,F~k=II,M~k=0Mk,Mk=diagMk,Mk,…,Mkis feasible, then H∞ performance can be satisfied.Proof.
Defining(20)ηk=ξkekthen the filtering error system can be expressed by(21)ηk+1=A~kηk+F~kg~k+D~kω~k+Y~kφ~y,z~k=M~kηk.
Defining(22)Jk=ηk+1TPk+1ηk+1-ηkTPkηkone can get from (9) that(23)Jk=ηk+1TPk+1ηk+1-ηkTPkηk=A~kηk+F~kg~k+D~kω~k+Y~kφ~yT·Pk+1A~kηk+F~kg~k+D~kω~k+Y~kφ~y-ηkTPkηk=ηkTA~kTPk+1A~kηk+g~kTF~kTPk+1F~kg~k+ω~kTD~kTPk+1D~kω~k+φ~yTY~kTPk+1Y~kφ~y+2ηkTA~kTPk+1F~kg~k+2ηkTA~kTPk+1D~kω~k+2ηkTA~kTPk+1Y~kφ~y+2g~kTF~kTPk+1D~kω~k+2g~kTF~kTPk+1Y~kφ~y+2ω~kTD~kTPk+1Y~kφ~y-ηkTPkηk=ψkTΩkψk,where (24)ψk=ηkTg~kTω~kTφ~yTT,Ωk=Ω11Ω12Ω13Ω14∗F~kTPk+1F~kF~kTPk+1D~kΩ24∗∗D~kTPk+1D~kΩ34∗∗∗Ω44,Ω11=A~kTPk+1A~k-Pk,Ω12=A~kTPk+1F~k,Ω13=A~kTPk+1D~k,Ω14=A~kTPk+1Y~k,Ω24=F~kTPk+1Y~k,Ω34=D~kTPk+1Y~k,Ω44=Y~kTPk+1Y~k.
Adding the zero term(25)z~kTz~k-γ2ω~kTω~k-z~kTz~k-γ2ω~kTω~kto both sides of (23), one can get(26)Jk=ψkTΛkψk-z~kTz~k-γ2ω~kTω~k,where (27)Λk=Ωk+diagM~kTM~k,0,-γ2I,0.
Next, by summing up (26) on both sides from 0 to N with respect to k, one can have(28)∑k=0TJk=ηT+1TPT+1ηT+1-η0TP0η0=∑k=0TψkTΛkψk-∑k=0Tz~kTz~k-γ2ω~kTω~kwhich is equivalent to(29)∑k=0Tz~kTz~k=∑k=0Tγ2ω~kTω~k+∑k=0TψkTΛkψk-ηT+1TPT+1ηT+1+η0TP0η0.
From (29), since PT+1>0 and the initial condition η0TP0η0≤γ2e0TΞe0, one can see that the H∞ performance can be satisfied if ψkTΛkψk≤0 is satisfied.
Next, one can see from the ellipsoidal nonlinear functiongk(xk) (2) that(30)ψkTΠkψk≤0,where (31)Πk=-δI000∗Sk-100∗∗00∗∗∗0with Sk=diag{Sk,Sk,…,Sk}.
Moreover, it also can be obtained from the sensor fault constraint (6) that(32)ψkTΠ¯kψk≤0,where (33)Π¯k=12000-C~kTKT∗000∗∗00∗∗∗2Iwith C~k=C1,k0C2,k0⋯CN,k0.
By Schur Complement Equivalence Lemma, the matrix inequality (18) can be rewritten as follows:(34)Λk-τ1kΠk-τ2kΠ¯k≤0.
By Lemma4, ψkTΛkψk≤0 can be satisfied, which means that the required H∞ performance can be satisfied. The proof is complete.Remark 8.
By means of iterative matrix inequality method, Theorem7 presents a sufficient condition for filtering error system to achieve the desired H∞ specification subject to the sensor saturations. Moreover, notice that the obtained condition is expressed by certain matrix inequalities including fractional parameter uncertainties ΔAk; we cannot solve the matrix inequality directly to obtain the explicit parametric expression of the required filter parameters G and H. As such, in the following, we will provide a theorem with which the addressed problem can be solved and the filter structure can then be obtained step by step.


### 3.2.  Filter Design

Theorem 9.
Let theH∞ performance γ>0 and the initial condition matrix Sk>0 be given. If there exist a sequence of matrices Gk(0≤k≤T), a sequence of matrices Hk(0≤k≤T), three sequences of positive scalars τ1k, τ2k, and ϵk(0≤k≤T), a sequence of positive definite matrices Xk(0≤k≤T+1), and a sequence of positive definite matrices Zk(0≤k≤T+1) satisfying η0Tdiag{X0-1,Z0-1}η0≤γ2e0Ξe0 such that(35)Λ¯k+ϵkλVTVUUT-ϵkI≤0,where (36)Λ¯k=-Xk+δτ1kI00012τ2kC~kTKTA^kTA˘kT∗-Zk+MkTMk+δτ1kI0000G¯kT∗∗-τ1kSk-100II∗∗∗-γ2I0DkTD¯kT∗∗∗∗-τ2kI0Y¯kT∗∗∗∗∗-Xk+10∗∗∗∗∗∗-Zk+1,A^k=diag⁡Ak,Ak,…,Ak,A˘k=A^k-Gk-∑i=1NΦiHkRiKk-Ck,U=00000U¯TU¯TT,U¯=diagU,U,…,U,V=V¯000000,V¯=diagV,V,…,V,with Xk=Xk-1 and Zk=Zk-1, and λ is the solution to the following optimization problem(37)minΣkλ=I-ΣkTJT-1I-JΣk-1s.t.ΣkTΣk<1,then the addressed H∞ filtering problem is solvable and the desired filter parameters at each time k can be obtained by solving the corresponding matrix inequality.Proof.
The proof is quite straightforward based on Theorem7. First, define Pk-1=diag{Xk-1,Zk-1}=diag{Xk,Zk}. One can obtain that the matrix inequality (18) is equivalent to the following inequality:(38)-Xk+δτ1kI00012τ2kC~kTKTAkTA¯kT∗-Zk+MkTMk+δτ1kI0000G¯kT∗∗-τ1kSk-100II∗∗∗-γ2I0DkTD¯kT∗∗∗∗-τ2kI0Y¯kT∗∗∗∗∗-Xk+10∗∗∗∗∗∗-Zk+1≤0.In what follows, we will try to prove that inequality (38) can be satisfied if inequality (35) is fulfilled. To this end, rewrite (38) as follows:(39)Λ¯k+UΞ~V+UΣ~VT≤0,where (40)Σ~=diagI-JΣ-1,I-JΣ-1,…,I-JΣ-1.
From Lemma6, it follows that(41)UΞ~V+UΞ~VT≤ϵ-1UUT+ϵλVTV.
Furthermore, one can obtain that(42)Λ¯k+ϵ-1UUT+ϵλVTV≤0,which, by Schur Complement Lemma, is equivalent to the linear matrix inequality (35). Thus, inequality (18) is satisfied if (35) is met, and, according to Theorem 7, the required H∞ filtering problem is solved.Remark 10.
Theorem9 outlines the methodology with which the addressed robust H∞ distributed filtering problem is solved for nonlinear systems subject to sensor saturations and fractional uncertainties. An iterative linear matrix inequality approach has been developed to calculate the desired filter parameters step by step. It should be pointed out that it will inevitably lead to the larger computing burdens. However, the algorithm we proposed in this paper is capable of dealing with the time-varying case, and the increase in computing burden can be largely reduced with the fast development of modern computer technology. In what follows, we will present a computing algorithm with which the numerical values of filter parameters can be obtained in each time step.


### 3.3.  Computational Algorithm

Algorithm 11 (H∞Filter Design Algorithm).
Step 1. Set the required γ, the initial condition Ξ, the initial η0, and T. Choose properly the values of U, V, and J. Select the initial values for X0 and Z0 satisfying the initial condition constraint.
Step 2. Set k=0. Solve (35) for G0, H0, X1, and Z1. Then the X1 and Z1 can also be obtained.
Step 3. Set k=k+1. Using the obtained Xk and Zk, solve (35) for Gk, Hk, Xk+1, and Zk+1.
Step 4. If k=N+1, then stop. Else go to Step 3.TheH∞Filter Design Algorithm gives a recursive way to obtain the numerical values of the desired filter parameters at each time point k. It should be pointed out that the existence of the filter is expressed as the feasibility of certain linear matrix inequalities that can be solved forward in time. The possible research topic in the future is to consider more performance indices and give the filtering schemes that are able to satisfy multiple requirements simultaneously.


## 3.1.H∞ Performance Analysis

The following theorem gives a sufficient condition under which the requiredH∞ performance can be satisfied.Theorem 7.
Given aH∞ performance γ>0 and the initial matrix Ξ>0. Let the filter parameters Gk and Hk be given. If there exist three sequences of positive scalars τ1k, τ2k(0≤k≤T), and a sequence of positive definite matrices Pk(0≤k≤T+1) satisfying η0TP0η0≤γ2e0TΞe0 such that the following matrix inequality(18)Λ1100τ2kC~kTKT2A~kT∗-τ1kSk-100F~kT∗∗-γ2I0D~kT∗∗∗-τ2kIY~kT∗∗∗∗-Pk+1-1≤0,where (19)Λ11=-Pk+M~kTM~k+τ1kδI,A~k=Ak0A¯kG¯k,A¯k=Ak-Gk-∑i=1NΦiHkRiKk-Ck,G¯k=Gk-∑i=1NΦiHkRiCk,Y~k=0Y¯k,Y¯k=-∑i=1NΦiHkRiD~k=DkD¯k,D¯k=Dk-∑i=1NΦiHkRiEk,F~k=II,M~k=0Mk,Mk=diagMk,Mk,…,Mkis feasible, then H∞ performance can be satisfied.Proof.
Defining(20)ηk=ξkekthen the filtering error system can be expressed by(21)ηk+1=A~kηk+F~kg~k+D~kω~k+Y~kφ~y,z~k=M~kηk.
Defining(22)Jk=ηk+1TPk+1ηk+1-ηkTPkηkone can get from (9) that(23)Jk=ηk+1TPk+1ηk+1-ηkTPkηk=A~kηk+F~kg~k+D~kω~k+Y~kφ~yT·Pk+1A~kηk+F~kg~k+D~kω~k+Y~kφ~y-ηkTPkηk=ηkTA~kTPk+1A~kηk+g~kTF~kTPk+1F~kg~k+ω~kTD~kTPk+1D~kω~k+φ~yTY~kTPk+1Y~kφ~y+2ηkTA~kTPk+1F~kg~k+2ηkTA~kTPk+1D~kω~k+2ηkTA~kTPk+1Y~kφ~y+2g~kTF~kTPk+1D~kω~k+2g~kTF~kTPk+1Y~kφ~y+2ω~kTD~kTPk+1Y~kφ~y-ηkTPkηk=ψkTΩkψk,where (24)ψk=ηkTg~kTω~kTφ~yTT,Ωk=Ω11Ω12Ω13Ω14∗F~kTPk+1F~kF~kTPk+1D~kΩ24∗∗D~kTPk+1D~kΩ34∗∗∗Ω44,Ω11=A~kTPk+1A~k-Pk,Ω12=A~kTPk+1F~k,Ω13=A~kTPk+1D~k,Ω14=A~kTPk+1Y~k,Ω24=F~kTPk+1Y~k,Ω34=D~kTPk+1Y~k,Ω44=Y~kTPk+1Y~k.
Adding the zero term(25)z~kTz~k-γ2ω~kTω~k-z~kTz~k-γ2ω~kTω~kto both sides of (23), one can get(26)Jk=ψkTΛkψk-z~kTz~k-γ2ω~kTω~k,where (27)Λk=Ωk+diagM~kTM~k,0,-γ2I,0.
Next, by summing up (26) on both sides from 0 to N with respect to k, one can have(28)∑k=0TJk=ηT+1TPT+1ηT+1-η0TP0η0=∑k=0TψkTΛkψk-∑k=0Tz~kTz~k-γ2ω~kTω~kwhich is equivalent to(29)∑k=0Tz~kTz~k=∑k=0Tγ2ω~kTω~k+∑k=0TψkTΛkψk-ηT+1TPT+1ηT+1+η0TP0η0.
From (29), since PT+1>0 and the initial condition η0TP0η0≤γ2e0TΞe0, one can see that the H∞ performance can be satisfied if ψkTΛkψk≤0 is satisfied.
Next, one can see from the ellipsoidal nonlinear functiongk(xk) (2) that(30)ψkTΠkψk≤0,where (31)Πk=-δI000∗Sk-100∗∗00∗∗∗0with Sk=diag{Sk,Sk,…,Sk}.
Moreover, it also can be obtained from the sensor fault constraint (6) that(32)ψkTΠ¯kψk≤0,where (33)Π¯k=12000-C~kTKT∗000∗∗00∗∗∗2Iwith C~k=C1,k0C2,k0⋯CN,k0.
By Schur Complement Equivalence Lemma, the matrix inequality (18) can be rewritten as follows:(34)Λk-τ1kΠk-τ2kΠ¯k≤0.
By Lemma4, ψkTΛkψk≤0 can be satisfied, which means that the required H∞ performance can be satisfied. The proof is complete.Remark 8.
By means of iterative matrix inequality method, Theorem7 presents a sufficient condition for filtering error system to achieve the desired H∞ specification subject to the sensor saturations. Moreover, notice that the obtained condition is expressed by certain matrix inequalities including fractional parameter uncertainties ΔAk; we cannot solve the matrix inequality directly to obtain the explicit parametric expression of the required filter parameters G and H. As such, in the following, we will provide a theorem with which the addressed problem can be solved and the filter structure can then be obtained step by step.


## 3.2.  Filter Design

Theorem 9.
Let theH∞ performance γ>0 and the initial condition matrix Sk>0 be given. If there exist a sequence of matrices Gk(0≤k≤T), a sequence of matrices Hk(0≤k≤T), three sequences of positive scalars τ1k, τ2k, and ϵk(0≤k≤T), a sequence of positive definite matrices Xk(0≤k≤T+1), and a sequence of positive definite matrices Zk(0≤k≤T+1) satisfying η0Tdiag{X0-1,Z0-1}η0≤γ2e0Ξe0 such that(35)Λ¯k+ϵkλVTVUUT-ϵkI≤0,where (36)Λ¯k=-Xk+δτ1kI00012τ2kC~kTKTA^kTA˘kT∗-Zk+MkTMk+δτ1kI0000G¯kT∗∗-τ1kSk-100II∗∗∗-γ2I0DkTD¯kT∗∗∗∗-τ2kI0Y¯kT∗∗∗∗∗-Xk+10∗∗∗∗∗∗-Zk+1,A^k=diag⁡Ak,Ak,…,Ak,A˘k=A^k-Gk-∑i=1NΦiHkRiKk-Ck,U=00000U¯TU¯TT,U¯=diagU,U,…,U,V=V¯000000,V¯=diagV,V,…,V,with Xk=Xk-1 and Zk=Zk-1, and λ is the solution to the following optimization problem(37)minΣkλ=I-ΣkTJT-1I-JΣk-1s.t.ΣkTΣk<1,then the addressed H∞ filtering problem is solvable and the desired filter parameters at each time k can be obtained by solving the corresponding matrix inequality.Proof.
The proof is quite straightforward based on Theorem7. First, define Pk-1=diag{Xk-1,Zk-1}=diag{Xk,Zk}. One can obtain that the matrix inequality (18) is equivalent to the following inequality:(38)-Xk+δτ1kI00012τ2kC~kTKTAkTA¯kT∗-Zk+MkTMk+δτ1kI0000G¯kT∗∗-τ1kSk-100II∗∗∗-γ2I0DkTD¯kT∗∗∗∗-τ2kI0Y¯kT∗∗∗∗∗-Xk+10∗∗∗∗∗∗-Zk+1≤0.In what follows, we will try to prove that inequality (38) can be satisfied if inequality (35) is fulfilled. To this end, rewrite (38) as follows:(39)Λ¯k+UΞ~V+UΣ~VT≤0,where (40)Σ~=diagI-JΣ-1,I-JΣ-1,…,I-JΣ-1.
From Lemma6, it follows that(41)UΞ~V+UΞ~VT≤ϵ-1UUT+ϵλVTV.
Furthermore, one can obtain that(42)Λ¯k+ϵ-1UUT+ϵλVTV≤0,which, by Schur Complement Lemma, is equivalent to the linear matrix inequality (35). Thus, inequality (18) is satisfied if (35) is met, and, according to Theorem 7, the required H∞ filtering problem is solved.Remark 10.
Theorem9 outlines the methodology with which the addressed robust H∞ distributed filtering problem is solved for nonlinear systems subject to sensor saturations and fractional uncertainties. An iterative linear matrix inequality approach has been developed to calculate the desired filter parameters step by step. It should be pointed out that it will inevitably lead to the larger computing burdens. However, the algorithm we proposed in this paper is capable of dealing with the time-varying case, and the increase in computing burden can be largely reduced with the fast development of modern computer technology. In what follows, we will present a computing algorithm with which the numerical values of filter parameters can be obtained in each time step.


## 3.3.  Computational Algorithm

Algorithm 11 (H∞Filter Design Algorithm).
Step 1. Set the required γ, the initial condition Ξ, the initial η0, and T. Choose properly the values of U, V, and J. Select the initial values for X0 and Z0 satisfying the initial condition constraint.
Step 2. Set k=0. Solve (35) for G0, H0, X1, and Z1. Then the X1 and Z1 can also be obtained.
Step 3. Set k=k+1. Using the obtained Xk and Zk, solve (35) for Gk, Hk, Xk+1, and Zk+1.
Step 4. If k=N+1, then stop. Else go to Step 3.TheH∞Filter Design Algorithm gives a recursive way to obtain the numerical values of the desired filter parameters at each time point k. It should be pointed out that the existence of the filter is expressed as the feasibility of certain linear matrix inequalities that can be solved forward in time. The possible research topic in the future is to consider more performance indices and give the filtering schemes that are able to satisfy multiple requirements simultaneously.


## 4. Simulation Example

In this section, an illustrative example is presented to show the effectiveness of the proposedH∞ Filtering Design Algorithm.Let us consider the nonlinear system (1) with the parameters given below: (43)Ak=0-0.40.50.7sin⁡6k-6,Dk=0.5-0.4,C1,k=0.30.1∗sin⁡6k-6,C2,k=0.30.2∗sin⁡6k-6,C3,k=0.30.3∗sin⁡6k-6,E1,k=E2,k=E3,k=0.3,Mk=1.50.75.Let g(xk)=[sin(10xk)0.5cos(10xk)]T and set Sk=2. Let ωk=e-k/35.ChooseU=0.1, V=0.2, and J=0.25, and we solve the optimaization problem given in Theorem 9 using Matlab toolbox and obtain λ=0.7. Set γ=1, ξ0=[0.30.2500]T, Ξ=I, and N=3. Set the initial value of the system state and its estimate by x0=[0.30.25]T and x^0=[00]T, and the initial values of X0 and Z0 are selected to be X0=Z0=2I. We can see the effectiveness and applicability of the proposed filtering algorithm from the simulation results (Figures 1-2).Figure 1
The statexk1 and its estimate.Figure 2
The statexk2 and its estimate.


## 5. Conclusion

The problem of distributedH∞ filtering for nonlinear stochastic systems in the presence of sensor saturations and fractional uncertainties. A sufficient condition is proposed under which the required H∞ performance can be satisfied even if the randomly occurring sensor failures happen. A recursive algorithm is given in order to obtain the required filter parameters iteratively. Finally, an illustrative example is presented to show the effectiveness of the proposed method. Noting that the nonlinearities considered in this paper are relatively simpler, one of the possible future research directions may be the robust distributed H∞ filtering problems with much more complex nonlinearities.



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*Source: 160683-2015-10-01.xml*