# H

Authors: Zhengwang Jia; Yongcheng Sun
Journal: Discrete Dynamics in Nature and Society
 (2013)

---

## Abstract

This paper deals with theH∞ filtering problem for nonlinear systems with randomly occurring output degradation phenomenon. Such a phenomenon is described by a stochastic variable which obeys the Bernoulli distribution with probability known priorly. A sufficient condition is derived for the nonlinear system to reach the required H∞ performance. 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.


---

## Body

## 1. Introduction

Many practical engineering systems, like the radar systems which are used for tracking the hostile weapon systems, are always encountering failures. For example, the radar systems will fail from time to time due to the electromagnetic interference from the enemy. Moreover, other reasons that lead the sensors to failures mainly include the external disturbance and changes of working conditions, to name just a few; see [1–3], for example. It is worth pointing out that, in the systems mentioned above, the sensor failure is not persistent all the time but is intermittent stochastically. In other words, the sensor failure occurs at random time points in a probabilistic way. Such phenomena are called the randomly occurring phenomena which would drastically degrade the system performance. Therefore, in recent years, the randomly occurring phenomena have stirred quite a lot of research interests due to its clear engineering insights and many results have been reported in the literature; see [4–11] for some latest publications. However, in spite of its clear physical insight and importance in engineering application, the filtering problem for nonlinear time-varying systems under the circumstance of randomly occurring output degradation has not yet been studied sufficiently.On another research frontier, it is well known that the nonlinearities are inevitable in practical engineering systems, and the analysis and synthesis of nonlinear systems have been attracting more and more research attention, among which the sector bound nonlinearity which could cover several class of well-studied nonlinearities has drawn particular research focus since many sensor failures like missing measurements, signal saturations can be easily converted into the nonlinearity belonging to a known sector; see [11–15] and the references therein. On the other hand, in recent years, time-varying systems have started to receive attention due to the fact that there are virtually no strictly time-invariant systems since the working circumstances, operating points, or equipment deterioration levels are inherently time-varying in nature; see [7, 8, 16, 17] for some latest results. For time-varying Markov jumping system, in [7], the fault detection problem has been solved in the presence of sensor saturation and randomly occurring nonlinearities. The distributed H∞ estimation algorithm has been proposed in [8] for a class of nonlinear time-varying stochastic systems. A distributed filter has been designed by [16] for a type of time-varying systems while quantization errors and packet dropouts are happening. Recently, in [10, 17], mixed H2/H∞ controllers have been designed for a special type of nonlinear stochastic systems over a finite horizon. Unfortunately, despite the importance of the time-varying nature in system modeling, the H∞ filtering problem for nonlinear time-varying systems with randomly occurring output degradation has not yet received enough research.Motivated by the above discussion and in order to meet the ever-increasing practical engineering requirements toward the system reliability, in this paper, we aim to design anH∞ filter for nonlinear time-varying stochastic systems in the presence of randomly occurring output degradation. The main contributions of this paper lie in the following two aspects: (1) for the first time, the phenomena of randomly occurring output degradation are considered to better illustrate the system's complex working environment and its impact on the performance. Moreover, such a model can stand for a wide range of output degradations, including the signal quantization phenomenon and saturation as special cases; (2) by virtue of the iterative linear matrix inequality method, the filtering problem is solved for a class of nonlinear time-varying systems which are much more general than those investigated in the existing literature. The rest of the paper is arranged as follows. Section 2 formulates the H∞ filtering problem for the nonlinear time-varying systems with randomly occurring output degradation. In Section 3, the H∞ performance is analyzed in terms of certain matrix inequality. Then, it gives the methodology to solve the addressed filtering problem and outlines the computational algorithm to recursively obtain the required filter parameters. A numerical example is presented in Section 4 to show the effectiveness and applicability of the proposed algorithm. Section 5 draws the conclusion.


## 2. Problem Formulation

Let us consider the discrete-time nonlinear stochastic system and measurement equations defined onk∈[0,N]:
(1)xk+1=Akxk+f(xk)+g(xk)+Dkωk,yk=β(Ckxk)+Ekωk,zk=Mkxk,
where xk, yk, zk, and ωk represent the state, measured output, estimated output, and disturbance belonging to l2, respectively. Ak, Ck, Dk, Ek, and Mk are known real time-varying matrices with appropriate dimensions. f(xk) and g(xk) are nonlinear functions.It is assumed that the nonlinear functiong(xk) has the following constraint:
(2)gT(xk)Sk-1g(xk)≤1,
where Sk is known positive-definite matrix sequence with appropriate dimensions describing the shape of the ellipsoids with 0 being the center of the ellipsoids.Remark 1.
The nonlinear functiong(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 always being a main origin for the degradation of the system performance.Definition 2.
A nonlinear functionψ(·) is said to satisfy the sector-bounded condition if
(3)(ψ(v)-V1v)T(ψ(v)-V2v)≤0
for some real matrices V1 and V2, where V=V2-V1 is a symmetric positive-definite matrix. In this case, we say that ψ(·) belongs to [V1,V2].The nonlinear functionf(xk) in (1) belongs to sector [R1,R2] where R1 and R2 are real matrices of appropriate dimensions; that is
(4)(f(xk)-R1xk)T(f(xk)-R2xk)≤0.Remark 3.
The nonlinear functionf(xk) is used to describe the nonlinearity which belongs to a specified sector. Such a kind of nonlinearity is frequently seen in both theoretical research and engineering practice and could cover quite a lot of nonlinearities as special cases, for example, the Lipschitz type nonlinearity.The nonlinear functionβ(·) has the following form:
(5)β(Ckxk)=αkσ(Ckxk)+(1-αk)Ckxk,
where αk is a Bernoulli distributed random sequence taking values between 0 and 1 with the following probability:
(6)Prob{αk=1}=𝔼{αk}=α-.Remark 4.
Equation (5) is introduced to illustrate the constraints on the output sampling when the output signal is confronted with degradation. The nonlinear function σ(·) which will be interpreted in detail later stands for the sector bound type output degradation phenomenon.Remark 5.
The Bernoulli distributed random variableαk is employed to describe the phenomenon of the randomly occurring output degradations. To be specific, when αk=0, the output signal will not be confronted with degradation; otherwise there might be possible signal missing during the output sampling. Compared to the traditional models where the output signals are assumed to be always available, such a model proposed in this paper could obviously better reflect the real-world engineering practice especially when the working condition is changing.As discussed in [11], there exist matrices K1 and K2 such that 0≤K1<I≤K2; the sensor fault function σ(Ckxk) can be written as
(7)σ(Ckxk)=K1Ckxk+φy(Ckxk),
where φy(Ckxk) is a nonlinear vector-valued function satisfying the sector-bounded condition with V1=0 and V2=K. In this case, φy(Ckxk) can be represented as follows:
(8)φyT(Ckxk)(φy(Ckxk)-KCkxk)≤0,
where K=K2-K1.The objective of this paper is to design a filter with the following form for system (1):
(9)x^k+1=Gkx^k+Nkyk,z^k=Mkx^k,
where x^k is the state estimate, z^k is the output to be estimated, and Gk and Nk are the filter parameters to be determined.Let us define the estimation error byz~k=zk-z^k and then obtain the following filtering error system:
(10)ξk+1=(A-k+A^k)ξk+Fkf(xk)+Fkg(xk)+(Φ-k+Φ^k)φy(Ckxk)+D~kωk,z~k=M~kξk,
where
(11)ξk=[xkx^k],A-k=[Ak0NkCk+α-Nk(K1-I)CkGk],A^k=[00α~kNk(K1-I)Ck0],Fk=[I0],D~k=[DkNkEk],Φ-k=[0α-Nk],Φ^k=[0α~kNk],A˘k=[00σαNk(K1-I)Ck0],Φ˘k=[0σαNk],M~k=[Mk-Mk].Given aH∞ performance parameter γ>0, the objective of this paper is to obtain Gk and Nk in (9) such that
(12)𝔼{∑k=0N‍z~kTz~k}≤γ2∑k=0N‍ωkTωk+γ2x0TRx0
are satisfied when the sensor faults are randomly happening.


## 3. Main Results

Before giving the main results, some useful lemmas are firstly introduced.Lemma 6.
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
(13)T0-∑i=1p‍τiTi≤0,
then the following is true:
(14)W1(η)≤0,W2(η)≤0,…,Wp(η)≤0⟹W0(η)≤0.Lemma 7 (Schur complement equivalence).
Given constant matricesQ1, Q2, and Q3 where Q1=Q1T and Q2=Q2T>0, then
(15)Q1+Q3TQ2-1Q3<0
if and only if
(16)[Q1Q3TQ3-Q2]<0
or equivalently
(17)[-Q2Q3Q3TQ1]<0.


### 3.1.H∞ Performance Analysis

The following theorem gives a sufficient condition under which the requiredH∞ performance can be satisfied.Theorem 8.
Given aH∞ performance γ>0 and the initial matrix R>0, let the filter parameters Gk and Nk be given. If there exist three sequences of positive scalars τ1k, τ2k, and τ3k  (0≤k≤N) and a sequence of positive definite matrices Pk  (0≤k≤N+1) satisfying ξ0TP0ξ0≤γ2ξ0TR^ξ0 such that the following matrix inequality
(18)[Ξ11τ1kR-1T+R-2T20τ3kFkCkTKT20A-kTA˘kT*-τ1kI000FkT0**-τ2kSk-100FkT0***-τ3kI0Φ-kTΦ˘kT****-γ2ID~kT0*****-Pk+1-10******-Pk+1-1]≤0,Ξ11=-Pk+M~kTM~k-τ1kR-1TR-2+R-2TR-12+τ2kI,R-1=[R10],R-2=[R20],R^=diag{R,0}
is feasible, then H∞ performance can be satisfied.Proof.
Defining(19)Jk=ξk+1TPk+1ξk+1-ξkTPkξk,
one can get from (10) that
(20)Jk=ξk+1TPk+1ξk+1-ξkTPkξk=((A-k+A^k)ξk+Fkf(xk)+Fkg(xk)+(Φ-k+Φ^k)φy(Ckxk)+D~kωk)TPk+1×((A-k+A^k)ξk+Fkf(xk)+Fkg(xk)+(Φ-k+Φ^k)φy(Ckxk)+D~kωk)-ξkTPkξk=ξkT(A-k+A^k)TPk+1((A-k+A^k)ξk)+fT(xk)FkTPk+1Fkf(xk)+gT(xk)FkTPk+1Fkg(xk)+((Φ-k+Φ^k)φy(Ckxk))T×Pk+1((Φ-k+Φ^k)φy(Ckxk))+ωkTD~kTPk+1D~kωk+2ξkT(A-k+A^k)TPk+1Fkf(xk)+2ξkT(A-k+A^k)TPk+1Fkg(xk)+2ξkT(A-k+A^k)TPk+1(Φ-k+Φ^k)φy(Ckxk)+2ξkT(A-k+A^k)TPk+1D~kωk+2fT(xk)FkTPk+1(Φ-k+Φ^k)φy(Ckxk)+2fT(xk)FkTPk+1Fkg(xk)+2fT(xk)FkTPk+1D~kωk+2gT(xk)FkTPk+1(Φ-k+Φ^k)φy(Ckxk)+2gT(xk)FkTPk+1D~kωk+2((Φ-k+Φ^k)φy(Ckxk))TPk+1D~kωk-ξkTPkξk=ηkTΘkηk,
where
(21)ηk=[ξkTfT(xk)gT(xk)φyT(Ckxk)ωkT]T,Θk=[Θ11Θ12Θ13Θ14Θ15*FkTPk+1FkFkTPk+1FkΘ24FkTPk+1D~k**FkTPk+1FkΘ34FkTPk+1D~k***Θ44Θ45****D~kTPk+1D~k],Θ11=(A-k+A^k)TPk+1(A-k+A^k)-Pk,Θ12=(A-k+A^k)TPk+1Fk,Θ13=(A-k+A^k)TPk+1Fk,Θ14=(A-k+A^k)TPk+1(Φ-k+Φ^k),Θ15=(A-k+A^k)TPk+1D~k,Θ24=FkTPk+1(Φ-k+Φ^k),Θ34=FkTPk+1(Φ-k+Φ^k),Θ44=(Φ-k+Φ^k)TPk+1(Φ-k+Φ^k),Θ45=(Φ-k+Φ^k)TPk+1D~k.
Adding the zero term(22)z~kTz~k-γ2ωkTωk-(z~kTz~k-γ2ωkTωk)
to both sides of (20), one can get
(23)Jk=ηkTΛkηk-(z~kTz~k-γ2ωkTωk),
where
(24)Λk=Θk+diag{M~kTM~k,0,0,0,-γ2I}.
Next, by summing up (23) on both sides from 0 to N with respect to k, one can have
(25)∑k=0N‍Jk=ξN+1TPN+1ξN+1-ξ0TP0ξ0=∑k=0N‍ηkTΛkηk-∑k=0N‍(z~kTz~k-γ2ωkTωk)
which is equivalent to
(26)∑k=0N‍z~kTz~k=-γ2ωkTωk+∑k=0N‍ηkTΛkηk-ξN+1TPN+1ξN+1+ξ0TP0ξ0.
Taking the property ofαk which is contained in random matrices A^k and Φ^k in Λk into consideration, by taking the mathematical expectation of both sides of (26), one can have
(27)𝔼{∑k=0N‍z~kTz~k}=-γ2ωkTωk+∑k=0N‍ηkTΛ-kηk-ξN+1TPN+1ξN+1+ξ0TP0ξ0,
where
(28)Λ-k=[Λ-11A-kTPk+1FkA-kTPk+1FkA-kTPk+1D~k*FkTPk+1FkFkTPk+1FkFkTPk+1Φ-kFkTPk+1D~k**FkTPk+1FkFkTPk+1Φ-kFkTPk+1D~k***Φ-kTPk+1D~k****D~kTPk+1D~k-γ2I],Λ-11=A-kTPk+1A-k+A˘kTPk+1A˘k-Pk+M~kTM~k,Λ-14=A-kTPk+1Φ-k+A˘kTPk+1Φ˘k,Λ-44=Φ-kTPk+1Φ-k+Φ˘kTPk+1Φ˘k.
From (27), since PN+1>0 and the initial condition ξ0TP0ξ0≤γ2ξ0TR^ξ0=γ2x0TRx0, one can see that the H∞ performance can be satisfied if ηkTΛ-kηk≤0 is satisfied.
Next, one can see from the sector-bounded condition of nonlinear functionf(xk) (4) that
(29)ηkTΥkηk≤0,
where
(30)Υk=12[R-1TR-2T+R-2TR-1-R-1T+R-2T000*2I000**000***00****0].
Thirdly, one can get from the ellipsoidal nonlinear functiong(xk) (2) that
(31)ηkTΩkηk≤0,
where
(32)Ωk=[-10000*0000**Sk-100***00****0].
Moreover, it also can be obtained from the sensor fault constraint (8) that
(33)ηkTΠkηk≤0,
where
(34)Πk=12[000-FkCkTKT0*0000**000***2I0****0].
By Schur complement equivalence lemma, the matrix inequality (18) can be rewritten as follows:
(35)Λ-k-τ1kΥk-τ2kΩk-τ3kΠk≤0.
By Lemma6, ηkTΛ-kηk≤0 can be satisfied, which means that the required H∞ performance can be satisfied. The proof is complete.


### 3.2. Filter Design

Theorem 9.
Let theH∞ performance γ>0 and the initial condition matrix R>0 be given. If there exist a sequence of matrices Gk  (0≤k≤N), a sequence of matrices Nk  (0≤k≤N), three sequences of positive scalars τ1k, τ2k, and τ3k  (0≤k≤N), a sequence of positive-definite matrices Xk  (0≤k≤N+1), and a sequence of positive-definite matrices Yk  (0≤k≤N+1) satisfying ξ0Tdiag{X0-1,Y0-1}ξ0≤γ2ξ0R^ξ0 such that the following matrix inequality
(36)[Γ1Γ2*Γ3]≤0,
where
(37)Γ1=[Γ11-MkTMkΓ13012τ3kCkTKT0*Γ220000**-τ1kI000***-τ2kSk-100****-τ3kI0*****-γ2I],Γ2=[AkTΓ180Γ110GkT000I000I0000α-NkT0σαNkTDkTEkTNkT00],Γ3=diag{-Xk+1,-Yk+1,-Xk+1,-Yk+1},Γ11=-Hk-12τ1k(R1TR2+R2TR1)+τ2kI+MkTMk,Γ13=12τ1k(R1T+R2T),Γ22=-Lk+τ2kI+MkTMk,Γ18=CkT((1-α-)I+α-K1)NkT,Γ110=σαCkT(K1-I)NkT
is satisfied, then the required H∞ filter parameters at each time k can be obtained by solving the corresponding matrix inequality.Proof.
The proof is quite straightforward. First, definePk-1=diag{Hk-1,Lk-1}=diag{Xk,Yk}. From Theorem 8, (36) can be obtained by simply expanding the matrix inequality (18).


### 3.3. Computational Algorithm

Algorithm 10 (H∞ filter design algorithm).
Step  1.
Set the requiredγ, the initial condition R, the initial ξ(0), and N. Select the initial values for H0 and L0 satisfying the initial condition constraint.
Step  2.
Setk=0. Solve (36) for G0, N0, X1, and Y1. Then H1 and L1 can also be obtained.
Step  3.
Setk=k+1. Using the obtained Hk and Lk, solve (36) for Gk, Nk, Xk+1, and Yk+1.
Step  4.
Ifk=N+1, then stop. Else go to Step  3.Remark 11.
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 8.
Given aH∞ performance γ>0 and the initial matrix R>0, let the filter parameters Gk and Nk be given. If there exist three sequences of positive scalars τ1k, τ2k, and τ3k  (0≤k≤N) and a sequence of positive definite matrices Pk  (0≤k≤N+1) satisfying ξ0TP0ξ0≤γ2ξ0TR^ξ0 such that the following matrix inequality
(18)[Ξ11τ1kR-1T+R-2T20τ3kFkCkTKT20A-kTA˘kT*-τ1kI000FkT0**-τ2kSk-100FkT0***-τ3kI0Φ-kTΦ˘kT****-γ2ID~kT0*****-Pk+1-10******-Pk+1-1]≤0,Ξ11=-Pk+M~kTM~k-τ1kR-1TR-2+R-2TR-12+τ2kI,R-1=[R10],R-2=[R20],R^=diag{R,0}
is feasible, then H∞ performance can be satisfied.Proof.
Defining(19)Jk=ξk+1TPk+1ξk+1-ξkTPkξk,
one can get from (10) that
(20)Jk=ξk+1TPk+1ξk+1-ξkTPkξk=((A-k+A^k)ξk+Fkf(xk)+Fkg(xk)+(Φ-k+Φ^k)φy(Ckxk)+D~kωk)TPk+1×((A-k+A^k)ξk+Fkf(xk)+Fkg(xk)+(Φ-k+Φ^k)φy(Ckxk)+D~kωk)-ξkTPkξk=ξkT(A-k+A^k)TPk+1((A-k+A^k)ξk)+fT(xk)FkTPk+1Fkf(xk)+gT(xk)FkTPk+1Fkg(xk)+((Φ-k+Φ^k)φy(Ckxk))T×Pk+1((Φ-k+Φ^k)φy(Ckxk))+ωkTD~kTPk+1D~kωk+2ξkT(A-k+A^k)TPk+1Fkf(xk)+2ξkT(A-k+A^k)TPk+1Fkg(xk)+2ξkT(A-k+A^k)TPk+1(Φ-k+Φ^k)φy(Ckxk)+2ξkT(A-k+A^k)TPk+1D~kωk+2fT(xk)FkTPk+1(Φ-k+Φ^k)φy(Ckxk)+2fT(xk)FkTPk+1Fkg(xk)+2fT(xk)FkTPk+1D~kωk+2gT(xk)FkTPk+1(Φ-k+Φ^k)φy(Ckxk)+2gT(xk)FkTPk+1D~kωk+2((Φ-k+Φ^k)φy(Ckxk))TPk+1D~kωk-ξkTPkξk=ηkTΘkηk,
where
(21)ηk=[ξkTfT(xk)gT(xk)φyT(Ckxk)ωkT]T,Θk=[Θ11Θ12Θ13Θ14Θ15*FkTPk+1FkFkTPk+1FkΘ24FkTPk+1D~k**FkTPk+1FkΘ34FkTPk+1D~k***Θ44Θ45****D~kTPk+1D~k],Θ11=(A-k+A^k)TPk+1(A-k+A^k)-Pk,Θ12=(A-k+A^k)TPk+1Fk,Θ13=(A-k+A^k)TPk+1Fk,Θ14=(A-k+A^k)TPk+1(Φ-k+Φ^k),Θ15=(A-k+A^k)TPk+1D~k,Θ24=FkTPk+1(Φ-k+Φ^k),Θ34=FkTPk+1(Φ-k+Φ^k),Θ44=(Φ-k+Φ^k)TPk+1(Φ-k+Φ^k),Θ45=(Φ-k+Φ^k)TPk+1D~k.
Adding the zero term(22)z~kTz~k-γ2ωkTωk-(z~kTz~k-γ2ωkTωk)
to both sides of (20), one can get
(23)Jk=ηkTΛkηk-(z~kTz~k-γ2ωkTωk),
where
(24)Λk=Θk+diag{M~kTM~k,0,0,0,-γ2I}.
Next, by summing up (23) on both sides from 0 to N with respect to k, one can have
(25)∑k=0N‍Jk=ξN+1TPN+1ξN+1-ξ0TP0ξ0=∑k=0N‍ηkTΛkηk-∑k=0N‍(z~kTz~k-γ2ωkTωk)
which is equivalent to
(26)∑k=0N‍z~kTz~k=-γ2ωkTωk+∑k=0N‍ηkTΛkηk-ξN+1TPN+1ξN+1+ξ0TP0ξ0.
Taking the property ofαk which is contained in random matrices A^k and Φ^k in Λk into consideration, by taking the mathematical expectation of both sides of (26), one can have
(27)𝔼{∑k=0N‍z~kTz~k}=-γ2ωkTωk+∑k=0N‍ηkTΛ-kηk-ξN+1TPN+1ξN+1+ξ0TP0ξ0,
where
(28)Λ-k=[Λ-11A-kTPk+1FkA-kTPk+1FkA-kTPk+1D~k*FkTPk+1FkFkTPk+1FkFkTPk+1Φ-kFkTPk+1D~k**FkTPk+1FkFkTPk+1Φ-kFkTPk+1D~k***Φ-kTPk+1D~k****D~kTPk+1D~k-γ2I],Λ-11=A-kTPk+1A-k+A˘kTPk+1A˘k-Pk+M~kTM~k,Λ-14=A-kTPk+1Φ-k+A˘kTPk+1Φ˘k,Λ-44=Φ-kTPk+1Φ-k+Φ˘kTPk+1Φ˘k.
From (27), since PN+1>0 and the initial condition ξ0TP0ξ0≤γ2ξ0TR^ξ0=γ2x0TRx0, one can see that the H∞ performance can be satisfied if ηkTΛ-kηk≤0 is satisfied.
Next, one can see from the sector-bounded condition of nonlinear functionf(xk) (4) that
(29)ηkTΥkηk≤0,
where
(30)Υk=12[R-1TR-2T+R-2TR-1-R-1T+R-2T000*2I000**000***00****0].
Thirdly, one can get from the ellipsoidal nonlinear functiong(xk) (2) that
(31)ηkTΩkηk≤0,
where
(32)Ωk=[-10000*0000**Sk-100***00****0].
Moreover, it also can be obtained from the sensor fault constraint (8) that
(33)ηkTΠkηk≤0,
where
(34)Πk=12[000-FkCkTKT0*0000**000***2I0****0].
By Schur complement equivalence lemma, the matrix inequality (18) can be rewritten as follows:
(35)Λ-k-τ1kΥk-τ2kΩk-τ3kΠk≤0.
By Lemma6, ηkTΛ-kηk≤0 can be satisfied, which means that the required H∞ performance can be satisfied. The proof is complete.


## 3.2. Filter Design

Theorem 9.
Let theH∞ performance γ>0 and the initial condition matrix R>0 be given. If there exist a sequence of matrices Gk  (0≤k≤N), a sequence of matrices Nk  (0≤k≤N), three sequences of positive scalars τ1k, τ2k, and τ3k  (0≤k≤N), a sequence of positive-definite matrices Xk  (0≤k≤N+1), and a sequence of positive-definite matrices Yk  (0≤k≤N+1) satisfying ξ0Tdiag{X0-1,Y0-1}ξ0≤γ2ξ0R^ξ0 such that the following matrix inequality
(36)[Γ1Γ2*Γ3]≤0,
where
(37)Γ1=[Γ11-MkTMkΓ13012τ3kCkTKT0*Γ220000**-τ1kI000***-τ2kSk-100****-τ3kI0*****-γ2I],Γ2=[AkTΓ180Γ110GkT000I000I0000α-NkT0σαNkTDkTEkTNkT00],Γ3=diag{-Xk+1,-Yk+1,-Xk+1,-Yk+1},Γ11=-Hk-12τ1k(R1TR2+R2TR1)+τ2kI+MkTMk,Γ13=12τ1k(R1T+R2T),Γ22=-Lk+τ2kI+MkTMk,Γ18=CkT((1-α-)I+α-K1)NkT,Γ110=σαCkT(K1-I)NkT
is satisfied, then the required H∞ filter parameters at each time k can be obtained by solving the corresponding matrix inequality.Proof.
The proof is quite straightforward. First, definePk-1=diag{Hk-1,Lk-1}=diag{Xk,Yk}. From Theorem 8, (36) can be obtained by simply expanding the matrix inequality (18).


## 3.3. Computational Algorithm

Algorithm 10 (H∞ filter design algorithm).
Step  1.
Set the requiredγ, the initial condition R, the initial ξ(0), and N. Select the initial values for H0 and L0 satisfying the initial condition constraint.
Step  2.
Setk=0. Solve (36) for G0, N0, X1, and Y1. Then H1 and L1 can also be obtained.
Step  3.
Setk=k+1. Using the obtained Hk and Lk, solve (36) for Gk, Nk, Xk+1, and Yk+1.
Step  4.
Ifk=N+1, then stop. Else go to Step  3.Remark 11.
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. An Illustrative 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:
(38)Ak=[0.5+0.1sin(k)-10.2-0.5],Dk=[0.2-0.4],Ck=[1-cos(5k)0.7],Ek=0.3,Mk=[1.50.75].
Let g(xk)=[sin(10xk)0.5cos(10xk)]T and set Sk=2. Let ωk=sin(5k)/3(k+1).Assume that the nonlinear functionf(xk) which is chosen as
(39)f(xk)=[-0.1x1k+0.25x2k+0.1x2ksin(x1k)x1k2+x2k2+10-0.05x1k+0.05x2k]
belongs to sector [R1,R2] with
(40)R1=[-0.40-0.2-0.3],R2=[0.20.30.10.4].Setγ=0.9, R=I, ξ0=[0.50.500]T, N=4, and α-=0.8. Set H0=0.8I and L0=I; hence the initial condition can be satisfied. The required filter parameters can be obtained by Algorithm 10, and some of the results are listed in Table 1.Table 1
The filter parameters.
G
k
N
k
k
=
0
[
0.25
-
0.1
0.23
0.7
]
[
0.54
1.25
]
k
=
1
[
-
0.65
-
0.03
0.38
1.12
]
[
1.40
-
0.25
]
k
=
2
[
-
0.16
0.45
-
0.28
0.07
]
[
1.25
0.17
]
k
=
3
[
1.65
-
0.39
-
0.87
0.05
]
[
1.90
0.15
]
⋮
⋮
⋮Set the initial value of the system state and its estimate byx0=[10]T and x^0=[00]T. We can see the effectiveness and applicability of the proposed filtering algorithm from the simulation results, Figures 1 and 2.Figure 1
The statexk1 and its estimate.Figure 2
The statexk2 and its estimate.


## 5. Conclusion

The problem ofH∞ filtering for nonlinear stochastic systems when the measurement output is confronted with possible randomly occurring degradation is considered. Such a phenomenon is described by a Bernoulli sequence with known probability. A sufficient condition is proposed under which the required H∞ performance can be satisfied even if the randomly occurring output degradation happens. 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.



---

*Source: 310915-2013-12-02.xml*