# H

Authors: Yan Li; Weihai Zhang; Xikui Liu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper discusses theH- index problem for stochastic linear discrete-time systems. A necessary and sufficient condition of H- index is given for such systems in finite horizon. It is proved that when the H- index is greater than a given value, the feasibility of H- index is equivalent to the solvability of a constrained difference equation. The above result can be applied to the fault detection observer design. Finally, some examples are presented to illustrate the proposed theoretical results.


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## Body

## 1. Introduction

Model-based fault detection has attracted increasing attention in recent years because of its importance in reliability, security, and fault tolerance of dynamic systems; see [1–3]. In general, model-based fault detection is related to residual generation, that is, constructing a residual signal and comparing it with a predefined threshold. If the residual exceeds the threshold, an alarm is ringed. However, the residual can change due to the effects of external disturbance and model uncertainty. So fault detection observers must be insensitive to external disturbance and model uncertainty. Some approaches have been given for the design of fault detection observers, such as H∞ norm, H2 norm, and H- index, which are to evaluate the effectiveness of a fault detection observer design [2, 3]. The H∞ norm characterizes the maximum effect of an input on an output, which plays an important role in robust control and was widely generalized by [4–9]. An upper bound of H∞ norm can be described by means of the bounded real lemma. On the contrary, H- index is used for measuring the sensitivity of residual to fault, which aims to maximize the minimum effect of fault on the residual output of a fault detection observer; see [3, 10–16] and the reference therein. In [3], H- index in zero frequency was defined by using the minimum nonzero singular value. In [13], H- index was defined as the minimum singular value over a given frequency range. A necessary and sufficient condition was given by LMIs for the infinite frequency range. The case for finite frequency range was obtained in terms of frequency weighting.In recent years, theH- index in time domain has attracted more attention. In [16], the authors developed a fault residual generator to maximize the fault sensitivity by H- index in the finite time domain. Based on H- index, the problems considered in [17, 18] were about the optimal fault detection for discrete time-varying linear systems. In [19], a necessary and sufficient condition of the H- index for linear continuous time-varying systems in finite horizon was given. Characterization of H- index for linear discrete time-varying systems was discussed in [20]. The H- index was described by the existence of the solution to a backward difference Riccati equation.Although there was much work on theH- index, little work was concerned with the the H- index in stochastic linear systems. In this paper, the characterization of the H- index for stochastic linear systems in finite horizon is presented. The definition of the H- index is extended to stochastic systems. New necessary and sufficient conditions are given for the H- index. The feasibility of the H- index is shown to be equivalent to the solvability of a constrained difference equation. As a special case, the stochastic H- of square systems is addressed in this paper. Our results can be viewed as the extensions of deterministic systems, which can be applied to the fault detection.The outline of the paper is organized as follows. Section2 is devoted to developing some efficient criteria for the linear stochastic H- index in finite horizon. Section 3 contains some examples provided to show the efficiency of the proposed results. Finally, we end this paper in Section 4 with a brief conclusion.Notations. R is the field of real numbers. Rm×n is the vector space of all m×n matrices with entries in R. Sn(R) is the set of all real symmetric matrices Rn×n. A′ denotes the transpose of the complex matrix A. A-1 is the inverse of A. Given a positive semidefinite (positive definite) matrix A, we denote it by A≥0 (A>0). Let E denote the mathematical expectation. In is n×n identity matrix. 0n is n×n zero matrix. Consider NT={0,1,…,T}, N={0,1,…}. A tall system refers to a system when the number of inputs is less than that of outputs. A wide system is the case of more inputs than outputs. A square system denotes a system when the number of inputs equals the output number.


## 2. Finite Horizon StochasticH- Index

In this section, we will discuss the finite horizon stochasticH- index problem. We give a necessary and sufficient condition for the finite horizon stochastic H- index.Consider the following stochastic system:(1)xt+1=Atxt+Btvt+A0txt+B0tvtwt,t∈NT,zt=Ctxt+Dtvt,x0=x0,where {w(t)}t≥0 is a sequence of one-dimensional independent real random variables defined on the complete probability space (Ω,F,P) with E[w(t)]=0 and E[w(s)w(t)]=δst, s,t∈NT, where δst is the Kronecker delta. Suppose v(t) and w(t) are independent. A∈Rn×n, B∈Rn×l, A0∈Rn×n, B0∈Rn×l, C∈Rq×n, and D∈Rq×l. x(t)∈Rn, v(t)∈Rl, and z(t)∈Rq are the state, input, and output, respectively.We define theσ-algebra generated by w(t), t∈Nt, Ft=σ{w(s),s∈Nt}, t∈N. {Ft}t∈N is an increasing sequence of σ-algebra Ft⊂F and w(t) is adapted to Ft for all t∈N. Let L2(Ω,Rk) be the space of Rk-valued random vectors ξ with Eξ2<∞. lw2(NT;Rk) denotes the space of all sequences y(t)∈L2(Ω,Rk) that are Ft-1 measurable for all t∈NT. The l2-norm of lw2(NT;Rk) is defined by y(·)lw2(NT;Rk)=(∑t∈NTE[y′(t)y(t)])1/2<∞. We suppose that x0 is deterministic. For any T∈N and (x0,v)∈Rn×lw2(NT;Rl), there exists a unique solution x(·)=x(·;x0,v)∈lw2(NT+1;Rn) of (1) with x(0)=x0.The finite horizon stochasticH- index problem of system (1) can be stated as follows.Definition 1.
For stochastic system (1), given 0≤T<∞, define (2)G-0,T=infv∈lw2NT;Rl,v≠0,x0=0⁡ztlw2NT;Rqvtlw2NT;Rl=infv∈lw2NT;Rl,v≠0,x0=0⁡∑t∈NTEz′tzt1/2∑t∈NTEv′tvt1/2,which is called H- index of (1) in NT.Remark 2.
Definition1 describes the smallest sensitivity of stochastic system (1) from input v to output z in time domain. Assume that v is fault signal and z is the residual; then G-[0,T] characterizes the minimal fault sensitivity.Remark 3.
When system (1) is wide, G-[0,T]=0 (see [20]). In this paper, we suppose that system (1) is tall or square.For any givenγ≥0, and T∈N, let (3)JTγx0,v=ztlw2NT;Rq2-γ2vtlw2NT;Rl2=∑t=0TEz′tzt-γ2v′tvt=∑t=0TECtxt+Dtvt′·Ctxt+Dtvt-γ2v′tvt,where x(·)=x(·;x0,v) is the solution of (1) and z(·)=z(·;x0,v) is the corresponding output. We will discuss the following optimal control problem: (4)min⁡JTγx0,vforv∈lw2NT;Rl.Remark 4.
Obviously,G-[0,T]>γ is equivalent to the following inequality: (5)JTγ0,v=ztlw2NT;Rq2-γ2vtlw2NT;Rl2=∑t=0TEz′tzt-γ2v′tvt>0,for allv∈lw2(NT;Rl), v≠0, x0=0.Remark 5.
WhenT=∞, (2) and (5) correspond to the infinite horizon H- index case.In the following, we present some useful lemmas, which play important roles throughout the paper.Lemma 6.
For givenT∈N, if (P(0),P(1),…,P(T+1)) is an arbitrary family of matrices in Sn(R), then for any x0∈Rn(6)JTγx0,v=x0′P0x0-Ex′T+1PT+1xT+1+∑t=0TExtvt′Mt,P·xtvt,where(7)Mt,P·=LPt+1-PtKPt+1K′Pt+1HγPt+1,LPt+1=A′tPt+1At+C′tCt+A0′tPt+1A0t,KPt+1=A′tPt+1Bt+C′tDt+A0′tPt+1B0t,HγPt+1=B′tPt+1Bt+D′tDt-γ2I+B0′tPt+1B0t.Proof.
Sincew(t) is independent of v(t), we conclude that A(t)x(t)+B(t)v(t) and A0(t)x(t)+B0(t)v(t) are Ft-1 measurable and independent of w(t), so (8)EAtxt+Btvt′Pt+1·A0txt+B0tvtwt=EA0txt+B0tvt′Pt+1·Atxt+Btvtwt=0.It follows that (9)Ex′t+1Pt+1xt+1-x′tPtxt=Extvt′Qt,P·xtvt,where (10)Qt,P·=Q1tQ2tQ2′tQ3t,where (11)Q1t=A′tPt+1At-Pt+A0′tPt+1A0t,Q2t=A′tPt+1Bt+A0′tPt+1B0t,Q3t=B′tPt+1Bt+B0′tPt+1B0t.Take summation from t=0 to T; it yields that (12)Ex′T+1PT+1xT+1-x′0P0x0=∑t=0TExtvt′Qt,P·xtvt.From (3), we get (13)JTγx0,v=∑t=0TEz′tzt-γ2v′tvt=∑t=0TECtxt+Dtvt′·Ctxt+Dtvt-γ2v′tvt+∑t=0TExtvt′Qt,P·xtvt-Ex′T+1PT+1xT+1+x′0P0x0=∑t=0TExtvt′Mt,P·xtvt-Ex′T+1PT+1xT+1+x0′P0x0,which completes the proof.Theorem 7.
For (1) and given γ≥0, if the following equation (14)Pt=LPt+1-KPt+1HγPt+1-1K′Pt+1,HγPt+1>0,PT+1=0has a solution PT(t), ∀t∈NT, then G-[0,T]>γ.Proof.
For anyv∈lw2(NT;Rl), v≠0, x0=0, by Lemma 6, we have (15)JTγ0,v=∑t=0TExtvt′Mt,PT·xtvt.
By completing squares and considering the first equality in (14), we obtain that (16)JTγ0,v=∑t=0TEx′t-PTt+LPTt+1-KPTT+1HγPTt+1-1K′PTT+1·xt+∑t=0TEvt+HγPTt+1-1K′PTt+1xt′·HγPTt+1vt+HγPTt+1-1K′PTt+1xt=∑t=0TEvt-v∗t′HγPTt+1vt-v∗t,where v∗(t)=-Hγ(PTt+1)-1K′(PT(t+1))x(t).
FromHγ(PT(t+1))>0, it is obvious that JTγ(0,v)≥0 and JTγ(0,v)=0 if and only if v(t)=v∗(t). Let us substitute v(t)=v∗(t) into system (1). It must be x(t)=0, t∈NT on the basis of the fact x0=0, which results in v∗(t)=0, t∈NT. Therefore, it is deduced that JTγ(0,v)=0 if and only if v(t)=v∗(t)=0, which contradicts the condition v(t)≠0. Without loss of generality, we assume that Hγ(PT(t+1))>ϵIl, ϵ>0. Then, (16) indicates that JTγ(0,v)>ϵv(t)-v∗(t)lw2(NT;Rl)2>0, which implies that G-[0,T]>γ. Theorem 7 is proved.The necessity of Theorem7 will be proved by a sequence of lemmas. To this end, we consider the following backward matrix equation: (17)Xt=LXt+1+KXt+1Ft+F′tHγXt+1Ft+F′tK′Xt+1,t∈NT,where F:NT→Rl×n is a given finite sequence of matrices. This equation has a unique solution X(t)=PFγ(t), t∈NT+1, satisfying X(T+1)=0.By the above,PFγ(t) is the solution of the following equation: (18)PFγt=InF′tLPFγt+1KPFγt+1K′PFγt+1HγPFγt+1InFt,PFγT+1=0,t∈NT.Lemma 8.
For the givenx0∈Rn, v∈lw2(NT;Rl), and F:NT→Rl×n, if the following equation (19)xFt+1=At+BtFtxFt+A0t+B0tFtxFtwt+B0tvtwt+Btvt,xF0=x0,t∈NTadmits a solution xF(t)=x(t;x0,F(t)xF(t)+v(t)), then the cost function J=JTγ(x0,F(t)xF(t)+v(t)) is given by (20)J=x0′PFγ0x0+∑t=0TExF′tG′tvt+v′tGtxFt+v′tHγPFγt+1vt,where PFγ(t), t∈NT+1 is the solution of (18) and (21)dssdsGt=K′PFγt+1+HγPFγt+1Ft,t∈NT.Furthermore, for v=0, (22)JTγx0,FtxFt=x0′PFγ0x0.Proof.
By Lemma6, we can derive that (23)JTγx0,FtxFt+vt=x0′PFγ0x0+∑t=0TExFtFtxFt+vt′Mt,PFγ·xFtFtxFt+vt=x0′PFγ0x0+∑t=0TExF′tInF′tLPFγt+1-PFγtKPFγt+1K′PFγt+1HγPFγt+1InFtxFt+∑t=0TEv′tGtxFt+xF′tG′tvt+v′tHγPFγt+1vt=x0′PFγ0x0+∑t=0TEv′tGtxFt+xF′tG′tvt+v′tHγPFγt+1vt.For v=0, (22) is obvious.Next, we will show that matricesHγ(PFγ(t+1)), t∈NT, are invertible.Lemma 9.
For system (1), assume that, for given γ≥0, G-[0,T]>γ. For given F:NT→Rl×n and T∈N, if PFγ(t) is the solution of (18), then (24)HγPFγt+1≥G-0,T2-γ2Il>0,t∈NT.Proof.
We first proveHγ(PFγ(t+1))≥0 on NT. Suppose that there exist t∗∈NT, η>0, u∈Rl, and u=1 such that u′Hγ(PFγ(t∗+1))u≤-η. Set F(t)=0, t∈NT, and (25)vt=0,t≠t∗,t∈NT;u,t=t∗.According to Lemma 8, we have (26)JTγ0,v=∑t=0TExF′tG′tvt+v′tGtxFt+v′tHγPFγt+1vt=ExF′t∗G′t∗·u+u′Gt∗xFt∗+u′HγPFγt∗+1u.From the definition of v(t), t∈NT, and (19), it follows that xF(t)=0 for t≤t∗. Additionally, in view of G-[0,T]>γ, we conclude that (27)0≤JTγ0,v=u′HγPFγt∗+1u≤-η,which leads to a contradiction. So, Hγ(PFγ(t+1))≥0 for any t∈NT.
Now let(G-[0,T])2>γ2+ρ2 for ρ>0 and λ=(γ2+ρ2)1/2. Replacing γ with λ in (18), we obtain the corresponding solution PFλ(t). As in the preceding proof, we have that Hγ(PFλ(t+1))≥0. For any t0 and t∈NT-t0, define Ft0(t)=F(t+t0). Let PFt0λ(t), t∈NT-t0, be the solution of (18) with γ and F replaced by λ and Ft0, respectively. Then PFt0λ(t)=PFλ(t+t0), t∈NT+1-t0. By (22), it follows that (28)x0′PFλt0x0=x0′PFt0λ0x0=JT-t0λx0,Ft0xFt0≤JT-t0γx0,Ft0xFt0=x0′PFγt0x0.Therefore, Hλ(PFγ(t0+1))≥Hλ(PFλ(t0+1))≥0, which means that Hγ(PFγ(t+1))≥ρ2I for all t∈NT and arbitrary ρ2<(G-[0,T])2-γ2. So Hγ(PFγ(t+1))>[(G-[0,T])2-γ2]Il.
This completes the proof.Remark 10.
From (24), for t=T, Hγ(PFγ(T+1))=D′(T)D(T)-γ2I>0. If system (1) is time-invariant and satisfies Lemma 9, then (29)D′D-γ2I>0.Remark 11.
IfI-AB or I-BA is invertible, then A(I-BA)-1=(I-AB)-1A. By this equality, we see that C′[I-D(D′D-γ2I)-1D′]C=C′[I-γ-2DD′]-1C. If system (1) is square and time-invariant, from (29), we can conclude that (30)C′I-DD′D-γ2I-1D′C=C′I-γ-2DD′-1C≤0.However, the above is not true for tall systems.Now, we discuss the necessity of Theorem7 and present the following theorem.Theorem 12.
For system (1), if G-[0,T]>γ for given γ≥0, then (14) has a unique solution PTγ(t), t∈NT+1, for any T≥0. Furthermore, JTγ(x0,v) is minimized with the optimal cost given by (31)minv∈lw2NT;Rl⁡JTγx0,v=x0′PTγ0x0and the optimal control is determined by (32)v∗t=F∗txF∗t,F∗t=-HγPTγt+1-1K′PTγt+1,where xF∗(t) satisfies (33)xF∗t+1=At+BtF∗txF∗t+A0t+B0tF∗txF∗twt,xF∗0=x0.Proof.
We first prove thatG-[0,T]>γ means that (14) admits a solution PTγ(t) on NT+1. As Hγ(PTγ(T+1))=D′(T)D(T)-γ2I>0, it is clear that there exists a solution to (14) at t=T; that is, (34)PTγT=LPTγT+1-KPTγT+1·D′TDT-γ2IK′PTγT+1.Suppose (14) does not have a solution on NT; then there must exist a minimum number T~∈NT, 0<T~≤T, such that (14) is solvable backward up to t=T~. That is to say, PTγ(T~),PTγ(T~+1),…,PTγ(T+1) satisfy (14) but PTγ(T~-1) does not, or Hγ(PTγ(T~)) is not a positive definite matrix.
SetF~T(t)=-Hγ(PTt+1)-1K′(PTγ(t+1)), t=T~,T~+1,…,T, and then F~T(t) is well defined. Let (35)F~t=0,t=0,1,…,T~-1,F~Tt,t=T~,T~+1,…,T.
Consider the following equation:(36)Pt=LPt+1+KPt+1F~t+F~′tK′Pt+1+F~′tHγPt+1F~t,HγPt+1>0,PT+1=0.Equation (36) admits a solution P~Tγ(t), t∈NT+1. Comparing (36) with (14), we arrive at PTγ(t)=P~Tγ(t) for t=T~,T~+1,…,T. Moreover, along the same line of Lemma 9, we have that Hγ(P~Tγ(t+1))>0 on NT. In particular, Hγ(PTγ(T~))=Hγ(P~Tγ(T~))>0. This is inconsistent with the nonpositiveness of Hγ(PTγ(T~)). Hence, (14) has a unique solution PTγ(t), t∈NT+1, for any T≥0.
Next, we suppose that the following equation(37)Xt=LXt+1-KXt+1HγXt+1-1K′Xt+1,t∈NT;XT+1=0admits a solution PTγ(t), t∈NT+1, and then F∗(t) is well defined by (32). If we replace F(t) in (17) by F∗(t), then (18) becomes (37) with X(t)=PF∗γ(t), so PF∗γ(t)=PTγ(t) for all t∈NT+1. By (21), when F(t)=F∗(t), it yields G(t)=0. By (20), we come to a conclusion that for v(t)∈lw2(NT;Rl)(38)JTγx0,F∗txFt+vt=x0′PTγ0x0+∑t=0TEv′tHγPTγt+1vt.By (24), we deduce that v∗(t)=F∗(t)xF∗(t) minimizes JTγ(x0,v) with the optimal value expressed by (31). This proof is complete.Lemma 13.
If system (1) is time-invaint, square and γ≥0, then, for any fixed t∈N, T¯+1>T+1>t, (39)PT¯γt≤PTγt.Proof.
Since system (1) is time-invaint, by the time invariance of (37), we have PTγ(t)=PT-tγ(0). Without loss of generality, we assume that t=0, T¯>T, and vT(·) is optimal for x0∈Rn on NT. Let (40)vt=vTt,t∈NT,-R-1D′Cxt,t∈T+1,T+2,…,T¯and R=D′D-γ2I. By (31) and Remark 11, (41)x0′PT¯γ0x0≤JT¯γx0,v=JTγx0,vT+∑t=T+1T¯Ez′t·zt-γ2v′tvt=JTγx0,vT+∑t=T+1T¯Ex′tC′I-DR-1D′Cxt+∑t=T+1T¯Evt+R-1D′Cxt′·Rvt+R-1D′Cxt=JTγx0,vT+∑t=T+1T¯Ex′tC′I-DR-1D′Cxt≤JTγx0,vT=x0′PTγ0x0.This implies (39).Based on Theorems7 and 12, it is easy to get the following main result.Theorem 14.
For system (1) and a given γ≥0, the following are equivalent. (a)
G-[0,T]>γ.(b)
The following equation(42)Pt=LPt+1-KPt+1HγPt+1-1KPt+1′,HγPt+1>0,PT+1=0admits a unique solution PTγ(t) on NT+1. Moreover, minv∈lw2(NT;Rl)⁡JTγ(x0,v)=x0′PTγ(0)x0.Remark 15.
The solution of (42) is not necessarily negative or positive definite.Theorem 16.
For givenγ≥0, if system (1) is time-invariant and square, then the following are equivalent. (a)
G-[0,T]>γ.(b)
The following equation(43)Pt=A′Pt+1A+A0′Pt+1A0+C′C-A′Pt+1B+A0′Pt+1B0+C′D·B′Pt+1B+B0′Pt+1B0+D′D-γ2I-1·A′Pt+1B+A0′Pt+1B0+C′D′,B′Pt+1B+B0′Pt+1B0+D′D-γ2I>0,PT+1=0admits a unique solution PTγ(t)≤0 on NT+1. Moreover, minv∈lw2NT;Rl⁡JTγ(x0,v)=x0′PTγ(0)x0.Proof.
By Theorem14, Theorem 16 is established as long as we prove PTγ(t)≤0.
For anyt0∈NT,x(t0)=x∈Rn, from Lemma 6 and (43), using completing squares method, it follows that (44)JTγx,v;t0=∑t=t0TEz′tzt-γ2v′tvt=x′PTγt0x+∑t=t0TExtvt′Mt,PTγ··xtvt=x′PTγt0x+∑t=t0TEvt-v∗t′·HγPTγt+1vt-v∗t,where v∗(t)=-Hγ(PTt+1)-1K′(PT(t+1))x(t).
SetR=D′D-γ2I. By completing squares, we have (45)JTγx,v;t0=∑t=t0TEz′tzt-γ2v′tvt=∑t=t0TEx′tC′I-DR-1D′Cxt+∑t=t0TEvt+R-1D′Cxt′·Rvt+R-1D′Cxt=∑t=t0TEx′t·C′I-DR-1D′Cxt+∑t=t0TEvt-v~t′·Rvt-v~t,where v~(t)=-R-1D′Cx(t).
Based on the above and Remark11, it is easy to see that (46)minv∈lw2NT;RlJTγx,v;t0=JTγx,v∗;t0=x′PTγt0x≤JTγx,v~;t0=∑t=t0TEx′tC′I-DR-1D′Cxt≤0for arbitrary x∈Rn. This implies PTγ(t)≤0, t∈NT+1.Remark 17.
For givenγ≥0, if system (1) is time-invariant and square, replacing B by Bδ=B0n×n, C by Cδ=CδIn, D by Dδ=D0l×n0n×l0n×n, v(t) by vδ(t)=v(t)0n×n, and z(t) by zδ(t) in (1), we have the corresponding H- index Gδ-[0,T] and the cost (47)JT,δγx0,v=∑t=0TEzδ′tzδt-γ2vδ′tvδt=∑t=0TEz′tzt-γ2v′tvt+δ2I.When G-[0,T]>γ, Gδ-[0,T]>γ. Applying Theorem 16 to the modified data, we find that the following equation (48)-Pt+A′Pt+1A+A0′Pt+1A0+C′C+δ2I-A′Pt+1B+A0′Pt+1B0+C′D·B0′Pt+1B0+B′Pt+1B+D′D-γ2I-1·A′Pt+1B+A0′Pt+1B0+C′D′=0,B0′Pt+1B0+B′Pt+1B+D′D-γ2I>0,PT+1=0has a unique solution Pδ,Tγ(t)≤0 on NT+1. Moreover, minv∈lw2(NT;Rl)⁡JT,δγ(x0,v)=x0′Pδ,Tγ(0)x0.


## 3. Examples

In this section, we present some simple examples to illustrate applications of the results developed in this paper.Example 1.
Consider system (1) with (49)At=422-t02,Bt=1122,A0t=1001,B0t=1002,Ct=0.10.50.30.432,Dt=341532-0.2×23-t,γ=1.5,T=2.By Theorem 14, we have (50)PTγ2=23.0431.6171.6176.29,PTγ1=247.7514014072,PTγ0=10549925392539084.We can see that PTγ(t) is not necessarily negative definite or positive definite.
Consider the following system:(51)xt+1=Atxt+Btft+A0txt+B0tftwt,zt=Ctxt+Dtft,x0=x0,t∈NT,where x(t)∈Rn is the state, z(t)∈Rq is the measurement output, and f(t)∈Rl is the fault input.
The fault detection observerF has the form (52)x^t+1=Atx^t+Lzt-z^t+A0tx^twt,z^t=Ctx^t,rt=Vzt-z^t,x0=x0,t∈NT,where x^(t)∈Rn is the state estimation, L∈Rn×q is the gain matrix to be designed, and V∈Rq×q is a nonsingular weighting matrix.
From the filterF and system (51), let e(t)=x(t)-x^(t), and we can express the residual error equation R as (53)et+1=A¯tet+B¯tft+A¯0tet+B¯0tftwt,t∈NTrt=C¯txt+D¯tft,x0=x0,where A¯(t)=A(t)-LC(t), B¯(t)=B(t)-LD(t), A¯0(t)=A0(t), B¯0(t)=B0(t), C¯(t)=VC(t), and D¯(t)=VD(t). We note that (53) is of the same form as (1). If we take f as input and r as the output, the worst-case fault sensitivity of system (53) is the H- index problem discussed in Section 2, and the H- index gives a guarantee on the performance of a fault detection observer.Example 2.
Consider system∑1 formed (53) with coefficients (54)A¯=4220-t02,B¯=1122,A¯0=1001,B¯0=1002,C¯=0.140.3532-0.1×220-t,D¯=30.510.432,T=20.
Forγ1=maxγ=1.5262, that is, G-[0,T]=1.5262, by Theorem 14, (42) admits a unique solution P1Tγ(t). Figure 1 shows the minimum eigenvalue of P1Tγ(t).
If system∑2 is the same as system ∑1 except A¯ and D¯,(55)A¯=43-0.220-t02,D¯=30.50.1×221-t531,with γ2=maxγ=1.9617, that is, G-[0,T]=1.9617, by Theorem 14, (42) admits a unique solution P2Tγ(t). Figure 2 shows the minimum eigenvalue of P2Tγ(t).
By comparing theH- indexes of system ∑1 and system ∑2, system ∑2 has higher fault detection ability as γ2>γ1.Figure 1
The minimum eigenvalue ofP1Tγ(t).Figure 2
The minimum eigenvalue ofP2Tγ(t).


## 4. Conclusion

In this paper, we have discussed the stochasticH- index of linear discrete-time systems with state and input dependent noise. A necessary and sufficient condition has been presented in finite time horizon. The condition is given by means of the solvability of a constrained difference equation. These results can be used in fault detection. The numerical examples are given to illustrate the proposed methods.



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*Source: 953831-2015-10-04.xml*