#### Abstract

This paper discusses the index problem for stochastic linear discrete-time systems. A necessary and sufficient condition of index is given for such systems in finite horizon. It is proved that when the index is greater than a given value, the feasibility of 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.

#### 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 norm, norm, and index, which are to evaluate the effectiveness of a fault detection observer design [2, 3]. The 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 norm can be described by means of the bounded real lemma. On the contrary, 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], index in zero frequency was defined by using the minimum nonzero singular value. In [13], 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, the index in time domain has attracted more attention. In [16], the authors developed a fault residual generator to maximize the fault sensitivity by index in the finite time domain. Based on 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 index for linear continuous time-varying systems in finite horizon was given. Characterization of index for linear discrete time-varying systems was discussed in [20]. The index was described by the existence of the solution to a backward difference Riccati equation.

Although there was much work on the index, little work was concerned with the the index in stochastic linear systems. In this paper, the characterization of the index for stochastic linear systems in finite horizon is presented. The definition of the index is extended to stochastic systems. New necessary and sufficient conditions are given for the index. The feasibility of the index is shown to be equivalent to the solvability of a constrained difference equation. As a special case, the stochastic 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. Section 2 is devoted to developing some efficient criteria for the linear stochastic 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*. is the field of real numbers. is the vector space of all matrices with entries in . is the set of all real symmetric matrices . denotes the transpose of the complex matrix . is the inverse of . Given a positive semidefinite (positive definite) matrix , we denote it by (). Let denote the mathematical expectation. is identity matrix. is zero matrix. Consider , . A 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 Stochastic Index

In this section, we will discuss the finite horizon stochastic index problem. We give a necessary and sufficient condition for the finite horizon stochastic index.

Consider the following stochastic system: where is a sequence of one-dimensional independent real random variables defined on the complete probability space with and , , where is the Kronecker delta. Suppose and are independent. , , , , , and . , , and are the state, input, and output, respectively.

We define the -algebra generated by , , , . is an increasing sequence of -algebra and is adapted to for all . Let be the space of -valued random vectors with . denotes the space of all sequences that are measurable for all . The -norm of is defined by . We suppose that is deterministic. For any and , there exists a unique solution of (1) with .

The finite horizon stochastic index problem of system (1) can be stated as follows.

*Definition 1. *For stochastic system (1), given , define which is called index of (1) in .

*Remark 2. *Definition 1 describes the smallest sensitivity of stochastic system (1) from input to output in time domain. Assume that is fault signal and is the residual; then characterizes the minimal fault sensitivity.

*Remark 3. *When system (1) is wide, (see [20]). In this paper, we suppose that system (1) is tall or square.

For any given , and , let where is the solution of (1) and is the corresponding output. We will discuss the following optimal control problem:

*Remark 4. *Obviously, is equivalent to the following inequality:

for all , , .

*Remark 5. *When , (2) and (5) correspond to the infinite horizon index case.

In the following, we present some useful lemmas, which play important roles throughout the paper.

Lemma 6. *For given , if is an arbitrary family of matrices in , then for any **where*

*Proof. *Since is independent of , we conclude that and are measurable and independent of , so It follows that where where Take summation from to ; it yields that From (3), we get which completes the proof.

Theorem 7. *For (1) and given , if the following equation **has a solution , , then .*

*Proof. *For any , , , by Lemma 6, we have By completing squares and considering the first equality in (14), we obtain that where .

From , it is obvious that and if and only if . Let us substitute into system (1). It must be , on the basis of the fact , which results in , . Therefore, it is deduced that if and only if , which contradicts the condition . Without loss of generality, we assume that , . Then, (16) indicates that , which implies that . Theorem 7 is proved.

The necessity of Theorem 7 will be proved by a sequence of lemmas. To this end, we consider the following backward matrix equation: where is a given finite sequence of matrices. This equation has a unique solution , , satisfying .

By the above, is the solution of the following equation:

Lemma 8. *For the given , , and , if the following equation **admits a solution , then the cost function is given by **where , is the solution of (18) and **Furthermore, for , *

*Proof. *By Lemma 6, we can derive that For , (22) is obvious.

Next, we will show that matrices , , are invertible.

Lemma 9. *For system (1), assume that, for given , . For given and , if is the solution of (18), then *

*Proof. *We first prove on . Suppose that there exist , , , and such that . Set , , and According to Lemma 8, we have From the definition of , , and (19), it follows that for . Additionally, in view of , we conclude that which leads to a contradiction. So, for any .

Now let for and . Replacing with in (18), we obtain the corresponding solution . As in the preceding proof, we have that . For any and , define . Let , , be the solution of (18) with and replaced by and , respectively. Then , . By (22), it follows that Therefore, , which means that for all and arbitrary . So .

This completes the proof.

*Remark 10. *From (24), for , . If system (1) is time-invariant and satisfies Lemma 9, then

*Remark 11. *If or is invertible, then . By this equality, we see that . If system (1) is square and time-invariant, from (29), we can conclude that However, the above is not true for tall systems.

Now, we discuss the necessity of Theorem 7 and present the following theorem.

Theorem 12. *For system (1), if for given , then (14) has a unique solution , , for any . Furthermore, is minimized with the optimal cost given by **and the optimal control is determined by **where satisfies *

*Proof. *We first prove that means that (14) admits a solution on . As , it is clear that there exists a solution to (14) at ; that is, Suppose (14) does not have a solution on ; then there must exist a minimum number , , such that (14) is solvable backward up to . That is to say, satisfy (14) but does not, or is not a positive definite matrix.

Set , , and then is well defined. Let Consider the following equation: Equation (36) admits a solution , . Comparing (36) with (14), we arrive at for . Moreover, along the same line of Lemma 9, we have that on . In particular, . This is inconsistent with the nonpositiveness of . Hence, (14) has a unique solution , , for any .

Next, we suppose that the following equation admits a solution , , and then is well defined by (32). If we replace in (17) by , then (18) becomes (37) with , so for all . By (21), when , it yields . By (20), we come to a conclusion that for By (24), we deduce that minimizes with the optimal value expressed by (31). This proof is complete.

Lemma 13. *If system (1) is time-invaint, square and , then, for any fixed , , *

*Proof. *Since system (1) is time-invaint, by the time invariance of (37), we have . Without loss of generality, we assume that , , and is optimal for on . Let and . By (31) and Remark 11, This implies (39).

Based on Theorems 7 and 12, it is easy to get the following main result.

Theorem 14. *For system (1) and a given , the following are equivalent. *(a)*.*(b)*The following equation **admits a unique solution on . Moreover, .*

*Remark 15. *The solution of (42) is not necessarily negative or positive definite.

Theorem 16. *For given , if system (1) is time-invariant and square, then the following are equivalent. *(a)*.*(b)*The following equation**admits a unique solution on . Moreover, .*

*Proof. *By Theorem 14, Theorem 16 is established as long as we prove .

For any , from Lemma 6 and (43), using completing squares method, it follows that where .

Set . By completing squares, we have where .

Based on the above and Remark 11, it is easy to see that for arbitrary . This implies , .

*Remark 17. *For given , if system (1) is time-invariant and square, replacing by , by , by , by , and by in (1), we have the corresponding index and the cost When , . Applying Theorem 16 to the modified data, we find that the following equation has a unique solution on . Moreover, .

#### 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 By Theorem 14, we have We can see that is not necessarily negative definite or positive definite.

Consider the following system: where is the state, is the measurement output, and is the fault input.

The fault detection observer has the form where is the state estimation, is the gain matrix to be designed, and is a nonsingular weighting matrix.

From the filter and system (51), let , and we can express the residual error equation as where , , , , , and . We note that (53) is of the same form as (1). If we take as input and as the output, the worst-case fault sensitivity of system (53) is the index problem discussed in Section 2, and the index gives a guarantee on the performance of a fault detection observer.

*Example 2. *Consider system formed (53) with coefficients For , that is, , by Theorem 14, (42) admits a unique solution . Figure 1 shows the minimum eigenvalue of .

If system is the same as system except and ,with , that is, , by Theorem 14, (42) admits a unique solution . Figure 2 shows the minimum eigenvalue of .

By comparing the indexes of system and system , system has higher fault detection ability as .

#### 4. Conclusion

In this paper, we have discussed the stochastic 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61174078, 61170054, and 61402265), the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the SDUST Research Fund (Grant no. 2011KYTD105).