Research Article | Open Access

# Fault Detection for a Class of Uncertain Linear Systems

**Academic Editor:**Bochao Zheng

#### Abstract

Design problem of the fault detection filter for a class of uncertain feedback systems is discussed in this paper. The system under consideration is the model with the nonstandard parameter uncertain regions. The fault detection filter design problem is reduced to an optimized filter design problem, which maximizes the sensitivity of the fault in the fault cases and meets the disturbance attenuation performance in both fault-free cases and fault cases, simultaneously. A numerical example is used to demonstrate the effectiveness of the proposed design method.

#### 1. Introduction

In recent years, fault detection technique is given more and more attentions for the higher demands of safety and reliability of the control systems. A considerable sum of results on fault detection, both in the light of the data-driven and the model-based detection techniques, have been achieved (see [1–3]). Among the model-based techniques, a number of results which depend on the linear matrix inequality (LMI) techniques have appeared to deal with the fault detection problem. References [4–6] investigate the estimation problem of the fault by designing a filter. Recently, a so-called design approach for the fault detection problem has received considerable attentions. Further, the fault detection problem associated with some certain performance indices is discussed in [7–10]. The fault models considered in the above papers are classical and can describe a large class of the fault types. However, it should be pointed out that the actuator stuck fault also needs to be investigated when the amplitude of the faults are not larger enough to be detected.

At the same time, most of the models are obtained by the identification experiments with some limited input/output data. Identification experiment in the prediction error (PE) framework delivers an uncertain region which contains the true system at some (user-chosen) probability level [11]. The uncertain region can result in an identified parametric ellipsoid set which is regarded as a nonstandard parameter uncertain structure in the related classical papers. The ellipsoidal uncertain structure is studied by some results; see [12–15] which reduce the uncertain region by employing the robust tools. Compared with the stability and the stabilization problems for this particular uncertainty, there are no results that to pay attention to deal with the fault detection problem with the models which have the ellipsoidal uncertainty. Moreover, since the existing approaches about fault detection (FD) are not appropriate for FD problem for this kind of uncertainty, the new FD technique should solve this case.

Motivated by the above reasons, the optimization design problem of the FD filter for uncertain linear feedback system is studied. The design objective is to maximize the sensitivity of the fault in the fault cases, which subjects to the disturbances attenuation performance in both fault-free cases and the fault cases. In practical, the frequency of the disturbance signals and the fault signals are usually in bound. Thus, a notion of the finite-frequency performance indices is introduced to describe the performance indices; LMIs-based sufficient conditions are provided using the finite-frequency approach proposed in [16, 17]. The optimal solution of gain matrices of the FD filter can be obtained.

The main contribution of this paper that the direct design approach of the FD filter proposed to the linear model with elliptical uncertain structure. Unlike other fault detection approaches for the uncertain model, in which the FD filter is achieved not only depending on the known lower and known upper bounds of the uncertainty but also extracting convex polygons of the uncertain region, a general parametrization for the set of multipliers is introduced to decrease the conservatism. To the best of our knowledge, such a framework for the fault detection with the elliptical parameter uncertainty structure has not been reported in the literature.

The paper is organized as follows. Section 2 introduces the problem under consideration and presents the design objectives. Section 3 illustrates a FD filter design approach in detail. The algorithm is given in Section 4. An illustrative example is given in Section 5 to demonstrate the proposed method. Conclusions to this work are given in the last section.

*Notation.* For a matrix , denotes its transpose. For a symmetric matrix, () and () denote positive-definiteness (positive semidefinite matrix) and negative-definiteness (negative semi-definite matrix), respectively. The Hermitian part of a square matrix is denoted by . The symbol within a matrix represents the symmetric entries. denotes minimum singular values of the transfer matrix . The symbol denotes the Kronecker product, and denotes zero matrix with appropriate dimension. Both and denote identity matrix with appropriate dimension. and denote the matrices and of appropriate dimensions, respectively.

#### 2. Problem Formulation

##### 2.1. System Model

Consider the following uncertain linear systems: where is the state is the output, is the input, and is an energy-bounded disturbance; is the reference input, is the state; , , , , , , , and are known constant matrices of appropriate dimensions. Consider that , where is defined as where is the number of in , then the dimension of the is , and is the parameter set of . The uncertain parameter with a certain use-chosen probability () is at the uncertain set . The uncertain set is an ellipsoid in the parameter space: where describes the form of the ellipsoid and is a constant computed by the desired probability in model identification.

*Remark 1. *The parameter uncertainty region in (1) is ellipsoidal. This kind of particular uncertainty is derived from classical prediction error identification, and it is a typical parameter uncertainty of the model which is identified. Although the parameter uncertainty can turn into the polytopic uncertainty by extracting convex polygons of the ellipsoidal region, this transformation method enlarges the the uncertain region and increases the conservatism. In this paper, the ellipsoidal uncertainty is disposed directly.

To detect the actuator stuck faults, the FD filter is designed such that the residual can be obtained as where is the state of the FD filter. Consider that , , , .

By combining (1) and (4), we have the following augmented systems: where ,

##### 2.2. Problem Formulation

The design problem of the FD filter to be addressed in this paper can be expressed as follows.

The design objective: consider a class of uncertain linear systems (1); the FD filter (4) is designed such that the augmented model (5) is stable; the disturbance effects on the residual are minimized in both the fault-free case and the fault cases, while the fault effects on the residual are maximized in the fault cases. To detect actuator faults, our design objective of the FD filter can now be formulated as the following optimization problem: where is the upper bounds on the frequency for the disturbance and is the upper bounds on the frequency for the fault. is the transfer function from the disturbance to the residual signal with fault-free case and fault cases. is the transfer function from the fault to the residual signal with fault cases.

*Remark 2. *Condition (8) describes the disturbance attenuation condition in both the fault-free case and fault cases. Condition (9) is formulated as the sensitiveness performance in the fault cases.

Before ending this section, the following lemmas and proposition shall be recalled to prove our main results for the FD filter design.

Lemma 3 (see [18]). *Given a symmetric matrix and two matrices and of column dimension , there exists matrix that satisfies if and only if the following two conditions hold:
**
where and denote arbitrarily bases of null space of and , respectively. *

Lemma 4 (see [19]). *Let , , and such that rank . The following statements are equivalent:*(1)*, for , subject to ;*(2)*, where is the kernel of ;*(3)*, for some scalar ;*(4)*, for some matrix .*

Proposition 5 (see [11]). *Consider the uncertainty with defined in (3), is defined as
**
Restrict the parametrization of matrices defined as
**
with as a positive complex Hermitian matrix of dimension . Consider that and
**
where describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization () can take any values provided. Consider that has the following structure that:
**
and , , . For , , .*

#### 3. Fault Detection Design

In this section, Lemma 6 and Proposition 7 are first given, and inequality conditions for performance indices (8) and (9) are also given. Based on these conditions, an algorithm based on LMIs is presented.

Lemma 6 (see [16]). *Consider the following uncertain system:
**
with . Let a real symmetric matrix and a positive scalar be given. The following robust finite-frequency condition
**
holds if there exist real symmetric matrices , , , and such that
**
where
*

Proposition 7. *Consider the uncertainty with defined in (3) and defined in (11). Restrict the parametrization of matrices defined as
**
where are positive complex Hermitian matrices of dimension ; , , and of the dimension have the structure with , , and , respectively. Moreover**
where describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization (, , , , , , , ) can take any values provided. For , , and , has the following structure:
**
and . Then, for , it has
**
where is defined in (11). *

*Proof. *Let all , have the structure in the statement of the proposition:
Therefore, for every , we can haveWhen and , we can have

##### 3.1. The Disturbance Attenuation Condition

Theorem 8. *Consider the system in (5) with . A real symmetric matrix is given. Condition
**
holds if there exist matrix variables , , , , , , , , , , , ,
**
of appropriate dimensions satisfying , , and such that**where
*

*Proof. *Substituting into (17), which is equivalent to (27). By applying Lemma 6, (27) holds if
where

Let
(32) is equivalent to
And on the other hand,

By combining (36) with (37), and applying Lemma 3, (32) holds if and only if
where introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, is defined as and partition into where and are nonsingular matrices variable. is a given matrix, defined as .

Let , and define the linearizing change of the control variables as follows:

By Pre- and postmultiplying (38) by and and choos satisfying Proposition 5, (38) is equivalent to

where , , , are the matrices which use the new defined variables , , in the matrices , , , .

By simple calculation, we can obtain that where , , . By applying the Schur complement formula, the following can be obtained:

Then, pre-and postmultiply to (42). Due to and defineing , , , , (29) hold. In addition, and pre- and postmultiplying (33) by and , respectively, (33) becomes (30).

Hence, if conditions (29) and (30) holds, the augmented uncertain system (5) is stable and guarantees the performance (27), which completes the proof.

##### 3.2. The Fault Sensitiveness Condition for Faulty Case

Theorem 9. *Consider the system in (5) with . A real symmetric matrix . The following condition
**
holds if there exist matrix variables , , , , , , , , , , , ,
**
of appropriate dimensions satisfying , , and such that**where
*

*Proof. *Substituting into (17), which is equivalent to (43). Applying Lemma 6, (43) holds if
hold.

Where

Let
(48) is equivalent to
and on the other hand,

Combining (52) with (53) and applying Lemma 3, (48) holds if and only if
where introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, is defined as , where , where and are the same as in Theorem 8. is an given matrix, defined as . Pre- and postmultiply (54) by and , where is defined as Theorem 8 and choose satisfying Proposition 5, (54) is equivalent towhere

Condition (55) is nonlinear owing to the product terms . To solve this problem, condition (55) is equivalent to
where

Defining satisfies . Using Lemma 4, and given a matrix , (57) is equivalent to
where one notes as
Defining , , , and and after some matrix manipulation, (59) becomes (45). By pre- and postmultiplying (49) by and , respectively, (49) becomes (46).

*Remark 10. *Theorem 8 considers the attenuation performance for the disturbance , and the conservatism comes from the special structure of and the same one in . Theorem 9 considers the sensitiveness performance for the fault , and the conservatism comes from the special structure of and the same one in . Then, the conservatism for Theorem 9 comes as the same as Theorem 8. However, in order to obtain the better detection performance, the sensitiveness performance is considered over the conservatism.

##### 3.3. Stability Condition for the Filter

Due to the fault tolerant controller, the closed-loop system is stable both in the fault-free cases and in the fault cases. Then, the condition that the augmented systems (1) is stable can translate into the condition that the FD filter is stable.

Lemma 11 (see [20]). * By considering the FD filter (4), the FD filter is stable, if there exist matrix variables , and satisfying
*

##### 3.4. Algorithm

In the precious sections, Theorems 8 and 9 and Lemma 11 have formulated the inequality conditions for the performance indexes (8) and (9) and the stable condition, respectively. Summarily, we have the following theorem.

Theorem 12. *By considering the uncertain system model (1), there exists a FD filter (4) such that the augmented system model (5) is stable and satisfies the performance indices (8) and (9) if inequality conditions (29), (30), (45), (46), and (61) hold.*

*Proof. * Combining Theorems 8–9 and Lemma 11, it is obviously that the theorem holds.

*Remark 13. *In Theorems 8–9, if the matrices , are given, all conditions (29), (30), (45), and (46) are the LMI conditions. Specially, if the disturbance and the fault are a scalar, respectively, is vector, and it can be easily chosen, which has been further confirmed in the simulink. In addition, Propositions 5 and 7 are used to directly deal with the ellipsoidal uncertainty; the variable structure is complex. However, all the variables in Propositions 5 and 7 is composed of linear matrices, and can be used for LMI conditions.

The gain matrices , , , and defining the FD filter can be derived by the means of a standard procedure. Denote , , , , and is the optimal solution of (7) with condition (29), (30), (45), (46), and (61):(1)compute the , by solving the following factorization problem ;(2)compute , by , and . Compute , by , and ; (3)finally, obtain the gain matrices of the FD filter

#### 4. Thresholds Computation

After the parameter matrices of the FD filter , , , and are designed, similar to that proposed in [21], the residual evaluation function can be selected as

Under fault-free condition, the residual becomes and namely, the Parseval Theorem (see [22]), one has that where is the upper bound on the disturbance. It is assumed that is the known. As a consequence, the following threshold results:

Based on this, the occurrence of faults can be detected by the following logic rule.