## Mathematical Modeling, Analysis, and Control of Hybrid Dynamical Systems

View this Special IssueResearch Article | Open Access

Jian Li, Qingyu Su, Lingfang Sun, Bo Li, "Fault Detection for Nonlinear Impulsive Switched Systems", *Mathematical Problems in Engineering*, vol. 2013, Article ID 815329, 12 pages, 2013. https://doi.org/10.1155/2013/815329

# Fault Detection for Nonlinear Impulsive Switched Systems

**Academic Editor:**Xiaojie Su

#### Abstract

This paper is concerned with the fault detection problem for nonlinear impulsive switched systems. Fault detection filters are designed such that the augmented systems are stable, and the residual signal generated by the filters achieves the -gain for disturbances and guarantees the performance for faults. Sufficient conditions for the solvability of this problem are formulated in terms of linear matrix inequalities; furthermore, the filter gains are characterized by a convex optimization problem. A simulation on a continuous stirred tank reactor control system is given to demonstrate the effectiveness of the proposed methods.

#### 1. Introduction

Switched systems are an important class of hybrid systems, which consist of a family of continuous-time or discrete-time subsystems and a switching law that specifies the switching between them. Study on this class of systems has attracted much attention in recent years for its theoretical significance [1â€“6] and engineering applications [7â€“9]. However, in a wide range of actual systems such as engineering, economics, and biology, there usually exist some impulses when the switched system is switching among subsystems. These special switched systems are called impulsive switched systems. As typical switched impulsive system, the circuit switching usually causes system state to change abruptly in some circuit systems [10]. The abrupt state at the time instant of switching could often lead to oscillations and instability and the lead poor performances. Recently, numbers of papers have focused on stability problem of switched impulsive systems [11â€“13].

On the other hand, fault detection (FD) is an important topic in system engineering from the viewpoint of the higher demands for safety and reliability of control systems [14â€“17]. Among these model-based approaches, the class of procedures are that observers or filters are firstly designed to construct a residual signal which is used to generate an alarm when the residual evaluation function has larger than the threshold. During the past decades, Many results, using robust technique [18â€“21] and technique, investigate this important issue [22, 23].

However, the problem of FD design in switched systems schemes is still in the early stage of development and a few results have been reported in the literatures [24â€“28]. It is worth noticing that all the aforementioned FD approaches for switched systems do not include impulsive switched systems. Moreover, since the FD approaches for switched systems without impulsive increments are not appropriate for switched systems with impulsive increments, the new FD technique need to solve the impulsive case. Because these impulsive switched systems exist widely, the FD problem for a general class of nonlinear impulsive switched systems is significant, both theoretically and practically.

In this paper, the problem of FD filters for a general class of nonlinear impulsive switched systems is presented. Different from [10, 29] which only involve linear impulsive switched systems with linear impulsive increments, nonlinear impulsive switched systems with nonlinear impulsive increments have been considered in this paper. The main contributions of this paper include (i) the performance of nonlinear impulsive switched systems to directly reflect the effect on the residual signal from faults and obtain the better detection results and (ii) the smaller conservatism of sufficient conditions for the performance and the performance. Finally, the filters gains are characterized in terms of the solution of a convex optimization problem.

The paper is organized as follows. Section 2 introduces the problem under consideration and presents the design objectives. Section 3 illustrates the FD filter design approach in detail. The thresholds are given in Section 4. Two examples are given in Section 5 to demonstrate the proposed method. Conclusions of this paper 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 semidefinite matrix), respectively. The Hermitian part of a square matrix is denoted by . The symbol within a matrix represents the symmetric entries.

#### 2. Problem Formulation

##### 2.1. System Model

Consider the following nonlinear impulsive switched systems: where is the state, is the measured output, and and are the disturbance input and the fault input, respectively. It is assumed that both the disturbance input and the fault input are energy bounded; then it is demanded that they belong to . , , are impulsive switching time points satisfying . The piecewise constant function is a switching rule which takes its values in the finite set ( is the number of subsystems). The index denotes the sequence of the active subsystem; that is, when , the subsystems is activated at the time point . , which is globally the Lipschitz continuous, and for all , with and ; that is, the solution is left continuous. is nonlinear function, and for all . The subsystem is denoted by the matrices , ,â€‰ , , â€‰, and with appropriate dimensions.

We introduce the following assumptions for nonlinear impulsive switched systems (1).

*Assumption 1. *There exist nonnegative scalars , , such that

*Assumption 2. *Denote as the spectral radius for each subsystem. for .

For the purpose of the fault detection, the following FD filters are designed: where is the state of the filter and is the residual signal. The matrices , and with appropriate dimensions are to be determined.

Combining (1) and (3), the nonlinear impulsive switched system can be written as where and when , To present the purpose of this paper more precisely, the following definition is introduced.

*Definition 3. *Let Assumptions 1 and 2 be satisfied and . Nonlinear impulsive switched system (4) under zero initial conditions is said to be stable with the -gain , if the condition holds that

##### 2.2. Problem Formulation

The design problem of the fault detection filters to be addressed in this paper can be expressed as follows.

The frameworks of FD filter design: given nonlinear impulsive switched system (1), the FD filters (3) are designed such that nonlinear impulsive switched system (4) is stable and the fault effects on the residual signal are maximized, while the disturbance effects on the residual signals are minimized. Our design objective of the FD filters can now be formulated as the following performances:

*Remark 4. *Condition (8) is expressed to maximize the effects of the fault on the residual output for impulsive switched system (4). That is, the residual output is sensitive for the fault . Condition (9) is used for the disturbance attenuation performance, which minimizes the disturbance effects on the all residual outputs and ensures that the disturbances are not disastrous.

#### 3. The Fault Detection Filter Design

Before beginning this section, the following lemmas are needed to present our main results.

Lemma 5 (see [30]). *Let be a given scalar and be a matrix such that , where is an identity matrix with appropriate dimension. Then for all and . *

Lemma 6 (see [31]). *Let be a given symmetric positive definite matrix and let be a given symmetric matrix. Then
**
for all , while and denote, respectively, the largest and the smallest eigenvalues of the matrix inside the brackets. *

In this section, sufficient conditions on the existence of the FD filters for nonlinear impulsive switched systems (4) would be given, and the desired filters can be obtained.

##### 3.1. The Fault Sensitiveness Performance (8)

Considering nonlinear impulsive switched system (4) with , we have performance for nonlinear impulsive switched system (11) is given.

Lemma 7. *Let , be constants satisfying , , , and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (11) switches from subsystem to th subsystem as switched time point . If there exists the Lyapunov functions candidate , satisfying the following inequalities
**
then nonlinear impulsive switched system (11) is stable with -gain for any switching signal satisfying
**
where . *

*Proof. *When and th subsystem is activated, evaluating the time derivative of along the trajectory of nonlinear impulsive switched system (11) gives
By Lemma 5, it is clear that
Therefore, when assuming the zero input (i.e., ) and using Assumptions 1, we have the following condition from (14) and (15):
where . If (12) holds, then , which implies that . Thus, from (16), we have . Integrating this inequality gives
At the impulsive switching time point , it has
By Lemma 6, it has
where , . Therefore, from (17) and (19), we have
where means the th switching impulsive value from th subsystem to th subsystems, and denotes the decay rate of the Lyapunov function for subsystem. Since (13) holds; that is, there exists such that , then
It follows that
Therefore, we conclude that converges to zero as then nonlinear impulsive switched system (11) with is stable.

Secondly, establish the performance defined in (7) for nonlinear impulsive switched system (11). Consider the following performance index:

For any nonzero and zero initial condition , let and consider the Lyapunov functions as . For subsystem, it has
where
Equation (12) is equivalent to . Then, it has . By iteration operation on the above inequality for , we have
Along the same lines as the impulsive switching time point , it has from (19) and (26) that
where and are defined as (21).

Under the zero initial condition, (27) implies . Because ,â€‰â€‰, it has ; that is,
When , nonlinear impulsive switched system (11) has performance, which completes the proof.

*Remark 8. *For switched systems, the Lyapunov function is discontinuous at the time instant of switching; that is to say, this Lyapunov function does not have its derivative in the whole time domain. Then the piecewise Lyapunov function was utilized in Lemma 7. Each subsystem has one corresponding Lyapunov function, and each Lyapunov function has its derivative in the corresponding subsystem time domain.

Based on Lemma 7, the following theorem is given to obtain the fault sensitiveness performance conditions in terms of linear matrix inequalities.

Theorem 9. *Let , , and be constants satisfying , , and Assumptions 1 and 2 hold. If there exist matrix variables , , , , , , , , , , , and symmetric positive-definite matrices
**
satisfying the following inequalities
**
where**then, for any switching signal satisfying (13), nonlinear impulsive switched system (11) is stable and guarantees the performance (8). Moreover, if (30) is feasible, then the FD filter gains in form of (3) can be given by
*

*Proof. *Now, to establish the convex condition, (12) can be rewritten as
where
On the other hand, we also can have

Based on Finslerâ€™s Lemma, it has
where introduced by Finslerâ€™s Lemma is the matrix variable of appropriate dimensions. Partition as , and
Due to , and in , then the condition (36) is non-convex. To establish the convex condition, (36) can be rewritten as
whereUsing Finslerâ€™s Lemma again, (38) holds if
holds for any with appropriate dimension and . Let and define , , , ; then (30) is reached by the Schur complement on (40). Hence if the conditions (30) holds, nonlinear impulsive switched system (11) is stable and guarantees the performance (8), which completes the proof.

*Remark 10. *Scalars are used to reduce conservatism for the same variable in matrix variables . On the other hand, the coefficient matrices are given to adjust the variable dimension. When we set , as the fixed parameters, the conditions in Theorem 9 become convex.

##### 3.2. The Disturbance Attenuation Performance (8)

Consider nonlinear impulsive switched system (4) with , we have

performance for nonlinear impulsive switched system (41) is given.

Lemma 11. *Let , be constants satisfying , , , and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (41) switches from subsystem to th subsystem as switched time point . If there exists the Lyapunov functions candidate , , satisfying
**
then nonlinear impulsive switched system (41) is stable with -gain for any switching signal satisfying (13). *

*Proof. *Following the same lines as those for Lemma 7, nonlinear impulsive switched system (41) is stable, for any nonzero and zero initial condition . Let and consider the Lyapunov function as Lemma 7; we can have
where
Equation (42) is equivalent to . Thus, it has
By iteration operation on the above inequality for , we have . As the same lines as those for Lemma 7, it has

Under the zero initial condition, (46) implies . Because , , it can further have

When , the nonlinear impulsive switched system (41) has performance, which completes the proof.

Based on Lemma 11, the following theorem is given to obtain sufficient conditions by linear matrix inequalities.

Theorem 12. *Let , , and be constants satisfying , , and and Assumptions 1 and 2 hold. If there exist matrix variables , , , , , , , , , and and symmetric positive-definite matrices
**
satisfying the following inequalities
**
where**then, for any switching signal satisfying (13), nonlinear impulsive switched system (41) is stable and guarantees the performance (9). Moreover, if (49) is feasible, then the FD filter gains in form of (3) can be given by
*

*Proof. *To establish the convex condition, (42) can be rewritten as follows:
Denote
From (52), we have
On the other hand,
Based on Projection Lemma, it follows from (54) and (55) that
where introduced by Projection Lemma is the matrix variable of appropriate dimensions. Partition as . By the Schur complement, (56) is equivalent toThen pre- and postmultiply to (57); one obtainswe partition , , respectively, as
and are set to be fixed parameters. Define , , , and ; then (58) becomes (49). Hence if the condition (30) holds, nonlinear impulsive switched system (41) is stable and guarantees the performance (8), which completes the proof.

*Remark 13. *Scalars are used to reduce conservatism for the same variable in matrix variables and . When we set as the fixed parameters, the conditions in Theorem 12 become convex.

##### 3.3. Algorithm

In the previous sections, Theorems 9 and 12 have formulated the inequality conditions for the performances (8)-(9), respectively. Summarily, we have the following algorithm.

It is noted that conditions (30) and (49) are all convex. Hence, the problem of FD filter design can directly translate into the following optimization problem:

*Remark 14. *The performance and performance are considered to describe the fault sensitiveness performance and the disturbance attenuation performance, respectively. Due to the multiobjective optimization problem, each performance has some LMI conditions. However, by analyzing the theorems and their proofs, it can be discovered that , in Theorem 9 and , in Theorem 12 can be defined as the same variable to decrease the computational complexity.

Finally, the gain matrices , , , and of the filters can be derived as

#### 4. Thresholds Computation

After the gain matrices of the filters , , , and are designed, similarly to that proposed in [32], the residual evaluation function can be chosen as where is square value which means the average energy of residual signal over a time interval , denotes the initial evaluation time instant, and stands for the evaluation time.

We propose to use the following threshold: Consequently, the occurrence of faults can be detected by the following logic rule:

#### 5. Examples

In this section, we present examples derived from a liquid level control system to illustrate the effectiveness of FD design approach.

Consider a continuous stirred tank reactor (CSTR) control system [33] shown in Figure 1. It consists of a constant-volume CSTR fed by a single inlet stream through a selector valve which is connected to two different source streams.

Assuming constant liquid volume, negligible heat losses, perfectly mixing, and a first-order reaction in reactant , the dynamical equations of the CSTR at each operating mode are described by where is the reactant concentration (moL/L), is the th modeâ€™s feed flow rate (L/min), is the th modeâ€™s concentration of the feed stream (moL/L), is the volume of the reactor , is the activation energy, is the gas constant (J/(moLÂ·K)), is the reactor temperature (K), is the th modeâ€™s reactor temperature of the feed stream , and is the coolant temperature (K). These nominal values of the parameters are described in [34], and we choose the nominal operating conditions corresponding to an unstable equilibrium point as , , and for both modes.

Since the objective in the example is to testify the FD filter design techniques for the switched system and sustain the main theoretical results, we make design as [33] and the closed-loop system can be obtained with matrices Suppose other system matrices in (1) as Let , , , , . By giving , , and and obtaining , , we solve the convex optimization problem (60) and get the optimal sensitivity performance gain . The gain matrices of the fault detection filters and the matrix in the Lyapunov function are Then by (13), the dwell time for each subsystem is obtained with :