Robust Fault Estimation for a Class of T-S Fuzzy Singular Systems with Time-Varying Delay via Improved Delay Partitioning Approach

The problem of delay-dependent robust fault estimation for a class of Takagi-Sugeno (T-S) fuzzy singular systems is investigated. By decomposing the delay interval into two unequal subintervals and with a new and tighter integral inequality transformation, an improved delay-dependent stability criterion is given in terms of linear matrix inequalities (LMIs) to guarantee that the fuzzy singular system with time-varying delay is regular, impulse-free, and stable firstly.Then, based on this criterion, by considering the system fault as an auxiliary disturbance vector and constructing an appropriate fuzzy augmented system, a fault estimation observer is designed to ensure that the error dynamic system is regular, impulse-free, and robustly stable with a prescribedH ∞ performance satisfied for all actuator and sensor faults simultaneously, and the obtained fault estimates can practically better depict the size and shape of the faults. Finally, numerical examples are given to show the effectiveness of the proposed approach.


Introduction
The demand for increased productivity leads to more challenging operating conditions for many modern engineering systems.The issue of fault detection and isolation (FDI) algorithms in dynamic systems and their applications to a wide range of industrial processes have been an active research area over the past two decades, as can be seen, in survey papers [1] for linear systems, [2] for multimodels representation and [3,4] for nonlinear systems.By using FDI procedures, the reliability can be achieved by faulttolerant control, which relies on early detection and isolation of faults.So FDI have become a popular topic and received considerable attention.However, it is generally difficult, in practice engineering, to obtain the exact information of the size of system fault from an FDI strategy only because of the existence of model uncertainties, time delays, and disturbances [5].As pointed out in [6], accurate and timely fault estimation is an important antecedent for satisfactory control reconfiguration.Therefore, the problem of fault estimation has stirred renewed research interest, and a variety of fault estimation approaches have been developed in the literatures; see, for example, [7][8][9][10][11] and the references therein.
On the other hand, singular systems have been extensively studied in the past years due to the fact that singular systems better describe physical systems than state-space ones, especially the T-S fuzzy singular systems, because they can combine the flexibility of fuzzy logic theory and fruitful linear singular system theory into a unified framework to approximate complex nonlinear singular systems.In fact, singular systems can be found in electrical circuits, economic systems, moving robots, and many other systems.Recently, many results about fault estimation have been reported on analysis and design of singular systems.For instance, by using online learning methodology, [12] proposed a fault estimation method for continuous-time nonlinear singular systems.Reference [7] used generalized unknown input observer to deal with the robust fault detection problem for linear singular systems.Reference [13] deals with actuator fault estimation for a class of discrete-time linear parametervarying singular systems.Reference [14] designed a robust fault detection filter for a class of nonlinear singular systems described by linear parameter-varying form with global Lipschitz term.However, it is known that time delays are frequently encountered in various engineering and communication systems, and a time delay in a dynamical system is often a primary source of instability and performance degradation.Therefore, it is important to develop fault estimation methods for time delay singular systems.But the fault diagnosis for singular systems with time-varying delay has not been well investigated yet [7,[12][13][14].More recently, [15] proposed a fault detection, isolation, and estimation scheme via unknown input proportional integral observers for linear descriptor systems.Reference [8] investigated fault detection for discrete-time switched singular systems with time-varying delays.It should be noticed that [8] deals with discrete-time switched singular systems while this paper focuses on the fuzzy continuous-time case.In [16], discretetime T-S systems with sensor faults are first formulated as a descriptor representation, and then a fault detection filter is designed based on the obtained descriptor system.However, this paper studies fault detection for regular systems by using the technique of descriptor systems.In [17], the author proposed a k-step fault estimation method for T-S fuzzy time delay system that only deals with regular systems.Moreover, our paper considers how to estimate the actuator and sensor faults simultaneously while attenuating the influence of the disturbance noise, which is not considered in [17].It is known that singular system representation is a generalization of the regular system.Therefore, the proposed method is more general than that in [16,17].
The aim of this paper is to develop a robust fault estimation method for a class of T-S fuzzy singular systems with time delays.The basic idea is to construct an augmented system by taking the fault as auxiliary disturbance vector and then design a fault estimation observer based on this augmented system.The main contribution of the proposed method lies in the following aspects.First, without ignoring any useful terms in the Lyapunov-Krasovskii functional (LKF), by decomposing the delay interval into two unequal subintervals, a new LKF is constructed; then the free weighting matrices approach is introduced to develop a new delay-dependent stability criteria, which ensure that the considered system is regular, impulse-free, and stable.Compared with some existing results, the approach to be proposed in this paper can be expected to give better results.Second, a new robust fault estimation observer with a novel structure is proposed for T-S fuzzy singular systems with time delays and actuator and sensor faults simultaneously, which is the main contribution of this paper.The proposed observer can be designed by solving a set of linear matrix inequalities and to attenuate the effect of unknown disturbance, fault variation on fault estimation.The effectiveness of the method is illustrated by some numerical examples.
The rest of this paper is organized as follows.The system description and preliminaries are presented in Section 2. Section 3 presents the main results on new stability criteria of fuzzy singular systems with time-varying delays and robust fault estimation observer design scheme.In Section 4, simulation results of numerical examples are presented to demonstrate the effectiveness and merits of the proposed methods.Finally, Section 5 concludes the paper.
Notations.Throughout the paper, R  denotes the dimensional real Euclidean space;  denotes the identity matrix; the superscripts  and −1 stand for the matrix transpose and inverse, respectively; notation  > 0( ≥ 0) means that matrix  is real symmetric positive definite (positive semidefinite); ‖⋅‖ is the spectral norm.If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations.The symbol " * " stands for matrix block induced by symmetry; sym() stands for  +   .

System Description and Preliminaries
Consider a nonlinear singular system which can be represented by the following extended T-S fuzzy time delay model with external disturbance, actuator and sensor faults, simultaneously.
where () ∈ R  is the state vector, () ∈ R  denotes the input vector, and () ∈ R  stands for the system output vector.() ∈ R  is the exogenous disturbance input that belongs to  2 [0, ∞); () ∈ R  represents the possible fault.The matrix  ∈ R × is a constant matrix, which may be singular; that is, rank() =  ≤ .  ,   ,   ,   ,   ,   ,   ,   ,   , and   are constant real matrices of appropriate dimensions.It is assumed that the pairs (,   ,   ) are of full column rank, where  = 1, 2, . . ., .  1 (), . . .,   () are the premise variables,   ( = 1, 2, . . ., ,  = 1, 2, . . ., ) are fuzzy sets, and   () is a vector-valued initial continuous function defined on the interval [− 2 , 0].In this paper, it is also assumed that the premise variables do not depend on the input variables (); () is the time-varying delay and satisfies Then, by fuzzy blending, the overall fuzzy singular system model is given by  ẋ () = where the fuzzy basis functions are given by where   (  ()) represents the grade of membership of   () in   .Here, it is easy to find that for all  > 0, we have Before proceeding further, we will introduce some definitions and assumptions to be needed in the development of main results throughout this paper.Consider an unforced singular time delay system described by Definition 1 (see [18]).(1) The pair (, ) is said to be regular if det( − ) is not identically zero.
(3) The pair (, ) is said to be stable, if all roots of det( − ) = 0 lie inside the unit disk with center at the origin.
(4) The delayed singular system ( 8) is said to be admissible if the pair (, ) is regular, impulse-free, and stable.
Definition 2 (see [19]).(1) The singular system ( 8) is said to be regular and impulse-free if the pair (, ) is regular and impulse-free.
The singular time delay system (8) may have an impulsive solution.However, the regularity and nonimpulse of (, ) guarantee the existence and uniqueness of impulse-free solution to (8) on [0, ∞).
Lemma 3 (see [20]).If a functional  :   [−, 0] → R is continuous and (, ) is a solution to (8), one defines V() = lim ℎ → 0 + sup(1/ℎ)((( + ℎ, ) − ())).Denote the system parameters of (8) as Assume that the singular system ( 8) is regular and impulsefree,  22 is invertible, and ( −1 22  22 ) < 1.Then, system ( 8) is stable if there exists positive numbers , , ], and a continuous function where In order to address the main results, the following assumptions are made.Assumption 4. Matrices  and   satisfy the following rank condition: Assumption 5.The triple matrix (,   ,   ) is -detectable [21] for ∀ = 1, 2, . . ., ; that is, Remark 6.Both Assumptions 4 and 5 are necessary conditions for the existence of the designed observer in the latter section.Similar assumptions can be also found in [21] and the references therein.Meanwhile, for T-S fuzzy system description (3), we can see that a general time-varying delay fuzzy singular system is considered in this paper, including possible actuator, sensor faults, and exogenous disturbance input simultaneously.

Delay-Dependent Stability.
In this subsection, we suggest developing a delay-dependent stability condition for the nominal unforced fuzzy singular system of ( 7), which can be written as Theorem 7.For the given  1 ,  2 ,   , the free fuzzy singular system (13) with () = 0 is admissible for any timevarying delay () satisfying ( 2), if there exists a nonsingular matrix , symmetric positive-definite matrices  1 > 0,  2 > 0,  1 > 0,  2 > 0, and  > 0, and any , such that the following set of inequalities hold: where with Proof.The proof of this theorem is divided into two parts.The first one is concerned with the regularity and the impulsefree characterizations, and the second one treats the stability property of system (13).
Since rank() =  ≤ , there must exist two invertible matrices  ∈ R × and  ∈ R × such that Similar to (19), we define Since Ψ  < 0 and  1 > 0 and  2 > 0 and  > 0, we can formulate the following inequality easily: Then, pre-and postmultiplying ϝ < 0 by   and , respectively, (21) yields Since ϝ11 and ϝ12 are irrelevant to the results of the following discussion, the real expression of these two variables is omitted here.From (22), it is easy to see that Since   (()) ≥ 0 and ∑  =1   (()) = 1, we have This implies that ∑  =1   (()) Ã22 is nonsingular.Therefore, the unforced fuzzy singular system ( 13) is regular and impulse-free.
Remark 8.In some existing literature, for example, [22,23], some delay-dependent criteria are given in terms of LMIs to guarantee that the fuzzy singular system is admissible by using LKF approach and integral inequality, such as Lemma 2 in [22] and Lemma 2.3 in [23].However, in the proof of our result of Theorem 7, we use one identical equality to estimate the upper bound of the derivative of () without any model transformation.Moreover, it is interesting to mention that this study presents criteria based on the free weighting matrix method, in which the bounding techniques on some cross product terms are not involved [11,23].The major feature of this method is to reduce the conservatism engendered by the system transformations and the bounding techniques.ẋ  () ẋ () was ignored [25][26][27], or some useful negative integral term was lost; see, for example, [28,29].Instead, in this paper all those terms ẋ  () ẋ (), which contain a great amount of useful information about systems, are preserved.Therefore, it is obvious to see that this method will lead to less conservatives than the existing ones in [25][26][27][28][29]. Furthermore, the introduction of parameter  ( ≥ 0) indicates that Lemma 3 can be suitable for time-varying delay () being unknown or not differentiable; that is, in the case of time-varying delay () not differentiable, one can set  = 0.
In order to estimate system faults, the following fault estimation observer is constructed: Therefore,  ∞ robust fault estimation observer design problem to be addressed in this paper can be formulated as follows: (i) The error dynamic system (37) with () = 0 is admissible for any time delay satisfying (2); (ii) for a given scalar , the following  ∞ performance is satisfied: for all  > 0 and () ∈  2 [0, ∞) under zero initial conditions.
Remark 10.From error dynamics (37), we can see that the new matrices (),   (), (),   (),   (), and   () are known matrices, while the matrices () contain two matrices () and () that have to be designed.Therefore, the proposed robust fault estimation observer design is converted to the problem of seeking the gain matrix ().
In the following, we will focus on the design of observer based on Lemma 11 and provide a new sufficient condition for the existence of robust fault estimation observer for fuzzy singular time delay system (3).Theorem 12.For the given positive scalars  1 ,  2 ,   , , and , the error dynamic system (37) is admissible with () = 0 while satisfying a prescribed  ∞ performance (39), if there exist appropriately dimensional matrices ,  1 > 0,  2 > 0,  1 > 0,  2 > 0,  > 0, and   and free weighting matrices , such that the following inequalities hold: where where with Then the observer gain matrices can be obtained as Proof.For any scalar , it follows from the fact ( − ) −1 ( − ) ≥ 0 that − −1  ≤ −2 +  2 .By Schur complement theorem, we can conclude that (42)-(43) hold if the following inequalities hold: where Φ 1 () is defined in Lemma 11.Then, if (52)-( 55) hold and with the changes of variables as   =     , we have which imply that the error dynamics (37) are stable with () = 0 while satisfying the prescribed  ∞ performance (39) by Lemma 11.The proof is completed.
Remark 13.Theorem 12 provides a criterion for designing  ∞ robust fault estimation observer of fuzzy singular time delay systems, which guarantees the stability of the resulting dynamic error system with  ∞ performance  > 0. As the delay term () is not simply enlarged, the proposed conditions are less conservative.Moreover, the proposed method is not only able to better depict the size and shape of the actuator fault but also able to estimate the sensor faults simultaneously.
Remark 14.Note that conditions (52)-(55) are LMIs.This indicates that the conditions ( 52)-( 55) can be included as an optimization variable problems, which can be exploited to reduce the attenuation level bound.Then, the minimum attenuation level of  ∞ performance can be obtained by the mincx function of Matlab toolbox.From the practical point of view, it is interesting to find an estimation law, which minimizes the disturbance rejection level  for the error dynamic system.This can be done by solving a convex optimization problem P: min  subject to (51)-( 55) with  =  2 .
Remark 15.In dealing with time-varying faults, there may be a time delay between the fault estimation and the system fault.This phenomenon results from the influence of fault variation.Theoretically, the attenuation level  min can be minimized so that the fault estimation is insensitive to the fault variation.However, the cost is that the fault estimation becomes less robust to disturbance noise.Therefore, the attenuation level of  min , the fault variation, and disturbance are a trade-off.

Numerical Examples
In this section, three examples are given to show the effectiveness of our results.All the numerical results are calculated via the Yalmip toolbox of Matlab.
Example 1.Consider a continuous fuzzy singular system composed of two rules and the following system matrices [22,23]: Assume that the delay () satisfies (2) and set  = 0.5.The obtained results are listed in Table 1, where () stands for the total number of decision variables.Table 1 tabulates a comparison of the maximum allowable upper delay bound  2 for a prescribed   .It can be seen from the table that our results are marked better than those obtained by the method in [22,23].Moreover, it is worth mentioning that the method proposed in this paper uses fewer number of LMIs scalar variables and fewer number of LMIs for stability computation; thus our method is more computationally efficient for improving the upper bound of delay; the stability criterion we derived is less conservative than those reported in the aforementioned papers.(62) To compare with the existing results, we assume that   = 0 and  = 0.5.Table 2 lists the comparison results on the maximum allowed time delay  2 via the methods in [10,20,[22][23][24] and Theorem 7 in this paper.From the comparison result we can see that the stability criterion we derived by using free weighting matrix approach in this work is less conservative than those reported in [10,20,[22][23][24].
Here, we consider the case where  = 2 and the time-varying delay is given as () = 0. ] .
The associate fault estimation observer gains in (36) are According to Theorem 12, we can consider different  to find the minimum index  for the given   = 0.2; see [25,28] for more details.The corresponding results are summarized in Table 3.
In order to illustrate the performance of robust fault estimation observer in dealing with fuzzy singular systems with time delay, first, an abrupt fault is simulated.It is assumed that the abrupt fault () is created as For simulation purpose, we choose the membership functions for Rules 1 and 2 to be  1 (()) = 1/(1 + exp( 1 () + 0.  is represented by the blue solid one.As shown in Figure 1, the robust fault estimation observer is insensitive to the model disturbance.Moreover, although there is estimation error, the fault estimate can quickly track the fault.This illustrates the fast convergence rate of the fault estimation observer in the face of initial estimation error.The simulation results shown in Figure 1 obviously illustrate that the proposed fault estimation has a good performance to estimate fault, and the error dynamic system is also stable in Figure 2. To illustrate the performance of robust fault estimation observer   (68) In this situation, the fault estimation result is depicted in Figures 3 and 5.It can be seen from Figures 3 and 5 that the fault is estimated with satisfactory accuracy and rapidity.The state of the error dynamic system is also stable in Figures 4 and 6.In [13], actuator fault estimation observer is designed for discrete-time linear parameter-varying descriptor systems, but the systems with time-varying delay case are not considered.In [9], pole assignment is used to ensure the fault estimation convergence speed while our method utilizes  ∞ technique to attenuate the effect of fault variation.From Figures 3 and 5, it can be seen that these methods have similar fault convergence speed.Nevertheless, [9] only deals with regular systems not with time delay.Moreover, our method considers the fuzzy singular system with actuator and sensor faults simultaneously.Therefore, the proposed method is more general than that in [9,13].

Conclusion
In this paper, the robust fault estimation problem for T-S fuzzy singular systems with time-varying delays is considered.By considering the fault as an auxiliary disturbance vector, based on the Lyapunov theorem and improved delay partitioning method with free weighting matrix approach, we give some less conservative criteria, which guarantee that the considered system is regular, impulse-free, and stable, while limiting the influence of disturbance despite the presence of actuator and sensor faults simultaneously.This paper proposes a novel fault estimation observer and presents an LMI-based design method for the fuzzy singular system.Finally, some numerical examples are used to demonstrate the effectiveness and performance of the proposed method.

Example 2 .
Consider the following singular time delay system:

Figure 1 :Figure 2 :
Figure 1: Fault estimation result of the robust fault estimation observer in abrupt fault case ().

Figure 3 :
Figure 3: Fault estimation result of the robust fault estimation observer in fault  1 ().

Figure 5 :Figure 6 :
Figure 5: Fault estimation result of the robust fault estimation observer in fault  2 ().

Table 1 :
Allowable upper bound of  2 for various   in Example 1.

Table 2 :
Comparison of maximum allowed delays  2 in Example 2.

Table 3 :
Minimum index  for various  in Example 3 with   = 0.2.