#### Abstract

The problem of observer-based robust fault reconstruction for a class of nonlinear sampled-data systems is investigated. A discrete-time Lipschitz nonlinear system is first established, and its Euler-approximate model is described; then, an Euler-approximate proportional integral observer (EPIO) is constructed such that simultaneous reconstruction of system states and actuator faults are guaranteed. The presented EPIO possesses the disturbance-decoupling ability because its architecture is similar to that of a nonlinear unknown input observer. The robust stability of the EPIO and convergence of fault-reconstructing errors are proved using Lyapunov stability theory together with techniques. The design of the EPIO is reformulated into convex optimization problem involving linear matrix inequalities (LMIs) such that its gain matrices can be conveniently calculated using standard LMI tools. In addition, to guarantee the implementation of the EPIO on the exact model, sufficient conditions of its semiglobal practical convergence are provided explicitly. Finally, a single-link flexible robot is employed to verify the effectiveness of the proposed fault-reconstructing method.

#### 1. Introduction

Increasing complexity of modern engineering systems and higher demand for system performance, particularly in safety-critical systems, will correspondingly raise the probability of system faults and/or failures. As a response to the requirement for system safety, reliability, and survivability, fault diagnosis of dynamic systems has been an attractive subject during the past few decades. In general, fault-diagnostic module consists of three essential tasks: fault detection, isolation, and reconstruction (also known as fault estimation or fault identification) [1–3]. In literature, fault detection and isolation (FDI) schemes are considered to be the most important and are of main focuses; nevertheless, fault reconstruction is an indispensable component in active fault-tolerant control (FTC) systems where faults can be effectively accommodated utilizing reconfigurable fault-tolerant controller with reconstructed fault information in real time.

During the past decade, observer-based methods for fault reconstruction have been increasingly attracting the attention of many researchers and different categories of fault reconstruction observers have been developed, such as adaptive observers [4, 5], sliding mode observers (SMOs) [6, 7], learning observers [8, 9], and proportional integral observers (PIOs) [10, 11], to name just a few. It is worth noting that most of existing observers are mainly applicable to continuous-time systems. On the other hand, it is well known that fault reconstruction for nonlinear discrete-time systems is considerably practical and challenging because the nonlinearities inherently exist in nature, and most continuous-time systems are implemented digitally in practical applications. Under the assumption that nonlinear discrete-time models are accurate, numerous fault-reconstructing strategies have been reported [12–15]. However, the availability of the exact models, originated from discretization of a continuous-time plant, is usually unrealistic. A practical solution to this difficult situation is to employ approximate discrete-time models, especially Euler-approximate models, instead of exact models.

More recently, many researchers have paid much attention to observer design and observer-based fault reconstruction for nonlinear discrete-time systems with Euler-approximate models. Reference [16] proposes a general framework for nonlinear observer design via the approximate discrete models. For Lipschitz nonlinear systems, the authors in [17] suggest a robust observer whose main advantage is that maximum admissible Lipschitz constant and robustness can be solved using LMI optimization techniques. Based on [16, 17], a Euler-approximate observer (EAO) is presented for fault reconstruction and active FTC in [18]; however, the robustness is not well guaranteed. For nonlinear networked control systems, an SMO-based fault reconstruction approach is investigated in [19] where the chattering phenomenon resulting from the signum function in the SMO is inevitable. In [20], a Euler-approximate unknown input observer- (UIO-) based robust fault detection strategy is developed. However, to the best of our knowledge, few results have been reported on observer-based robust fault reconstruction for nonlinear discrete-time systems via Euler-approximate models. In addition, how to solve the error problem caused by the approximate models such that the observer designed under Euler-approximate models can be implemented on the exact models has still been an interesting and challenging issue. All of these motivate us to pursue this investigation.

The main objective of this work is to design and analyze an observer-based robust fault-reconstructing strategy for a class of nonlinear discrete-time systems using Euler-approximate models. First, a sampled-data nonlinear system, satisfying the Lipschitz condition, is formulated into a Euler-approximate model; then, on the basis of our previous results in [20], a discrete-time Euler-approximate proportional integral observer (EPIO) is constructed to simultaneously reconstruct system states and actuator faults. The EPIO has an architecture similar to that of a nonlinear UIO. Compared with the EAO proposed in [18], the designed EPIO is able to partially decouple external disturbances and is robust to the reminding part of external disturbances and measurement noises. As a result, the accuracy of the EPIO-based fault reconstruction can be guaranteed. In addition, to guarantee implementation of the EPIO on the exact models, sufficient conditions for* semiglobal practical convergence*, which is defined in [16], are explicitly provided. Besides, systematic observer synthesis with an technique is effectively solved using LMI optimization techniques. Simulation results on a single-link flexible robot are also presented to verify the effectiveness of the proposed fault-reconstructing method.

The rest of this paper is organized as follows. In Section 2, an Euler-approximate model of a Lipschitz nonlinear system is described, and main problems are formulated. In Section 3, a discrete-time EPIO is constructed, and robust stability and semiglobal practical convergence of the proposed observer are analyzed. Simulation studies are reported in Section 4, and conclusions are drawn in the last section.

#### 2. System Description and Problem Formulation

A continuous-time nonlinear system subject to actuator faults, external disturbances, and measurement noises is described aswhere , and , representing system state vector, control input vector and measurable output vector, respectively. Variable denotes actuator faults with . Vectors and , representing external disturbances and measurement noises, respectively. Without loss of generality, matrices and are assumed to be of full column rank and of full row rank, respectively. The symbol is a continuous nonlinear function that is assumed to satisfy the Lipschitz condition (at least local), that is, , where is a Lipschitz constant.

Herein, system control input is taken to be a piecewise constant signal as during the sampling intervals with a zero-order hold, where is the sampling period. Therefore, the Euler-approximate discrete model of continuous-time system ((1a), (1b)) can be formulated as where discrete nonlinear term still satisfies Lipschitz constraint, namely,

Before finishing this section, the following lemma is provided for proof of theorems in the latter section.

Lemma 1. *For any positive scalar , there exists a positive-definite symmetric matrix such that the following inequality holds:*

Throughout the paper, denotes that is positive (negative) definite, and are the minimum and maximum eigenvalues of , represents the maximum singular value of matrix , and represent Euclidean norm and infinity norm of a vector or matrix, means symmetric term, is an identity matrix of size , is a zero matrix of size , represents a pseudoinverse of a matrix, and denotes the set of all complex numbers.

#### 3. Fault Reconstruction via Euler-Approximate Proportional Integral Observers

In this section, we will construct a discrete-time EPIO to achieve robust actuator-fault reconstruction; then, robust stability and semiglobal practical convergence of the presented EPIO will be discussed in detail.

##### 3.1. Design of a Euler-Approximate Proportional Integral Observer

In order to perform fault reconstruction for actuators in Lipschitz nonlinear system ((1a), (1b)), a discrete-time nonlinear EPIO is established for the Euler-approximate model ((2a), (2b)) as follows:where , , and ; they represent the observer state vector, the estimated state vector, and measurement output vector, respectively. Fault reconstruction signal is recursively updated by both itself and output estimation errors at the sampling time . Here, , , , and are gain matrices with appropriate dimensions to be determined later.

Defining , , and and letting , estimation error dynamics can be easily obtained aswhere .

Since the nonlinear function satisfies the Lipschitz condition, thenInspired by disturbance-decoupling principle of UIOs [20, 21], we will design gain matrix such that state estimation error can be partially decoupled from external disturbances. As such, state error dynamics (6a) can be reorganized aswhere and is the maximum number of columns of the matrix for which the condition, , is satisfied. Matrix is composed of columns of the matrix while matrix represents the remaining part of the matrix .

Matrix is selected such that the condition, , holds. Herein, we define matrix as . Hence, state error dynamics (6a) can be further manipulated as

Considering (5d), fault-reconstructing errors can be expressed in the following equation:where fault variation vector . If constant actuator faults are considered in system ((1a), (1b)), then . Therefore, one obtains

*Remark 2. *Equations (8) and (9) display that the designed EPIO is only partially decoupled from external disturbance . It is because either the number of measurement outputs is not greater than that of external disturbances or fault-reconstructing ability of the presented EPIO should be guaranteed when coupling problem exists between external disturbances and actuator faults. To tackle the above disturbance-decoupling problem, gain matrices and should be designed such that matrix equations, , and , are satisfied simultaneously. Augmented matrix equation composing of these two constraints can be written asTherefore, a general solution of (12) can be determined byIn addition, to guarantee actuator-fault detectability by the proposed EPIO, the constraint, , should be also satisfied.

According to (9) and (10), the following augmented error dynamics can be constructed as

Denote

Further, augmented error dynamics (14) can be reorganized into a compact formBesides, fault reconstruction error , where .

For nonlinear function , the following inequality holds:where .

It is noticed that augmented error system (16) can be treated as an observation-error equation of the following nonlinear system:where An augmented state observer can be established for (18) as follows:

*Remark 3. *If both the nonlinear term and augmented disturbance are not considered in (16), we can obtainThe linear error dynamics (21) is asymptotically stable if and only if the pair is detectable. It is equivalent toand one obtains(1)If ,(2)If ,Thus, the pair is detectable if and only ifwhen and . Therefore, the above two conditions can be regarded as necessary conditions for existence of the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)).

*Remark 4. *Equations (14)–(20) imply that the design of the proposed nonlinear EPIO is now converted into the analysis of robust stability of the augmented error dynamics (16), that is, the design of robust observer (20) for the augmented nonlinear system (18). Therefore, to guarantee the robustness of the proposed EPIO against the remaining part of external disturbance and measurement noise , we shall design augmented matrix , which is composed of two unknown matrices and , such that the robust stability of the augmented error dynamics (16) is guaranteed with a prescribed performance specification.

##### 3.2. Robust Stability Analysis of the Euler-Approximate Proportional Integral Observer

In this subsection, we will foucs on the robust stability analysis of the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)). Equation (14) shows that different components of the augmented disturbance have different influences on fault reconstruction performance. To guarantee better robustness of the designed nonlinear EPIO against external disturbance , measurement noise , and fault variation vector , multiple attenuation levels in an performance specification will be adopted in the EPIO synthesis. In what follows, a theorem is presented to characterize the robust stability of the proposed EPIO ((5a), (5b), (5c), and (5d)).

Theorem 5. *Consider the Euler-approximate discrete model ((2a), (2b)). For a given positive scalar , , if there exist a positive-definite symmetric matrix , a matrix , and positive scalars , such that**where , , , and , then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can realize uniform ultimate boundedness of state-estimating error and fault-reconstructing error with an performance criterion:**Moreover, augmented matrix can be determined by .*

*Proof. *Consider the Lyapunov function candidate . The external disturbance and measurement noise are temporarily dropped; then, the difference is derived aswhere .

Considering Lemma 1, the following inequalities hold:where and .

Substituting (17), (31) into (30) leads towhere and with and .

If , then we havewhere and .

Further, the condition can be rewritten asBy the Scular complement lemma [22], (34) is equivalent to (27).

Therefore, if (27) holds, then for , which means that the trajectory of that is outside of the set will converge to the set according to the Lyapunov stability theory. Therefore, the uniform ultimate boundedness of the state-estimating error and the fault-reconstructing error can be guaranteed.

To guarantee that the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) is robust to , , and , an performance index function is chosen as [23]Under zero-initial conditions, the following is obtained:Considering external disturbance and measurement noise , it follows from (17), (30), and (36) thatwherewith , , , , , , , , , , , , , , , , , , , , and .

Further, the inequality, , can be transformed intoUsing the Schur complement lemma again, (39) is equivalent to (28), which means that, for all nonzero , , and , one obtains . Therefore, we can conclude from (35) that the performance criterion (29) is satisfied. This completes the proof.

In Theorem 5, multiple attenuation levels in the performance criterion (29) are prescribed for better restricting each component of the augmented disturbance on fault reconstruction performance. If only a single attenuation level is considered for the augmented disturbance , the following corollary can be readily obtained based on Theorem 5.

Corollary 6. *Consider the Euler-approximate discrete model ((2a), (2b)). For a given positive scalar , if there exist a positive-definite symmetric matrix , a matrix , and positive scalars , satisfying (27) and**then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can realize the uniform ultimate boundedness of state-estimating error and fault-reconstructing error with an performance criterion: . Moreover, augmented matrix can be determined by .*

*Proof. *The proof of Corollary 6 is similar to that of Theorem 5; it is thus omitted here.

*Remark 7. *In Corollary 6, the performance criterion with an attenuation level is adopted such that the designed nonlinear EPIO is robust to the augmented disturbance . However, this design inevitably results in conservatism such that the EPIO has a limited robust performance. To obtain less conservative result in the EPIO design, the criterion with attenuation levels , is adopted in Theorem 5 such that the impact of , , and on can be effectively attenuated. Therefore, Theorem 5 is more flexible than Corollary 6. Compared with Corollary 6, one may obtain smaller attenuation levels using Theorem 5; that is, , . Additionally, to guarantee accurate reconstruction of time-varying faults, especially fast-varying faults, we can select a sufficiently small attenuation level to restrict variation vector at the expense of the robustness against external disturbance and measurement noise .

Considering constant actuator faults in the system ((1a), (1b)), the results proposed in Theorem 5 and in Corollary 6 can be readily particularized in the following corollaries.

Corollary 8. *Consider the Euler-approximate discrete model ((2a), (2b)). For given positive scalars , , and , if there exist a positive-definite symmetric matrix , a matrix , and a positive scalar such that the following LMI holds:**where , , and , then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can asymptotically reconstruct constant actuator faults, and fault-reconstructing error satisfies the following inequality:**Moreover, augmented matrix can be determined by .*

Corollary 9. *Consider the Euler-approximate model ((2a), (2b)). For a given positive scalar , if there exist a positive-definite symmetric matrix , a matrix , and a positive scalar such that the following LMI holds:**where , , and , then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can asymptotically reconstruct constant actuator faults, and fault reconstruction error satisfies that , where . Moreover, augmented matrix can be determined by .*

*Remark 10. *Theorem 5 and Corollary 6 can guarantee the uniform ultimate boundedness of state estimation error and fault reconstruction error such that the designed EPIO can achieve accurate reconstruction of constant and time-varying actuator faults. If constant actuator faults are considered in the system 1, then one obtains . Therefore, using Corollaries 8 and 9, the designed EPIO can asymptotically reconstruct constant actuator faults.

*Remark 11. *Since the Lipschitz condition (32) is introduced in the proof of Theorem 5, there may be no feasible solution for (27) and (28), especially for a large Lipschitz constant. However, the condition (17) is not overly restrictive because the Lipschitz constant might be reduced via coordinate transformation techniques, as discussed in [14, 17, 24]. For multiobjective observer design, the increased number of LMI constraints inevitably results in conservatism. It is worth noting that slack matrix variable technique can be adopted to reduce this conservatism; the interested readers can be referred to [14, 25, 26] for detailed information.

According to Theorem 5, the design procedure of the proposed EPIO will be summarized as follows:

*Step 1. *Find the maximum number of columns of matrix for which the condition is satisfied.

*Step 2. *Solve (13) to obtain and , and verify that .

*Step 3. *Choose appropriate scalars , ; then, based on (27) and (40), performance level is computed using the Solver* mincx* in the LMI toolbox of Matlab.

*Step 4. *Choose , ; then, solving (27) and (28) using the Solver* feasp* in the LMI toolbox of Matlab, matrices and can be obtained.

*Step 5. *Compute , then obtain gain matrices and .

*Step 6. *Construct the EPIO based on the above calculated matrices.

Using the above theorem and corollaries, we can design observer gain matrices such that the stability and convergence of the state estimation error and fault reconstruction error between the nonlinear EPIO and the Euler-approximate model are guaranteed. However, it might be impossible for the EPIO designed under the Euler-approximate model to be implemented on the exact system models. To solve this problem, the EPIO will be required to satisfy semiglobal practical convergence property introduced by [16] such that the EPIO can track actuator faults with satisfactory accuracy. This is what we shall address in next subsection.

##### 3.3. Semiglobal Practical Convergence Analysis

In this subsection, we will derive sufficient conditions for existence of observer gain matrices and the sampling interval to achieve semiglobal practical convergence under the Euler-approximate model without considering external disturbances and measurement noises. The definition of semiglobal practical convergence [16] is given as follows.

*Definition 12. *An observer is said to be semiglobal practical in the sampling period if there exists a class function such that, for any and compact sets , , we can find a number with the property that, for all ,imply

In reference to the method proposed by [16], the following theorem presents the semiglobal practical convergence of the EPIO ((5a), (5b), (5c), and (5d)).

Theorem 13. *The proposed nonlinear EPIO under the Euler-approximate model ((2a), (2b)) is semiglobal practical in the sampling period if there exist a positive scalar , a fixed positive-definite symmetric matrix , and a positive-definite symmetric matrix such that*

*Proof. *Consider the Lyapunov function candidate, ; then,Since the convergence of the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) to the Euler-approximate model has been established using the aforementioned theorem and corollaries, then one obtainsThus,With the help of Lemma 1, the following inequalities can be easily derived:where ,where , andwhere .

Substituting (52)–(54) into (51) leads towhere , , and with .

According to (46) and (47), (55) can be reorganized and simplifiedIn addition, the following functions are defined:Therefore, the following results can be derived:where , and are in class and , , and are nondecreasing functions.

Based on the theorem and corollaries discussed in Subsection 3.2, it can be known that fault reconstruction error are uniformly bounded; herein, they are considered as the external disturbances. According to [17, 27], the Euler-approximate model is consistent with the exact discrete-time model. Conditions (50), (58) together display that all the conditions for semiglobal practical convergence as required by [16] are satisfied; hence, the designed nonlinear EPIO is semiglobal practical convergence in the sampling time in . This completes the proof.

Motivated by [18], we will present another sufficient conditions for semiglobal practical convergence of the proposed nonlinear EPIO ((5a), (5b), (5c) and (5d)).

Theorem 14. *The proposed nonlinear EPIO ((5a), (5b), (5c) and (5d)) designed under the Euler-approximate model ((2a), (2b)) is semiglobal practical in the sampling period if there exist positive-definite symmetric matrices and , a matrix , and a positive scalar , such that*

*Proof. *The proof of Theorem 14 is similar to that of Theorem in [18]; it is thus omitted here.

*Remark 15. *Theorem 5 and Corollaries 6–9 imply that observer gain matrices and can be conveniently calculated using standard LMI optimization technique; furthermore, using Theorem 13 or Theorem 14, whether or not semiglobal practical convergence property of the designed EPIO can be guaranteed can be checked. If yes, the EPIO designed under Euler-approximate models can be implemented on the exact discrete-time models.

#### 4. Simulation Studies

In this section, a single-link robot with a flexible joint borrowed from [28] is employed to illustrate the effectiveness of the proposed fault-reconstructing method. The considered system dynamics is described in a form of (1a), (1b) with where system states are the motor position, link position, and velocities. Measurement outputs are the first three system states. In this work, we consider actuator faults that usually occur in input channels. Therefore, it is reasonable to assume that fault distribution matrix .

The Lipschitz constant for the system ((1a), (1b)) is selected as and the sampling period is selected as . Therefore, an Euler-approximate model of the considered system ((1a), (1b)) can be established for the flexible-joint robot.

In order to achieve robust actuator-fault reconstruction, observer gain matrices and are chosen as follows:

Letting , and solving (27) and (40) in Corollary 6 using the Solver* mincx* in the LMI toolbox of Matlab, one can obtain the attenuation level . Further, choosing , , , and and solving (27) and (28) in Theorem 5 using the LMI toolbox of Matlab, a feasible solution for gain matrices and can be obtained as