Robust Fault Reconstruction in Discrete-Time Lipschitz Nonlinear Systems via Euler-Approximate Proportional Integral Observers

The problem of observer-based robust fault reconstruction for a class of nonlinear sampled-data systems is investigated. A discretetime 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 H ∞ 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 faultreconstructing method.


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, faultdiagnostic module consists of three essential tasks: fault detection, isolation, and reconstruction (also known as fault estimation or fault identification) [1][2][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 2 Mathematical Problems in Engineering been reported [12][13][14][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 Eulerapproximate 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 Eulerapproximate 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 Eulerapproximate models.First, a sampled-data nonlinear system, satisfying the Lipschitz condition, is formulated into a Eulerapproximate 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.

System Description and Problem Formulation
A continuous-time nonlinear system subject to actuator faults, external disturbances, and measurement noises is described as where () ∈ R  , () ∈ R  , and () ∈ R  , representing system state vector, control input vector and measurable output vector, respectively.Variable () denotes actuator faults with () ∈ R  .Vectors  1 ∈ R  and  2 ∈ R  , representing external disturbances and measurement noises, respectively.Without loss of generality, matrices  1 and  are assumed to be of full column rank and of full row rank, respectively.The symbol Φ((), ) is a continuous nonlinear function that is assumed to satisfy the Lipschitz condition (at least local), that is, ‖Φ((), ) − Φ( x(), )‖ ≤   ‖() − x()‖, where   is a Lipschitz constant.
Herein, system control input () is taken to be a piecewise constant signal as () during the sampling intervals [, ( + 1)] 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  ( + 1) = (  + )  () + Φ ( () , ) +  () where discrete nonlinear term ‖Φ((), )‖ 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 positivedefinite symmetric matrix  such that the following inequality holds: Throughout the paper,  > 0 ( < 0) denotes that  is positive (negative) definite,  min () and  max () 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 , 0  is a zero matrix of size , † represents a pseudoinverse of a matrix, and C denotes the set of all complex numbers.

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.
Here, , , , and  are gain matrices with appropriate dimensions to be determined later.
Since the nonlinear function Φ((), ) satisfies the Lipschitz condition, then Inspired by disturbance-decoupling principle of UIOs [20,21]  Considering (5d), fault-reconstructing errors   () can be expressed in the following equation: (10) where fault variation vector Δ() := ( + 1) − ().If constant actuator faults are considered in system ((1a), (1b)), then Δ() ≡ 0. Therefore, one obtains Remark 2. Equations ( 8) and ( 9) display that the designed EPIO is only partially decoupled from external disturbance  1 ().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  11 = 0, are satisfied simultaneously.Augmented matrix equation composing of these two constraints can be written as Therefore, a general solution of ( 12) can be determined by In addition, to guarantee actuator-fault detectability by the proposed EPIO, the constraint, rank() = rank(), should be also satisfied.
According to ( 9) and ( 10), the following augmented error dynamics can be constructed as Mathematical Problems in Engineering Denote Further, augmented error dynamics ( 14) can be reorganized into a compact form Besides, fault reconstruction error   () =   (), where For nonlinear function ΔΦ(), 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 ΔΦ() and augmented disturbance () are not considered in (16), we can obtain The linear error dynamics ( 21) is asymptotically stable if and only if the pair and one obtains (1) (2) Thus, the pair when  = −1 and || > 1.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  1 () and measurement noise  2 (), 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.

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  1 (), measurement noise  2 (), and fault variation vector Δ(), 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)).
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.
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   , ∀ ∈ {1, 2, 3, 4} is adopted in Theorem 5 such that the impact of  1 (),  2 (), and Δ() 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,  ≥   , ∀ ∈ {1, 2, 3, 4}.Additionally, to guarantee accurate reconstruction of time-varying faults, especially fast-varying faults, we can select a sufficiently small attenuation level  4 to restrict variation vector Δ() at the expense of the robustness against external disturbance  1 () and measurement noise  2 ().
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.
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  1 for which the condition rank( 11 ) = rank( 11 ) is satisfied.
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.

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 Eulerapproximate 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-KL function (⋅, ⋅) such that, for any  >  > 0 and compact sets X ⊂ R  , U ⊂ R  , we can find a number  * > 0 with the property that, for all  ∈ (0,  * ], 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 Eulerapproximate model ((2a), (2b)
) is semiglobal practical in the sampling period  if there exist a positive scalar  1 > 1, a fixed positive-definite symmetric matrix  1 ∈ R × , and a positive- 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 obtains With the help of Lemma 1, the following inequalities can be easily derived: where where  2 > 0, and where  3 > 0.
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.
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.

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 0.333 and the sampling period  is selected as 0.01 s.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   = 0.
and  * = 0.6162 s, which is greater than 0.01 s.We can now check that the designed observer gain matrices and the sampling period 0.01 s can make the Theorems 13 and 14 hold.Hence, it is concluded that the designed EPIO ((5a), (5b), (5c), and (5d)) is semiglobal practical convergent in the sampling period 0.01 s.According to Corollary 9, let  = 0.1; then, (43) can be solved to obtain  = 1.3007.Further, choosing  1 = 0. Using Theorems 13 and 14, it is easily checked that the EPIO designed using Corollary 8 is also semiglobal practical convergent in 0.01 s.
In the simulation, the control input signal is assigned as () = 0.05 sin 5; the initial values of the estimated states are assumed as x(0) = [−0.1 0.5 − 0.1 0.2]  ; external disturbance  1 () and measurement noise  2 () are chosen as 0.001 sin(10) and a Gaussian-distributed random signal with zero mean and variance of 10 −5 , respectively.Additionally, actuator faults are assumed to be changes of bus voltage, and two fault scenarios are employed as

Conclusions
In this paper, a robust fault-reconstructing scheme for Lipschitz nonlinear systems with Euler-approximate models is investigated.A new discrete-time EPIO is constructed to achieve robust actuator-fault reconstruction.An important advantage of the newly designed EPIO is its disturbancedecoupling ability such that the accuracy of the reconstructed faults can be guaranteed.To ensure convergence of the estimated states to the system states of the continuoustime model, semiglobal practical convergence of the EPIO is proved.Another advantage of the proposed approach is that a systematic observer synthesis with  ∞ techniques is effectively solved using a standard LMI tool.Simulated results on a flexible-joint robot have clearly verified the effectiveness of the proposed fault-reconstruction method.With mismatching errors caused by approximate discrete models and reconstructed fault signals, active FTC for Lipschitz nonlinear systems with Euler-approximate models will be our future research target.

( 1 )
a constant fault that is described as  () = { 0.2 2 s ≤  ≤ 6 s EPIO via Theorem 5, the estimated system states in fault-free conditions are exhibited in Figures 1, 2, 3, and 4 where the estimated states can track the original states of system ((1a), (1b)) with satisfactory accuracy.Further, reconstruction signals for the constant and time-varying faults obtained by the EPIO designed using Theorem 5 are demonstrated in Figures5 and 6while constant-fault reconstruction signal by the EPIO designed using Corollary 8 is illustrated in Figure7.The above figures reveal that the proposed EPIO can accurately reconstruct actuator faults.All of simulation results claim that the EPIObased fault-reconstructing method is effective for Lipschitz nonlinear systems with Euler-approximate models.
state x 2 (t) Estimated state x 2 (k)

Figure 6 :
Figure 6: Reconstruction signal of a time-varying fault via the EPIO designed using Theorem 5.

Figure 7 :
Figure 7: Reconstruction signal of a constant fault via the EPIO designed using Corollary 8.
and  is the maximum number of columns of the matrix  1 for which the condition, rank( 11 ) = rank( 11 ), is satisfied.Matrix  11 is composed of  columns of the matrix  1 while matrix  12 represents the remaining part of the matrix  1 .Matrix  is selected such that the condition,  11 = 0, holds.Herein, we define matrix  as [0   12 ].