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

Authors: Qingxian Jia; Wen Chen; Yingchun Zhang; Yu Jiang
Journal: Mathematical Problems in Engineering
 (2015)

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## 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 withH∞ 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.


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## Body

## 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 H∞ 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 forsemiglobal practical convergence, which is defined in [16], are explicitly provided. Besides, systematic observer synthesis with an H∞ 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 Section2, 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 as(1a)x˙t=Axt+Φxt,t+But+D1d1t+Fft,(1b)yt=Cxt+D2d2t,where x(t)∈Rn,u(t)∈Rm, and y(t)∈Rp, representing system state vector, control input vector and measurable output vector, respectively. Variable f(t) denotes actuator faults with f(t)∈Rm. Vectors d1∈Rd and d2∈Rs, representing external disturbances and measurement noises, respectively. Without loss of generality, matrices D1 and C are assumed to be of full column rank and of full row rank, respectively. The symbol Φ(x(t),t) is a continuous nonlinear function that is assumed to satisfy the Lipschitz condition (at least local), that is, Φ(x(t),t)-Φ(x^(t),t)≤Lgx(t)-x^(t), where Lg is a Lipschitz constant.Herein, system control inputu(t) is taken to be a piecewise constant signal as u(k) during the sampling intervals [kτ,(k+1)τ] with a zero-order hold, where τ is the sampling period. Therefore, the Euler-approximate discrete model of continuous-time system ((1a), (1b)) can be formulated as (2a)xk+1=In+τAxk+τΦxk,k+τBuk+τD1d1k+τFfk,(2b)yk=Cxk+D2d2k,where discrete nonlinear term Φ(x(k),k) still satisfies Lipschitz constraint, namely,(3)Φ(x(k),k)-Φ(x^(k),k)≤Lgx(k)-x^(k).Before finishing this section, the following lemma is provided for proof of theorems in the latter section.Lemma 1.
For any positive scalarδ, there exists a positive-definite symmetric matrix N such that the following inequality holds:(4)2aTNb≤δaTNa+1δbTNb,∀(a,b)∈Rn.Throughout the paper,A>0(A<0) denotes that A is positive (negative) definite, λmin(A) and λmax(A) are the minimum and maximum eigenvalues of A, σ¯(A) represents the maximum singular value of matrix A, · and ·∞ represent Euclidean norm and infinity norm of a vector or matrix, ∗ means symmetric term, In is an identity matrix of size n, 0n is a zero matrix of size n, † represents a pseudoinverse of a matrix, and C 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:(5a)zk+1=T(In+τA)x^(k)+τTΦ(x^(k),k)+τTBu(k)+Lyk-y^k+τTFf^k,(5b)x^k=zk+Hyk,(5c)y^k=Cx^k,(5d)f^k+1=f^k+Kyk-y^k,where z(k)∈Rn, x^(k)∈Rn, and y^(k)∈Rp; they represent the observer state vector, the estimated state vector, and measurement output vector, respectively. Fault reconstruction signal f^(k) is recursively updated by both itself and output estimation errors at the sampling time k-1. Here, T, H, L, and K are gain matrices with appropriate dimensions to be determined later.Definingex(k):=x(k)-x^(k), ey(k):=y(k)-y^(k), and ef(k):=f(k)-f^(k) and letting T+HC=In, estimation error dynamics can be easily obtained as(6a)exk+1=TIn+τA-LCexk+τTΔΦk+τTD1d1(k)-LD2d2(k)-HD2d2(k+1)+τTFef(k)(6b)eyk=Cexk+D2d2k,where ΔΦ(k)=Φ(x(k),k)-Φ(x^(k),k).Since the nonlinear functionΦ(x(k),k) satisfies the Lipschitz condition, then(7)ΔΦT(k)ΔΦ(k)-Lg2exT(k)ex(k)≤0.Inspired by disturbance-decoupling principle of UIOs [20, 21], we will design gain matrix T such that state estimation error can be partially decoupled from external disturbances. As such, state error dynamics (6a) can be reorganized as(8)exk+1=TIn+τA-LCexk+τTΔΦk+τTD11D12︸D1×d11Tkd12TkT︸d1k-LD2d2k-HD2d2k+1+τTFefk,where d11∈Rq and q is the maximum number of columns of the matrix D1 for which the condition, rank(CD11)=rank(D11), is satisfied. Matrix D11 is composed of q columns of the matrix D1 while matrix D12 represents the remaining part of the matrix D1.MatrixT is selected such that the condition, TD11=0, holds. Herein, we define matrix E as 0qD12. Hence, state error dynamics (6a) can be further manipulated as(9)exk+1=TIn+τA-LCexk+τTΔΦk+τTEd1k-LD2d2k-HD2d2(k+1)+τTFef(k).Considering (5d), fault-reconstructing errors ef(k) can be expressed in the following equation:(10)efk+1=efk-KCexk-KD2d2k+Δfk,where fault variation vector Δf(k):=f(k+1)-f(k). If constant actuator faults are considered in system ((1a), (1b)), then Δf(k)≡0. Therefore, one obtains(11)efk+1=efk-KCexk-KD2d2k.Remark 2.
Equations (8) and (9) display that the designed EPIO is only partially decoupled from external disturbance d1(k). 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 T and H should be designed such that matrix equations, T+HC=In, and TD11=0, are satisfied simultaneously. Augmented matrix equation composing of these two constraints can be written as(12)THInD11C0=In0.Therefore, a general solution of (12) can be determined by(13)TH=In0InD11C0†.In addition, to guarantee actuator-fault detectability by the proposed EPIO, the constraint, rank(TF)=rank(F), should be also satisfied.According to (9) and (10), the following augmented error dynamics can be constructed as(14)exk+1efk+1=T(In+τA)-LCτTF-KCImex(k)ef(k)+τTΔΦ(k)0+τTE-LD2-HD200-KD20Imd1(k)d2(k)d2(k+1)Δf(k).Denote(15)e¯k=exkefk,ωk=d1kd2kd2k+1Δfk,T¯=T00Im,A¯=In+τAτF0Im,ΔΦ¯k=ΔΦk0,D¯1=-τTE0HD20000-Im,L¯=LK,C¯=C0,D¯2=0-D200.Further, augmented error dynamics (14) can be reorganized into a compact form(16)e¯k+1=T¯A¯-L¯C¯e¯k+τT¯ΔΦ¯k+L¯D¯2-D¯1ω(k)ey(k)=C¯e¯(k)-D¯2ω(k).Besides, fault reconstruction error ef(k)=Cfe¯(k), where Cf=0Im.For nonlinear functionΔΦ¯(k), the following inequality holds:(17)Γ=ΔΦ¯T(k)ΔΦ¯(k)-e¯T(k)L¯gTL¯ge¯(k)≤0,where L¯g=LgIn000m.It is noticed that augmented error system (16) can be treated as an observation-error equation of the following nonlinear system:(18)x¯(k+1)=T¯A¯x¯(k)+τT¯Φ¯(x¯(k),k)+D¯1ω(k)yk=C¯x¯k-D¯2ωk,where (19)x¯k=xTkfTkT,Φ¯x¯k,k=ΦTxk,k0T.An augmented state observer can be established for (18) as follows:(20)x¯^k+1=T¯A¯x¯^k+τT¯Φ¯x¯^k,k+L¯yk-y^k,y^k=C¯x¯^k.Remark 3.
If both the nonlinear termΔΦ¯(k) and augmented disturbance ω(k) are not considered in (16), we can obtain(21)e¯k+1=T¯A¯-L¯C¯e¯k,ey(k)=C¯e¯(k).The linear error dynamics (21) is asymptotically stable if and only if the pair (T¯A¯,C¯) is detectable. It is equivalent to(22)rankzIn+m-T¯A¯C¯=n+m,∀z∈C,z≥1,and one obtains(23)rankzIn+m-T¯A¯C¯=rankzIn-T(In+τA)-τTF0(z-1)ImC0.(1)
Ifz=1,(24)rankzIn-TIn+τA-τTF0z-1ImC0=rankIn-TIn+τA-τTFC0=rankIn-In-HCIn+τA-τTFC0=rankATFC0.(2)
Ifz≠1,(25)rankzIn-TIn+τA-τTF0z-1ImC0=rankzIn-TIn+τAC+m=rankzIn-In-HCIn+τAC+m=rankzIn-(In+τA)C+m.Thus, the pair (T¯A¯,C¯) is detectable if and only if(26)rankATFC0=n+m,rankzIn-(In+τA)C=nwhen z=-1 and |z|>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 d1(k) and measurement noise d2(k), we shall design augmented matrix L¯, which is composed of two unknown matrices L and K, such that the robust stability of the augmented error dynamics (16) is guaranteed with a prescribed H∞ 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 ω(k) have different influences on fault reconstruction performance. To guarantee better robustness of the designed nonlinear EPIO against external disturbance d1(k), measurement noise d2(k), and fault variation vector Δf(k), multiple attenuation levels in an H∞ 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 γi, i={1,2,3,4}, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and positive scalars δi, i={1,2,3,4} such that(27)-P¯1+δ1(P¯T¯A¯-Y¯C¯)τ1+δ2P¯T¯∗-P¯+δ3L¯gTL¯g0∗∗-δ3In+m<0,(28)-P¯P¯T¯A¯-Y¯C¯τP¯T¯τP¯T¯E¯-Y¯D2-P¯D¯hP¯I¯m∗-P¯+δ4L¯gTL¯g+CfTCf00000∗∗-δ4In+m0000∗∗∗-γ12Id000∗∗∗∗-γ22Is00∗∗∗∗∗-γ32Is0∗∗∗∗∗0-γ42Im<0,where L¯g=LgIn000m, E¯=E0, D¯h=HD20, and I¯m=0Im, then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can realize uniform ultimate boundedness of state-estimating error ex(k) and fault-reconstructing error ef(k) with an H∞ performance criterion:(29)efk2≤γ12d1k2+(γ22+γ32)d2(k)2+γ42Δf(k)2.Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Proof.
Consider the Lyapunov function candidateV(k)=e¯T(k)P¯e¯(k). The external disturbance d1(k) and measurement noise d2(k) are temporarily dropped; then, the difference ΔV(k)=V(k+1)-V(k) is derived as(30)ΔVk=e¯kΔΦ¯kTRτT¯A¯-L¯C¯TP¯T¯∗τ2T¯TP¯T¯e¯kΔΦ¯k+2e¯T(k)T¯A¯-L¯C¯TP¯I¯mΔf(k)+2τΔΦ¯TkT¯TP¯I¯mΔfk+ΔfTkI¯mTP¯I¯mΔfk,where R=(T¯A¯-L¯C¯)TP¯(T¯A¯-L¯C¯)-P¯.
Considering Lemma1, the following inequalities hold:(31)2e¯T(k)T¯A¯-L¯C¯TP¯I¯mΔf(k)≤δ1e¯T(k)T¯A¯-L¯C¯TP¯(T¯A¯-L¯C¯)e¯(k)+1δ1ΔfT(k)I¯mTP¯I¯mΔf(k),2τΔΦ¯TkT¯TP¯I¯mΔfk≤δ2τ2ΔΦ¯TkT¯TP¯T¯ΔΦ¯k+1δ2ΔfTkI¯mTP¯I¯mΔfk,where δ1>0 and δ2>0.
Substituting (17), (31) into (30) leads to(32)ΔVk≤e¯kΔΦ¯kTRτT¯A¯-L¯C¯TP¯T¯∗τ2T¯TP¯T¯e¯kΔΦ¯k-δ3Γ+δ1e¯T(k)T¯A¯-L¯C¯TP¯(T¯A¯-L¯C¯)e¯(k)+1δ1ΔfTkI¯mTP¯I¯mΔfk+δ2τ2ΔΦ¯TkT¯TP¯T¯ΔΦ¯k+1δ2ΔfTkI¯mTP¯I¯mΔfk+ΔfTkI¯mTP¯I¯mΔfk≤e¯kΔΦ¯kTR1+δ3L¯gTL¯gτT¯A¯-L¯C¯TP¯T¯∗1+δ2τ2T¯TP¯T¯-δ3In+m×e¯kΔΦ¯k+1+1δ1+1δ2ΔfTkI¯mTP¯I¯mΔfk≤ξTkΩξk+1+1δ1+1δ2ΔfTkI¯mTP¯I¯mΔfk,where ξ(k)=e¯(k)ΔΦ¯(k) and Ω=R1+δ3L¯gTL¯gτ(T¯A¯-L¯C¯)TP¯T¯∗(1+δ2)τ2T¯TP¯T¯-δ3In+m with R1=(1+δ1)(T¯A¯-L¯C¯)TP¯(T¯A¯-L¯C¯)-P¯ and δ3>0.
IfΩ<0, then we have(33)ΔVk≤-ϵe¯k2+ν,where ϵ=λmin(-Ω) and ν=(1+1/δ1+1/δ2)λmax(P)∥Δf(k)∥∞.
Further, the conditionΩ<0 can be rewritten as(34)1+δ1T¯A¯-L¯C¯TP¯τ1+δ2T¯TP¯P¯-11+δ1T¯A¯-L¯C¯TP¯τ1+δ2T¯TP¯T+-P+δ3L¯gTL¯g0∗-δ3In+m<0.By the Scular complement lemma [22], (34) is equivalent to (27).
Therefore, if (27) holds, then ΔV(k)<0 for ϵe¯(k)2>ν, which means that the trajectory of e¯(k) that is outside of the set Φ={e¯(k)∣e¯(k)2≤ν/ϵ} will converge to the set Φ according to the Lyapunov stability theory. Therefore, the uniform ultimate boundedness of the state-estimating error ex(k) and the fault-reconstructing error ef(k) can be guaranteed.
To guarantee that the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) is robust to d1(k), d2(k), and Δf(k), an H∞ performance index function is chosen as [23](35)J=∑k=0∞‍efTkefk-γ12d1Tkd1k-γ22d2Tkd2k-γ32d2T(k+1)d2(k+1)-γ42ΔfT(k)Δf(k).
Under zero-initial conditions, the following is obtained:(36)J≤∑k=0∞‍ΔV(k)+efT(k)ef(k)-γ12d1T(k)d1(k)-γ22d2Tkd2k-γ32d2Tk+1×d2k+1-γ42ΔfTkΔfk.
Considering external disturbanced1(k) and measurement noise d2(k), it follows from (17), (30), and (36) that(37)ΔVk+efTkefk-γ12d1Tkd1k-γ22d2Tkd2k-γ32d2Tk+1d2k+1-γ42ΔfTkΔfk≤e¯kΔΦ¯kTR+δ4L¯gTL¯gτT¯A¯-L¯C¯TP¯T¯∗τ2T¯TP¯T¯-δ4In+me¯kΔΦ¯k+2τe¯T(k)T¯A¯-L¯C¯TP¯T¯E¯d1(k)-2e¯T(k)T¯A¯-L¯C¯TP¯L¯D2d2(k)-2e¯T(k)T¯A¯-L¯C¯TP¯HD¯hd2(k+1)+2e¯T(k)T¯A¯-L¯C¯TP¯I¯mΔf(k)+2τ2ΔΦ¯T(k)T¯TP¯T¯E¯d1(k)-2τΔΦ¯TkT¯TP¯L¯D2d2k-2τΔΦ¯TkT¯TP¯HD¯hd2k+1+2τΔΦ¯T(k)T¯TP¯I¯mΔf(k)+τ2d1T(k)T¯E¯TP¯T¯E¯d1(k)-2τd1T(k)T¯E¯TP¯L¯D2d2(k)-2τd1T(k)T¯E¯TP¯HD¯hd2(k+1)+2τd1T(k)T¯E¯TP¯I¯mΔf(k)+d2T(k)L¯D2TP¯L¯D2d2(k)+2d2T(k)L¯D2TP¯HD¯hd2(k+1)-2d2T(k)L¯D2TP¯I¯mΔf(k)+d2T(k+1)HD¯hTP¯HD¯hd2k+1-2d2T(k+1)HD¯hTP¯I¯mΔf(k)+ΔfT(k)I¯mTP¯I¯mΔf(k)-γ12d1Tkd1k-γ22d2Tkd2k-γ32d2Tk+1d2k+1-γ42ΔfTkΔfk=e¯kΔΦ¯kd1kd2kd2k+1ΔfkTΠ11Π12Π13Π14Π15Π16∗Π22Π23Π24Π25Π26∗∗Π33Π34Π35Π36∗∗∗Π44Π45Π46∗∗∗∗Π55Π56∗∗∗∗∗Π66×e¯kΔΦ¯kd1kd2kd2k+1Δfk=ζTkΠζk,where(38)ζk=e¯T(k)ΔΦ¯T(k)d1T(k)d2T(k)d2T(k+1)ΔfT(k)T,Π=Π11Π12Π13Π14Π15Π16∗Π22Π23Π24Π25Π26∗∗Π33Π34Π35Π36∗∗∗Π44Π45Π46∗∗∗∗Π55Π56∗∗∗∗∗Π66with Π11=R+δ4L¯gTL¯g+CfTCf, Π12=τ(T¯A¯-L¯C¯)TP¯T¯, Π13=τ(T¯A¯-L¯C¯)TP¯T¯E¯, Π14=-(T¯A¯-L¯C¯)TP¯L¯D2, Π15=-(T¯A¯-L¯C¯)TP¯HD¯h, Π16=(T¯A¯-L¯C¯)TP¯I¯m, Π22=τ2T¯TP¯T¯-δ4In+m, Π23=τ2T¯TP¯T¯E¯, Π24=-τT¯TP¯L¯D2, Π25=-τT¯TP¯HD¯h, Π26=τT¯TP¯I¯m, Π33=τ2(T¯E¯)TP¯T¯E¯-γ12Id, Π34=-τ(T¯E¯)TP¯L¯D2, Π35=-τ(T¯E¯)TP¯HD¯h, Π36=τ(T¯E¯)TP¯I¯m, Π44=(L¯D2)TP¯L¯D2-γ22Is, Π45=(L¯D2)TP¯HD¯h, Π46=-(L¯D2)TP¯I¯m, Π55=(HD¯h)TP¯HD¯h-γ32Is, Π56=-(HD¯h)TP¯I¯m, and Π66=I¯mTP¯I¯m-γ42Im.
Further, the inequality,Π<0, can be transformed into(39)T¯A¯-L¯C¯TP¯τT¯TP¯τT¯E¯TP¯L¯TP¯HD¯hTP¯I¯mTP¯P¯-1T¯A¯-L¯C¯TP¯τT¯TP¯τT¯E¯TP¯L¯TP¯HD¯hTP¯I¯mTP¯T+-P¯+δ4L¯gTL¯g+CfTCf00000∗-δ4In+m0000∗∗-γ12Id000∗∗∗-γ22Is00∗∗∗∗-γ32Is0∗∗∗∗0-γ42Im<0.
Using the Schur complement lemma again, (39) is equivalent to (28), which means that, for all nonzero d1(k)∈L2[0,∞), d2(k)∈L2[0,∞), and Δf(k)∈L2[0,∞), one obtains J<0. Therefore, we can conclude from (35) that the H∞ performance criterion (29) is satisfied. This completes the proof.In Theorem5, multiple attenuation levels in the H∞ performance criterion (29) are prescribed for better restricting each component of the augmented disturbance ω(k) on fault reconstruction performance. If only a single H∞ attenuation level is considered for the augmented disturbance ω(k), 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 γ, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and positive scalars δi, i={1,2,3,4} satisfying (27) and(40)-P¯P¯T¯A¯-Y¯C¯τP¯T¯Y¯D¯2-P¯D¯1∗-P¯+δ4L¯gTL¯g+CfTCf00∗∗-δ4In+m0∗∗∗-γ2I2s+d+m<0,then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can realize the uniform ultimate boundedness of state-estimating error ex(k) and fault-reconstructing error ef(k) with an H∞ performance criterion: ef(k)≤γω(k). Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Proof.
The proof of Corollary6 is similar to that of Theorem 5; it is thus omitted here.Remark 7.
In Corollary6, the H∞ performance criterion with an attenuation level γ is adopted such that the designed nonlinear EPIO is robust to the augmented disturbance ω(k). 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 H∞ criterion with attenuation levels γi, ∀i∈{1,2,3,4} is adopted in Theorem 5 such that the impact of d1(k), d2(k), and Δf(k) on ef(k) can be effectively attenuated. Therefore, Theorem 5 is more flexible than Corollary 6. Compared with Corollary 6, one may obtain smaller H∞ attenuation levels using Theorem 5; that is, γ≥γi, ∀i∈{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 γ4 to restrict variation vector Δf(k) at the expense of the robustness against external disturbance d1(k) and measurement noise d2(k).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 γ1, γ2, and γ3, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and a positive scalar δ such that the following LMI holds:(41)-P¯P¯T¯A¯-Y¯C¯τP¯T¯τP¯T¯E¯-Y¯D2-P¯D¯h∗-P¯+δL¯gTL¯g+CfTCf0000∗∗-δIn+m000∗∗∗-γ12Id00∗∗∗∗-γ22Is0∗∗∗∗∗-γ32Is<0,where L¯g=LgIn000m, E¯=E0, and D¯h=HD20, then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can asymptotically reconstruct constant actuator faults, and fault-reconstructing error ef(k) satisfies the following inequality:(42)ef(k)2≤γ12d1(k)2+(γ22+γ32)d2(k)2.Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Corollary 9.
Consider the Euler-approximate model ((2a), (2b)). For a given positive scalar γ, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and a positive scalar δ such that the following LMI holds:(43)-P¯P¯T¯A¯-Y¯C¯τP¯T¯Y¯D¯23-P¯D¯13∗-P¯+δL¯gTL¯g+CfTCf00∗∗-δIn+m0∗∗∗-γ2I2s+d<0,where L¯g=LgIn000m, D¯13=-τTE0HD2000, and D¯23=0-D20, then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can asymptotically reconstruct constant actuator faults, and fault reconstruction error ef(k) satisfies that ef(k)≤γωd(k), where ωd(k)=d1T(k)d2T(k)d2T(k+1)T. Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Remark 10.
Theorem5 and Corollary 6 can guarantee the uniform ultimate boundedness of state estimation error ex(k) and fault reconstruction error ef(k) 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 Δf(k)≡0. 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 Theorem5, the design procedure of the proposed EPIO will be summarized as follows:Step 1.
Find the maximum number of columnsq of matrix D1 for which the condition rank(CD11)=rank(D11) is satisfied.Step 2.
Solve (13) to obtain T and H, and verify that rank(TF)=rank(F).Step 3.
Choose appropriate scalarsδi, i={1,2,3,4}; then, based on (27) and (40), H∞ performance level γ is computed using the Solvermincx in the LMI toolbox of Matlab.Step 4.
Chooseγi≤γ, i={1,2,3,4}; then, solving (27) and (28) using the Solverfeasp in the LMI toolbox of Matlab, matrices P¯ and Y¯ can be obtained.Step 5.
ComputeL¯=P¯-1Y¯, then obtain gain matrices L and K.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 intervalτ 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 periodτ if there exists a class-KL function β(·,·) such that, for any D>d>0 and compact sets X⊂Rn, U⊂Rm, we can find a number τ∗>0 with the property that, for all τ∈(0,τ∗],(44)x^(0)-x(0)≤D,x(k)∈X,u(k)∈U,∀k≥0imply(45)x^(k)-x(k)≤β(x^(0)-x(0),kτ)+d.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 τ if there exist a positive scalar σ1>1, a fixed positive-definite symmetric matrix Q1∈Rn×n, and a positive-definite symmetric matrix P1∈Rn×n such that(46)σ1TIn+τA-LCTP1(T(In+τA)-LC)-P1=-Q1,(47)λmin(Q1)=τ.Proof.
Consider the Lyapunov function candidate,V(k)=exT(k)P1ex(k); then,(48)Vk1-Vk2=exTk1P1exk1-exTk2P1exk2=exk1+exk2TP1[ex(k1)-ex(k2)]≤λmax(P1)ex(k1)+ex(k2)ex(k1)-ex(k2).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(49)∃M∈(0,∞),λmax(P1)ex(k1)+ex(k2)≤M.Thus,(50)V(k1)-V(k2)≤Mexk1-exk2,(51)ΔVk≤exT(k)TIn+τA-LCT×P1TIn+τA-LCexk-exTkP1exk+2τexT(k)TIn+τA-LCTP1TΔΦ(k)+2τexT(k)TIn+τA-LCTP1TFef(k)+τ2ΔΦTkTTP1TΔΦk+2τ2ΔΦTkTTP1TFefk+τ2efT(k)TFTP1(TF)ef(k).
With the help of Lemma1, the following inequalities can be easily derived:(52)2τexT(k)TIn+τA-LCTP1TΔΦ(k)≤ϵ1exT(k)TIn+τA-LCT×P1TIn+τA-LCexk+τ2ϵ1ΔΦTkTTP1TΔΦk,where ϵ1>0,(53)2τexT(k)TIn+τA-LCTP1TFef(k)≤ϵ2exT(k)TIn+τA-LCT×P1(T(In+τA)-LC)ex(k)+τ2ϵ2efTTFTP1TFefk,where ϵ2>0, and(54)2τΔΦTkTTP1TFefk≤ϵ3ΔΦTkTTP1TΔΦk+τ2ϵ3efTTFTP1TFefk,where ϵ3>0.
Substituting (52)–(54) into (51) leads to(55)ΔVk≤exT(k)×σ1TIn+τA-LCTP1TIn+τA-LC-P1×ex(k)+σ2τ2ΔΦTkTTP1TΔΦk+σ3τ2efT(k)TFTP1TFef(k)≤exTk×σ1TIn+τA-LCTP1TIn+τA-LC-P1×ex(k)+σ2τ2Lg2σ¯2TλmaxP1exk2+σ3τ2σ¯2TFλmaxP1efk2,where σ1=1+ϵ1+ϵ2, σ2=1+1/ϵ1+ϵ3, and σ3=1+1/ϵ2+1/ϵ3 with ϵi>0,i=1,2,3.
According to (46) and (47), (55) can be reorganized and simplified(56)ΔVkτ≤-exk2+σ2τLg2σ¯2TλmaxP1exk2+σ3τσ¯2(TF)λmax(P1)efk2.In addition, the following functions are defined:(57)α1(ex(k))≜λmin(P1)exk2,α2(exk)≜λmax(P1)exk2,α3(exk)≜exk2,ρ0(τ)≜τ,ν0(exk)≜σ2τ2Lg2σ¯2(T)λmax(P1)exk2,ν1(xk)≜0,ν2(u(k))≜0,ν3(efk)=σ3τσ¯2(TF)λmax(P1)efk2.
Therefore, the following results can be derived:(58)α1(ex(k))≤V(k)≤α2(ex(k)),Vk+1-Vkτ≤-α3exk+ρ0(τ)+ν2(u(k))+ν3(ef(k))ν0(ex(k))+ν1(x(k))+ν2(u(k))+ν3(ef(k)),where α1(·),α2(·),α3(·), and ρ0(·) are in class-K∞ and ν0(·),ν1(·), ν2(·), and ν3(·) are nondecreasing functions.
Based on the theorem and corollaries discussed in Subsection3.2, it can be known that fault reconstruction error ef(k) 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 τ. 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 τ if there exist positive-definite symmetric matrices P2∈Rn×n and Q2∈Rn×n, a matrix L∈Rn×p, and a positive scalar ϵ4, such that(59)TIn+τA-LCTP2(T(In+τA)-LC)-P2=-Q2,(60)τ≤τ∗=λmax(Q2)-ϵ4[λmax(P2)+λmax(Q2)]2Lgσ¯(T)λmax(P2)1+λmax(Q2)/λmax(P2).Proof.
The proof of Theorem14 is similar to that of Theorem  2 in [18]; it is thus omitted here.Remark 15.
Theorem5 and Corollaries 6–9 imply that observer gain matrices L and K 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.


## 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:(5a)zk+1=T(In+τA)x^(k)+τTΦ(x^(k),k)+τTBu(k)+Lyk-y^k+τTFf^k,(5b)x^k=zk+Hyk,(5c)y^k=Cx^k,(5d)f^k+1=f^k+Kyk-y^k,where z(k)∈Rn, x^(k)∈Rn, and y^(k)∈Rp; they represent the observer state vector, the estimated state vector, and measurement output vector, respectively. Fault reconstruction signal f^(k) is recursively updated by both itself and output estimation errors at the sampling time k-1. Here, T, H, L, and K are gain matrices with appropriate dimensions to be determined later.Definingex(k):=x(k)-x^(k), ey(k):=y(k)-y^(k), and ef(k):=f(k)-f^(k) and letting T+HC=In, estimation error dynamics can be easily obtained as(6a)exk+1=TIn+τA-LCexk+τTΔΦk+τTD1d1(k)-LD2d2(k)-HD2d2(k+1)+τTFef(k)(6b)eyk=Cexk+D2d2k,where ΔΦ(k)=Φ(x(k),k)-Φ(x^(k),k).Since the nonlinear functionΦ(x(k),k) satisfies the Lipschitz condition, then(7)ΔΦT(k)ΔΦ(k)-Lg2exT(k)ex(k)≤0.Inspired by disturbance-decoupling principle of UIOs [20, 21], we will design gain matrix T such that state estimation error can be partially decoupled from external disturbances. As such, state error dynamics (6a) can be reorganized as(8)exk+1=TIn+τA-LCexk+τTΔΦk+τTD11D12︸D1×d11Tkd12TkT︸d1k-LD2d2k-HD2d2k+1+τTFefk,where d11∈Rq and q is the maximum number of columns of the matrix D1 for which the condition, rank(CD11)=rank(D11), is satisfied. Matrix D11 is composed of q columns of the matrix D1 while matrix D12 represents the remaining part of the matrix D1.MatrixT is selected such that the condition, TD11=0, holds. Herein, we define matrix E as 0qD12. Hence, state error dynamics (6a) can be further manipulated as(9)exk+1=TIn+τA-LCexk+τTΔΦk+τTEd1k-LD2d2k-HD2d2(k+1)+τTFef(k).Considering (5d), fault-reconstructing errors ef(k) can be expressed in the following equation:(10)efk+1=efk-KCexk-KD2d2k+Δfk,where fault variation vector Δf(k):=f(k+1)-f(k). If constant actuator faults are considered in system ((1a), (1b)), then Δf(k)≡0. Therefore, one obtains(11)efk+1=efk-KCexk-KD2d2k.Remark 2.
Equations (8) and (9) display that the designed EPIO is only partially decoupled from external disturbance d1(k). 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 T and H should be designed such that matrix equations, T+HC=In, and TD11=0, are satisfied simultaneously. Augmented matrix equation composing of these two constraints can be written as(12)THInD11C0=In0.Therefore, a general solution of (12) can be determined by(13)TH=In0InD11C0†.In addition, to guarantee actuator-fault detectability by the proposed EPIO, the constraint, rank(TF)=rank(F), should be also satisfied.According to (9) and (10), the following augmented error dynamics can be constructed as(14)exk+1efk+1=T(In+τA)-LCτTF-KCImex(k)ef(k)+τTΔΦ(k)0+τTE-LD2-HD200-KD20Imd1(k)d2(k)d2(k+1)Δf(k).Denote(15)e¯k=exkefk,ωk=d1kd2kd2k+1Δfk,T¯=T00Im,A¯=In+τAτF0Im,ΔΦ¯k=ΔΦk0,D¯1=-τTE0HD20000-Im,L¯=LK,C¯=C0,D¯2=0-D200.Further, augmented error dynamics (14) can be reorganized into a compact form(16)e¯k+1=T¯A¯-L¯C¯e¯k+τT¯ΔΦ¯k+L¯D¯2-D¯1ω(k)ey(k)=C¯e¯(k)-D¯2ω(k).Besides, fault reconstruction error ef(k)=Cfe¯(k), where Cf=0Im.For nonlinear functionΔΦ¯(k), the following inequality holds:(17)Γ=ΔΦ¯T(k)ΔΦ¯(k)-e¯T(k)L¯gTL¯ge¯(k)≤0,where L¯g=LgIn000m.It is noticed that augmented error system (16) can be treated as an observation-error equation of the following nonlinear system:(18)x¯(k+1)=T¯A¯x¯(k)+τT¯Φ¯(x¯(k),k)+D¯1ω(k)yk=C¯x¯k-D¯2ωk,where (19)x¯k=xTkfTkT,Φ¯x¯k,k=ΦTxk,k0T.An augmented state observer can be established for (18) as follows:(20)x¯^k+1=T¯A¯x¯^k+τT¯Φ¯x¯^k,k+L¯yk-y^k,y^k=C¯x¯^k.Remark 3.
If both the nonlinear termΔΦ¯(k) and augmented disturbance ω(k) are not considered in (16), we can obtain(21)e¯k+1=T¯A¯-L¯C¯e¯k,ey(k)=C¯e¯(k).The linear error dynamics (21) is asymptotically stable if and only if the pair (T¯A¯,C¯) is detectable. It is equivalent to(22)rankzIn+m-T¯A¯C¯=n+m,∀z∈C,z≥1,and one obtains(23)rankzIn+m-T¯A¯C¯=rankzIn-T(In+τA)-τTF0(z-1)ImC0.(1)
Ifz=1,(24)rankzIn-TIn+τA-τTF0z-1ImC0=rankIn-TIn+τA-τTFC0=rankIn-In-HCIn+τA-τTFC0=rankATFC0.(2)
Ifz≠1,(25)rankzIn-TIn+τA-τTF0z-1ImC0=rankzIn-TIn+τAC+m=rankzIn-In-HCIn+τAC+m=rankzIn-(In+τA)C+m.Thus, the pair (T¯A¯,C¯) is detectable if and only if(26)rankATFC0=n+m,rankzIn-(In+τA)C=nwhen z=-1 and |z|>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 d1(k) and measurement noise d2(k), we shall design augmented matrix L¯, which is composed of two unknown matrices L and K, such that the robust stability of the augmented error dynamics (16) is guaranteed with a prescribed H∞ 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 ω(k) have different influences on fault reconstruction performance. To guarantee better robustness of the designed nonlinear EPIO against external disturbance d1(k), measurement noise d2(k), and fault variation vector Δf(k), multiple attenuation levels in an H∞ 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 γi, i={1,2,3,4}, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and positive scalars δi, i={1,2,3,4} such that(27)-P¯1+δ1(P¯T¯A¯-Y¯C¯)τ1+δ2P¯T¯∗-P¯+δ3L¯gTL¯g0∗∗-δ3In+m<0,(28)-P¯P¯T¯A¯-Y¯C¯τP¯T¯τP¯T¯E¯-Y¯D2-P¯D¯hP¯I¯m∗-P¯+δ4L¯gTL¯g+CfTCf00000∗∗-δ4In+m0000∗∗∗-γ12Id000∗∗∗∗-γ22Is00∗∗∗∗∗-γ32Is0∗∗∗∗∗0-γ42Im<0,where L¯g=LgIn000m, E¯=E0, D¯h=HD20, and I¯m=0Im, then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can realize uniform ultimate boundedness of state-estimating error ex(k) and fault-reconstructing error ef(k) with an H∞ performance criterion:(29)efk2≤γ12d1k2+(γ22+γ32)d2(k)2+γ42Δf(k)2.Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Proof.
Consider the Lyapunov function candidateV(k)=e¯T(k)P¯e¯(k). The external disturbance d1(k) and measurement noise d2(k) are temporarily dropped; then, the difference ΔV(k)=V(k+1)-V(k) is derived as(30)ΔVk=e¯kΔΦ¯kTRτT¯A¯-L¯C¯TP¯T¯∗τ2T¯TP¯T¯e¯kΔΦ¯k+2e¯T(k)T¯A¯-L¯C¯TP¯I¯mΔf(k)+2τΔΦ¯TkT¯TP¯I¯mΔfk+ΔfTkI¯mTP¯I¯mΔfk,where R=(T¯A¯-L¯C¯)TP¯(T¯A¯-L¯C¯)-P¯.
Considering Lemma1, the following inequalities hold:(31)2e¯T(k)T¯A¯-L¯C¯TP¯I¯mΔf(k)≤δ1e¯T(k)T¯A¯-L¯C¯TP¯(T¯A¯-L¯C¯)e¯(k)+1δ1ΔfT(k)I¯mTP¯I¯mΔf(k),2τΔΦ¯TkT¯TP¯I¯mΔfk≤δ2τ2ΔΦ¯TkT¯TP¯T¯ΔΦ¯k+1δ2ΔfTkI¯mTP¯I¯mΔfk,where δ1>0 and δ2>0.
Substituting (17), (31) into (30) leads to(32)ΔVk≤e¯kΔΦ¯kTRτT¯A¯-L¯C¯TP¯T¯∗τ2T¯TP¯T¯e¯kΔΦ¯k-δ3Γ+δ1e¯T(k)T¯A¯-L¯C¯TP¯(T¯A¯-L¯C¯)e¯(k)+1δ1ΔfTkI¯mTP¯I¯mΔfk+δ2τ2ΔΦ¯TkT¯TP¯T¯ΔΦ¯k+1δ2ΔfTkI¯mTP¯I¯mΔfk+ΔfTkI¯mTP¯I¯mΔfk≤e¯kΔΦ¯kTR1+δ3L¯gTL¯gτT¯A¯-L¯C¯TP¯T¯∗1+δ2τ2T¯TP¯T¯-δ3In+m×e¯kΔΦ¯k+1+1δ1+1δ2ΔfTkI¯mTP¯I¯mΔfk≤ξTkΩξk+1+1δ1+1δ2ΔfTkI¯mTP¯I¯mΔfk,where ξ(k)=e¯(k)ΔΦ¯(k) and Ω=R1+δ3L¯gTL¯gτ(T¯A¯-L¯C¯)TP¯T¯∗(1+δ2)τ2T¯TP¯T¯-δ3In+m with R1=(1+δ1)(T¯A¯-L¯C¯)TP¯(T¯A¯-L¯C¯)-P¯ and δ3>0.
IfΩ<0, then we have(33)ΔVk≤-ϵe¯k2+ν,where ϵ=λmin(-Ω) and ν=(1+1/δ1+1/δ2)λmax(P)∥Δf(k)∥∞.
Further, the conditionΩ<0 can be rewritten as(34)1+δ1T¯A¯-L¯C¯TP¯τ1+δ2T¯TP¯P¯-11+δ1T¯A¯-L¯C¯TP¯τ1+δ2T¯TP¯T+-P+δ3L¯gTL¯g0∗-δ3In+m<0.By the Scular complement lemma [22], (34) is equivalent to (27).
Therefore, if (27) holds, then ΔV(k)<0 for ϵe¯(k)2>ν, which means that the trajectory of e¯(k) that is outside of the set Φ={e¯(k)∣e¯(k)2≤ν/ϵ} will converge to the set Φ according to the Lyapunov stability theory. Therefore, the uniform ultimate boundedness of the state-estimating error ex(k) and the fault-reconstructing error ef(k) can be guaranteed.
To guarantee that the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) is robust to d1(k), d2(k), and Δf(k), an H∞ performance index function is chosen as [23](35)J=∑k=0∞‍efTkefk-γ12d1Tkd1k-γ22d2Tkd2k-γ32d2T(k+1)d2(k+1)-γ42ΔfT(k)Δf(k).
Under zero-initial conditions, the following is obtained:(36)J≤∑k=0∞‍ΔV(k)+efT(k)ef(k)-γ12d1T(k)d1(k)-γ22d2Tkd2k-γ32d2Tk+1×d2k+1-γ42ΔfTkΔfk.
Considering external disturbanced1(k) and measurement noise d2(k), it follows from (17), (30), and (36) that(37)ΔVk+efTkefk-γ12d1Tkd1k-γ22d2Tkd2k-γ32d2Tk+1d2k+1-γ42ΔfTkΔfk≤e¯kΔΦ¯kTR+δ4L¯gTL¯gτT¯A¯-L¯C¯TP¯T¯∗τ2T¯TP¯T¯-δ4In+me¯kΔΦ¯k+2τe¯T(k)T¯A¯-L¯C¯TP¯T¯E¯d1(k)-2e¯T(k)T¯A¯-L¯C¯TP¯L¯D2d2(k)-2e¯T(k)T¯A¯-L¯C¯TP¯HD¯hd2(k+1)+2e¯T(k)T¯A¯-L¯C¯TP¯I¯mΔf(k)+2τ2ΔΦ¯T(k)T¯TP¯T¯E¯d1(k)-2τΔΦ¯TkT¯TP¯L¯D2d2k-2τΔΦ¯TkT¯TP¯HD¯hd2k+1+2τΔΦ¯T(k)T¯TP¯I¯mΔf(k)+τ2d1T(k)T¯E¯TP¯T¯E¯d1(k)-2τd1T(k)T¯E¯TP¯L¯D2d2(k)-2τd1T(k)T¯E¯TP¯HD¯hd2(k+1)+2τd1T(k)T¯E¯TP¯I¯mΔf(k)+d2T(k)L¯D2TP¯L¯D2d2(k)+2d2T(k)L¯D2TP¯HD¯hd2(k+1)-2d2T(k)L¯D2TP¯I¯mΔf(k)+d2T(k+1)HD¯hTP¯HD¯hd2k+1-2d2T(k+1)HD¯hTP¯I¯mΔf(k)+ΔfT(k)I¯mTP¯I¯mΔf(k)-γ12d1Tkd1k-γ22d2Tkd2k-γ32d2Tk+1d2k+1-γ42ΔfTkΔfk=e¯kΔΦ¯kd1kd2kd2k+1ΔfkTΠ11Π12Π13Π14Π15Π16∗Π22Π23Π24Π25Π26∗∗Π33Π34Π35Π36∗∗∗Π44Π45Π46∗∗∗∗Π55Π56∗∗∗∗∗Π66×e¯kΔΦ¯kd1kd2kd2k+1Δfk=ζTkΠζk,where(38)ζk=e¯T(k)ΔΦ¯T(k)d1T(k)d2T(k)d2T(k+1)ΔfT(k)T,Π=Π11Π12Π13Π14Π15Π16∗Π22Π23Π24Π25Π26∗∗Π33Π34Π35Π36∗∗∗Π44Π45Π46∗∗∗∗Π55Π56∗∗∗∗∗Π66with Π11=R+δ4L¯gTL¯g+CfTCf, Π12=τ(T¯A¯-L¯C¯)TP¯T¯, Π13=τ(T¯A¯-L¯C¯)TP¯T¯E¯, Π14=-(T¯A¯-L¯C¯)TP¯L¯D2, Π15=-(T¯A¯-L¯C¯)TP¯HD¯h, Π16=(T¯A¯-L¯C¯)TP¯I¯m, Π22=τ2T¯TP¯T¯-δ4In+m, Π23=τ2T¯TP¯T¯E¯, Π24=-τT¯TP¯L¯D2, Π25=-τT¯TP¯HD¯h, Π26=τT¯TP¯I¯m, Π33=τ2(T¯E¯)TP¯T¯E¯-γ12Id, Π34=-τ(T¯E¯)TP¯L¯D2, Π35=-τ(T¯E¯)TP¯HD¯h, Π36=τ(T¯E¯)TP¯I¯m, Π44=(L¯D2)TP¯L¯D2-γ22Is, Π45=(L¯D2)TP¯HD¯h, Π46=-(L¯D2)TP¯I¯m, Π55=(HD¯h)TP¯HD¯h-γ32Is, Π56=-(HD¯h)TP¯I¯m, and Π66=I¯mTP¯I¯m-γ42Im.
Further, the inequality,Π<0, can be transformed into(39)T¯A¯-L¯C¯TP¯τT¯TP¯τT¯E¯TP¯L¯TP¯HD¯hTP¯I¯mTP¯P¯-1T¯A¯-L¯C¯TP¯τT¯TP¯τT¯E¯TP¯L¯TP¯HD¯hTP¯I¯mTP¯T+-P¯+δ4L¯gTL¯g+CfTCf00000∗-δ4In+m0000∗∗-γ12Id000∗∗∗-γ22Is00∗∗∗∗-γ32Is0∗∗∗∗0-γ42Im<0.
Using the Schur complement lemma again, (39) is equivalent to (28), which means that, for all nonzero d1(k)∈L2[0,∞), d2(k)∈L2[0,∞), and Δf(k)∈L2[0,∞), one obtains J<0. Therefore, we can conclude from (35) that the H∞ performance criterion (29) is satisfied. This completes the proof.In Theorem5, multiple attenuation levels in the H∞ performance criterion (29) are prescribed for better restricting each component of the augmented disturbance ω(k) on fault reconstruction performance. If only a single H∞ attenuation level is considered for the augmented disturbance ω(k), 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 γ, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and positive scalars δi, i={1,2,3,4} satisfying (27) and(40)-P¯P¯T¯A¯-Y¯C¯τP¯T¯Y¯D¯2-P¯D¯1∗-P¯+δ4L¯gTL¯g+CfTCf00∗∗-δ4In+m0∗∗∗-γ2I2s+d+m<0,then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can realize the uniform ultimate boundedness of state-estimating error ex(k) and fault-reconstructing error ef(k) with an H∞ performance criterion: ef(k)≤γω(k). Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Proof.
The proof of Corollary6 is similar to that of Theorem 5; it is thus omitted here.Remark 7.
In Corollary6, the H∞ performance criterion with an attenuation level γ is adopted such that the designed nonlinear EPIO is robust to the augmented disturbance ω(k). 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 H∞ criterion with attenuation levels γi, ∀i∈{1,2,3,4} is adopted in Theorem 5 such that the impact of d1(k), d2(k), and Δf(k) on ef(k) can be effectively attenuated. Therefore, Theorem 5 is more flexible than Corollary 6. Compared with Corollary 6, one may obtain smaller H∞ attenuation levels using Theorem 5; that is, γ≥γi, ∀i∈{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 γ4 to restrict variation vector Δf(k) at the expense of the robustness against external disturbance d1(k) and measurement noise d2(k).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 γ1, γ2, and γ3, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and a positive scalar δ such that the following LMI holds:(41)-P¯P¯T¯A¯-Y¯C¯τP¯T¯τP¯T¯E¯-Y¯D2-P¯D¯h∗-P¯+δL¯gTL¯g+CfTCf0000∗∗-δIn+m000∗∗∗-γ12Id00∗∗∗∗-γ22Is0∗∗∗∗∗-γ32Is<0,where L¯g=LgIn000m, E¯=E0, and D¯h=HD20, then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can asymptotically reconstruct constant actuator faults, and fault-reconstructing error ef(k) satisfies the following inequality:(42)ef(k)2≤γ12d1(k)2+(γ22+γ32)d2(k)2.Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Corollary 9.
Consider the Euler-approximate model ((2a), (2b)). For a given positive scalar γ, if there exist a positive-definite symmetric matrix P¯∈R(n+m)×(n+m), a matrix Y¯∈R(n+m)×p, and a positive scalar δ such that the following LMI holds:(43)-P¯P¯T¯A¯-Y¯C¯τP¯T¯Y¯D¯23-P¯D¯13∗-P¯+δL¯gTL¯g+CfTCf00∗∗-δIn+m0∗∗∗-γ2I2s+d<0,where L¯g=LgIn000m, D¯13=-τTE0HD2000, and D¯23=0-D20, then the proposed nonlinear EPIO ((5a), (5b), (5c), and (5d)) can asymptotically reconstruct constant actuator faults, and fault reconstruction error ef(k) satisfies that ef(k)≤γωd(k), where ωd(k)=d1T(k)d2T(k)d2T(k+1)T. Moreover, augmented matrix L¯ can be determined by L¯=P¯-1Y¯.Remark 10.
Theorem5 and Corollary 6 can guarantee the uniform ultimate boundedness of state estimation error ex(k) and fault reconstruction error ef(k) 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 Δf(k)≡0. 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 Theorem5, the design procedure of the proposed EPIO will be summarized as follows:Step 1.
Find the maximum number of columnsq of matrix D1 for which the condition rank(CD11)=rank(D11) is satisfied.Step 2.
Solve (13) to obtain T and H, and verify that rank(TF)=rank(F).Step 3.
Choose appropriate scalarsδi, i={1,2,3,4}; then, based on (27) and (40), H∞ performance level γ is computed using the Solvermincx in the LMI toolbox of Matlab.Step 4.
Chooseγi≤γ, i={1,2,3,4}; then, solving (27) and (28) using the Solverfeasp in the LMI toolbox of Matlab, matrices P¯ and Y¯ can be obtained.Step 5.
ComputeL¯=P¯-1Y¯, then obtain gain matrices L and K.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 intervalτ 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 periodτ if there exists a class-KL function β(·,·) such that, for any D>d>0 and compact sets X⊂Rn, U⊂Rm, we can find a number τ∗>0 with the property that, for all τ∈(0,τ∗],(44)x^(0)-x(0)≤D,x(k)∈X,u(k)∈U,∀k≥0imply(45)x^(k)-x(k)≤β(x^(0)-x(0),kτ)+d.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 τ if there exist a positive scalar σ1>1, a fixed positive-definite symmetric matrix Q1∈Rn×n, and a positive-definite symmetric matrix P1∈Rn×n such that(46)σ1TIn+τA-LCTP1(T(In+τA)-LC)-P1=-Q1,(47)λmin(Q1)=τ.Proof.
Consider the Lyapunov function candidate,V(k)=exT(k)P1ex(k); then,(48)Vk1-Vk2=exTk1P1exk1-exTk2P1exk2=exk1+exk2TP1[ex(k1)-ex(k2)]≤λmax(P1)ex(k1)+ex(k2)ex(k1)-ex(k2).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(49)∃M∈(0,∞),λmax(P1)ex(k1)+ex(k2)≤M.Thus,(50)V(k1)-V(k2)≤Mexk1-exk2,(51)ΔVk≤exT(k)TIn+τA-LCT×P1TIn+τA-LCexk-exTkP1exk+2τexT(k)TIn+τA-LCTP1TΔΦ(k)+2τexT(k)TIn+τA-LCTP1TFef(k)+τ2ΔΦTkTTP1TΔΦk+2τ2ΔΦTkTTP1TFefk+τ2efT(k)TFTP1(TF)ef(k).
With the help of Lemma1, the following inequalities can be easily derived:(52)2τexT(k)TIn+τA-LCTP1TΔΦ(k)≤ϵ1exT(k)TIn+τA-LCT×P1TIn+τA-LCexk+τ2ϵ1ΔΦTkTTP1TΔΦk,where ϵ1>0,(53)2τexT(k)TIn+τA-LCTP1TFef(k)≤ϵ2exT(k)TIn+τA-LCT×P1(T(In+τA)-LC)ex(k)+τ2ϵ2efTTFTP1TFefk,where ϵ2>0, and(54)2τΔΦTkTTP1TFefk≤ϵ3ΔΦTkTTP1TΔΦk+τ2ϵ3efTTFTP1TFefk,where ϵ3>0.
Substituting (52)–(54) into (51) leads to(55)ΔVk≤exT(k)×σ1TIn+τA-LCTP1TIn+τA-LC-P1×ex(k)+σ2τ2ΔΦTkTTP1TΔΦk+σ3τ2efT(k)TFTP1TFef(k)≤exTk×σ1TIn+τA-LCTP1TIn+τA-LC-P1×ex(k)+σ2τ2Lg2σ¯2TλmaxP1exk2+σ3τ2σ¯2TFλmaxP1efk2,where σ1=1+ϵ1+ϵ2, σ2=1+1/ϵ1+ϵ3, and σ3=1+1/ϵ2+1/ϵ3 with ϵi>0,i=1,2,3.
According to (46) and (47), (55) can be reorganized and simplified(56)ΔVkτ≤-exk2+σ2τLg2σ¯2TλmaxP1exk2+σ3τσ¯2(TF)λmax(P1)efk2.In addition, the following functions are defined:(57)α1(ex(k))≜λmin(P1)exk2,α2(exk)≜λmax(P1)exk2,α3(exk)≜exk2,ρ0(τ)≜τ,ν0(exk)≜σ2τ2Lg2σ¯2(T)λmax(P1)exk2,ν1(xk)≜0,ν2(u(k))≜0,ν3(efk)=σ3τσ¯2(TF)λmax(P1)efk2.
Therefore, the following results can be derived:(58)α1(ex(k))≤V(k)≤α2(ex(k)),Vk+1-Vkτ≤-α3exk+ρ0(τ)+ν2(u(k))+ν3(ef(k))ν0(ex(k))+ν1(x(k))+ν2(u(k))+ν3(ef(k)),where α1(·),α2(·),α3(·), and ρ0(·) are in class-K∞ and ν0(·),ν1(·), ν2(·), and ν3(·) are nondecreasing functions.
Based on the theorem and corollaries discussed in Subsection3.2, it can be known that fault reconstruction error ef(k) 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 τ. 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 τ if there exist positive-definite symmetric matrices P2∈Rn×n and Q2∈Rn×n, a matrix L∈Rn×p, and a positive scalar ϵ4, such that(59)TIn+τA-LCTP2(T(In+τA)-LC)-P2=-Q2,(60)τ≤τ∗=λmax(Q2)-ϵ4[λmax(P2)+λmax(Q2)]2Lgσ¯(T)λmax(P2)1+λmax(Q2)/λmax(P2).Proof.
The proof of Theorem14 is similar to that of Theorem  2 in [18]; it is thus omitted here.Remark 15.
Theorem5 and Corollaries 6–9 imply that observer gain matrices L and K 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 (61)A=0100-48.6-1.2548.60000119.50-19.5-1.325,Φxt,t=000-0.333sin⁡x3t,B=021.6000,C=100001000010,D1=0.1000.10000,D2=0.1I3,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 F=B.The Lipschitz constantLg for the system ((1a), (1b)) is selected as 0.333 and the sampling period τ is selected as 0.01s. 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 matricesT and H are chosen as follows: (62)T=000000.500000.500001.0,H=1.00000.50000.5000.Lettingδi=0.1,∀i={1,2,3,4}, and solving (27) and (40) in Corollary 6 using the Solvermincx in the LMI toolbox of Matlab, one can obtain the H∞ attenuation level γ=2.0230. Further, choosing γ1=0.5, γ2=2.0230, γ3=2.0230, and γ4=2.0230 and solving (27) and (28) in Theorem 5 using the LMI toolbox of Matlab, a feasible solution for gain matrices L and K can be obtained as (63)L=0.01740.00140.19510.00141.64010.03590.0358-0.00300.32991.3353-0.28608.8951,K=-0.006710.6257-0.4187T.Therefore, the EPIO designed using Theorem 5 has better robustness against external disturbances and measurement noises.ChoosingQ1=0.01I4 and σ1=1.01, matrix P1 can be obtained by solving (46): (64)P1=0.0961-0.01191.0957-0.0976-0.01190.0020-0.16110.01641.0957-0.161115.0770-1.3221-0.09760.0164-1.32210.1317.Further, choosingϵ4=0.01 and solving (59) lead to (65)P2=102×3.5324-0.00810.2805-0.0369-0.00810.0006-0.03740.00470.2805-0.03746.6747-0.3458-0.03690.0047-0.34580.0376,Q2=102×3.5024-0.0044-0.0268-0.0077-0.00440.00010.00690.0006-0.02680.00692.34260.0201-0.00770.00060.02010.0042,and τ∗=0.6162s, which is greater than 0.01s. We can now check that the designed observer gain matrices and the sampling period 0.01s 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 0.01s.According to Corollary9, let δ=0.1; then, (43) can be solved to obtain γ=1.3007. Further, choosing γ1=0.3, γ2=1.3007, and γ3=1.3007, and solving (41) in Corollary 8 leads to (66)L=-0.0167-0.1111-0.0104-0.11340.38380.00230.027660.17530.58990.45781.516814.4412,K=0.01210.6371-0.1017T.Using Theorems 13 and 14, it is easily checked that the EPIO designed using Corollary 8 is also semiglobal practical convergent in 0.01s.In the simulation, the control input signal is assigned asuk=0.05sin5k; the initial values of the estimated states are assumed as x^0=[-0.10.5-0.10.2]T; external disturbance d1(k) and measurement noise d2(k) are chosen as 0.001sin(10k) 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(1)
a constant fault that is described as(67)f(k)=0.22s≤kτ≤6s0others.(2)
a time-varying fault that is described as(68)f(k)=0.05sin(5kτ)+0.05sin(3kτ)+0.06sin(2kτ)+0.2,kτ≥2s0others.Using the designed EPIO via Theorem5, 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 Figures 5 and 6 while constant-fault reconstruction signal by the EPIO designed using Corollary 8 is illustrated in Figure 7. The above figures reveal that the proposed EPIO can accurately reconstruct actuator faults. All of simulation results claim that the EPIO-based fault-reconstructing method is effective for Lipschitz nonlinear systems with Euler-approximate models.Figure 1
Actual system statex1(t) and its estimate.Figure 2
Actual system statex2(t) and its estimate.Figure 3
Actual system statex3(t) and its estimate.Figure 4
Actual system statex4(t) and its estimate.Figure 5
Reconstruction signal of a constant fault via the EPIO designed using Theorem5.Figure 6
Reconstruction signal of a time-varying fault via the EPIO designed using Theorem5.Figure 7
Reconstruction signal of a constant fault via the EPIO designed using Corollary8.


## 5. 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 disturbance-decoupling 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 continuous-time model, semiglobal practical convergence of the EPIO is proved. Another advantage of the proposed approach is that a systematic observer synthesis withH∞ 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.



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*Source: 741702-2015-07-26.xml*