# Sliding Mode Observers Based-Simultaneous Actuator and Sensor Faults Estimation for Linear Parameter Varying Systems

Authors: Ali Ben Brahim; Slim Dhahri; Fayçal Ben Hmida; Anis Sallami
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper proposes a scheme to estimate actuator and sensor faults simultaneously for a class of linear parameter varying system expressed in polytopic structure where its parameters evolve in the hypercube domain. Transformed coordinate system design is adopted to decouple faults in actuators and sensors during the course of the system’s operation coincidentally, and then two polytopic subsystems are constructed. The first subsystem includes the effect of actuator faults but is free from sensor faults and the second one is affected only by sensor faults. The main contribution is to conceive two polytopic sliding mode observers in order to estimate the system states and actuator and sensor faults at the same time. Meanwhile, in linear matrix inequality optimization formalism, sufficient conditions are derived withH∞ performances to guarantee the stability of estimation error and to minimize the effect of disturbances. Therefore, all parameters of observers can be designed by solving these conditions. Finally, simulation results are given to illustrate the effectiveness of the proposed simultaneous actuator and sensor faults estimation.


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## Body

## 1. Introduction

Fault disturbs the normal system operations, thus causing an unacceptable deterioration of these performances or leading to wrong dangerous situations. Fault can occur in any part of the system such as actuators and sensors. Sensor fault degrades the feedback system performances and so touches the control stability. Similarly, actuator fault may lead to minimizing the performances of design controllers or even the overall system execution. To this end, determining the size, location, and dynamic behavior of the fault along a system trajectories becomes a powerful alternative active research. Indeed, fault estimation is considered a major problem in the modern control theory that received a considerable amount of attention during the past few years. In the context of actuator faults estimation, constructing diagnosis model in order to estimate faults is not possible if sensor faults occur simultaneously. The same difficulty is present when trying to estimate sensor faults. Several design methods have been developed in a precise and effective way when actuator and sensor faults estimation is divided into two steps.Step 1.
Actuator faults estimation is proven without considering sensor faults [1–11].Step 2.
Sensor faults estimation is solved without considering actuator faults [12–19].Nevertheless, in practical systems, it is often the case when actuator and sensor faults occur simultaneously. In this framework, simultaneous faults estimation is highly important. So far, [20] considers the problem of simultaneous actuator and sensor fault estimation for Lipschitz nonlinear systems. The idea of this work based on the transformation of sensor fault obtained an augmented actuator fault vector. In [21], the authors propose simultaneously estimating actuator and sensor faults study. Two subsystems are constructed where each one contains a particular class of faults (actuator or sensor). More recently, [22, 23] consider the problem of robust simultaneous actuator and sensor faults estimation study for a class of uncertain Takagi-Sugeno nonlinear system with unmeasurable premise variables. The main contribution is to develop a sliding mode observer (SMO) with two discontinuous terms to solve the problem of simultaneous faults.On the other hand, very interesting approaches have represented actual physical systems under linear parameter varying (LPV) representation expressed in polytopic structure where its parameters evolve in the hypercube domain. Roughly speaking, the feature is to understand the overall system behavior by a set of local linear models scheduled by convex weighting functions. Taking the polytopic LPV representation, several attempts have been oriented to the diagnosis of nonlinear systems; see, for instance, [24–28]. More recently, [29] addresses the problem of simultaneous actuator and sensor faults estimation for the polytopic LPV systems. The idea is based on the descriptor approach by extending the sensor faults as an auxiliary states vector. This approach consists in transforming a standard LPV system into a descriptor form to estimate sensor faults. Furthermore, this approach is not applicable directly to real descriptor systems; the fact is that these systems have more complicated structure and usually have three types of modes: dynamic, impulse, and static, which are not considered in this approach. Reference [30] investigates simultaneous actuator and sensor faults estimation for polytopic LPV systems by introducing a filter in order to transform sensor faults as a “pseudoactuator” faults. Appropriate filtering of the original system output yields “augmented fictitious systems” with augmented actuator faults comprising actuator and sensor faults. Nevertheless, this leads to some conservatism. This approach is based on some rank restrictive assumptions, which must be verified for both original and augmented systems to ensure the observer-based fault estimation.Regarding the fact that simultaneous actuator and sensor faults scenario has not yet been fully tackled, we will develop robust fault estimation scheme for LPV system subject to both faults affecting actuators and sensor faults and disturbances in the present study. It should be noticed that the LPV system is expressed in polytopic structure where its parameters evolve in the hypercube domain. Motivated by the success of SMO based methods of fault estimation, the present study focuses on developing effective and robust simultaneous actuator and sensor faults estimation for a class of LPV systems. The main contributions are as follows:(i)
Designing state and output coordinate transformation such that actuator and sensor faults are completely separated, hence two polytopic LPV subsystems being introduced where each one contains only a particular class of faults (actuator or sensor).(ii)
Constructing of two sliding mode observers, thanks to their robustness against disturbances, which are designed to exactly simultaneous polytopic LPV system states, faults affecting actuators, and sensor faults estimation.(iii)
Developing convex Linear Matrix Inequalities (LMIs) optimization approach in which the disturbance attenuation level is minimizedThe outline of this paper is organized as follows: in Section2, we describe the uncertain LPV system expressed by the polytopic representation. In Section 3, we propose the H∞ faults estimation scheme using two sliding mode observers. Section 4 is devoted to simultaneous actuator and sensor faults estimation. Simulation examples are described in Section 5, illustrating the effectiveness of the proposed method. Finally, Section 6 presents some concluding remarks.


## 2. Problem Formulation

Consider the LPV system governed by(1)x˙t=Aθtxt+Bθtut+Dθtξt+Mθtfat,yt=Cθtxt+Nθtfst,where x(t)∈Rn is the state vector, u(t)∈Rm is the control input, and y(t)∈Rp denotes the output vector. fa(t)∈Rq and fs(t)∈Rh represent actuator and sensor faults, respectively, which are assumed unknown but bounded by known constants such as fa(t)≤ρ1 and fs(t)≤ρ2. ξ(t)∈Rl stands for the unknown disturbances such that ξ(t)≤ξ0.A
(
θ
(
t
)
)
∈
R
n
×
n, B(θ(t))∈Rn×m, M(θ(t))∈Rn×q, D(θ(t))∈Rn×l, C(θ(t))∈Rp×n, and N(θ(t))∈Rp×h are continuous functions that depend affinely on the time-varying parameter vector θ(t)∈Rν where n>p≥q.In what follows, the parameter vectorθ(t) is considered bounded and lies into a hypercube Θ as(2)θt∈Θ=θ∣θ_t≤θt≤θ¯t,∀t≥0.Based on parameter affine dependence [25, 28], the elements of the representation (1) can be written as(3)Πθt=Π0+∑i=1kθitΠi,∀θt∈Θ,where(4)Πθt=AθtBθtDθtMθtCθt00Nθt.Consequently, the LPV system (1) can be transformed into a convex interpolation of the vertices of Θ such as(5)x˙t=∑i=1kμiθtAixt+Biut+Diξt+Mifatyt=∑i=1kμiθtCixt+Nifst,where Ai, Bi, Di, Mi, Ci, and Ni are time invariant known matrices with appropriate dimensions defined for the ith vertex of Θ.μ
i
(
θ
(
t
)
) denotes the weighting functions which vary into the convex set Ω:(6)Ω=μθt=μ1θt⋯μkθtT,μθt≥0,∑i=1kμiθt=1.Herein,k=2ν is the total number of subsystems. It is noted that system (5) refers to a polytopic LPV system where its parameters evolve in the hypercube Θ. Each local model must verify the following assumptions.Assumption  A1. The actuator fault distribution matrix Mi is of full-column rank, ∀i∈1,…,k, that is,(7)rankCiMi=rankMi=q.Assumption  A2. System (5) is with minimum phase; that is, the invariant zeros of the triplet (Ai,Ci,Mi),∀i∈1,…,k, are all in the open left-hand complex plant, or(8)ranksIn-AiMiCi0=n+qholds for all complex number s with Re(s)≥0.Assumptions  A1 and A2 imply that the system is of relative degree one and minimum phase. These conditions are necessary and sufficient for the sliding mode observer design based simultaneous fault estimation.Under Assumption  A1, there exists linear transformation [11] such that the polytopic LPV system matrices from (5) yield, ∀i∈1,…,k,(9)TiAiTi-1=A1,iA2,iA3,iA4,i,TiBi=B1,iB2,i,TiDi=D1,iD2,i,TiMi=M1,i0,SiCiTi-1=C1,i00C4,i,SiNi=0N2,i,where(10)Ti=T1,iTT2,iTT∈Rn×n,Si=S1,iTS2,iTT∈Rp×p,A1,i∈Rq×q,B1,i∈Rq×m,M1,i∈Rq×q,D1,i∈Rq×l,C1,i∈Rq×q,C4,i∈Rp-q×n-q,N2,i∈Rp-q×h,r
a
n
k
(
M
1
,
i
)
=
q and C1,i is invertible. The matrices N2,i have a structure N2,i=0N22,i with N22,i∈Rh×h.Lemma 1.
The pair matrix(A4,i,C4,i) is detectable, ∀i∈1,…,k, if and only if each local model for the polytopic LPV system (5) is minimum phase; that is, Assumption  A2 holds.Proof (see [21]).
Based on the preceding state and output transformations,x1T(t)x2T(t)T=Tix(t) and y1T(t)y2T(t)T=Siy(t), the polytopic LPV system (5) is converted into two submodels:(11)x˙1t=∑i=1kμiθtA1,ix1t+A2,ix2t+B1,iut+D1,iξt+M1,ifat,y1t=∑i=1kμiθtC1,ix1t,(12)x˙2t=∑i=1kμiθtA3,ix1t+A4,ix2t+B2,iut+D2,iξt,y2t=∑i=1kμiθtC4,ix2t+N2,ifst,where (11) is referred to as the first subsystem which contains only actuator faults fa(t) and (12) is referred to as the second subsystem which results in only sensor faults fs(t).
We define a new system state vectorx3(t)=∫0ty2(τ)dτ where x˙3(t)=∑i=1kμi(θ(t))C4,ix2(t)+N2,ifs(t). Thus, an augmented polytopic LPV submodel of order (n+p-2q) can be expressed as(13)x˙0t=∑i=1kμiθtA0,ix0t+A01,ix1t+B0,iut+D0,iξt+M0,ifst,y0t=∑i=1kμiθtC0,ix0t,such that (14)x0t=x2tx3t∈Rn+p-2q,y0t∈Rp-q,A0,i=A4,i0C4,i0∈Rn+p-2q×n+p-2q,A01,i=A3,i0∈Rn+p-2q×q,B0,i=B2,i0∈Rn+p-2q×m,D0,i=D2,i0∈Rn+p-2q×l,M0,i=0N2,i∈Rn+p-2q×h,C0,i=0Ip-q∈Rp-q×n+p-2q.
The following lemma express the observability properties of the polytopic LPV subsystem (13).Lemma 2.
The pair(A0,i,C0,i) is observable, ∀i∈1,…,k, if the pair (A4,i,C4,i) is detectable, ∀i∈1,…,k. Then, there exists matrix L, having the special structure L=L10, such that A4,i+LC4,i is stable ∀i∈1,…,k.Proof (see [21]).
Consider system (13). Introduce a coordinate transformation z(t)=TLx0(t) where(15)TL≔In-qL0Ip-q.
It follows that, in the new coordinate systemz(t), system (13) becomes(16)z˙t=∑i=1kμiθtAz,izt+TLA01,ix1t+Bz,iut+Dz,iξt+Mz,ifst,y0t=∑i=1kμiθtCz,izt,where(17)Az,i=A4,i+LC4,i-A4,i+LC4,iLC4,i-C4,iL,Bz,i=B2,i0,Dz,i=D2,i0,Mz,i=0N2,i,Cz,i=0Ip-q.
The following section is dedicated to the design of the main results for this paper.


## 3. Sliding Mode Observers Design

We propose to conceive two sliding mode observers for the polytopic LPV system (5) expressed in two subsystems (11) and (16) based simultaneous actuator and sensor faults estimation design with minimizing the effect of disturbances using H∞ performances.As a result, in the new coordinate systemsz(t), (11) and (16) can be rewritten, respectively, as(18)x˙1t=∑i=1kμiθtA1,ix1t+A2,iz1t+B1,iut-A2,iLy0t+D1,iξt+M1,ifat,y1t=∑i=1kμiθtC1,ix1t,(19)z˙1t=∑i=1kμiθtA4,i+LC4,iz1t-A4,i+LC4,iLz2t+A3,ix1t+B2,iut+D2,iξt,z˙2t=∑i=1kμiθtC4,iz1t-C4,iLz2t+N2,ifst,y0t=z2t.For subsystems (18) and (19), we construct the following two sliding mode observers, respectively,(20)x^˙1t=∑i=1kμiθtA1,ix^1t+A2,iz^1t-A2,iLy0t+B1,iut+M1,iv1,it+A1,i-A1,isC1,i-1y1t-y^1t,y^1t=∑i=1kμiθtC1,ix^1t,(21)z^˙1t=∑i=1kμiθtA4,i+LC4,iz^1t-A4,i+LC4,iLy0t+B2,iut+A3,iC1,i-1y1t,z^˙2t=∑i=1kμiθtC4,iz^1t-C4,iLz^2t-C4,iL+Kiy0t-y^0t+N2,iv2,it,y^0t=z^2t.A
1
,
i
s
∈
R
q
×
q and Ki∈R(p-q)×(p-q) are the observer gains.The discontinuous output error injection signalsv1,i(t) and v2,i(t) are expressed as, respectively,(22)v1,it=ρ1+η1,iM1,iTP1C1,i-1S1,iyt-x^1tM1,iTP1C1,i-1S1,iyt-x^1tifC1,i-1S1,iyt-x^1t≠00otherwise,(23)v2,it=ρ2+η2,iN2,iTP¯03y0t-y^0tN2,iTP¯03y0t-y^0tify0t-y^0t≠00otherwise,where P1∈Rq×q and P¯03∈R(p-q)×(p-q) are symmetric definite Lyapunov matrices. η1,i and η2,i are two positive scalars to be determined.Lete1(t)=x1(t)-x^1(t), e2(t)=z1(t)-z^1(t), and e3(t)=z2(t)-z^2(t). Then, from (18)-(19) and (20)-(21), the state estimation errors dynamics are described by(24)e˙1t=∑i=1kμiθtA1,ise1t+A2,ie2t+D1,iξt+M1,ifat-M1,iv1,it,(25)e˙2t=∑i=1kμiθtA4,i+LC4,ie2t+D2,iξt,(26)e˙3t=∑i=1kμiθtC4,ie2t+Kie3t+N2,ifst-N2,iv2,it.The objective now is to present the sufficient conditions for the stability withH∞ performances of the observer errors (24)–(26) by using Lyapunov stability and LMIs technique.


### 3.1. Sliding Motion Stability

Suppose that(27)gt=Het=He1te2te3tis a linear error variables rearrangement. H is a prespecified weight matrix and assumed to have full rank:(28)H≔H1000H2000H3.The following theorem provides sufficient conditions to ensure the properties of sliding motion stability with specifiedH∞ performances to be defined as(29)H∞=supξtL2≠0⁡gtL22ξtL22≤γ,where γ is a small positive constant. So that, small H∞ gain of the transfer function from the disturbances ξ(t) to the estimation errors combination g(t) means a small influence of ξ(t) on e(t).Theorem 3.
Consider the polytopic LPV systems (18)-(19) and the sliding mode observers (20)-(21). The observer estimation errors (24)–(26) are stable with attenuation level γ>0 subject to g(t)L2<γξ(t)L2, if there exist the matrices P1>0, P01∈R(n-q)×(n-q)>0, P¯03>0, X1,i<0 such that the following optimization problem holds, ∀i∈1,…,k, (30)minimizeγs.t.X1,i+X1,iT+H1TH1P1A2,i0P1D1,iA2,iTP1X2,i+YC4,i+C4,iTYT+H2TH2C4,iTP¯03P01D2,i0P¯03C4,iX3,i+X3,iT+H3TH30D1,iTP1D2,iTP010-γIl<0,where X1,i=P1A1,is, X2,i=A4,iTP01+P01A4,i.Once the LMI problem is solved, it can provide design freedom sliding mode observers (20)-(21), such that (31)L=P01-1Y,A1,is=P1-1X1,i,Ki=P¯03-1X3,i.Proof.
The proof of this theorem is performed by using the following Lyapunov function:(32)Vt=V1t+V2t+V3t,where V1(t)=e1T(t)P1e1(t), V2(t)=e2T(t)P01e2(t), V3(t)=e3T(t)P¯03e3(t).
The time derivation ofV1(t) along a system trajectories is expressed as(33)V˙1t=∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+2e1TtP1A2,ie2t+2e1TtP1D1,iξt+2e1TtP1M1,ifat-v1,it.
From the definition ofv1,i(t) in (22) and using the bound of fa(t), it follows that(34)e1TtP1M1,ifat-v1,it=e1TtP1M1,ifat-ρ1+η1,ie1TtP1M1,iM1,iTP1e1tM1,iTP1e1t≤-η1,iM1,iTP1e1t<0.
Thus,(35)V˙1t≤∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+2e1TtP1A2,ie2t+2e1TtP1D1,iξt.
Likewise, the time derivations ofV2(t) and V3(t) along a system trajectories take the following equalities:(36)V˙2t=∑i=1kμiθte2TtA4,i+LC4,iTP01+P01A4,i+LC4,ie2t+2e2TtP01D2,iξt,V˙3t=∑i=1kμiθte3TtP¯03Ki+KiTP¯03e3t+2e3TtP¯03C4,ie2t+2e3TtP¯03N2,ifst-2e3TtP¯03N2,iv2,it.
Manipulating the definition ofv2,i(t) in (23) and using the bound of fs(t), it turns out that(37)e3TtP¯03N2,ifst-v2,it≤-η2,iN2,iTP¯03y0t-y^0t<0.
Therefore, the time derivation ofV(t) is handled as(38)V˙t=e˙1TtP1e1t+e1TtP1e˙1t+e˙2TtP01e2t+e2TtP01e˙2t+e˙3TtP¯03e3t+e3TtP¯03e˙3t=∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+e2TtA4,i+LC4,iTP01+P01A4,i+LC4,ie2t+e3TtP¯03Ki+KiTP¯03e3t+2e1TtP1A2,ie2t+2e3TtP¯03C4,ie2t+2e1TtP1D1,iξt+2e2TtP01D2,iξt.
In order to achieve the robustness of the proposed sliding mode observers to the disturbancesξ(t) in the L2 sense, we define(39)Jt≔V˙t+gTtgt-γξTtξt.
As a result, it yields that (41) is satisfied where, ∀i∈1,…,k,(40)Q1,i=A1,isTP1+P1A1,is,Q2,i=A4,iTP01+P01A4,i+P01LC4,i+C4,iTLTP01,Q3,i=P¯03Ki+KiTP¯03.
IfΔi<0,∀i∈1,…,k, then J(t)≤0 along system trajectories with disturbances attenuation level γ:(41)Jt≤∑i=1kμiθte1te2te3tξtTQ1,i+H1TH1P1A2,i0P1D1,iA2,iTP1Q2,i+H2TH2C4,iTP¯03P01D2,i0P¯03C4,iQ3,i+H3TH30D1,iTP1D2,iTP010-γIl︸Δie1te2te3tξt.
Integrating the expression in (39) from 0 to ∞, we can obtain with respect to t over time period 0∞ the following:(42)∫0∞V˙t+∫0∞gTtgt-∫0∞γξTtξt<0,
together with(43)Vt=0>0,Vt⟶∞=e1T∞P1e1∞+e2T∞P01e2∞+e3T∞P¯03e3∞⟶0.
Therefore, we obtain(44)∫0∞gTtgt∫0∞ξTtξt≤γ,namely,(45)gtL2≤γξtL2.
This completes the proof.


### 3.2. Sliding Motion Reachability

For systems (24)–(26), consider the sliding surface(46)Sg=e1t,e2t,e3t∣e1t=0,e3t=0.In order to confine stable sliding motion for the polytopic LPV system on sliding surface, it is necessary to use the switching gainsv1,i(t) and v2,i(t). The objective now is to determine the scalars gains η1,i in (22) and η2,i in (23) such that the errors system can be driven to the sliding surface Sg in finite time.The main challenge of this subsection will be proved.Theorem 4.
The error dynamics (24) and (26) can be driven to the sliding surface Sg (46) in finite time and remain on it if the gains η1,i and η2,i are chosen to satisfy(47)η1,i≥M1,i-TA2,ie2t+D2,iξ0+η1,i0,(48)η2,i≥N2,i-TC4,ie2t+η2,i0,where η1,i0 and η2,i0 are positive scalars ∀i∈1,…,k.Proof.
LetV1(t)=e1T(t)P1e1(t). From the expression of the estimation error in (24), it turns out that(49)V˙1t=∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+2e1TtP1A2,ie2t+2e1TtP1D1,iξt+2e1TtP1M1,ifat-2e1TtP1M1,iv1,it.
Since by designA1,is is stable matrices, ∀i∈1,…,k, it follows that (A1,is)TP1+P1A1,is<0. Then, from the Cauchy-Schwartz inequality and by applying (22), we can obtain that(50)V˙1t≤∑i=1kμiθt2e1TtP1A2,ie2t+D1,iξt+2e1TtP1M1,ifat-2e1TtP1M1,iv1,it≤∑i=1kμiθt2P1e1tA2,ie2t+2P1e1tD1,iξt+2e1TtP1M1,ifat-2ρ1+η1,ie1TtP1M1,iM1,iTP1e1tM1,iTP1e1t≤∑i=1kμiθt2P1e1tA2,ie2t+2P1e1tD1,iξt+2M1,iTP1e1tρ1-2ρ1+η1,iM1,iTP1e1t2M1,iTP1e1t≤∑i=1kμiθt2P1e1tA2,ie2t+D1,iξt-2η1,iM1,iTP1e1t.
It follows from (47) that(51)V˙1t≤∑i=1kμiθt-2η1,i0M1,iTP1e1t≤∑i=1kμiθt-2η1,i0M1,iTλminP1V11/2.
Similarly, it can be verified from (23) that if (48) is satisfied, then(52)V˙3t≤∑i=1kμiθt-2η2,i0N2,iTP¯03e3t≤∑i=1kμiθt-2η2,i0N2,iTλminP¯03V31/2.
Thus, it shows that the reachability condition is satisfied and an ideal sliding motion is achieved and maintained after some finite timetf≥0.
This completes the proof.


## 3.1. Sliding Motion Stability

Suppose that(27)gt=Het=He1te2te3tis a linear error variables rearrangement. H is a prespecified weight matrix and assumed to have full rank:(28)H≔H1000H2000H3.The following theorem provides sufficient conditions to ensure the properties of sliding motion stability with specifiedH∞ performances to be defined as(29)H∞=supξtL2≠0⁡gtL22ξtL22≤γ,where γ is a small positive constant. So that, small H∞ gain of the transfer function from the disturbances ξ(t) to the estimation errors combination g(t) means a small influence of ξ(t) on e(t).Theorem 3.
Consider the polytopic LPV systems (18)-(19) and the sliding mode observers (20)-(21). The observer estimation errors (24)–(26) are stable with attenuation level γ>0 subject to g(t)L2<γξ(t)L2, if there exist the matrices P1>0, P01∈R(n-q)×(n-q)>0, P¯03>0, X1,i<0 such that the following optimization problem holds, ∀i∈1,…,k, (30)minimizeγs.t.X1,i+X1,iT+H1TH1P1A2,i0P1D1,iA2,iTP1X2,i+YC4,i+C4,iTYT+H2TH2C4,iTP¯03P01D2,i0P¯03C4,iX3,i+X3,iT+H3TH30D1,iTP1D2,iTP010-γIl<0,where X1,i=P1A1,is, X2,i=A4,iTP01+P01A4,i.Once the LMI problem is solved, it can provide design freedom sliding mode observers (20)-(21), such that (31)L=P01-1Y,A1,is=P1-1X1,i,Ki=P¯03-1X3,i.Proof.
The proof of this theorem is performed by using the following Lyapunov function:(32)Vt=V1t+V2t+V3t,where V1(t)=e1T(t)P1e1(t), V2(t)=e2T(t)P01e2(t), V3(t)=e3T(t)P¯03e3(t).
The time derivation ofV1(t) along a system trajectories is expressed as(33)V˙1t=∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+2e1TtP1A2,ie2t+2e1TtP1D1,iξt+2e1TtP1M1,ifat-v1,it.
From the definition ofv1,i(t) in (22) and using the bound of fa(t), it follows that(34)e1TtP1M1,ifat-v1,it=e1TtP1M1,ifat-ρ1+η1,ie1TtP1M1,iM1,iTP1e1tM1,iTP1e1t≤-η1,iM1,iTP1e1t<0.
Thus,(35)V˙1t≤∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+2e1TtP1A2,ie2t+2e1TtP1D1,iξt.
Likewise, the time derivations ofV2(t) and V3(t) along a system trajectories take the following equalities:(36)V˙2t=∑i=1kμiθte2TtA4,i+LC4,iTP01+P01A4,i+LC4,ie2t+2e2TtP01D2,iξt,V˙3t=∑i=1kμiθte3TtP¯03Ki+KiTP¯03e3t+2e3TtP¯03C4,ie2t+2e3TtP¯03N2,ifst-2e3TtP¯03N2,iv2,it.
Manipulating the definition ofv2,i(t) in (23) and using the bound of fs(t), it turns out that(37)e3TtP¯03N2,ifst-v2,it≤-η2,iN2,iTP¯03y0t-y^0t<0.
Therefore, the time derivation ofV(t) is handled as(38)V˙t=e˙1TtP1e1t+e1TtP1e˙1t+e˙2TtP01e2t+e2TtP01e˙2t+e˙3TtP¯03e3t+e3TtP¯03e˙3t=∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+e2TtA4,i+LC4,iTP01+P01A4,i+LC4,ie2t+e3TtP¯03Ki+KiTP¯03e3t+2e1TtP1A2,ie2t+2e3TtP¯03C4,ie2t+2e1TtP1D1,iξt+2e2TtP01D2,iξt.
In order to achieve the robustness of the proposed sliding mode observers to the disturbancesξ(t) in the L2 sense, we define(39)Jt≔V˙t+gTtgt-γξTtξt.
As a result, it yields that (41) is satisfied where, ∀i∈1,…,k,(40)Q1,i=A1,isTP1+P1A1,is,Q2,i=A4,iTP01+P01A4,i+P01LC4,i+C4,iTLTP01,Q3,i=P¯03Ki+KiTP¯03.
IfΔi<0,∀i∈1,…,k, then J(t)≤0 along system trajectories with disturbances attenuation level γ:(41)Jt≤∑i=1kμiθte1te2te3tξtTQ1,i+H1TH1P1A2,i0P1D1,iA2,iTP1Q2,i+H2TH2C4,iTP¯03P01D2,i0P¯03C4,iQ3,i+H3TH30D1,iTP1D2,iTP010-γIl︸Δie1te2te3tξt.
Integrating the expression in (39) from 0 to ∞, we can obtain with respect to t over time period 0∞ the following:(42)∫0∞V˙t+∫0∞gTtgt-∫0∞γξTtξt<0,
together with(43)Vt=0>0,Vt⟶∞=e1T∞P1e1∞+e2T∞P01e2∞+e3T∞P¯03e3∞⟶0.
Therefore, we obtain(44)∫0∞gTtgt∫0∞ξTtξt≤γ,namely,(45)gtL2≤γξtL2.
This completes the proof.


## 3.2. Sliding Motion Reachability

For systems (24)–(26), consider the sliding surface(46)Sg=e1t,e2t,e3t∣e1t=0,e3t=0.In order to confine stable sliding motion for the polytopic LPV system on sliding surface, it is necessary to use the switching gainsv1,i(t) and v2,i(t). The objective now is to determine the scalars gains η1,i in (22) and η2,i in (23) such that the errors system can be driven to the sliding surface Sg in finite time.The main challenge of this subsection will be proved.Theorem 4.
The error dynamics (24) and (26) can be driven to the sliding surface Sg (46) in finite time and remain on it if the gains η1,i and η2,i are chosen to satisfy(47)η1,i≥M1,i-TA2,ie2t+D2,iξ0+η1,i0,(48)η2,i≥N2,i-TC4,ie2t+η2,i0,where η1,i0 and η2,i0 are positive scalars ∀i∈1,…,k.Proof.
LetV1(t)=e1T(t)P1e1(t). From the expression of the estimation error in (24), it turns out that(49)V˙1t=∑i=1kμiθte1TtA1,isTP1+P1A1,ise1t+2e1TtP1A2,ie2t+2e1TtP1D1,iξt+2e1TtP1M1,ifat-2e1TtP1M1,iv1,it.
Since by designA1,is is stable matrices, ∀i∈1,…,k, it follows that (A1,is)TP1+P1A1,is<0. Then, from the Cauchy-Schwartz inequality and by applying (22), we can obtain that(50)V˙1t≤∑i=1kμiθt2e1TtP1A2,ie2t+D1,iξt+2e1TtP1M1,ifat-2e1TtP1M1,iv1,it≤∑i=1kμiθt2P1e1tA2,ie2t+2P1e1tD1,iξt+2e1TtP1M1,ifat-2ρ1+η1,ie1TtP1M1,iM1,iTP1e1tM1,iTP1e1t≤∑i=1kμiθt2P1e1tA2,ie2t+2P1e1tD1,iξt+2M1,iTP1e1tρ1-2ρ1+η1,iM1,iTP1e1t2M1,iTP1e1t≤∑i=1kμiθt2P1e1tA2,ie2t+D1,iξt-2η1,iM1,iTP1e1t.
It follows from (47) that(51)V˙1t≤∑i=1kμiθt-2η1,i0M1,iTP1e1t≤∑i=1kμiθt-2η1,i0M1,iTλminP1V11/2.
Similarly, it can be verified from (23) that if (48) is satisfied, then(52)V˙3t≤∑i=1kμiθt-2η2,i0N2,iTP¯03e3t≤∑i=1kμiθt-2η2,i0N2,iTλminP¯03V31/2.
Thus, it shows that the reachability condition is satisfied and an ideal sliding motion is achieved and maintained after some finite timetf≥0.
This completes the proof.


## 4. Simultaneous Actuator and Sensor Faults Estimation

It is assumed in this section that the sliding mode observers (20)-(21) have been designed. The objective is to simultaneously estimate actuator and sensor faults. From Theorems 3 and 4, it follows that, in finite time, the error dynamics (24) and (26) will be driven to the sliding surface Sg and an ideal sliding motion is maintained therefore. During the sliding motion,(53)e1t=e˙1t=0,e3t=e˙3t=0.One can prove that (24) becomes(54)0=∑i=1kμiθtA2,ie2t+D1,iξt+M1,ifat-M1,iv1eq,it,where v1eq,i(t) denotes the equivalent output error injection signal required to maintain the sliding motion [2]. The signal v1eq,i(t) can be approximated by(55)v1eq,it=ρ1+η1,i×M1,iTP1C1,i-1S1,iyt-x^1tM1,iTP1C1,i-1S1,iyt-x^1t+δ1,where δ1 is a small positive scalar designed to reduce the chattering effect.It remains to prove, from (54), that ∀i∈1,…,k,(56)fat-v1eq,itL2=M1,i-1A2,ie2t+M1,i-1D1,iξtL2≤σmaxM1,i-1A2,ie2tL2+σmaxM1,i-1D1,iξtL2≤σmaxM1,i-1A2,ietL2+σmaxM1,i-1D1,iξtL2.So, sincee(t)L2≤σmax(H-1)γξ(t)L2,(57)supξtL2≠0⁡fat-v1eq,itL2ξtL2=γβ11,i+β12,i,where β11,i=σmax(M1,i-1A2,i)σmax(H-1) and β12,i=σmax(M1,i-1D1,i),∀i∈1,…,k.Thus, for smallβ11,i and β12,i,∀i∈1,…,k, the actuator fault signal can be obtained as(58)f^at≅∑i=1kμiθtρ1+η1,i×M1,iTP1C1,i-1S1,iyt-x^1tM1,iTP1C1,i-1S1,iyt-x^1t+δ1.Similarly, we can deduce that(59)fst-v2eq,itL2=N2,i-1C4,ie2tL2≤σmaxN2,i-1C4,ie2tL2≤σmaxN2,i-1C4,ietL2,namely,(60)supξtL2≠0⁡fst-v2eq,itL2ξtL2=γβ2,i,where β2,i=σmax(N2,i-1C4,i)σmax(H-1) will be small scalars, ∀i∈1,…,k.Therefore, the sensor fault estimation is handled as follows:(61)f^st≅∑i=1kμiθtρ2+η2,i×N2,iTP¯03y0t-y^0tN2,iTP¯03y0t-y^0t+δ2,where δ2 is a small positive scalar.


## 5. Illustrative Example

### 5.1. Model Expression

The performances of the proposed sliding mode observer-based simultaneous actuator and sensor faults estimation is performed by considering a polytopic LPV representation of the Vertical Take-Off and Landing (VTOL) Aircraft defense system taken from [31]. The LPV state-space model is given by (62)x˙t=Aθtxt+Bθtut+Dθtξt+Mθtfat,yt=Cθtxt+Nθtfst,where the state vector x(t)=[VhVfqφ]T represents, respectively, horizontal velocity, vertical velocity, pitch rate, and pitch angle. ut=[ucul]T are, respectively, pitch angle, collective pitch control, and longitudinal cyclic pitch control. ξ(t) denotes unknown external disturbances vector. It is worth pointing that simulation results are illustrated with simultaneous additive faults fs(t) affecting measurement outputs and actuator faults fa(t). The LPV system matrices are expressed as(63)Aθt=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09222.6361+θ1t-4.1975-19.2774+θ2t0010,Bθt=0.44220.17613.5446+θ2t-7.5922-5.52004.490000,Dθt=D=0.100.10.1,Mθt=M=0.1761-7.59224.49000,Cθt=C=100001000010,Nθt=N=001.The varying parameter functionsθ1(t) and θ2(t) are bounded as θ1(t)∈-0.5,0.5 and θ2(t)∈-2,2. Therefore, corresponding to the extreme values of the parameters θ1(t) and θ2(t), an obtained system evolves in a polytope of 4 vertices described as (5). The polytopic LPV system matrices are given as (64)A1=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09222.1361-4.1975-21.27740010,A2=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09222.1361-4.1975-17.27740010,A3=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09223.1361-4.1975-21.27740010,A4=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09223.1361-4.1975-17.27740010,B1=0.44220.17611.5446-7.5922-5.52004.490000,B2=0.44220.17615.5446-7.5922-5.52004.490000,B3=0.44220.17611.5446-7.5922-5.52004.490000,B4=0.44220.17615.5446-7.5922-5.52004.490000.The convex weighing functionsμiθ(t) are(65)μ1θt=θ1t-θ_1θ¯1-θ_1θ2t-θ_2θ¯2-θ_2=θ1t+0.5θ2t+24,μ2θt=θ1t-θ_1θ¯1-θ_1θ¯2-θ2tθ¯2-θ_2=θ1t+0.5-θ2t+24,μ3θt=θ¯1-θ1tθ¯1-θ_1θ2t-θ_2θ¯2-θ_2=0.5-θ1tθ2t+24,μ4θt=θ¯1-θ1tθ¯1-θ_1θ¯2-θ2tθ¯2-θ_2=0.5-θ1t2-θ2t4.These functions of each local model are depicted in Figure1.Figure 1
Weighing functions.Based on the polytopic LPV system matrices, it follows that(i)
Rank(CM)=Rank(M)=1;(ii)
the invariant zeros of the triplet system(Ai,M,C) are in the open left-hand complex plane.Therefore, Assumptions  A1 and A2 are satisfied.


### 5.2. Simulation Results

The aim is to design the proposedH∞ sliding mode observers-based actuator and sensor faults estimation in the simultaneous scenario assuming that the matrix H is chosen as H1=1, H2=I3, and H3=I2.The disturbances attenuation level is determined asγ=0.3720. By solving the LMI conditions given in Theorem 3, the sliding mode observers (20)-(21) design is achieved as follows:(66)A1,1s=-12.547,A1,2s=-12.145,A1,3s=-12.557,A1,4s=-12.169,P1=0.184,P01=0.1530.033-0.40.0330.107-0.505-0.4-0.50510.748,P¯03=0.863000.604,L=10.290-53.0640-4.50,K1=-2.350.0570.077-3.481,K2=-2.3410.0840.115-3.4,K3=-2.4130.0140.015-3.487,K4=-2.3910.0440.058-3.422.The corresponding simulations are done by considering initial conditionsx(t=0)=[0;0;0;0] and x^(t=0)=[-0.2;1;1;-0.25], respectively, for the LPV system and observers. We assume also ρ1=10, ρ2=20, δ1=0.05, δ2=0.02 and simultaneous actuator and sensor faults scenario to leave t = 5 s. The simulation was carried out with the input signal u(t)=2sin⁡(t) and uncertainty ξ(t)=0.1sin⁡(0.2t).Figure2 displays the states estimation errors to prove a comparison between x1t, x2t, x3t, and x4t against the estimated values x^1t, x^2t, x^3t, and x^4t respectively. It can be seen that the estimated states can converge towards the original states.Figure 2
States estimation errors.In order to highlight the effectiveness of the proposed robust and simultaneous actuator and sensor faults estimation for LPV systems compared to descriptor approach [29], we consider different scenarios of faults occurrence.


#### 5.2.1. Simulation Results under Incipient Actuator Faults

The objective of the first scenario is to illustrate the sliding mode observer performances under an incipient actuator fault. Incipient faults are important in nonlinear complex system, and some common manifestations include zero drift, reduced precision, and delayed response. Consequently, the study of incipient faults in nonlinear systems has practical significance.As can be observed, the proposed sliding mode observers are capable of estimating incipient actuator fault after a short time of fault generation, as well as sensor faults (sinusoidal signal) with good performances and small error compared to the estimation based on descriptor approach [29]. So, one can see, from Figures 3 and 4, that the external disturbances are faithfully reduced by integrating the H∞ attenuation levels into the sliding mode observers design.Figure 3
Actuator fault estimation under incipient actuator fault.Figure 4
Sensor fault estimation under incipient actuator fault.


#### 5.2.2. Simulation Results under Intermittent Actuator Faults

The second scenario has the objective of illustrating the proposed sliding mode observers performances under intermittent actuator faults. Figures5 and 6 show simultaneous actuator and sensor faults estimation under intermittent actuator faults. The results display that the proposed sliding mode observers present better performances under intermittent actuator fault compared to descriptor approach [29]. The proposed method is capable of decoupling the effects of simultaneous actuator and sensor faults and managing to accomplish the main objective of this paper (both actuator and sensor fault estimation for the polytopic LPV system) with good performances in terms of precision and robustness against the external disturbances.Figure 5
Actuator fault estimation under intermittent actuator fault.Figure 6
Sensor fault estimation under intermittent actuator fault.


## 5.1. Model Expression

The performances of the proposed sliding mode observer-based simultaneous actuator and sensor faults estimation is performed by considering a polytopic LPV representation of the Vertical Take-Off and Landing (VTOL) Aircraft defense system taken from [31]. The LPV state-space model is given by (62)x˙t=Aθtxt+Bθtut+Dθtξt+Mθtfat,yt=Cθtxt+Nθtfst,where the state vector x(t)=[VhVfqφ]T represents, respectively, horizontal velocity, vertical velocity, pitch rate, and pitch angle. ut=[ucul]T are, respectively, pitch angle, collective pitch control, and longitudinal cyclic pitch control. ξ(t) denotes unknown external disturbances vector. It is worth pointing that simulation results are illustrated with simultaneous additive faults fs(t) affecting measurement outputs and actuator faults fa(t). The LPV system matrices are expressed as(63)Aθt=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09222.6361+θ1t-4.1975-19.2774+θ2t0010,Bθt=0.44220.17613.5446+θ2t-7.5922-5.52004.490000,Dθt=D=0.100.10.1,Mθt=M=0.1761-7.59224.49000,Cθt=C=100001000010,Nθt=N=001.The varying parameter functionsθ1(t) and θ2(t) are bounded as θ1(t)∈-0.5,0.5 and θ2(t)∈-2,2. Therefore, corresponding to the extreme values of the parameters θ1(t) and θ2(t), an obtained system evolves in a polytope of 4 vertices described as (5). The polytopic LPV system matrices are given as (64)A1=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09222.1361-4.1975-21.27740010,A2=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09222.1361-4.1975-17.27740010,A3=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09223.1361-4.1975-21.27740010,A4=-9.9477-0.74760.26325.033752.16592.74525.5532-24.422126.09223.1361-4.1975-17.27740010,B1=0.44220.17611.5446-7.5922-5.52004.490000,B2=0.44220.17615.5446-7.5922-5.52004.490000,B3=0.44220.17611.5446-7.5922-5.52004.490000,B4=0.44220.17615.5446-7.5922-5.52004.490000.The convex weighing functionsμiθ(t) are(65)μ1θt=θ1t-θ_1θ¯1-θ_1θ2t-θ_2θ¯2-θ_2=θ1t+0.5θ2t+24,μ2θt=θ1t-θ_1θ¯1-θ_1θ¯2-θ2tθ¯2-θ_2=θ1t+0.5-θ2t+24,μ3θt=θ¯1-θ1tθ¯1-θ_1θ2t-θ_2θ¯2-θ_2=0.5-θ1tθ2t+24,μ4θt=θ¯1-θ1tθ¯1-θ_1θ¯2-θ2tθ¯2-θ_2=0.5-θ1t2-θ2t4.These functions of each local model are depicted in Figure1.Figure 1
Weighing functions.Based on the polytopic LPV system matrices, it follows that(i)
Rank(CM)=Rank(M)=1;(ii)
the invariant zeros of the triplet system(Ai,M,C) are in the open left-hand complex plane.Therefore, Assumptions  A1 and A2 are satisfied.


## 5.2. Simulation Results

The aim is to design the proposedH∞ sliding mode observers-based actuator and sensor faults estimation in the simultaneous scenario assuming that the matrix H is chosen as H1=1, H2=I3, and H3=I2.The disturbances attenuation level is determined asγ=0.3720. By solving the LMI conditions given in Theorem 3, the sliding mode observers (20)-(21) design is achieved as follows:(66)A1,1s=-12.547,A1,2s=-12.145,A1,3s=-12.557,A1,4s=-12.169,P1=0.184,P01=0.1530.033-0.40.0330.107-0.505-0.4-0.50510.748,P¯03=0.863000.604,L=10.290-53.0640-4.50,K1=-2.350.0570.077-3.481,K2=-2.3410.0840.115-3.4,K3=-2.4130.0140.015-3.487,K4=-2.3910.0440.058-3.422.The corresponding simulations are done by considering initial conditionsx(t=0)=[0;0;0;0] and x^(t=0)=[-0.2;1;1;-0.25], respectively, for the LPV system and observers. We assume also ρ1=10, ρ2=20, δ1=0.05, δ2=0.02 and simultaneous actuator and sensor faults scenario to leave t = 5 s. The simulation was carried out with the input signal u(t)=2sin⁡(t) and uncertainty ξ(t)=0.1sin⁡(0.2t).Figure2 displays the states estimation errors to prove a comparison between x1t, x2t, x3t, and x4t against the estimated values x^1t, x^2t, x^3t, and x^4t respectively. It can be seen that the estimated states can converge towards the original states.Figure 2
States estimation errors.In order to highlight the effectiveness of the proposed robust and simultaneous actuator and sensor faults estimation for LPV systems compared to descriptor approach [29], we consider different scenarios of faults occurrence.


### 5.2.1. Simulation Results under Incipient Actuator Faults

The objective of the first scenario is to illustrate the sliding mode observer performances under an incipient actuator fault. Incipient faults are important in nonlinear complex system, and some common manifestations include zero drift, reduced precision, and delayed response. Consequently, the study of incipient faults in nonlinear systems has practical significance.As can be observed, the proposed sliding mode observers are capable of estimating incipient actuator fault after a short time of fault generation, as well as sensor faults (sinusoidal signal) with good performances and small error compared to the estimation based on descriptor approach [29]. So, one can see, from Figures 3 and 4, that the external disturbances are faithfully reduced by integrating the H∞ attenuation levels into the sliding mode observers design.Figure 3
Actuator fault estimation under incipient actuator fault.Figure 4
Sensor fault estimation under incipient actuator fault.


### 5.2.2. Simulation Results under Intermittent Actuator Faults

The second scenario has the objective of illustrating the proposed sliding mode observers performances under intermittent actuator faults. Figures5 and 6 show simultaneous actuator and sensor faults estimation under intermittent actuator faults. The results display that the proposed sliding mode observers present better performances under intermittent actuator fault compared to descriptor approach [29]. The proposed method is capable of decoupling the effects of simultaneous actuator and sensor faults and managing to accomplish the main objective of this paper (both actuator and sensor fault estimation for the polytopic LPV system) with good performances in terms of precision and robustness against the external disturbances.Figure 5
Actuator fault estimation under intermittent actuator fault.Figure 6
Sensor fault estimation under intermittent actuator fault.


## 5.2.1. Simulation Results under Incipient Actuator Faults

The objective of the first scenario is to illustrate the sliding mode observer performances under an incipient actuator fault. Incipient faults are important in nonlinear complex system, and some common manifestations include zero drift, reduced precision, and delayed response. Consequently, the study of incipient faults in nonlinear systems has practical significance.As can be observed, the proposed sliding mode observers are capable of estimating incipient actuator fault after a short time of fault generation, as well as sensor faults (sinusoidal signal) with good performances and small error compared to the estimation based on descriptor approach [29]. So, one can see, from Figures 3 and 4, that the external disturbances are faithfully reduced by integrating the H∞ attenuation levels into the sliding mode observers design.Figure 3
Actuator fault estimation under incipient actuator fault.Figure 4
Sensor fault estimation under incipient actuator fault.


## 5.2.2. Simulation Results under Intermittent Actuator Faults

The second scenario has the objective of illustrating the proposed sliding mode observers performances under intermittent actuator faults. Figures5 and 6 show simultaneous actuator and sensor faults estimation under intermittent actuator faults. The results display that the proposed sliding mode observers present better performances under intermittent actuator fault compared to descriptor approach [29]. The proposed method is capable of decoupling the effects of simultaneous actuator and sensor faults and managing to accomplish the main objective of this paper (both actuator and sensor fault estimation for the polytopic LPV system) with good performances in terms of precision and robustness against the external disturbances.Figure 5
Actuator fault estimation under intermittent actuator fault.Figure 6
Sensor fault estimation under intermittent actuator fault.


## 6. Conclusion

This paper presents a robust simultaneous actuator and sensor faults estimation scheme for LPV systems with disturbances. The developed strategy is based on polytopic representation where the parameters evolve in the hypercube domain. State and output transformations are introduced in order to formulate two polytopic LPV subsystems where each one includes a particular class of faults (actuator or sensor). To this end, two sliding mode observers are conceived to exactly estimate simultaneously: incipient and intermittent actuator faults, sensor faults, and polytopic LPV system states. Sufficient conditions to guarantee the stability of the two observers has been derived withH∞ performances for minimizing the effect of disturbances on estimation error dynamics and solved within LMIs optimization design. The simulation results display that the proposed scheme can successfully estimate actuator and sensor faults simultaneously despite the existence of disturbances.



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*Source: 1747095-2017-12-04.xml*