fault-tolerant control for a class of non-linear systems with ......robust adaptive control was...

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Fault-tolerant control for a class of non-linear systemswith dead-zoneMou Chena, Bin Jianga & William W. Guoba College of Automation Engineering, Nanjing University of Aeronautics and Astronautics,Nanjing, Chinab School of Engineering and Technology, Central Queensland University, North Rockhampton,AustraliaPublished online: 08 Aug 2014.

To cite this article: Mou Chen, Bin Jiang & William W. Guo (2014): Fault-tolerant control for a class of non-linear systemswith dead-zone, International Journal of Systems Science, DOI: 10.1080/00207721.2014.945984

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International Journal of Systems Science, 2014http://dx.doi.org/10.1080/00207721.2014.945984

Fault-tolerant control for a class of non-linear systems with dead-zone

Mou Chena,∗, Bin Jianga and William W. Guob

aCollege of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China; bSchool of Engineering andTechnology, Central Queensland University, North Rockhampton, Australia

(Received 14 November 2013; accepted 25 April 2014)

In this paper, a fault-tolerant control scheme is proposed for a class of single-input and single-output non-linear systems withthe unknown time-varying system fault and the dead-zone. The non-linear state observer is designed for the non-linear systemusing differential mean value theorem, and the non-linear fault estimator that estimates the unknown time-varying systemfault is developed. On the basis of the designed fault estimator, the observer-based fault-tolerant tracking control is thendeveloped using the backstepping technique for non-linear systems with the dead-zone. The stability of the whole closed-loop system is rigorously proved via Lyapunov analysis and the satisfactory tracking control performance is guaranteed inthe presence of the unknown time-varying system fault and the dead-zone. Numerical simulation results are presented toillustrate the effectiveness of the proposed backstepping fault-tolerant control scheme for non-linear systems.

Keywords: non-linear systems; fault estimator; state observer; fault-tolerant control; tracking control

1. Introduction

With the development of the complex system, a key chal-lenge is how to achieve the satisfactory control performancein the presence of the time-varying system fault (Zhang,Polycarpoub, & Parisini, 2010). As is well known, faultsmay lead to control system performance degradation andcause instability and even disastrous accidents (Alwi & Ed-wards, 2008a; Gao, Jiang, Shi, Liu, & Xu, 2012; Xu, Jiang,& Tao, 2011; Ye & Yang, 2006). Specially, to the safety crit-ical systems, such as passenger aircraft and modern fighteraircraft, the control system must maintain stability and ac-ceptable performance during faults including actuator fault,sensor fault, and so on. Thus, fault-tolerant control has be-come an important research topic in the control area (Alwi& Edwards, 2008b; Gao, Jiang, Qi, & Xu, 2011). In Liand Yang (2012), robust adaptive fault-tolerant control wasdeveloped for uncertain linear systems with actuator fail-ures. Adaptive fuzzy fault-tolerant output-feedback controlwas proposed for uncertain non-linear systems with actua-tor faults in Huo, Tong, and Li (2013) and Tong, Hou, andLi (2014). In Wang, Xie, and Zhang (2012), sliding modefault-tolerant control was studied to deal with modellinguncertainties and actuator faults. In practice, many controlsystems have non-linear characteristic. Thus, the tolerantcontrol scheme should be further investigated for the non-linear system with the unknown time-varying system fault.

For the fault-tolerant control design of the non-linearsystem, the significant research activity is the fault-estimator design (Panagi & Polycarpou, 2011). To design

∗Corresponding author. Email: [email protected]

the fault estimator, first the non-linear state observer shouldbe developed (Wang, Zhou, & Gao, 2007). However, manyresearch results of the state observer are effective for linearsystems (Darouach, Zasadzinski, & Xu, 1994; Jo & Seo,2000). For non-linear systems, the state observer designneeds to be further investigated (Li, Cao, & Ding, 2011; Liu,Tong, & Li, 2011). In Ekramian, Sheikholeslam, Hossein-nia, and Yazdanpanah (2013), adaptive state observer wasdeveloped for Lipschitz non-linear systems. State observer-based fuzzy adaptive output tracking control was proposedfor non-linear systems in Tong, Wang, and Tang (2000).In Dong and Mei (2011), state observer was presented fora class of multi-output non-linear dynamic systems. Re-cently, a new state observer design was developed for a classof systems with monotonic non-linearities (Zemouche,Boutayeb, & Bara, 2005). In this new design, the globalLipschitz restriction is removed. Moreover, the basic ideaof observer design is to use the well-known differentialmean value theorem (DMVT) (Zemouche, Boutayeb, &Bara, 2006). In Zemouche, Boutayeb, and Bara (2008),the state observers were proposed for a class of Lipschitzsystems with extension to H∞ performance analysis usingthe DMVT technique. Furthermore, the fault estimatorneeds to be developed for the non-linear system based onthe designed non-linear state observer for the appearedunknown time-varying system fault. Based on the outputof the fault estimator, the observer-based fault-toleranttracking control should be designed for the non-linearsystem to maintain the tracking control performance

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2 M. Chen et al.

under the influence of time-varying unknown fault. InHuo, Li, and Tong (2012) and Tong, Hou, and Li (2014),observer-based fuzzy adaptive fault-tolerant output-feedback control schemes were proposed for single-inputand single-output (SISO) and multi-input and multi-output(MIMO) non-linear systems in the strict-feedback form.Finite-time fault-tolerant control was developed for rigidspacecrafts with actuator saturations in Lu and Xia (2013).Considering the state observer and the fault estimator, thefault-tolerant control design will become more complicatedfor non-linear systems.

The coupled interaction between the fault estimator andthe fault-tolerant controller in a closed-loop configurationis very complex, especially for non-linear systems. Back-stepping control as an efficient control method of non-linearsystems can provide a systematic framework for the track-ing control, which is suitable for a class of special non-linear systems (Li, Qiang, Zhuang, & Kaynak, 2004). Dueto the design flexibility, the robust adaptive backsteppingcontrol has been extensively studied in the non-linear con-trol system design (Chen, Jiang, & Wu, 2008; Chen, Jiao,Li, & Li, 2010; Tong, Li, & Shi, 2012). Adaptive fuzzyoutput-feedback control was proposed for a class of MIMOnon-linear uncertain systems with time-varying delays andunknown backlash-like hysteresis using the backsteppingmethod in Li, Tong, and Li (2012a). In Tong, Li, Li, andLiu (2011), observer-based adaptive fuzzy backsteppingcontrol was developed for a class of stochastic non-linearstrict-feedback systems. At the same time, some backstep-ping fault-tolerant control results have been studied for thenon-linear system. In Jiang, Hu, and Ma (2010), adaptivebackstepping fault-tolerant control was proposed for flex-ible spacecrafts with unknown bounded disturbances andactuator failures. Adaptive neural network output-feedbackcontrol was proposed for stochastic non-linear systems withunknown dead-zone and unmodelled dynamics in Tong,Wang, Li, and Zhang (2014). However, the actuator can-not promptly respond to the command signal resulting inan actuator dead-zone for a practical system. Thus, thebackstepping fault-tolerant control scheme should be fur-ther developed for the non-linear system with the unknowntime-varying system fault and the dead-zone.

Dead-zone as one of the non-smooth non-linearitiescan severely degrade system control performance, if it isignored in the tolerant control design. Thus, the degra-dation of fault-tolerant control performance caused bythe dead-zone of actuators needs to be considered in thecontrol design stage (Corradini & Orlando, 2002; Luo,Wu, & Guan, 2010). To handle the actuator dead-zoneproblem, various control schemes have been proposed inCorradini and Orlando (2003), Zhou, Wen, and Zhang(2006), Li and Yang (2009), and Tong and Li (2013). Adap-tive output control was proposed for uncertain non-linearsystems with non-symmetric dead-zone input in Ma andYang (2010). In Zhang and Ge (2008), adaptive dynamic

surface control was proposed to handle a class of pure-feedback non-linear systems with unknown dead-zone andperturbed uncertainties. Robust adaptive control was de-veloped for a class of non-linear systems with unknowndead-zone in Wang, Su, and Hong (2004). In Tong andLi (2012) and Tong, Wang, and Li (2014), observer-basedadaptive fuzzy output-feedback tracking backstepping con-trol approaches were first proposed for strict-feedbackSISO/MIMO non-linear systems with the unknown dead-zones, and the stability of the control systems were proved.The presence of dead-zone and unknown time-varying sys-tem fault will further enhance the design difficulty of thefault-tolerant tracking control scheme for non-linear sys-tems, which should be further derived.

This work is motivated by the design of fault-tolerantcontrol scheme to follow the desired trajectories of non-linear systems with the unknown time-varying system faultand the dead-zone. The organisation of the paper is as fol-lows. The problem statement is detailed in Section 2. Thefault-tolerant control of the non-linear system is proposedin Section 3 including the state observer design and the faultestimator design. Simulation results of an example are pre-sented in Section 4 to demonstrate the effectiveness of thedeveloped fault-tolerant control of the non-linear systems,followed by some concluding remarks in Section 5.

2. Problem statement

To develop the fault-tolerant control, let us consider thefollowing non-linear systems in the form of:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ẋi = xi+1...ẋn−1 = xnẋn = f (x) + gu + η(t)y = x1,

(1)

where x = [x1, x2, . . . , xn]T = [x1, ẋ1, . . . , x(n−1)1 ] ∈ Rn isthe state vector, and u ∈ R is the control input of the non-linear system. f(x) ∈ R is a known continuous function,and g ∈ R is a known control gain constant. In order tobe controllable for (1), it is required that g �= 0. η(t) ∈ Rdenotes the (additive) unknown system time-varying fault.

In accordance with (1), the non-linear system with theunknown time-varying system fault can be equivalentlywritten as{

ẋ = Ax + Ly + B(f (x) + gu + η(t))y = CT x , (2)

where A =

⎡⎢⎢⎣

−l1 1 0 . . . 0−l2 0 1 . . . 0...

. . .. . .

. . . 0−ln−1 0 0 . . . 1−ln 0 0 . . . 0

⎤⎥⎥⎦, L =

⎡⎢⎢⎣

l1l2...ln−1ln

⎤⎥⎥⎦, B =

⎡⎢⎢⎣

00...01

⎤⎥⎥⎦

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International Journal of Systems Science 3

and C =

⎡⎢⎢⎣

10...00

⎤⎥⎥⎦. li, i = 1, 2, . . . , n are the designed con-

stants. In the backstepping fault-tolerant control design, Lshould be properly chosen, such that A is a strict Hurwitzmatrix.

In practice, the actuator usually exists the dead-zonenon-linearity. Thus, the control input u can be expressedas

u(t) = D(v(t)), (3)

where v is the input of the dead-zone and D(.) denotes adead-zone operator.

In this paper, the considered dead-zone is described as(Wang et al., 2004)

u(t) = D(v(t)) =⎧⎨⎩

m(v(t) − br ), for v(t) ≥ br0, for bl < v(t) < brm(v(t) − bl), for v(t) ≤ bl,

(4)

where m > 0, br > 0, and bl < 0 are the known dead-zoneparameters.

To develop the fault-tolerant control scheme, the dead-zone model (4) is rewritten as (Wang et al., 2004)

D(v(t)) = mv(t) + d(v(t)), (5)

where d(v(t)) is given by

d(v(t)) =⎧⎨⎩

−mbr, for v(t) ≥ br−mv(t), for bl < v(t) < br−mbl, for v(t) ≤ bl.

(6)

For a practical system, we know that the parameter mof the dead-zone is bounded. Thus, from (6), we have

|d(v(t))| ≤ dm, (7)

where dm = max {mmax brmax , −mmax blmin }, mmax is theupper bound of the parameter m, brmax is the upper boundof the parameter br and blmin is the lower bound of theparameter bl.

Invoking (2) and (4) yields{ẋ = Ax + Ly + B(f (x) + gD(v(t)) + η(t))y = CT x . (8)

In this paper, the state observer and the fault estimatorwill be proposed for the non-linear system (1) with the un-known time-varying system fault and the dead-zone. Then,the backstepping fault-tolerant control design is presentedfor the studied non-linear system. The control objective is

that the backstepping fault-tolerant control scheme can ren-der the system output to follow a given desired output ofthe non-linear system in the presence of the unknown time-varying system fault. For the desired system output yd, theproposed backstepping fault-tolerant control must ensurethat all closed-loop system signals are convergent. To fa-cilitate and proceed the backstepping fault-tolerant controldesign for the non-linear system (1), the following lemmasand assumptions are required:

Lemma 1 (Ge & Wang, 2004): For bounded initial con-ditions, if there exists a C1 continuous and positive defi-nite Lyapunov function v(x) satisfying π1(‖x‖) ≤ V(x) ≤π2(‖x‖), such that V̇ (x) ≤ −c1V (x) + c2, where π1, π2:Rn → R are the class K functions, and c1, c2 are the posi-tive constants, then the solution x(t) is uniformly bounded.

Lemma 2 (Zemouche et al., 2005, 2006): Let f : Rn −→R

q and a, b ∈ Rn. Assume that f is differentiable onCo(a, b). Then, there exist constant vectors c1, . . . , cq ∈Co(a, b), ci �= a, ci �= b for i = 1, . . . , q, such that

f (a) − f (b) =( q,n∑

i,j=1eq(i)e

Tn (j )

∂fi

∂xj(ci)

)(a − b), (9)

where eq(i) = (0, . . . , 0, 1︸︷︷︸ith

, 0, . . . , 0) ∈ Rq and Co(a, b)

= λa + (1 − λ)b, 0 ≤ λ ≤ 1.Assumption 1 (Ge & Wang, 2004): There exists two pos-itive constants g and ḡ to make g ≤ |g| ≤ ḡ valid.Assumption 2: For the desired system trajectory yd, thereexist unknown positive constants τ i, such that |y(i)d | ≤τi, i = 1, . . . , n.Assumption 3 (Jiang, Wang, & Soh, 2002): For the un-known time-varying system fault η(t), we assume that itsatisfies |η̇| ≤ δ with δ > 0.Remark 1: In this paper, the fault-tolerant control schemewill be developed for the non-linear system (1) in the pres-ence of the unknown time-varying fault. Generally speak-ing, the studied system (1) can denote a large varietyof non-linear systems, such as the single-link robot sys-tem, the chaotic system, and so on. To the system faultη(t), it is important to develop the fault-tolerant controlscheme for achieving the satisfactory tracking control per-formance. Although the states of non-linear system (1)are in a cascade form, the fault-tolerant control analy-sis of more general non-linear systems can be similarlydeveloped. On the other hand, η(t) in non-linear system(1) can denote the actuator/component fault of the sys-tem. Since the actuator/component fault η0 appears for thesubsystem ẋn = f (x) + gu, the subsystem can be rewrit-ten as ẋn = f (x) + gu + gη0. Defining η = gη0, we haveẋn = f (x) + gu + η(t).

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4 M. Chen et al.

3. Backstepping fault tolerant control for non-linearsystems

3.1. Design of state observer and fault estimator

In this subsection, the state observer and the system faultestimator will be developed for the non-linear system, re-spectively. Motivated by the state observer in Li, Tong, andLi (2012b), the state observer of the non-linear system (1)is proposed as

{˙̂x = Ax̂ + Ly + B(f (x̂) + gD(v(t)) + η̂(t))ŷ = CT x̂, (10)

where η̂(t) is the estimate of the unknown system fault η(t).Define x̃ = x − x̂ and η̃ = η − η̂. Considering (10) and

(2) yields

˙̃x = Ax̃ + B(f (x) − f (x̂) + η̃). (11)

By invoking Lemma 2, there exists ξ (t) ∈Co(x(t), x̂(t)), such that (Zemouche et al., 2005, 2006),

f (x) − f (x̂) =( n∑

i=1eTn (i)

∂f

∂xi(ξ )

)x̃. (12)

Define hi(t) = eTn (i) ∂f∂xi (ξ ) and h(t) =∑n

i=1 hi(t). Then,(12) can be written as

f (x) − f (x̂) = h(t)x̃. (13)

Invoking (13), (11) can be expressed as

˙̃x = Āx̃ + Bη̃, (14)

where Ā = A + Bh(t).For the matrix Ā, by choosing a matrix P = PT > 0,

there always exists a positive definite matrix Q = QT > 0,such that the following matrix inequality hold:

ĀT P + P Ā ≤ −Q. (15)

To achieve the estimate η̂(t) of the unknown time-varying system fault η, we define z = η − γ xn withγ > 0. Then, considering (1), we have

ż = η̇ − γ ẋn = η̇ − γ (f (x) + gD(v(t)) + η(t))= η̇ − γ (f (x) + gD(v(t)) + z + γ xn). (16)

Furthermore, the estimate of intermediate variable z isdesigned as

˙̂z = −γ ẑ − γ (f (x̂) + gD(v(t)) + γ x̂n). (17)

According to z = η − γ xn, the estimate of the unknowntime-varying system fault η can be written as

η̂ = ẑ + γ x̂n. (18)

Defining z̃ = z − ẑ and invoking (18) yields

z̃ = z − ẑ = η − η̂ − γ (xn − x̂n) = η̃ − γ x̃n. (19)

Differentiating (19), and considering (13), (16), and(17) yields

˙̃z = ż − ˙̂z = η̇ − γ z̃ − γ (f (x) − f (x̂)) − γ 2x̃n= η̇ − γ z̃ − γ h(t)x̃ − γ 2x̃n. (20)

Consider the Lyapunov candidate V0 for the estimateerrors of the state observer and the system fault estimatoras follows:

V0 = x̃T P x̃ + 12z̃2. (21)

Considering (11) and (20), the time derivative of V0 isgiven by

V̇0 = ˙̃xT P x̃ + x̃T P ˙̃x + z̃ ˙̃z= x̃T (ĀT P + P Ā)x̃ + 2x̃T PBη̃

+ z̃(η̇ − γ z̃ − γ h(t)x̃ − γ 2x̃n)≤ −x̃T Qx̃ − γ z̃2 + 2x̃T PBη̃

+ z̃(η̇ − γ h(t)x̃ − γ 2x̃n). (22)

Considering the following facts

2x̃T PBη̃ ≤ β0‖x̃‖2 + η̃2 (23)

η̃2 = (z̃ + γ x̃n)2 ≤ 2z̃2 + 2γ 2‖x̃‖2 (24)

2z̃η̇ ≤ z̃2 + δ2 (25)

− 2γ z̃h(t)x̃ ≤ z̃2 + γ 2h2‖x̃‖2 (26)

− 2γ 2z̃x̃n ≤ z̃2 + γ 4‖x̃‖2, (27)

we have

V̇0 ≤ −x̃T (Q − βIn×n)x̃ − (γ − 3.5)z̃2 + 0.5δ2, (28)

where β0 = ‖PB‖2 and β = β0 + 2γ 2 + 0.5γ 2h2 +0.5γ 4.

3.2. Design of backstepping fault tolerant control

In this subsection, the backstepping fault-tolerant controlscheme will be proposed, which is based on the developed

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the state observer and the system fault estimator using thestandard backstepping technique. At the same time, the sta-bility of the whole closed-loop system will be analysed viathe Lyapunov method. To explicitly show the backsteppingfault-tolerant control design, the detailed design process isdescribed as follows.

Step 1: To design the fault-tolerant control strategy, letus define

e1 = x̂1 − yd (29)

e2 = x̂2 − α1, (30)

where α1 is a virtual control law. x̂1 and x̂2 are the estimatevalues of x1 and x2, respectively.

Considering (10), (30), and differentiating e1 with re-spect to time yields

ė1 = x̂2 + l1(y − x̂1) − ẏd= e2 + α1 + l1(y − x̂1) − ẏd . (31)

The virtual control law α1 is designed as

α1 = −k1e1 + ẏd − l1(y − x̂1), (32)

where k1 > 0.Substituting (32) into (31) yields

ė1 = −k1e1 + e2. (33)

Consider the Lyapunov function candidate as

V1 = 12e21. (34)

Invoking (33), the time derivative of V1 is given by

V̇1 = e1ė1 = e1(−k1e1 + e2) = −k1e21 + e1e2. (35)

Step i (2 ≤ i ≤ n − 1): Define

ei = x̂i − αi−1 (36)

ei+1 = x̂i+1 − αi, (37)

where αi − 1 and αi are the virtual control laws. x̂i and x̂i+1are the estimates of xi and xi + 1, respectively.

Considering (10), (36), and differentiating ei with re-spect to time yields

ėi = x̂i+1 + li(y − x̂1) − α̇i−1= ei+1 + αi + li(y − x̂1) − α̇i−1. (38)

The virtual control law αi is designed as

αi = −kiei − ei−1 + α̇i−1 − li(y − x̂1), (39)

where ki > 0.Substituting (39) into (38), we have

ėi = −kiei − ei−1 + ei+1. (40)

Choose the Lyapunov function candidate as

Vi = 12e2i . (41)

Invoking (40), the time derivative of Vi is given by

V̇i = ei ėi = ei(−kiei − ei−1 + ei+1)= −kie2i − ei−1ei + eiei+1. (42)

Step n: Define

en = x̂n − αn−1, (43)

where αn − 1 is the virtual control law of the (n − 1)th stepand x̂n is the estimate of xn.

Considering (5), (10), and differentiating en with respectto time yields

ėn = ln(y − x̂1) + f (x̂) + gD(v(t)) + η̂(t) − α̇n−1= ln(y − x̂1) + f (x̂) + gmv(t) + gd(v(t))

+ η̂(t) − α̇n−1. (44)

The system control law v is designed as

v = 1mg

(−knen − en−1 − f (x̂) + α̇n−1 − ln(y − x̂1)− η̂ − ḡdmsign{en}), (45)

where kn > 0.Substituting (45) into (44), we have

ėn = −knen − en−1 + gd(v(t)) − ḡdmsign{en}. (46)

Consider the Lyapunov function candidate

Vn = 12e2n. (47)

Invoking (46), the time derivative of Vn is given by

V̇n = enėn = en(−knen − en−1 + gd(v(t)) − ḡdmsign{en})≤ −kne2n − enen−1 + ḡdm|en| − ḡdmensign{en})= −kne2n − enen−1. (48)

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6 M. Chen et al.

The above design procedure and analysis of the fault-tolerant control can be summarised in the following theo-rem, which includes the fault-tolerant control results for thenon-linear system (1) with unknown time-varying systemfault and dead-zone based on the backstepping technique.

Theorem 3.1: Considering the non-linear system (1) withunknown time-varying system fault and the dead-zone, thestate observer is proposed as (10), and the fault estimatoris designed according to (17) and (18). Based on the out-puts of the state observer and the fault estimator, the back-stepping tolerant control law is proposed as (45). Then,all closed-loop system signals are semi-globally uniformlystable under the proposed backstepping tolerant controlscheme.

Proof: To analyse the convergence of all signals in theclosed-loop system, the Lyapunov function candidate of thewhole closed-loop fault-tolerant control system is chosenas

V =n∑

i=0Vi. (49)

Differentiating V and considering (28), (35), (42), and(48), we have

V̇ ≤ ≤ −x̃T (Q − βIn×n)x̃ − (γ − 3.5)z̃2

−n∑

i=1kie

2i + 0.5δ2,

≤ −κV + C, (50)where κ and C are given by

κ := min(

λmin(Q−βIn×n)λmax(P )

, 2(γ − 3.5), 2ki)

C := 0.5δ2. (51)

To ensure the closed-loop stability, all design parametersli, P, Q, γ , and ki should be chosen, such that Q − βIn × n> 0 and γ − 3.5 > 0. From (50) and Lemma 1, we canknow that the closed-loop system signals ei, x̃i , and η̃ arebounded. Since e1 and η̃ are bounded, the tracking error andthe estimate error of the unknown system fault are bounded.This concludes the proof. �Remark 2: In the developed backstepping tolerant controllaw, all parameters of the dead-zone for the studied non-linear system is known. For a practical system, the dead-zone can be known when the system actuator is determined.It is our future work to study the backstepping tolerantcontrol scheme for the non-linear system with unknowntime-varying system fault and the unknown dead-zone.

Remark 3: In the ith step, the virtual control law αi in-cludes α̇i−1. However, α̇i−1 cannot be available due to theexisting α̇i−2 and x2. However, αi − 1 is known. Thus, thefirst-order sliding mode differentiator can be employed to

approximate α̇i−1. To produce the corresponding deriva-tives, the general form of the nth-order higher order slid-ing mode differentiator (HOSMD) is designed as (Levant,2003)

ż0 = ζ0 = −�0|z0 − p(t)|n/(n+1) sign(z0 − p(t)) + z1...

żi = ζi = −�i |zi − ζi−1|(n−i)/(n−i+1) sign(zi − ζi−1) + zi+1...

żn−1 = ζn−1 = −�n−1|zn−1 − ζn−2|1/2 sign(zn−1 − ζn−2) + znżn = −�n sign(zn − ζn−1), (52)

where zi and ζ i are the states of the system (52), �0, �1, . . . ,�n are designed parameters, p(t) is the known function. Theaim of HOSMD is to make ζ j approximate the differentialterm p(t)(j + 1) to arbitrary accuracy, i = 0, 1, . . . , n, j = 0, 1,. . . , n − 1. To obtain the estimate of differential term α̇i−1,the order of HOSMD should be only 1 and p(t) = αi − 1.

4. Simulation study

In this section, simulation results of an example are givento illustrate the effectiveness of the proposed backsteppingfault-tolerant control scheme for a non-linear system withunknown time-varying system fault and the dead-zone. Inthe simulation study, let us consider the single-link robotsystem which is shown in Figure 1. Its motion dynamicscan be described as (Wang, Chai, & Zhangh, 2010)

Mq̈ + 0.5m0gl sin(q) = τ + η0(t)y = q, (53)

where g = 9.8 m/s2 is the acceleration due to gravity, M isthe inertia, q is the angle position, q̇ is the angle velocity, q̈

Figure 1. A single-link robot system.

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0 5 10 15 200

0.5

1

1.5

2

2.5

Time [s]

y

yd

Figure 2. Tracking control result of case 1.

is the angle acceleration, l is the length of the link, m0 is themass of the link, τ is the control force, and η0(t) denotesthe actuator/component fault in the system. Setting x1 = q,x2 = q̇, and u = τ , then Equation (53) can be rewritten inthe following non-linear form:

ẋ1 = x2ẋ2 = f (x) + gu + ηy = x1, (54)

where f (x) = −0.5m0gl sin(x1)M

, g = 1M

, u = D(v(t)), and η =η0(t)M

is the unknown time-varying system fault.For the single-link robot system (54) with unknown

system fault and the dead-zone, the state observer is pro-posed as (10) and the fault estimator is designed accordingto (17) and (18). The backstepping tolerant control law isproposed as (45). In the backstepping tolerant control law,

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

Time [s]

Tra

ckin

g er

ror

Figure 3. Tracking error of case 1.

0 5 10 15 20−60

−40

−20

0

20

40

60

Time [s]

u

Figure 4. Control input of case 1.

the first-order sliding mode differentiator is employed toestimate α̇1. In this simulation, all system parameters of thesingle-link robot are chosen as m0 = 1 kg, M = 0.5 kg m2,l = 1 m and η = sin (0.5t) + 0.2cos (t). The parameters ofdead-zone are chosen as m = 1, br = 0.2 and bl = −0.3. Atthe time, the input saturation is considered for the single-link robot system (54) with umax = 50 and umin = −50. Toproceed the design of backstepping fault-tolerant control,all design parameters are chosen as l1 = 8, l2 = 15, γ = 5,k1 = 10 and k2 = 20. The initial state conditions are chosenas x1(0) = 0.1, x2(0) = 0.2, and η̂(0) = 0.Case 1. For the constant tracking signalHere, the desired trajectory of the single-link robot system(54) is taken as yd = 2. Under the proposed fault-tolerantcontrol (45), from Figures 2 and 3, the satisfactory trackingperformance is noted and the tracking error is convergentfor the single-link robot system (54) in the presence of theunknown system fault and the dead-zone. From Figure 4,

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

Time [s]

Fau

lt es

timat

e

η

η̂

Figure 5. Fault estimate result of case 1.

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8 M. Chen et al.

0 5 10 15 20−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time [s]

Fau

lt es

timat

e er

ror

Figure 6. Fault estimate error of case 1.

0 5 10 15 20−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

yy

d

Figure 7. Tracking control result of case 2.

0 5 10 15 20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Time [s]

Tra

ckin

g er

ror

Figure 8. Tracking error of case 2.

0 5 10 15 20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Time [s]

Tra

ckin

g er

ror

Figure 9. Control input of case 2.

the control input is bounded and convergent. The faultapproximate ability of the developed fault estimator isshown in Figures 5 and 6. From Figure 5, the satisfactory es-timate performance is obtained by using the developed faultestimator. The estimate error of the unknown system fault ηis presented in Figure 6, which converges to zero with timechange. Thus, the developed fault-tolerant control (45) canaccurately track the given constant desired trajectory.

Case 2. For the time-varying tracking signalIn this case, the desired trajectory is chosen as yd = cos (t)+ sin (0.5t). From Figure 7, we note that the tracking perfor-mance is satisfactory for the single-link robot system (54)in the presence of the unknown system fault. The trackingerror is presented in Figure 8 which is convergent. Accord-ing to Figure 9, the bounded control input is obtained. Thegood fault estimate ability of the developed fault estimatoris shown in Figures 10 and 11. According to Figures 10and 11, we can see that the fault estimate error is bounded

0 5 10 15 20−30

−20

−10

0

10

20

30

40

50

60

Time [s]

u

Figure 10. Fault estimate result of case 2.

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0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

Time [s]

Fau

lt es

timat

e

η

η̂

Figure 11. Fault estimate error of case 2.

and convergent. From these simulation results, we can con-clude that the proposed backstepping fault-tolerant controlscheme has a good tracking control performance for thenon-linear system with the unknown system fault and thedead-zone.

In accordance with above simulation results, the satis-factory tracking control performances are achieved underthe developed backstepping fault-tolerant control scheme ofthe single-link robot system with the unknown system faultand the dead-zone. Thus, the developed backstepping fault-tolerant control scheme is valid for the non-linear system.

5. Conclusion

Backstepping fault-tolerant control approach has been de-veloped for a class of non-linear systems in the presence ofthe unknown time-varying system fault and the dead-zonein this paper. The state observer and the fault estimator havebeen proposed for the studied non-linear system. Then, thebackstepping fault-tolerant control has been developed us-ing the outputs of the state observer and the fault estimator.By utilising the Lyapunov method, the uniformly asymptoti-cal convergence of all closed-loop signals have been proved.Finally, simulation results have been given to illustrate theeffectiveness of the developed backstepping fault-tolerantcontrol approach. In our developed fault-tolerant controlscheme, the dead-zone is required as known. The futurework is to study the backstepping tolerant control schemefor the non-linear system with unknown time-varying sys-tem fault and the unknown dead-zone. Furthermore, the in-put saturation can be considered for the studied non-linearsystem (1) and the developed fault-tolerant control can beextended to the MIMO non-linear system.

AcknowledgementsThe authors gratefully acknowledge the helpful commentsand suggestions of the reviewers, which have improved thepresentation.

FundingThis work is partially supported by National Natural ScienceFoundation of China [grant number 61174102]; Program for NewCentury Excellent Talents in University of China [grant numberNCET-11-0830]; Jiangsu Natural Science Foundation of China[grant number SBK20130033], [grant number SBK2011069];Specialised Research Fund for the Doctoral Program of HigherEducation [grant number 20133218110013].

Notes on contributorsMou Chen was born in Sichuan, China, in1975. He received his BSc degree in mate-rial science and engineering at Nanjing Uni-versity of Aeronautics & Astronautics, Nan-jing, China, in 1998, the MSc and the PhDdegrees in automatical control engineeringat Nanjing University of Aeronautics & As-tronautics, Nanjing, China, in 2004. He iscurrently a full professor in College of Au-

tomation Engineering at Nanjing University of Aeronautics & As-tronautics, China. He was an academic visitor at the Department ofAeronautical and Automotive Engineering, Loughborough Uni-versity, UK, from November 2007 to February 2008. From June2008 to September 2009, he was a research fellow in the De-partment of Electrical and Computer Engineering, the NationalUniversity of Singapore. His research interests include non-linearsystem control, intelligent control, and flight control.

Bin Jiang was born in Jiangxi, China, in1966. He obtained the PhD degree in auto-matic control from Northeastern University,Shenyang, China, in 1995. He had ever beena postdoctoral fellow or a research fellow inSingapore, France and USA, respectively.Now he is a professor and dean of Col-lege of Automation Engineering in NanjingUniversity of Aeronautics and Astronautics,

China. He currently serves as associate editor or editorial boardmember for a number of journals such as IEEE Transactions onControl Systems Technology; International Journal of SystemsScience; International Journal of Control, Automation and Sys-tems; International Journal of Innovative Computing, Informationand Control; International Journal of Applied Mathematics andComputer Science; Acta Automatica Sinica; Journal of Astronau-tics. He is a senior member of IEEE, a member of IFAC TechnicalCommittee on Fault Detection, Supervision, and Safety of Tech-nical Processes. His research interests include fault diagnosis andfault-tolerant control and their applications.

William Guo received his PhD degree fromThe University of Western Australia in1999. From 1999–2001 as postdoctoral re-search fellow at Curtin University and EdithCowan University in Australia, he has beena faculty member at Edith Cowan University(2002–2007), and then Central QueenslandUniversity (2007–). Professor Guo is cur-rently the senior deputy dean of the School

of Engineering and Technology at Central Queensland Univer-sity. He was the Chair of ICT Programs Committee and regu-larly served as the acting dean of the School of Information andCommutation Technology at Central Queensland University in2010–2012. He has significant experience in academic governancethrough his services in various committees and boards since 2009,

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including memberships of Faculty Education Committee (2009–2012), University Education Committee (2011–2012), UniversityAcademic Board (2013–), and Australian Council of Deans ofICT (2013–), and as an executive member of Australian Councilof Professors and Heads of IS (2012–). He is the editor-in-chief ofinternational journal Progress in Intelligent Computing and Appli-cations (PICA), and guest editor for Journal of Networks, Journalof Software, Journal of Computers and Mathematical Problems inEngineering.

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Abstract1. Introduction2. Problem statement3. Backstepping fault tolerant control for non-linear systems3.1. Design of state observer and fault estimator3.2. Design of backstepping fault tolerant control

4. Simulation study5. ConclusionAcknowledgementsFundingReferences

fault-tolerant control for a class of non-linear systems with ......robust adaptive control was...

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