IEEE TRANSACTIONS ON CYBERNETICS 1 Adaptive Fault … · uncertain nonlinear large-scale systems under the integrated effects of the unknown dead zone, actuator fault, and unknown

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IEEE TRANSACTIONS ON CYBERNETICS 1

Adaptive Fault-Tolerant Control of UncertainNonlinear Large-Scale Systems With

Unknown Dead ZoneMou Chen, Member, IEEE, and Gang Tao, Fellow, IEEE

Abstract—In this paper, an adaptive neural fault-tolerant con-trol scheme is proposed and analyzed for a class of uncertainnonlinear large-scale systems with unknown dead zone and exter-nal disturbances. To tackle the unknown nonlinear interactionfunctions in the large-scale system, the radial basis functionneural network (RBFNN) is employed to approximate them.To further handle the unknown approximation errors and theeffects of the unknown dead zone and external disturbances,integrated as the compounded disturbances, the correspondingdisturbance observers are developed for their estimations. Basedon the outputs of the RBFNN and the disturbance observer,the adaptive neural fault-tolerant control scheme is designedfor uncertain nonlinear large-scale systems by using a decen-tralized backstepping technique. The closed-loop stability of theadaptive control system is rigorously proved via Lyapunov anal-ysis and the satisfactory tracking performance is achieved underthe integrated effects of unknown dead zone, actuator fault,and unknown external disturbances. Simulation results of amass–spring–damper system are given to illustrate the effec-tiveness of the proposed adaptive neural fault-tolerant controlscheme for uncertain nonlinear large-scale systems.

Index Terms—Adaptive fault control, backstepping control,disturbance observer, large-scale systems, neural network (NN).

I. INTRODUCTION

MANY physical systems such as power system,aerospace system, chemical engineering system, and

telecommunication network are composed of interconnectionsof lower-dimensional subsystems [1]–[4]. Unique technicalissues naturally arise from control of the large-scale systemsto achieve the corresponding control goals under subsysteminteractions. At the same time, the large-scale system usuallypossesses nonlinear and uncertain characteristic which willfurther enhance the control design difficulty [5]. It is well

Manuscript received February 9, 2015; revised May 5, 2015; acceptedJune 25, 2015. This work was supported in part by the Jiangsu NaturalScience Foundation of China under Grant SBK20130033, in part by theNational Natural Science Foundation of China under Grant 61374130 andGrant 61573184, in part by the Program for New Century Excellent Talentsin University of China under Grant NCET-11-0830, and in part by the SixTalents Peak Project of Jiangsu Province under Grant 2012-XXRJ-010. Thispaper was recommended by Associate Editor T. H. Lee.

M. Chen is with the College of Automation Engineering, NanjingUniversity of Aeronautics and Astronautics, Nanjing 210016, China (e-mail:[email protected]).

G. Tao is with the Department of Electrical and Computer Engineering,University of Virginia, Charlottesville, VA 22903 USA.

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCYB.2015.2456028

known that the decentralized control structure naturally alle-viates the computational burden associated with a centralizedcontrol scheme. Hence, the decentralized robust control as aneffective control approach has been extensively studied forthe uncertain nonlinear large-scale system [6]. Up to now,there has been an increasing interest in the development ofdecentralized control theories for large-scale systems [7], [8].In [9], a fuzzy adaptive decentralized output-feedback con-trol design was proposed for large-scale nonlinear systemswith dynamical uncertainties. Adaptive fuzzy decentralizedcontrol was studied for large-scale nonlinear systems withtime-varying delays and unknown high-frequency gain signin [10]. In [11], a global decentralized robust stabilizationwas studied for interconnected uncertain nonlinear systemswith multiple inputs. In the practice, the large-scale sys-tem may suffer the integrated effects coming from actu-ator fault, external disturbance, and unknown dead zone.Thus, the adaptive decentralized fault-tolerant control shouldbe further investigated for uncertain nonlinear large-scalesystems.

Control systems are often subjected to faults which can becaused by actuators, sensors, or system faults [12]. It is impor-tant to design the efficient fault-tolerant control scheme to keepthe stable and acceptable control performance when faultsoccur [13]. An adaptive fault diagnosis and fault-tolerant con-trol was developed for multi-input and multi-output (MIMO)nonlinear uncertain systems in [14]. Fault detection and fault-tolerant control were studied for a civil aircraft using thesliding mode method in [15]. Actuator faults as a com-mon fault widely exist in control systems, which should beefficiently tackled to guarantee closed-loop system stabilityand tracking performance when actuator faults occur [16].An adaptive actuator failure compensation controller wasproposed using output feedback in [17]. In [18], a virtualgrouping based adaptive actuator failure compensation tech-nique was studied for MIMO nonlinear systems. In particular,some fault-tolerant control results have been achieved forthe nonlinear large-scale system. In [19], an observer-basedadaptive decentralized fuzzy fault-tolerant controller was pro-posed for nonlinear large-scale systems with actuator failures.Decentralized fault-tolerant control was studied for a classof interconnected nonlinear systems in [20]. However, thefault-tolerant control considering the unknown dead zoneand the unknown external disturbance needs to be furtherinvestigated.

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2 IEEE TRANSACTIONS ON CYBERNETICS

The dead zone characteristic as one of the most importantnonsmooth input nonlinearity widely exists in lots of practi-cal systems. The related robust control problem has attractedextensive attentions [21]–[24]. An adaptive dead zone com-pensation method was proposed for output-feedback canonicalsystems in [25]. In [26], a neural-hybrid control was developedfor systems with sandwiched dead zones. Adaptive outputcontrol schemes were developed for nonlinear systems withuncertain dead zone nonlinearity in [27] and for uncertainnonlinear systems with nonsymmetric dead zone input in [28].For a large-scale system with actuator faults, the input deadzone will further degrade the tracking control performance.Thus, the decentralized control considering the unknown deadzone should be further developed for the uncertain nonlinearlarge-scale system. In [29], a decentralized variable struc-ture controller was proposed for uncertain large-scale systemscontaining a dead zone. Adaptive fuzzy decentralized outputfeedback control was studied for nonlinear large-scale sys-tems with unknown dead zone inputs in [30]. However, theunknown external disturbance needs to be efficiently tack-led in the adaptive fault-tolerant control design for uncertainnonlinear large-scale systems.

One of the key challenges in decentralized control foruncertain nonlinear large-scale systems is the developmentof techniques for dealing with the interaction uncertain-ties and the unknown disturbances. In this paper, the radialbasis function neural network (RBFNN) is employed toapproximate the unknown nonlinear continuous interconnec-tion functions [31]–[33]. In many existing works of the con-trol area, universal function approximators [such as fuzzylogic systems and neural networks (NNs)] are employed totackle the system uncertainty [34]–[39]. To efficiently handlethe unknown approximation error and the unknown distur-bance, the disturbance observer is developed to estimatethe compounded disturbance which consists of the unknownapproximation error, the effect of the unknown dead zone,and the unknown disturbance. In the past decades, variousdisturbance observer design and disturbance-observer-basedcontrol schemes have been extensively studied for systemwith external disturbances [40]. In [41], a sliding mode con-troller using disturbance observer was proposed for systemswith mismatched uncertainties. An adaptive fuzzy trackingcontroller was developed for a class of uncertain MIMOnonlinear systems using disturbance observer in [42]. Thedisturbance attenuation and rejection issue were investigatedfor a class of MIMO nonlinear systems based on the distur-bance observer [43]. However, the disturbance observer needsto be further developed for fault-tolerant control of uncertainnonlinear large-scale systems with unknown dead zone.

To develop fault-tolerant control scheme of uncertain non-linear large-scale systems, a decentralized backstepping con-trol technique is employed in this paper. During the lastdecades, various backstepping control schemes have beendeveloped [44]–[49]. Decentralized dynamic surface controlwas studied for large-scale interconnected systems in strict-feedback form using NNs in [50]. In [51], an adaptive NNdecentralized backstepping output-feedback control was pro-posed for nonlinear large-scale systems with time delays.

However, there are few backstepping control results for uncer-tain nonlinear large-scale systems with unknown dead zone,actuator fault, and unknown disturbances.

This paper aims at developing an adaptive neural fault-tolerant control scheme to track the desired trajectories ofuncertain nonlinear large-scale systems under the integratedeffects of the unknown dead zone, actuator fault, and unknownexternal disturbances. The organization of this paper is asfollows. The problem statement is given in Section II. Theadaptive neural fault-tolerant control scheme based on the dis-turbance observer and the RBFNN is developed using thebackstepping method in Section III. Simulation studies areprovided in Section IV to demonstrate the effectiveness ofthe proposed adaptive neural fault-tolerant control approach,followed by some concluding remarks in Section V.

II. PROBLEM FORMULATION

Consider a nonlinear large-scale system that is composedof N subsystems and the ith subsystem �i(i = 1, . . . ,N) canbe described by

�i

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

xi,1 = xi,2 + fi,1( y, xi,1)+ di,1(t)xi,2 = xi,3 + fi,2( y, xi,2)+ di,2(t)...

xi,ni−1 = xi,ni + fi,ni−1( y, xi,ni−1)+ di,ni−1(t)xi,ni = fi,ni( y, xi,ni)+ gi,ni( y, xi,ni)ui + di,ni(t)yi = xi,1

(1)

where xi,j, i = 1, . . . ,N, j = 1, . . . , ni are the subsystemstates; xi,j = [xi,1, xi,2, . . . , xi,j]T ∈ Rj is the group statevector; ui ∈ R is the control input; yi ∈ R is the subsys-tem output and y = [y1, . . . , yN] is the whole system outputvector; fi,j( y, xi,j) ∈ R are the unknown smooth functions;gi,ni( y, xi,ni) ∈ R is an unknown control gain function; anddi,j(t) ∈ R are the system unknown external disturbances. Thesystem states xi,j are assumed to be measurable.

However, in practical engineering, actuators may becomefaulty. Bias faults and gain faults are two kinds of actuatorfaults that are commonly occurring in the practice. An actuatorbias fault can be described as

u fi (t) = ui(t)+ ζi(t) (2)

where ζi(t) denotes a bounded signal, and an actuator gainfault can be described as

u fi (t) = ρi(xi,ni)ui(t) (3)

where 0 < ρi(xi,ni) ≤ 1, which denotes the remaining controlrate. Here, ρi(xi,ni) at the failure time instant tf is assumed tobe unknown.

The two actuator faults can be uniformly described as

u fi (t) = ρiui(t)+ ζi(t). (4)

On the other hand, there may exist the actuator dead zonenonlinearity in an uncertain nonlinear large-scale system. Insuch case, the control input ui can be expressed as

ui(t) = DZ(vi(t)) (5)


CHEN AND TAO: ADAPTIVE FAULT-TOLERANT CONTROL OF UNCERTAIN NONLINEAR LARGE-SCALE SYSTEMS 3

where vi is the input of the dead zone and DZ(·) denotes adead zone operator.

In this paper, the considered dead zone is described as [23]

ui(t) = DZ(vi(t))

=⎧⎨

⎩

mi(vi(t)− bir), for vi(t) ≥ bir

0, for bil < vi(t) < bir

mi(vi(t)− bil), for vi(t) ≤ bil

(6)

where mi > 0, bir > 0, and bil < 0 are the unknown deadzone parameters.

To develop a robust fault-tolerant control scheme, the deadzone model (6) is rewritten as [23]

DZ(vi(t)) = mivi(t)+ ηi(vi(t)) (7)

where ηi(vi(t)) is given by

ηi(vi(t)) =⎧⎨

⎩

−mibir, for vi(t) ≥ bir

−mivi(t), for bil < vi(t) < bir

−mibil, for vi(t) ≤ bil.

(8)

For a practical system, we know that the parameter mi ofthe dead zone is bounded. Thus, from (8), we have

|ηi(vi(t)| ≤ ηim (9)

where ηim = max{mi maxbir max,−mi maxbil min}, mi max is theupper-bound of the parameter mi, bir max is the upper-boundof the parameter bir, and bil min is the lower-bound of theparameter bil.

Considering (4), the uncertain nonlinear large-scale sys-tem (1) with the unknown dead zone and the actuator faultcan be rewritten as

�i

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

xi,1 = xi,2 + fi,1( y, xi,1)+ di,1(t)xi,2 = xi,3 + fi,2( y, xi,2)+ di,2(t)...

xi,ni−1 = xi,ni + fi,ni−1( y, xi,ni−1)+ di,ni−1(t)xi,ni = fi,ni( y, xi,ni)+ gi,ni( y, xi,ni)vi

+ Di,ni(t)yi = xi,1

(10)

where gi,ni( y, xi,ni) = gi,ni( y, xi,ni)ρimi and Di,ni(t) =di,ni(t)+ gi,ni( y, xi,ni)ρiηi + gi,ni( y, xi,ni)ζi.

For the given desired trajectory yi,d, the control objective isthat the adaptive fault-tolerant control input vi is designed toensure the tracking error of the uncertain nonlinear large-scalesystem convergent in the presence of the system uncertain-ties, unknown dead zone, actuator faults, and time-varyingunknown external disturbances.

In this paper, RBFNNs are employed to approximate theuncertain interconnection functions fi,j( y, xi,j) of the non-linear large-scale system (1) and the nonlinear disturbanceobserver is developed to estimate the compounded disturbanceby combing the NN approximation error with the unknownexternal disturbance. To facilitate proceed the design of adap-tive neural fault-tolerant control scheme, the following lemmasand assumptions are required.

Lemma 1 [52]: As a class of linearly parameterized NN,RBFNNs are employed to approximate the continuous func-tion f (Z) : Rq → R which can be written as

f (Z) = θTφ(Z)+ ε (11)

where Z = [z1, z2, . . . , zq]T ∈ Rq is the input vector ofthe NN, θ ∈ Rp is a weight vector of the NN, φ(Z) =[φ1(Z), φ2(Z), . . . , φp(Z)]T ∈ Rp is the basis function vec-tor, and ε is the approximation error of the NN. The optimalweight value θ of RBFNN is expressed as

θ = arg minθ∈f

[

supz∈SZ

∣∣∣ f(

Z|θ)

− f (Z)∣∣∣

]

(12)

where f = {θ : ‖θ‖ ≤ M} is a valid field of the estimateparameter θ , M is a design parameter, and SZ ⊂ Rn is anallowable set of the state vector. Using the optimal weightvalue, we have

f (Z) = θTφ(Z)+ ε∗

|ε∗| ≤ ε (13)

where ε∗ is the optimal approximation error and ε > 0 is theupper bound of the approximation error.

Lemma 2 [52]: For bounded initial conditions, if thereexists a C1 continuous and positive definite Lyapunov func-tion V(x) satisfying π1(‖x‖) ≤ V(x) ≤ π2(‖x‖), such thatV(x) ≤ −c1V(x)+c2, where π1, π2 : Rn → R are class K func-tions and c1 and c2 are positive constants, then the solutionx(t) is uniformly bounded.

Assumption 1 [52]: There exist positive constants gi,

i = 1, . . . ,N and gi, such that gi

≤ |gi,ni( y, xi,ni)| ≤ gi.Without losing generality, we shall assume that gi,ni arepositive in the adaptive fault-tolerant control design.

Assumption 2: There exist the unknown positive constantsβ0i,j, i = 1, . . . ,N, j = 1, . . . , ni and β1i,j such that theexternal disturbance satisfy |di,j| ≤ β0i,j and |di,j| ≤ β1i,j.

Assumption 3 [52]: There exist constants gdi > 0, i =

1, 2, . . . , n such that | ˙gi,ni(.)| ≤ gdi in the compact set j.

Assumption 4: For the desired system trajectory yi,d, thereexist unknown positive constants τi such that |y(ni)

i,d | ≤ τi,i = 1, . . . ,N.

Remark 1: Considering 0 < ρi < 1, we have |gi,ni(.)| =|gi,ni( y, xi,ni)ρimi| ≤ |gi,ni( y, xi,ni)mi|. Invoking Assumption 1yields g

i0≤ |gi,ni | ≤ gi0, where g

i0and gi0 are positive con-

stants. For a practical system, the remaining control rate ρi

satisfies ρi ≤ ρi0 with ρi0 > 0. Considering the unknown con-stant mi of dead zone and Assumption 3, we have |gi,ni | ≤ ϑi

with ϑi > 0. In Assumption 2, we assume that the externaldisturbance and the time derivative of the external disturbanceare bounded. Usually, the time-dependent disturbance di,j(t) isbounded. At the same time, if the external disturbance is fastchanging, the system can become uncontrollable in the prac-tice. Hence, it is reasonable to assume that the time derivationof the external disturbance di,j(t) is bounded. Otherwise, thenonlinear larger scale system may be uncontrollable.

Remark 2: Comparing with the existing actuator faultmodel, the state-dependent nonlinear actuator gain fault modelis considered using ρi(xi,ni) in this paper. Generally speaking,the actuator fault will degrade the control performance and fur-ther affect the system states. On the contrary, the undesirablesystem states may lead to more serious actuator fault. Namely,there exists interaction between the fault and the system states.



Thus, the nonlinear remaining control rate ρi given in (3) ismore suitable for a practical control problem.

III. ADAPTIVE NEURAL FAULT-TOLERANT CONTROL

DESIGN USING DISTURBANCE OBSERVER

In this section, the adaptive neural fault-tolerant controlscheme will be proposed based on the output of the designeddisturbance observer and RBFNN by using the standard back-stepping technique. The detailed design process of adaptiveneural fault-tolerant control consists of: 1) the disturbanceobserver design in each step for the ith subsystem; 2) thevirtual control law design; and 3) the stability analysis oferror systems. The detailed design of the ith subsystem�i(i = 1, . . . ,N) is presented as follows.

Step 1: To design the adaptive neural fault-tolerant control,we define

ei,1 = xi,1 − yi,d (14)

ei,2 = xi,2 − αi,1 − yi,d (15)

where αi,1 is a designed virtual control law.Considering (10) and differentiating ei,1 with respect to time

yields

ei,1 = xi,1 − yi,d = xi,2 + fi,1( y, xi,1)+ di,1(t)− yi,d. (16)

The RBFNN is employed to approximate the unknown termλi,1fi,1( y, xi,1). According to (13) yields

ei,1 = xi,2 + λ−1i,1 θ

Ti,1φ(Zi,1)+ λ−1

i,1 ε∗i,1 + di,1(t)− yi,d (17)

where λi,1 > 0 is a design parameter and Zi,1 = [y, xi,1]T .Defining Di,1 = λ−1

i,1 ε∗i,1 + di,1(t), we have

ei,1 = xi,2 + λ−1i,1 θ

Ti,1φ(Zi,1)+ Di,1(t)− yi,d. (18)

A. Disturbance Observer Design

For enhancing the disturbance rejection ability, the nonlin-ear disturbance observer is adopted to estimate the unknowncompounded disturbance in this paper. From the approxima-tion ability of the RBFNN, we know that the time derivative ofε∗i,1 is bounded. At the same time, the time derivative of di,1(t)is also bounded according to Assumption 2. Thus, the timederivative of the compounded disturbance Di,1(t) is bounded.Namely, we have

|Di,1| ≤ βi,10, |Di,1| ≤ βi,11 (19)

where βi,10 and βi,11 are unknown positive constants.The nonlinear disturbance observer is proposed as

Di,1 = λi,1(ei,1 − ξi,1)

ξi,1 = xi,2 + λ−1i,1 θ

Ti,1φ(Zi,1)+ Di,1 − yi,d. (20)

Considering (18) and (20), we obtain

˙Di,1 = λi,1(ei,1 − ξi,1

)

= λi,1

(Di,1 − Di,1 − λ−1

i,1 θTi,1φ(Zi,1)+ λ−1

i,1 θTi,1φ(Zi,1)

)

= λi,1Di,1 − θTi,1φ(Zi,1) (21)

where θi,1 = θi,1 − θi,1, and Di,1 = Di,1 − Di,1.

Considering (21) yields

˙Di,1 = Di,1 − ˙Di,1 = Di,1 − λi,1Di,1 + θTi,1φ(Zi,1). (22)

B. Adaptive Virtual Control Law

The parameter updated law of θi,1 is designed as

˙θi,1 = γi,1

(λ−1

i,1 φ(Zi,1

)ei,1 − δi,1θi,1

)(23)

where γi,1 > 0 and δi,1 > 0 are design parameters.Invoking (15) and (18), we have

ei,1 = ei,2 + αi,1 + λ−1i,1 θ

Ti,1φ(Zi,1)+ Di,1(t). (24)

The virtual control law αi,1 is proposed as

αi,1 = −ki,1ei,1 − λ−1i,1 θ

Ti,1φ(Zi,1)− Di,1(t) (25)

where ki,1 > 0 is a designed parameter.Substituting (25) into (24), we obtain

ei,1 = −ki,1ei,1 + ei,2 + λ−1i,1 θ

Ti,1φ(Zi,1)

− λ−1i,1 θ

Ti,1φ(Zi,1)+ Di,1(t)− Di,1(t)

= −ki,1ei,1 + ei,2 − λ−1i,1 θ

Ti,1φ(Zi,1)+ Di,1. (26)

Step j (2≤ j ≤ ni−1): Define

ei,j = xi,j − αi,j−1 − y( j−1)i,d (27)

ei,j+1 = xi,j+1 − αi,j − y( j)i,d (28)

where y(i−1)i,d is the (i − 1)th order time-derivative of yi,d and

y(i)i,d is the ith order time-derivative of yi,d, respectively. αi,j−1and αi,j are the virtual control laws of the ( j − 1)th step andthe jth step of the ith subsystem, respectively.

Considering (10) and differentiating ei,j with respect to timeyields

ei,j = xi,j − αi,j−1 − y( j)i,d = xi,j+1 + fi,j

(y, xi,j

)

+ di,j(t)− αi,j−1 − y( j)i,d . (29)

The RBFNN is employed to approximate the unknown termλi,j fi,j( y, xi,j). According to (13) yields

ei,j = xi,j+1 + λ−1i,j θ

Ti,jφ

(Zi,j

) + λ−1i,j ε

∗i,j

+ di,j(t)− αi,j−1 − y( j)i,d (30)

where λi,j > 0 is a design parameter and Zi,j = [y, xi,j]T .

C. Disturbance Observer Design

Defining Di,j = λ−1i,j ε

∗i,j + di,j(t), we have

ei,j = xi,j+1 + λ−1i,j θ

Ti,jφ

(Zi,j

) + Di,j(t)− αi,j−1 − y( j)i,d . (31)

From the definition of the compounded disturbance Di,j(t),we know that its time derivative is bounded. Namely, we have

∣∣Di,j

∣∣ ≤ βi,j0,

∣∣Di,j

∣∣ ≤ βi,j1 (32)

where βi,j0 and βi,j1 are unknown positive constants.



The nonlinear disturbance observer is proposed as

Di,j = λi,j(ei,j − ξi,j)

ξi,j = xi,j+1 + λ−1i,j θ

Ti,jφ

(Zi,j

) + Di,j

− αi,j−1 − y( j)i,d . (33)

Considering (31) and (33), we obtain

˙Di,j = λi,j(ei,j − ξi,j

)

= λi,j

(Di,j − Di,j − λ−1

i,j θTi,jφ

(Zi,j

) + λ−1i,j θ

Ti,jφ

(Zi,j

))

= λi,jDi,j − θTi,jφ

(Zi,j

)(34)

where θi,j = θi,j − θi,j and Di,j = Di,j − Di,j.Considering (34) yields

˙Di,j = Di,j − ˙Di,j = Di,j − λi,jDi,j + θTi,jφ

(Zi,j

). (35)

D. Adaptive Virtual Control Law

The parameter updated law of θi,j is designed as

˙θi,j = γi,j

(λ−1

i,j φ(Zi,j

)ei,j − δi,jθi,j

)(36)

where γi,j > 0 and δi,j > 0 are design parameters.Invoking (28) and (31), we have

ei,j = ei,j+1 + αi,j + λ−1i,j θ

Ti,jφ

(Zi,j

) + Di,j(t)− αi,j−1. (37)

The virtual control law αi,j is proposed as

αi,j = −ki,jei,j − λ−1i,j θ

Ti,jφ

(Zi,j

) − Di,j(t)− ei,j−1 + αi,j−1

(38)

where ki,j > 0 is a design constant.Substituting (38) into (37), we obtain

ei,j = −ki,jei,j + ei,j+1 − ei,j−1 + λ−1i,j θ

Ti,jφ

(Zi,j

)

− λ−1i,j θ

Ti,jφ

(Zi,j

) + Di,j(t)− Di,j(t)

= −ki,jei,j + ei,j+1 − ei,j−1

− λ−1i,j θ

Ti,jφ

(Zi,j

) + Di,j. (39)

Step ni: Define

ei,ni = xi,ni − αi,ni−1 − y(ni−1)i,d (40)

where y(ni−1)i,d is the (ni − 1)th order time-derivative of yi,d.

αi,ni−1 is the virtual control law of the (ni − 1)th step of theith subsystem.

Considering (10) and (40) and differentiating ei,ni withrespect to time yields

ei,ni = xi,ni − αi,ni−1 − y(n)i,d = fi,ni

(y, xi,ni

)

+ gi,ni

(y, xi,ni

)vi + Di,ni − αi,ni−1 − y(ni)

i,d . (41)

Considering Di,ni(t) = di,ni(t) + gi,ni( y, xi,ni)ρiηi +gi,ni( y, xi,ni)ζi and Remark 1, we have

∣∣Di,ni

∣∣ ≤ βi,ni0,

∣∣Di,ni

∣∣ ≤ βi,ni1 (42)

where βi,ni0 and βi,ni1 are unknown positive constants.

E. Desired Control Law

The desired fault-tolerant control v∗i can be designed as

v∗i = −ki,ni ei,ni − Di,ni − ei,ni−1

− 1

gi,ni

(fi,ni

(y, xi,ni

) − αi,ni−1 − y(ni)i,d

)(43)

where ki,ni > 0 is a design constant.Define hi(x) = 1/(gi,ni)( fi,ni( y, xi,ni) − αi,ni−1 − y(ni)

i,d ).By employing RBFNNs to approximate hi(x) and consider-ing (13), v∗

i can be expressed as

v∗i = −ki,ni ei,ni − θT

i,niφ(Zi,ni

) − ε∗i,ni− Di,ni − ei,ni−1 (44)

where Zi,ni = [xi,ni , y, αi,ni−1, y(ni)id ]T .

F. Adaptive Control Law

Considering the unknown θi,ni , ε∗i,ni

, and Di,ni , the adaptivefault-tolerant control law v is proposed as

vi = −ki,ni ei,ni − θTi,niφ(Zi,ni

) − Di,ni − ei,ni−1 (45)

where Di,ni is the estimate of Di,ni and θi,ni is the estimate ofθi,ni which is updated by

˙θi,ni = γi,ni

(φ(Zi,ni

)ei,ni − δi,ni θi,ni

)(46)

where γi,ni > 0 and δi,ni > 0 are design parameters.Considering (41) and (45), we obtain

ei,ni = gi,ni

(y, xi,ni

)(θT

i,niφ(Zi,ni

) + ε∗i,ni

)

+ gi,ni

(y, xi,ni

)(−ki,ni ei,ni − θTi,niφ(Zi,ni

))

+ gi,ni

(y, xi,ni

)(−Di,ni − ei,ni−1

)+ Di,ni . (47)

Define θi,ni = θi,ni − θi,ni , we have

ei,ni = gi,ni

(y, xi,ni

)(−θTi,niφ(Zi,ni

) + ε∗i,ni− ki,niei,ni

)

+ gi,ni

(y, xi,ni

)(−Di,ni − ei,ni−1

)+ Di,ni . (48)

G. Disturbance Observer Design

To facilitate the design of nonlinear disturbanceobserver, (41) can be also written as

ei,ni = l−1i,niϕi(xi,ni , vi

) + Di,ni − αi,ni−1 − y(ni)i,d

= l−1i,niθT

iϕφiϕ(xi,ni , vi

) + l−1i,niε∗iϕ

+ Di,ni − αi,ni−1 − y(ni)i,d (49)

where ϕi(xi,ni , vi) = li,ni( fi,ni( y, xi,ni)+ gi,ni( y, xi,ni)vi), θiϕ isoptimal weight value of the NN, ε∗iϕ is an optimal approxima-tion error of the RBFNN, and li,ni > 0 is a design parameterof the developed nonlinear disturbance observer.

For the error system (49), an auxiliary variable isdefined as [53]

si = ei,ni − ξi,ni (50)

and the intermedial variable ξi,ni is proposed as

ξi,ni = qisi + l−1i,niθT

iϕφiϕ(xi,ni , vi)− αi,ni−1 − y(ni)i,d (51)

where qi > 0 is a designed parameter and θiϕ is the estimateof the optimal weight value θiϕ .



Differentiating (50) and considering (51), we have

si = ei,ni − ξi,ni

= l−1i,niθT

iϕφiϕ(xi,ni , vi

) + l−1i,niε∗iϕ + Di,ni

−(

qisi + l−1i,niθT

iϕφiϕ(xi,ni , vi

)) = −qisi

− l−1i,niθT

iϕφiϕ(xi,ni , vi

) + l−1i,niε∗iϕ + Di,ni (52)

where θiϕ = θiϕ − θiϕ .Considering (52) yields

sisi = −qis2i − l−1

i,nisiθ

Tiϕφiϕ

(xi,ni , vi

)

+ l−1i,ni

siε∗iϕ + siDi,ni

≤ −(qi − 1.0)s2i − l−1

i,nisiθ

Tiϕφiϕ

(xi,ni , vi

)

+ 0.5l−2i,niε∗2

iϕ + 0.5β2i,ni0. (53)

Using the auxiliary variable si, the nonlinear disturbance isdesigned as

Di,ni = li,ni(si − ψi) (54)

and the intermedial variable ψi is given by

ψi = −qisi + Di,ni . (55)

Define, Di,ni = Di,ni − Di,ni . Differentiating (54), andconsidering (52) and (55) yield

˙Di,ni = li,ni

(si − ψi

)

= li,ni

(−qisi − l−1

i,niθT

iϕφiϕ(xi,ni , vi

))

+ li,ni

(l−1i,niε∗iϕ + Di,ni −

(−qisi + Di,ni

))

= −θTiϕφiϕ

(xi,ni , vi

) + ε∗iϕ + li,ni

(Di,ni − Di,ni

)

= −θTiϕφiψ

(xi,ni , vi

) + ε∗iϕ + li,ni Di,ni . (56)

Considering (56) yields

˙Di,ni = Di,ni − ˙Di,ni

= Di,ni − li,ni Di,ni + θTiϕφiϕ

(xi,ni , vi

) − ε∗iϕ. (57)

Invoking (57), we have

Di,ni˙Di,ni = Di,ni Di,ni − li,ni D

2i,ni

+ Di,ni θTiϕφiϕ

(xi,ni , vi

) − Di,niε∗iϕ. (58)

Considering the following fact:

2Di,ni θTiϕφiϕ

(xi,ni , vi

) ≤ 2∣∣Di,ni

∣∣∥∥∥θiϕ

∥∥∥∥∥φiϕ

(xi,ni , vi

)∥∥

≤ τi,niμ2i,ni

D2i,ni

+ 1

τi,ni

∥∥∥θiϕ

∥∥∥

2(59)

yields

Di,ni˙Di,ni ≤ D2

i,ni+ 0.5D2

i,ni− li,ni D

2i,ni

+ 0.5τi,niμ2i,ni

D2i,ni

+ 1

2τi,ni

∥∥∥θiϕ

∥∥∥

2 + 0.5ε∗2iϕ

≤ −(

li,ni −(

1.0 + 0.5τi,niμ2i,ni

))D2

i,ni

+ 1

2τi,ni

∥∥∥θiϕ

∥∥∥

2 + 0.5β2i,ni1 + 0.5ε∗2

iϕ (60)

where ‖φiϕ(xi,ni , vi)‖ ≤ μi,ni and τi,ni > 0 is a designparameter.

The parameter updated law θiϕ is designed as

˙θiϕ = γiϕ

(l−1i,niϕiϕ

(xi,ni , vi

)si − δiϕθiϕ

)(61)

where γiϕ > 0 and δiϕ > 0 are the design parameters.

H. Main Results

The above adaptive fault-tolerant control design procedureand stability analysis can be summarized in the following the-orem, which contains the results for the uncertain nonlinearlarge-scale systems (1) with the external disturbance, unknowndead zone, and actuator fault using the nonlinear disturbanceobserver and backstepping technique.

Theorem 1: Consider a class of uncertain large-scale sys-tems (1) with the external disturbance, unknown dead zone,and actuator fault and suppose that full-state informationis available. The nonlinear disturbance observers of eachsubsystems are designed as (20), (33), (50), (51), (54),and (55). The updated laws of the NN weight are cho-sen as (23), (36), (46), and (61). The nonlinear disturbanceobserver based fault-tolerant control scheme is proposedas (45). Given for all initial conditions, the appropriatedesign parameters ki,j, qi, li,j, δi,j, δiϕ , and τi,j can be cho-sen according to (A.23) such that all closed-loop signalsuniformly ultimately bounded under the proposed adaptivefault-tolerant control based on the nonlinear disturbanceobserver.

IV. SIMULATION RESULTS

In this section, simulation results of the mass–spring–damper system are presented to illustrate the effectiveness ofthe proposed fault-tolerant control scheme using the distur-bance observer. The considered mass–spring–damper systemis illustrated in Fig. 1 and the motion equations of mechanicalsystem can be expressed as [54]

⎧⎨

⎩

M1y1 = u1 − fK1(x)− fB1(x)+ fK2(x)+ fB2(x)− fC1(x)+ fC2(x)+�1

M2y2 = u2 − fK2(x)− fB1(x)− fC2(x)+�2

(62)

where x = [y1, y1, y2, y2]T are the state variables: fK1(x) =K10y1 +�K1y1

3 and fK2(x) = K20(y2 − y1)+�K2(y2 − y1)3

denote the spring forces; fB1(x) = 2y1 + 0.2y21 and fB2(x) =

2.2(y2 − y1) + 0.15(y2 − y1)2 denote the friction forces.

fC1(x) = 0.02sgn(y1) and fC2(x) = 0.02sgn(y2 − y1)

are the coulomb friction forces. The parameters �1 and�2 are given as �1 = 0.2 sin(3t) exp(−0.2t) and �2 =0.2 cos(3t) exp(−0.1t), respectively. The parameters are givenas M1 = 0.25 kg, M2 = 0.2 kg, K10 = 1(N/m), and K20 =2(N/m). The perturbations are given as �K1 = 0.1 and�K2 = 0.12 [54].



Fig. 1. Mass–spring–damper system.

Fig. 2. Tracking results of x1 and x1d of case 1.

Let x11 = y1, x12 = y1, x21 = y2, and x22 = y2. Then, (62)can be transformed into the following form:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

x11 = x12

x12 = 1M1

u1 + 1M1

[−fK1(x)− fB1(x)+ fK2(x)+ fB2(x)− fC1(x)+ fC2(x)+�1

]

y1 = x11x21 = x22

x22 = 1M2

u2 + 1M2

[ − fK2(x)− fB1(x)− fC2(x)+�2]y2 = x21.

(63)

The control objective is that the system outputs can track thereference signals

y1,d ={

2 8k ≤ t ≤ 4(2k + 1)0 4(2k + 1) ≤ t ≤ 8(k + 1)

Fig. 3. Tracking errors of case 1.

Fig. 4. Control inputs of case 1.

and y2,d = sin(t) to a bounded compact set and all the signalsin the closed-loop system are guaranteed to be bounded. Thedead zone parameters are chosen as bi,r = 0.5, bi,l = −0.5,and m1 = 1. At the same time, the control input boundariesare assumed as ui max = 30 and ui max = −30 to satisfy thepractical system requirement.



Fig. 5. Tracking results of x1 and x1d of case 2.

In the simulation study, the nonlinear disturbance observersof each subsystems are designed as (20), (33), (50), (51), (54),and (55). The updated laws of the NN weight are cho-sen as (23), (36), (46), and (61). The nonlinear disturbanceobserver based fault-tolerant control scheme is proposedas (45). To design the adaptive fault-tolerant control scheme,all design parameters are chosen as k1 = 10, k2 = 40,λ1 = λ2 = 10, δ1 = δ2 = 0.2, l2 = 2, q2 = 3, and δϕ = 0.5.The initial state conditions are chosen as x11 = 0, x22 = 0,and θ1 = θ2 = 0.

To illustrate the effectiveness of the proposed adaptive fault-tolerant control scheme, the tracking control simulation resultsof two cases are given. In two cases, the actuator bias fault ischosen as ζi = 0.2 sin(0.5t) and the actuator fault appear attf = 10 s. At the same time, the unknown nonlinear remainingcontrol rate coefficient ρi are chosen as ρi = 1/(1 + e−0.2) andρi = 1/(1 + e−2[ sin(x11)

2+cos(x21)2]), respectively.

Case 1: The nonlinear remaining control rate coefficient ρi

is chosen as a constant ρi = 1/(1 + e−0.2).In this case, under the proposed adaptive fault-tolerant con-

troller based on the disturbance observer and the NN, thetracking control result of the uncertain nonlinear plant (63)is shown in Fig. 2 and the tracking error is given in Fig. 3.If all actuators do not have any faults for t < 10 s, wecan see that the satisfactory control performance has beenobtained. Meanwhile, there is not the overshooting for the

Fig. 6. Tracking errors of case 2.

Fig. 7. Control inputs of case 2.

tracking performance when actuators lose constant efficiencyat 10 s. We can note that the satisfactory tracking controlperformance is always guaranteed under the time-varying



external disturbance, the dead zone, and the actuator faultsfrom Figs. 2 and 3. In Fig. 4, the control input is presentedwhich is bounded.

Case 2: The nonlinear remaining control rate coeffi-cient ρi is chosen as a time-varying parameter ρi =1/(1 + e−2[ sin(x11)

2+cos(x21)2]).

In this case, under the proposed adaptive fault-tolerant con-troller based on the disturbance observer and the NN, thetracking control result of the uncertain nonlinear plant (63)is shown in Fig. 5 and the tracking error is given in Fig. 6.If all actuators do not have any faults for t < 10 s, we notethat the satisfactory control performance has been obtained,while the small overshooting will appear when actuators havetime-varying loss efficiency at 10 s. From Figs. 5 and 6, thesatisfactory tracking control performance is achieved underthe time-varying external disturbance, the dead zone, and theactuator fault. The bounded control input is presented in Fig. 7.

Based on the simulation results of the simulation exam-ple, we can conclude that the developed adaptive fault-tolerantcontrol based on the disturbance observer and the RBFNN iseffective for the uncertain nonlinear large-scale system withthe time-varying external disturbance and unknown dead zone.

V. CONCLUSION

An adaptive fault-tolerant control scheme has been devel-oped for uncertain nonlinear large-scale systems with unknowndead zones, unknown external disturbances, and actuator faultsin this paper. The RBFNN has been employed to approximatethe unknown nonlinear continuous interconnection functionterms and the nonlinear disturbance observer has been usedto estimate the compounded disturbance consisting of theunknown approximation error, the unknown dead zone, andthe unknown disturbance. Using outputs of the RBFNN andthe disturbance observer, the adaptive fault-tolerant controllerhas the desired capacity to control uncertain nonlinear large-scale systems with the unknown dead zone, unknown externaldisturbances, and actuator faults including the time-varyingbias fault and the nonlinear gain fault. The uniformly ulti-mately bounded convergence of all closed-loop signals hasbeen established. Finally, simulation results have illustratedthe effectiveness of the proposed adaptive fault-tolerant controlscheme. In the future, the application of the developed adaptivefault-tolerant control scheme should be further studied.

APPENDIX

A. Error System Analysis in Step 1

Proof: Choose the Lyapunov function candidate as

Vi,1 = 1

2e2

i,1 + 1

2γi,1θT

i,1θi,1 + 1

2D2

i,1. (A.1)

Invoking (22) and (26), the time derivative of Vi,1 isgiven by

Vi,1 = ei,1ei,1 + γ−1i,1 θ

Ti,1

˙θi,1 + Di,1

˙Di,1

= −ki,1e2i,1 + ei,1ei,2 − λ−1

i,1 ei,1θTi,1φ

(Zi,1

) + ei,1Di,1

+ Di,1Di,1 − λi,1D2i,1 + Di,1θ

Ti,1φ

(Zi,1

)

+ γ−1i,1 θ

Ti,1

˙θi,1. (A.2)

Substituting (23) into (A.2) yields

Vi,1 = −ki,1e2i,1 + ei,1ei,2 − δi,1θ

Ti,1θi,1 + ei,1Di,1

+ Di,1Di,1 − λi,1D2i,1 + Di,1θ

Ti,1φ

(Zi,1

). (A.3)

Considering (19) and the following facts:

2θTi,1θi,1 =

∥∥∥θi,1

∥∥∥

2 +∥∥∥θi,1

∥∥∥

2 − ∥∥θi,1

∥∥2

≥∥∥∥θi,1

∥∥∥

2 − ∥∥θi,1

∥∥2 (A.4)

2Di,1θTi,1φ

(Zi,1

) ≤ τi,1μ2i,1D2

i,1 +∥∥∥θi,1

∥∥∥

2

τi,1(A.5)

we have

Vi,1 ≤ −(ki,1 − 0.5

)e2

i,1 −(δi,1

2− 1

2τi,1

)∥∥∥θi,1

∥∥∥

2

−(λi,1 − 1 − 0.5τi,1μ

2i,1

)D2

i,1 + 0.5β2i,11

+ δi,1

2

∥∥θi,1

∥∥2 + ei,1ei,2 (A.6)

where τi,1 > 0 is a design parameter and ‖φ(Zi,1)‖ ≤ μi,1.This concludes the proof.

B. Error System Analysis in Step j

Proof: Choose the Lyapunov function candidate as

Vi,j = 1

2e2

i,j + 1

2γi,jθT

i,jθi,j + 1

2D2

i,j. (A.7)

Invoking (35) and (39), the time derivative of Vi,j is given by

Vi,j = ei,jei,j + γ−1i,j θ

Ti,j

˙θi,j + Di,j

˙Di,j

= −ki,je2i,j + ei,jei,j+1 − λ−1

i,j ei,jθTi,jφ

(Zi,j

)

+ ei,jDi,j + Di,jDi,j − λi,jD2i,j + Di,jθ

Ti,jφ

(Zi,j

)

+ γ−1i,j θ

Ti,j

˙θi,j − ei,jei,j−1. (A.8)

Substituting (36) into (A.8) yields

Vi,j = −ki,je2i,j + ei,jei,j+1 − δi,jθ

Ti,jθi,j

+ ei,jDi,j + Di,jDi,j − λi,jD2i,j

+ Di,jθTi,jφ

(Zi,j

) − ei,jei,j−1. (A.9)

Considering (32) and the following facts:

2θTi,jθi,j =

∥∥∥θi,j

∥∥∥

2 +∥∥∥θi,j

∥∥∥

2 − ∥∥θi,j

∥∥2

≥∥∥∥θi,j

∥∥∥

2 − ∥∥θi,j

∥∥2 (A.10)

2Di,jθTi,jφ

(Zi,j

) ≤ τi,jμ2i,jD

2i,j +

∥∥∥θi,j

∥∥∥

2

τi,j(A.11)

we have

Vi,j ≤ −(ki,j − 0.5

)e2

i,j −(δi,j

2− 1

2τi,j

)∥∥∥θi,j

∥∥∥

2

−(λi,j − 1 − 0.5τi,jμ

2i,j

)D2

i,j + 0.5β2i,j1

+ ei,jei,j+1 − ei,jei,j−1 + δi,j

2

∥∥θi,j

∥∥2 (A.12)

where τi,j > 0 is a design parameter and ‖φ(Zi,j)‖ ≤ μi,j. Thisconcludes the proof.



C. Error System Analysis in Step ni

Proof: Consider the Lyapunov function candidate as

Vi,ni = 1

2gi,ni

e2i,ni

+ 1

2γi,ni

θTi,niθi,ni

+ 1

2γiϕθT

iϕθiϕ + 1

2s2

i + 1

2D2

i,ni. (A.13)

Invoking (48), (53), (60) and Remark 1, the time derivativeof Vi,ni is

Vi,ni = 1

gi,ni

ei,ni ei,ni − gi,ni

2g2i,ni

e2i,ni

+ 1

γi,ni

θTi,ni

˙θi,ni

+ 1

γiϕθT

iϕ˙θiϕ + sisi + Di,ni

˙Di,ni

≤ ei,ni

(−ki,ni ei,ni − θT

i,niφi,ni

(Zi,ni

))

− ei,ni

(Di,ni − ε∗i,ni

+ ei,ni−1

)+ ei,ni Di,ni

gi,ni

+ ϑi

2g2i0

e2i,ni

+ 1

γi,ni

θTi,ni

˙θi,ni + 1

γiϕθT

iϕ˙θiϕ

− (qi − 1.0)s2i − l−1

i,nisiθ

Tiϕφiϕ

(xi,ni , vi

) + 0.5l−2i,niε∗2

iϕ

+ 0.5β2i,ni0 −

(li,ni −

(1.0 + 0.5τi,niμ

2i,ni

))D2

i,ni

+ 1

2τi,ni

∥∥∥θiϕ

∥∥∥

2 + 0.5β2i,ni1 + 0.5ε∗2

iϕ . (A.14)

Considering Di,ni = Di,ni − Di,ni and Remark 1, we have

Vi,ni ≤ ei,ni

(−ki,ni ei,ni − θT

i,niφi,ni

(Zi,ni

) + ε∗i,ni

)

+ ϑi

2g2i0

e2i,ni

+ ei,ni Di,ni + ei,niDi,ni

(1

gi,ni

− 1

)

− ei,ni ei,ni−1 + 1

γi,ni

θTi,ni

˙θi,ni + 1

γiϕθT

iϕ˙θiϕ

− (qi − 1.0)s2i − l−1

i,nisiθ

Tiϕφiϕ

(xi,ni , vi

) + 0.5l−2i,niε∗2

iϕ

+ 0.5β2i,ni0 −

(li,ni −

(1.0 + 0.5τi,niμ

2i,ni

))D2

i,ni

+ 1

2τi,ni

∥∥∥θiϕ

∥∥∥

2 + 0.5β2i,ni1 + 0.5ε∗2

iϕ

≤ −(

ki,ni − 1.5 − ϑi

2g2i0

)

e2i,ni

− ei,ni θTi,niφi,ni

(Zi,ni

)

− ei,ni ei,ni−1 + 1

γi,ni

θTi,ni

˙θi,ni + 1

γiϕθT

iϕ˙θiϕ

− (qi − 1.0)s2i − l−1

i,nisiθ

Tiϕφiϕ

(xi,ni , vi

)

−(

li,ni −(

1.5 + 0.5τi,niμ2i,ni

))D2

i,ni

+ 1

2τi,ni

∥∥∥θiϕ

∥∥∥

2 +⎛

⎝0.5 + 0.5

∣∣∣∣∣

1

gi0

− 1

∣∣∣∣∣

2⎞

⎠β2i,ni0

+(

0.5 + 0.5l−2i,ni

)ε∗2

iϕ + 0.5β2i,ni1 + 0.5ε∗2

i,ni. (A.15)

Substituting (46) and (61) into (A.15), we obtain

Vi,ni ≤ −(

ki,ni − 1.5 − ϑi

2g2i0

)

e2i,ni

− ei,ni ei,ni−1 − δi,ni θTi,niθi,ni − δiϕθ

Tiϕθiϕ

− (qi − 1.0)s2i −

(li,ni −

(1.5 + 0.5τi,niμ

2i,ni

))D2

i,ni

+ 1

2τi,ni

∥∥∥θiϕ

∥∥∥

2 +⎛

⎝0.5 + 0.5

∣∣∣∣∣

1

gi0

− 1

∣∣∣∣∣

2⎞

⎠β2i,ni0

+(

0.5 + 0.5l−2i,ni

)ε∗2

iϕ + 0.5β2i,ni1 + 0.5ε∗2

i,ni. (A.16)

Considering the following facts:

2θTi,niθi,ni =

∥∥∥θi,ni

∥∥∥

2 +∥∥∥θi,ni

∥∥∥

2 − ∥∥θi,ni

∥∥2

≥∥∥∥θi,ni

∥∥∥

2 − ∥∥θi,ni

∥∥2 (A.17)

and

2θTiϕθiϕ =

∥∥∥θiϕ

∥∥∥

2 +∥∥∥θiϕ

∥∥∥

2 − ∥∥θiϕ

∥∥2

≥∥∥∥θiϕ

∥∥∥

2 − ∥∥θiϕ

∥∥2 (A.18)

we have

Vi,ni ≤ −(

ki,ni − 1.5 − ϑi

2g2i0

)

e2i,ni

− ei,niei,ni−1

− δi,ni

2

∥∥∥θi,ni−1

∥∥∥

2 −(δiϕ

2− 1

2τi,ni

)∥∥∥θiϕ

∥∥∥

2

− (qi − 1.0)s2i −

(li,ni −

(1.5 + 0.5τi,niμ

2i,ni

))D2

i,ni

+⎛

⎝0.5 + 0.5

∣∣∣∣∣

1

gi0

− 1

∣∣∣∣∣

2⎞

⎠β2i,ni0

+0.5β2i,ni1 +

(0.5 + 0.5l−2

i,ni

)ε∗2

iϕ

+ δi,ni

2

∥∥θi,ni

∥∥2 + δiϕ

2

∥∥θiϕ

∥∥2 + 0.5ε∗2

i,ni. (A.19)

Thus, this concludes the proof.

D. Proof of Theorem 1

Proof: For the large-scale systems (1) consisting of Nsubsystems, the Lyapunov function candidate is chosen as

V =N∑

i=1

ni∑

j=1

Vi,j. (A.20)

Invoking (A.6), (A.12), and (A.21), we have

V ≤ −N∑

i=1

ni−1∑

j=1

(ki,j − 0.5

)e2

i,j

−N∑

i=1

ni−1∑

j=1

(δi,j

2− 1

2τi,j

)∥∥∥θi,j

∥∥∥

2

−N∑

i=1

ni−1∑

j=1

(λi,j − 1 − 0.5τi,jμ

2i,j

)D2

i,j



−N∑

i=1

(

ki,ni − 1.5 − ϑi

2g2i0

)

e2i,ni

−N∑

i=1

δi,ni

2

∥∥∥θi,ni−1

∥∥∥

2 −N∑

i=1

(δiϕ

2− 1

2τi,ni

)∥∥∥θiϕ

∥∥∥

2

−N∑

i=1

(qi − 1.0)s2i −

N∑

i=1

(li,ni −

(1.5 + τi,niμ

2i,ni

))D2

i,ni

+N∑

i=1

⎡

⎣

⎛

⎝0.5 + 0.5

∣∣∣∣∣

1

gi0

− 1

∣∣∣∣∣

2⎞

⎠β2i,ni0

+(

0.5 + 0.5l−2i,ni

)ε∗2

iϕ + 0.5ε∗2i,ni

⎤

⎦

+N∑

i=1

ni∑

j=1

(

0.5β2i,j1 + δi,j

2

∥∥θi,j

∥∥2

)

+N∑

i=1

δiϕ

2

∥∥θiϕ

∥∥2

≤ −κV + C (A.21)

where κ and C are given by

κ: = min

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

ki,j − 0.5, λi,j − 1 − 0.5τi,jμ2i,j,

ki,ni − 1.5 − ϑi

2g2i0

qi − 1.0, li,ni −(

1.5 + τi,niμ2i,ni

)

δi,j

γi,j− 1

γi,jτi,j,δi,ni

γi,ni

,δiϕ

γiϕ− 1

γiϕτi,ni

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

C: =N∑

i=1

⎡

⎣

⎛

⎝0.5 + 0.5

∣∣∣∣∣

1

gi0

− 1

∣∣∣∣∣

2⎞

⎠β2i,ni0

⎤

⎦

=N∑

i=1

[(0.5 + 0.5l−2

i,ni

)ε∗2

iϕ + 0.5ε∗2i,ni

]+

N∑

i=1

δiϕ

2

∥∥θiϕ

∥∥2

+N∑

i=1

ni∑

j=1

(

0.5β2i,j1 + δi,j

2

∥∥θi,j

∥∥2

)

. (A.22)

To ensure the closed-loop system stability, the correspond-ing design parameters ki,j, λi,j, τi,j, ki,ni , qi, li,ni , δi,j, and δiϕ

should be chosen to make the following inequalities hold:

ki,j − 0.5 > 0

λi,j − 1 − 0.5τi,jμ2i,j > 0

ki,ni − 1.5 − ϑi

2g2i0

> 0

qi − 1.0 > 0

li,ni − (1.5 + τi,ni > 0δi,j

γi,j− 1

γi,jτi,j> 0

δiϕ

γiϕ− 1

γiϕτi,ni

> 0. (A.23)

According to (A.21), we have

0 ≤ V ≤ C

κ+

[V(0)− C

�

]e−κt. (A.24)

From (A.21), we can know that V is exponentially conver-gent, i.e., limt−→∞ V = (C/κ). According to (A.23), it maydirectly show that the signals ei,j and Di,j are semiglobally

uniformly bounded when t → 0. Hence, the tracking errorsei,1 and the approximation error Di,j of the closed-loop systemare bounded. This concludes the proof.

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Mou Chen (M’10) received the B.Sc. degree inmaterial science and engineering and the M.Sc.and Ph.D. degrees in automatic control engineer-ing from the Nanjing University of Aeronautics andAstronautics, Nanjing, China, in 1998 and 2004,respectively.

He is currently a Professor with the Collegeof Automation Engineering, Nanjing University ofAeronautics and Astronautics. He was an AcademicVisitor with the Department of Aeronautical andAutomotive Engineering, Loughborough University,

Loughborough, U.K., from 2007 to 2008. From 2008 to 2009, he was aResearch Fellow with the Department of Electrical and Computer Engineering,National University of Singapore, Singapore. In 2014, he was a SeniorAcademic Visitor with the School of Electrical and Electronic Engineering,University of Adelaide, Adelaide, SA, Australia, for seven months. His cur-rent research interests include nonlinear system control, intelligent control,and flight control.

Gang Tao (S’84–M’89–SM’96–F’07) received theB.S. degree from the University of Science andTechnology of China, Hefei, China, in 1982, andthe M.S. and Ph.D. degrees from the Universityof Southern California, Los Angeles, CA, USA,in 1984 and 1989, respectively, all in electricalengineering.

He is currently a Professor with the University ofVirginia, Charlottesville, VA, USA. His research wasmainly in adaptive control, with particular interest inadaptive control of systems with multiple inputs and

multiple outputs, with nonsmooth nonlinearities or with uncertain faults, instability and robustness of adaptive control systems, and in system passivitycharacterizations. His current research interests include adaptive control ofsystems with actuator failures and structural damage, adaptive approximationbased control, and resilient aircraft and spacecraft control applications. Hehas authored and co-authored or co-edited six books and over 350 papers onsome related topics.

Prof. Tao served as an Associate Editor for Automatica, the InternationalJournal of Adaptive Control and Signal Processing, and the IEEETRANSACTIONS ON AUTOMATIC CONTROL, a Guest Editor for the Journalof Systems Engineering and Electronics, and an Editorial Board Member ofthe International Journal of Control, Automation and Systems.

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IEEE TRANSACTIONS ON CYBERNETICS 1 Adaptive Fault … · uncertain nonlinear large-scale systems under the integrated effects of the unknown dead zone, actuator fault, and unknown