2086 IEEE TRANSACTIONS ON NEURAL NETWORKS ...2086 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING...

2086 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 9, SEPTEMBER 2015

Dynamic Surface Control Using Neural Networksfor a Class of Uncertain Nonlinear Systems

With Input SaturationMou Chen, Member, IEEE, Gang Tao, Fellow, IEEE, and Bin Jiang, Senior Member, IEEE

Abstract— In this paper, a dynamic surface control (DSC)scheme is proposed for a class of uncertain strict-feedbacknonlinear systems in the presence of input saturation andunknown external disturbance. The radial basis function neuralnetwork (RBFNN) is employed to approximate the unknownsystem function. To efficiently tackle the unknown external dis-turbance, a nonlinear disturbance observer (NDO) is developed.The developed NDO can relax the known boundary requirementof the unknown disturbance and can guarantee the disturbanceestimation error converge to a bounded compact set. UsingNDO and RBFNN, the DSC scheme is developed for uncertainnonlinear systems based on a backstepping method. Using aDSC technique, the problem of explosion of complexity inherentin the conventional backstepping method is avoided, which isspecially important for designs using neural network approxima-tions. Under the proposed DSC scheme, the ultimately boundedconvergence of all closed-loop signals is guaranteed via Lyapunovanalysis. Simulation results are given to show the effectivenessof the proposed DSC design using NDO and RBFNN.

Index Terms— Backstepping control, dynamic surface control(DSC), nonlinear disturbance observer (NDO), robust control,uncertain nonlinear system.

I. INTRODUCTION

IN PRACTICAL engineering, lots of plants possess nonlin-ear and uncertain characteristics. On the other hand, themagnitude of control signal is always limited due to actuatorphysical constraints. Thus, it is very important to developeffective robust control techniques for uncertain nonlinear sys-tems with input saturation. Saturation as one of the commonnonsmooth nonlinear constraint of control input should beexplicitly considered in the control design to enhance robustcontrol performance. If the input saturation is ignored inthe control design, the closed-loop control performance willbe severely degraded, and instability may occur. In recent

Manuscript received July 31, 2013; revised August 21, 2014; acceptedSeptember 21, 2014. Date of publication December 4, 2014; date of currentversion August 17, 2015. This work was supported in part by the JiangsuNatural Science Foundation of China under Grant SBK20130033, in part bythe National Natural Science Foundation of China under Grant 61374130 andGrant 61174102, in part by the Program for New Century Excellent Talentsin the University of China under Grant NCET-11-0830, and in part by theJiangsu Province Blue Project through the Innovative Research Team.

M. Chen and B. Jiang are with the College of Automation Engineering,Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China(e-mail: [email protected]; [email protected]).

G. Tao is with the Department of Electrical and ComputerEngineering, University of Virginia, Charlottesville, VA 22904 USA(e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TNNLS.2014.2360933

years, there have been extensive studies on various systemswith input saturation in [1]–[3]. Neural network (NN)-basednear-optimal control was developed for a class of discrete-time affine nonlinear systems with control constraints in [4].In [5], a robust adaptive control scheme was proposed foruncertain nonlinear systems in the presence of input saturationand external disturbance. Robust adaptive neural networkcontrol was proposed for a class of uncertain multi-input andmulti-output (MIMO) nonlinear systems with input nonlin-earities [6]. Backstepping control was studied for hoveringunmanned aerial vehicle, including input saturations in [7].In [8], an adaptive tracking control scheme was developedfor uncertain MIMO nonlinear systems with input saturation.Adaptive control was studied for minimum phase single-inputand single-output plants with input saturation [9]. However,there are few existing research results for the dynamic surfacecontrol (DSC) scheme of uncertain strict-feedback nonlinearsystems with input saturation and unknown external distur-bance.

On the other hand, robust adaptive backstepping con-trol as an efficient control method has been extensivelyused for nonlinear control system design due to its designflexibility [10]–[13]. At the same time, NNs and fuzzy logicalsystems as the universal approximators have been widelyemployed to tackle the system uncertainty [14]–[19]. In [20],an adaptive sliding-mode control was proposed for nonlinearactive suspension vehicle systems using Takagi–Sugeno fuzzyapproach. Robust adaptive tracking control scheme was pro-posed for nonlinear systems based on the fuzzy approximatorin [21]. A combined backstepping and small-gain approachwas developed for the robust adaptive fuzzy output feedbackcontrol design in [22]. In [23], a globally stable adaptivebackstepping fuzzy control scheme was studied for output-feedback systems with unknown high-frequency gain sign.Adaptive backstepping fuzzy control was proposed for nonlin-early parameterized systems with periodic disturbance in [24].In [25], an observer-based adaptive decentralized fuzzy fault-tolerant control scheme was studied for nonlinear large-scalesystems with actuator failures. Furthermore, backstepping con-trol has been extensively used in many practical systems.Nonlinear adaptive flight control was proposed using back-stepping method and NNs in [26]. In [27], a fuzzy adaptivecontrol design was studied for hypersonic vehicles via back-stepping method. Robust attitude control was developed forhelicopters with actuator dynamics using NNs in [28]. In [29],

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CHEN et al.: DSC USING NNs FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS 2087

an observer-based adaptive fuzzy backstepping control schemewas proposed for a class of stochastic nonlinear strict-feedbacksystems. However, there are few backstepping control resultsfor uncertain nonlinear systems using disturbance observers.To tackle the unknown time-varying disturbance for effectivebackstepping control design, the robust adaptive backsteppingcontrol based on disturbance observation should be furtherdeveloped.

With conventional backstepping, a possible issue is theproblem of explosion of complexity. That is, the complexityof the controller grows drastically as the order n of the systemincreases. This explosion of complexity is caused by therepeated differentiations of certain nonlinear functions. To effi-ciently handle the system uncertainty in each subsystem, radialbasis function NN (RBFNN) with the universal approximationcapability is employed in [30] and [31]. Since RBFNN is used,we need to take derivatives of those radial basis functions,which further lead to heavier calculation burden in each stepdesign. Recently, the DSC method was employed to solve thisproblem and many research results were presented [32]. In[33], an adaptive DSC design was proposed using adaptivebackstepping for nonlinear systems. DSC was presented for aclass of nonlinear systems in [34]. In [35], NN-based adaptiveDSC was developed for nonlinear systems in strict-feedbackform. A robust adaptive NN tracking control design wasproposed for strict-feedback nonlinear systems using DSCapproach in [36]. In [37], a NN-based adaptive DSC schemewas studied for uncertain nonlinear pure-feedback systems.Simultaneous quadratic stabilization was studied for a class ofnonlinear systems with input saturation using DSC in [38]. In[39], an output feedback adaptive DSC scheme was developedfor a class of nonlinear systems with input saturation. Recently,L∞-type criteria are used in the DSC design to enhancethe control performance [40]–[42]. However, DSC shouldbe further investigated for uncertain strict-feedback nonlinearsystems in the presence of input saturation and unknownexternal disturbance.

In recent years, disturbance observer design and applicationhave attracted considerable interest for robust control ofuncertain nonlinear systems. Thus, different disturbanceobservers have been developed [43]–[47] and robust controlschemes were proposed using disturbance observers. Ageneral framework was given for nonlinear systems usingdisturbance observer based control (DOBC) techniques in[48]. In [49], composite DOBC and terminal sliding modecontrol were investigated for uncertain structural systems.The disturbance attenuation and rejection problem wasinvestigated for a class of MIMO nonlinear systems usinga DOBC framework in [50]. In [51], composite DOBC andH∞ control designs were proposed for complex continuousmodels. Adding robustness to nominal output feedbackcontrollers was studied for uncertain nonlinear systems usinga disturbance observer in [52]. Although significant progresshas been made for the disturbance observer design, thereare still some open problems that need to be solved. Inalmost all approaches reported in the literature, the unknowndisturbance is assumed as a slowly changeable disturbancefor the disturbance observer design that implies the derivative

of the disturbance approaching to zero. It is apparent thatthis assumption is restrictive for a practical system. TheNDO can provide the estimation of the bounded unknowndisturbance and can be employed in the robust control designto compensate for the unknown disturbance. At the sametime, the NDO does not rely on complete knowledge of thedisturbance mathematical model, as an efficient disturbanceobserver. In this paper, the NDO is proposed for the uncertainnonlinear systems for which the known upper boundaryassumption of the unknown disturbance is canceled and theconvergence of the disturbance estimation error is proved.

This paper develops a new NDO-based DSC design foruncertain nonlinear systems with unknown external distur-bance and input saturation. The control objective is that theproposed DSC can track a desired trajectory in the presenceof unknown time-varying external disturbance and input satu-ration. The main contributions of this paper are as follows.

1) An NDO is developed to estimate the unknown distur-bance. Especially, the known upper boundary requirementof the unknown disturbance is eliminated for the designof NDO.

2) DSC is implemented using the output of the developedNDO for uncertain nonlinear systems with input satura-tion and unknown external disturbances to enhance therobust control performance of the closed-loop system.

3) Closed-loop system stability is guaranteed using Lya-punov method, which shows that all closed-loop systemsignals are semiglobal uniformly ultimately bounded.

The organization of this paper is as follows. Section IIdetails the problem formulation. Section III presents theDSC scheme with NDO. Simulation studies are presented inSection IV to demonstrate the effectiveness of the developednonlinear disturbance observer-based DSC, followed by someconcluding remarks in Section V.

Throughout this paper, (·̃) = (·̂) − (·)∗, || · || denotes the l2norm, and λmin(·) and λmax(·) denote the smallest and largesteigenvalues of a square matrix ·, respectively.

II. PROBLEM STATEMENT AND PRELIMINARIESA. Problem Statement

Consider a class of uncertain strict-feedback nonlinear sys-tems with input saturation and unknown disturbance which aredescribed by

ẋi = fi (x̄i ) + gi(x̄i )xi+1, i = 1, . . . , n − 1ẋn = fn(x̄n) + gn0(x̄n)u(v(t)) + d(t)y = x1 (1)

where x̄i = [x1, x2, . . . , xi ]T ∈ Ri , i = 1, 2, . . . , n, arestate vectors which are assumed to be measurable; y ∈ R isthe output of the uncertain nonlinear system; function termsfi (x̄i) : Ri → R, i = 1, 2, . . . , n, gi(x̄i ) : Ri → R, i =1, 2, . . . , n − 1, and gn0(x̄n) : Rn → R are unknown andcontinuous; d ∈ R is an unknown and bounded disturbance;v(t) ∈ R is the control input and u(·) denotes the plant inputwhich is subject to saturation nonlinearity described by [5]

u(v(t)) = sat(v(t)) ={

sign(v(t))uM , |v(t)| ≥ uMv(t), |v(t)| < uM (2)

where uM is a bound of u(t).


To efficiently tackle the saturation u(v(t)) in the DSC, it isapproximated by the following smooth function [5]:

h(v) = uM tanh(

v

uM

)= uM e

v/uM − e−v/uMev/uM + e−v/uM . (3)

It is apparent that there exists a difference �(v) betweensat(v(t)) and h(v). Then, we have

�(v) = sat(v(t)) − h(v). (4)Since the bounded property of the tanh function and the

sat function, we can see that the difference �(v) is boundedwhich satisfies the following condition [5]:

|�(v)| = |sat(v(t)) − h(v)| ≤ uM (1 − tanh(1)) = d̄. (5)Consider the saturation characteristic and the corresponding

approximation error, the uncertain nonlinear system (1) can bewritten as

ẋi = fi (x̄i) + gi (x̄i)xi+1, i = 1, . . . , n − 1ẋn = fn(x̄n) + gn0(x̄n)h(v(t)) + gn0(x̄n)�(v) + d(t)y = x1. (6)

To facilitate the DSC design for the uncertain nonlinearsystem (6), invoking the mean-value theorem [53], we canexpress h(v(t)) in (6) as follows:

h(v(t)) = h(v0) + ∂h(v)∂v

∣∣∣∣v=vμ

(v − v0) (7)

where vμ = μv + (1 − μ)v0 with 0 < μ < 1.By choosing v0 = 0, we obtain

h(v(t)) = h(0) + ∂h(v)∂v

∣∣∣∣v=vμ

v. (8)

Considering h(0) = 0, we have

h(v(t)) = ∂h(v)∂v

∣∣∣∣v=vμ

v. (9)

Define gn(x̄n) = gn0(x̄n)(∂h(v)/∂v)|v=vμ and D(t) = d(t)+gn0(x̄n)�(v). Then, the uncertain nonlinear system (6) can berewritten as

ẋi = fi (x̄i ) + gi(x̄i )xi+1, i = 1, . . . , n − 1ẋn = fn(x̄n) + gn(x̄n)v + D(t)y = x1. (10)

B. Neural Networks

In many references of robust adaptive control for uncertainnonlinear systems, RBFNNs are usually employed asapproximation models for the unknown nonlinear and con-tinuous function terms using their inherent approximationcapabilities [54]. As a class of linearly parameterized NNs,RBFNNs are adopted to approximate the unknown and con-tinuous function f (Z) : Rq → R can be written as follows:

f (Z) = Ŵ T S(Z) + ε(Z) (11)where Z = [z1, z2, . . . , zq ]T ∈ Rq is an input vector ofNN, Ŵ ∈ R p is a weight vector of the NN, S(Z) =[s1(Z), s2(Z), . . . , sp(Z)]T ∈ R p is a basis function, ε is the

approximation error which satisfies |ε| ≤ |ε̄|, and ε̄ is a boundunknown parameter.

In general, RBFNN can smoothly approximate any contin-uous function f (Z) over the compact set �Z ∈ Rq to anyarbitrary accuracy as [55]

f (Z) = W∗T S(Z) + ε∗(Z) ∀Z ∈ �Z ⊂ Rq (12)where W∗ is the optimal weight value and ε∗(Z) is thesmallest approximation error. The Gaussian function is writtenin the form of

si (Z)=exp[−(Z − ci )T (Z − ci )/b2i ], i =1, 2, . . . , p (13)where ci and bi are the center and width of the neural cell ofthe i th hidden layer.

The optimal weight value of RBFNN is given by [55]

W∗ = arg minŴ∈� f

[ supz∈SZ

| f̂ (Z |Ŵ ) − f (Z)|] (14)

where � f = {Ŵ : ‖Ŵ‖ ≤ M} is a valid field of the parameterand M is a design parameter. SZ ⊂ Rn is an allowable set ofthe state vector.

Using the optimal weight value yields

| f (Z) − W∗T S(Z)| = |ε∗(Z)| ≤ |ε̄|. (15)In this paper, the NDO is employed to estimate the unknown

compounded disturbance D(t) which consists of d(t) andgn0(x̄n)�(v). The RBFNNs are used to approximate theunknown continuous functions. Based on estimated outputsof the developed NDO and the RBFNN, the DSC scheme isproposed for uncertain nonlinear systems. The control objec-tive is that the developed DSC scheme can make the systemoutput follow a given desired system output yd of the nonlinearsystem in the presence of the unknown external disturbanceand the input saturation for all initial conditions satisfying�i := {∑ij=1(z2j + (W̃ Tj � j W̃ j )) + ∑ij=2 η2j < 2 p}, i =1, . . . , n with p > 0, z1 = x1 − yd , zi = xi −λi , i = 2, . . . , n,W̃ j = Ŵ j −W∗j , j = 1, . . . , n, λi and ηi will be given. For thedesired system output yd , the proposed nonlinear disturbanceobserver-based DSC should ensure that all closed-loop signalsare convergent.

To proceed with the design of the nonlinear disturbanceobserver-based DSC for the uncertain nonlinear system (1),the following assumptions are required.

Assumption 1 [56]: For all t > 0, the reference signalyd(t) is a sufficiently smooth function of t , and yd , ẏd , andÿd are bounded, that is, there exists a positive constant B0such that �0 := {(yd , ẏd , ÿd ) : (yd)2 + (ẏd)2 + (ÿd)2 ≤ B0}.

Assumption 2 [57]: The signs of gi , i = 1, . . . , n − 1 andgn0 are known. Furthermore, there exist positive constants giand ḡi , such that gi ≤ |gi | ≤ ḡi . At the same time, there existtwo positive constants g

0and ḡ0 to render g0 ≤ |gn0| ≤ ḡ0

valid. Without losing generality, we shall assume that gi andgn0 are positive in the DSC design.

Assumption 3: There exist the unknown positive constantsβ0 and β1 such that the external disturbance satisfy |d| ≤ β0and |ḋ| ≤ β1.

Assumption 4 [57]: There exist constants gdi > 0, i =1, 2, . . . , n such that |ġi(.)| ≤ gdi in the compact set � j .


At the same time, there exists a positive constant gdn0 suchthat |ġn0(.)| ≤ gdn0.

Assumption 5: For a practical system described by theuncertain strict-feedback nonlinear system (1) subject to theinput saturation (2) and the desired reference signal yd , thereshould exist a feasible actual control input v which can achievethe given tracking control objective.

Remark 1: Due to the control input saturation u(v(t)) andthe unknown external disturbance d(t), the control design ofthe uncertain nonlinear system (1) becomes more complicated.In accordance with the characteristic of the DSC, the referencesignal yd(t) and its time derivatives ẏd(t), ÿd (t) are assumedto be bounded in Assumption 1. Assumption 2 implies thatsmooth functions are strictly either positive or negative. ToAssumption 3, the external disturbance is assumed as boundedand the boundary is unknown. Since the time-dependent distur-bance d(t) can be largely attributed to the exogenous effects, ithas finite energy. Hence, it is bounded and the time derivationis also bounded. On the other hand, the approximation error�(v) of the control input saturation is bounded which equalsto �(v) = sat(v(t)) − h(v). For a practical system, the timederivation of sat(v(t)) is bounded when the actuator is deter-mined. Furthermore, the time derivation of tanh function h(v)is also bounded. Thus, the time derivation of �̇(v) is bounded.At the same time, Ḋ(t) = ḋ(t)+(ġn0(x̄n)�(v)+gn0(x̄n)�̇(v)).According to Assumptions 2 and 4, we know that gn0 andġn0 are bounded. From above analysis, we know that thecompounded disturbance D(t) satisfies |D| ≤ θ0 and |Ḋ| ≤ θ1with the unknown constants θ0 > 0 and θ1 > 0.

Remark 2: For a given practical system, the input satura-tion should meet the physical requirement of system control.In other words, there should exist a DSC that can track thegiven desired output of the nonlinear system in the presenceof the unknown external disturbance and the input saturationfor all given initial conditions. Many practical systems arecontrollable under the control input saturation, such as aflight control system. For an aircraft, the deflexion anglesof control surfaces are limited, which lead to the boundedcontrol forces and control moments. However, there usuallyexists a possible control to meet the flight control requirementunder the limited control forces and control moments. Thus,for a given practical system, the input saturation should meetthe physical requirement of system control. Namely, thereshould exist a DSC that can track the given desired outputof the nonlinear system in the presence of the unknownexternal disturbance and the input saturation for all giveninitial conditions.

Remark 3: To tackle the control input saturation of theuncertain strict-feedback nonlinear system, the saturation func-tion is approximated by the tanh function in the DSC design.To facilitate the DSC design for the uncertain nonlinearsystem (6), we introduce gn0(x̄n)(∂h(v)/∂v) |v=vμ to be asa control gain function by invoking the mean-value theorem.In general, ∂h(v)/∂v goes to zero as v → ∞ which may leadto gn(x̄n) = gn0(x̄n)(∂h(v)/∂v)|v=vμ going to zero. However,from Assumption 5, we know that the difference between thedesigned control input v and the actual control input u shouldbe bounded to meet the controllable requirement. Due to the

bounded actual control input u, the designed control input vdoes not go to infinite which means gn without going to zeroin our DSC design.

III. DSC USING NONLINEAR DISTURBANCE OBSERVERAND BACKSTEPPING TECHNIQUE

In this section, the NN-based DSC scheme will bedeveloped for the uncertain strict-feedback nonlinearsystem (1) using the NDO. The detailed design process isdescribed as follows.

Step 1: Consider the first equation in (10) when n = 1 anddefine the error variable as

z1 = x1 − yd . (16)Invoking (10) and differentiating z1 with respect to time

yields

ż1 = ẋ1 − ẏd = f1(x1) + g1(x1)x2 − ẏd . (17)Assuming x2 as a virtual control input, the desired feedback

control α∗2 can be designed as

α∗2 = −k1z1 −1

g1( f1 − ẏd) (18)

where k1 is a positive design constant. f1 and g1 are unknownsmooth functions of x1.

Define ρ1(Z1) = (1/g1(x1))( f1(x1) − ẏd) with Z1 =[x1, ẏd ]T . By employing the RBFNN to approximate ρ1(Z1)and considering (12), α∗2 can be expressed as

α∗2 = −k1z1 − W∗T1 S1(Z1) − ε∗1 . (19)Since W∗1 and ε∗1 are unknown, the virtual control law α2

is proposed as

α2 = −k1z1 − Ŵ T1 S1(Z1) (20)where Ŵ1 is the estimation of W∗1 which is updated by

˙̂W1 = �1(S1(Z1)z1 − σ1Ŵ1) (21)where �1 = �T1 > 0 and σ1 > 0 are the design parameters.

To avoid repeatedly differentiating α2, which leads to theso-called explosion of complexity in the sequel steps, the DSCtechnique can be employed to solve it. Introducing a first-orderfilter λ2, and letting α2 pass through it with time constant τ2yields

τ2λ̇2 + λ2 = α2, λ2(0) = α2(0). (22)Defining z2 = x2 − λ2 and η2 = λ2 − α2, we have

λ̇2 = −η2/τ2 and x2 = z2+η2+α2. Considering (17) and (20),we obtain

ż1 = f1 + g1(z2 + η2 + α2) − ẏd= g1ρ1 + g1(z2 + η2 + α2)= g1

(W∗T1 S1(Z1) + ε∗1

)+ g1

(z2 + η2 − k1z1 − Ŵ T1 S1(Z1)

)= g1

(z2 + η2 − k1z1 − W̃ T1 S1(Z1) + ε∗1

)(23)

where W̃1 = Ŵ1 − W∗1 .


For η2, we have

η̇2 = λ̇2 − α̇2= −η2

τ2+

(− ∂α2

∂x1ẋ1 − ∂α2

∂z1ż1 − ∂α2

∂Ŵ1− ∂α2

∂ydẏd

)

= −η2τ2

+ M2(z1, z2, η2, Ŵ1, yd , ẏd , ÿd) (24)

where M2(z1, z2, η2, Ŵ1, yd , ẏd , ÿd) = −(∂α2/∂x1)ẋ1 −(∂α2/∂z1)ż1 − ∂α2/∂Ŵ1 − (∂α2/∂yd)ẏd is a continuous func-tion. For any B0 and p, the sets �0 := {(yd , ẏd , ÿd) : (yd)2 +(ẏd)2 + (ÿd)2 ≤ B0} and �2 := {∑2j=1 z2j + W̃ T1 �1W̃1 +η22 <2 p} are compact in R3 and RN1+3, respectively, where N1is the dimension of W̃1. Thus, �0 × �2 is also compact.Considering the continuous property, the function M2(.) hasa maximum value B2 for the given initial conditions in thecompact set �0 × �2 [35].

Consider the Lyapunov function candidate

V1 = 12g1

z21 +1

2η22 +

1

2W̃ T1 �

−11 W̃1. (25)

Invoking (21), (23), and (24), the time derivative of V1 isgiven by

V̇1 = 1g1

z1 ż1 − ġ12g21

z21 + η2η̇2 + W̃ T1 �−11 ˙̃W1

≤ z1(z2 + η2 − k1z1 − W̃ T1 S1(Z1) + ε∗1

) + gd12g2

1

z21

+ η2(

− η2τ2

+ M2)

+ W̃ T1 �−11 ˙̃W1

= − k1z21 + z1z2 + z1η2 + z1ε∗1 +gd1

2g21

z21

−η22

τ2+ η2 M2 − σ1W̃ T1 Ŵ1 (26)

where M2 denotes M2(z1, z2, η2, Ŵ1, yd , ẏd , ÿd).Considering the following fact:

2W̃ T1 Ŵ1 = ‖W̃1‖2 + ‖Ŵ1‖2 − ‖W∗1 ‖2≥ ‖W̃1‖2 − ‖W∗1 ‖2 (27)

we have

V̇1 ≤ −(

k1 − 1.5 − gd1

2g21

)z21 −

(1

τ2− 1

)η22 −

σ1

2‖W̃1‖2

+ 0.5z22 + 0.5ε∗21 + 0.5B22 +σ1

2‖W∗1 ‖2. (28)

Step i (2 ≤ i ≤ n − 1): In the i th step, we define the errorvariable as

zi = xi − λi (29)where λi is obtained from the (i − 1)th step.

Considering (10) and differentiating zi with respect to timeyields

żi = ẋi − λ̇i = fi (x̄i ) + gi(x̄i )xi+1 − λ̇i . (30)

Assuming xi+1 as a virtual control input, the desired feed-back control α∗i+1 can be designed as

α∗i+1 = −ki zi −1

gi( fi − λ̇i ) (31)

where ki is a positive design constant. fi and gi are unknownsmooth functions of x̄i .

Define ρi (Zi ) = (1/gi (x̄i))( fi (x̄i )−λ̇i ) with Zi = [x̄i , ˙̄λi ]T.By employing the RBFNN to approximate ρi (Zi ) andconsidering (12), α∗i+1 can be expressed as

α∗i+1 = −ki zi − W∗Ti Si (Zi ) − ε∗i . (32)Since W∗i and ε∗i are unknown, the virtual control law αi+1

is proposed as

αi+1 = −ki zi − Ŵ Ti Si (Zi ) (33)where Ŵi is the estimation of W∗i which is updated by

˙̂Wi = �i (Si (Zi )zi − σi Ŵi ) (34)where �i = �Ti > 0 and σi > 0 are the design parameters.

To avoid repeatedly differentiating αi+1, which leads to theso-called explosion of complexity in the sequel steps, the DSCtechnique can be employed to solve it. Introducing a first-orderfilter λi+1, and letting αi+1 pass through it with time constantτi+1 yields

τi+1λ̇i+1 + λi+1 = αi+1, λi+1(0) = αi+1(0). (35)Defining zi+1 = xi+1 − λi+1 and ηi+1 = λi+1 − αi+1, we

have λ̇i+1 = −ηi+1/τi+1 and xi+1 = zi+1 + ηi+1 + αi+1.Considering (30) and (33), we obtain

żi = fi + gi(zi+1 + ηi+1 + αi+1) − λ̇i= giρi + gi (zi+1 + ηi+1 + αi+1)= gi

(W∗Ti Si (Zi ) + ε∗i

)+ gi

(zi+1 + ηi+1 − ki zi − Ŵ Ti Si (Zi )

)= gi

(zi+1 + ηi+1 − ki zi − W̃ Ti Si (Zi ) + ε∗i

)(36)

where W̃i = Ŵi − W∗i .For ηi+1, we have

η̇i+1 = λ̇i+1 − α̇i+1 = −ηi+1τi+1

+(

− ∂α∂xi

ẋi − ∂αi+1∂zi

żi − ∂αi+1∂Ŵi

− ∂αi+1∂λi

λ̇i

)

= −ηi+1τi+1

+ Mi+1 (37)

where Mi+1 denotes Mi+1(z1, . . . , zi+1, η1, . . . , ηi , Ŵ1, . . . ,Ŵi , yd , ẏd , ÿd) and Mi+1 = −(∂α/∂xi )ẋi − (∂αi+1/∂zi )żi −∂αi+1/∂Ŵi − (∂αi+1/∂λi )λ̇i is a continuous function. For anyB0 and p, the sets �0 := {(yd , ẏd , ÿd) : (yd)2+(ẏd)2+(ÿd)2 ≤B0} and �i := {∑ij=1(z2j + (W̃ Tj � j W̃ j )) + ∑ij=2 η2j <2 p}, i = 1, . . . , n − 1 are compact in R3 and R

∑ij=1 Ni +2i−1,

respectively, where Ni is the dimension of W̃i . Thus, �0 ×�iis also compact. Considering the continuous property, thefunction Mi+1 has a maximum value Bi+1 for the given initialconditions in the compact set �0 × �i [35].


Consider the Lyapunov function candidate

Vi = 12gi

z2i +1

2η2i+1 +

1

2W̃ Ti �

−1i W̃i . (38)

Invoking (34), (36), and (37), the time derivative of V1 isgiven by

V̇i = 1gi

zi żi − ġi2g2i

z2i + ηi+1η̇i+1 + W̃ Ti �−1i ˙̃Wi

≤ zi(zi+1 + ηi+1 − ki zi − W̃ Ti Si (Zi ) + ε∗i

) + gdi2g2

i

z2i

+ ηi+1(

− ηi+1τi+1

+ Mi+1)

+ W̃ Ti �−1i ˙̃Wi= − ki z2i + zi zi+1 + ziηi+1 + ziε∗i

+ gdi

2g2i

z2i −η2i+1τi+1

− σi W̃ Ti Ŵi + ηi+1 Mi+1. (39)

Considering the following fact:2W̃ Ti Ŵi = ‖W̃i‖2 + ‖Ŵi ‖2 − ‖W∗i ‖2

≥ ‖W̃i‖2 − ‖W∗i ‖2 (40)we have

V̇i ≤ −(

ki − 1.5 − gdi

2g2i

)z2i −

(1

τi+1− 1

)η2i+1 −

σi

2‖W̃i‖2

+ 0.5z2i+1 + 0.5ε∗2i + 0.5B2i+1 +σi

2‖W∗i ‖2. (41)

Step n: In this step, the error variable is defined as

zn = xn − λn (42)where λn is obtained from the (n − 1)th step.

Considering (10) and differentiating zn with respect to timeyields

żn = ẋn − λ̇n = fn(x̄n) + gn(x̄n)v + D − λ̇n . (43)The desired feedback control v∗ can be designed as

v∗ = −knzn − 1gn

( fn − λ̇n) − D (44)where kn is a positive design constant. fn and gn are unknownsmooth functions of x̄n .

Define ρn(Zn) = 1/gn(x̄n)( fn(x̄n) − λ̇n) with Zn =[x̄n, ˙̄λn]T. By employing the RBFNN to approximate ρn(Zn)and considering (12), v∗ can be expressed as

v∗ = −knzn − W∗Tn Sn(Zn) − ε∗n − D. (45)Since W∗n , ε∗n , and D are unknown, the control law v is

proposed as

v = −knzn − Ŵ Tn Sn(Zn) − D̂ (46)where D̂ is the estimation of D and Ŵn is the estimation ofW∗n which is updated by

˙̂Wn = �n(Sn(Zn)zn − σn Ŵn) (47)where �n = �Tn > 0 and σn > 0 are the design parameters.

Considering (43) and (46), we obtain

żn = fn + gnv + D(t) − λ̇n = gn(W∗Tn Sn(x̄n) + ε∗n

)+ gn

( − knzn − Ŵ Tn Sn(Zn) − D̂) + D(t)= gn

( − knzn − W̃ Tn Sn(Zn) − D̂ + ε∗n) + D(t) (48)where W̃n = Ŵn − W∗n .

To facilitate the design of the NDO, (43) can be also writtenas

żn = l−1ρ(x̄n, v) + D − λ̇n= l−1W∗Tρ Sρ(x̄n) + l−1ερ + D − λ̇n (49)

where ρ(x̄n, v) = l( fn(x̄n) + gn(x̄n)v), W∗ρ is optimal weightvalue of the RBFNN, ερ is the approximation error of theRBFNN, and l > 0 is a design parameter of the developedNDO.

Invoking (49), an auxiliary variable is given by

s = zn − ξ (50)and the intermedial variable ξ is proposed as

ξ̇ = cs + l−1Ŵ Tρ Sρ(x̄n) − λ̇n (51)where c > 0 is a designed parameter and Ŵρ is the estimateof the optimal weight value W∗ρ .

Differentiating (50) and considering (49) and (51), we have

ṡ = żn − ξ̇ = l−1W∗Tρ Sρ(x̄n) + l−1ερ + D − λ̇n− (cs + l−1Ŵ Tρ Sρ(x̄n) − λ̇n)

= − cs − l−1W̃ Tρ Sρ(x̄n) + l−1ερ + D (52)where W̃ρ = Ŵρ − W∗ρ .

Considering (52) yields

sṡ = −cs2 − l−1sW̃ Tρ Sρ(x̄n) + l−1sερ + s D≤ − (c − 1.0)s2−l−1sW̃ Tρ Sρ(x̄n)+0.5l−2ε2ρ +0.5θ20 .

(53)

On the basis of the auxiliary variable s, the NDO isdesigned as

D̂ = l(s − φ) (54)and the intermedial variable φ is given by

φ̇ = −cs + D̂. (55)Define D̃ = D − D̂. Differentiating (54), and considering

(52) and (55) yield

˙̂D = l(ṡ−φ̇)= l((−cs−l−1W̃ Tρ Sρ(x̄n)+l−1ερ +D)−(−cs+ D̂))= −W̃ Tρ Sρ(x̄n) + ερ + l(D − D̂)= −W̃ Tρ Sρ(x̄n) + ερ + l D̃. (56)Considering (56), we have

˙̃D = Ḋ − ˙̂D = Ḋ − l D̃ + W̃ Tρ Sρ(x̄n) − ερ. (57)Invoking (57), we obtain

D̃ ˙̃D = D̃ Ḋ − l D̃2 + D̃W̃ Tρ Sρ(x̄n) − D̃ερ. (58)


Considering the following fact:2D̃W̃ Tρ Sρ(x̄n) ≤ 2|D̃|||W̃ρ ||||Sρ(x̄n)||

≤ γ0ϑ2 D̃2 + 1γ0

||W̃ρ ||2 (59)yields

D̃ ˙̃D ≤ D̃2 + 0.5Ḋ2 − l D̃2 + γϑ2 D̃2 + 1γ

||W̃ρ ||2 + 0.5ε2ρ≤ −(l − (1.0 + γϑ2))D̃2 + 1

γ||W̃ρ ||2 + 0.5θ21 + 0.5ε2ρ

(60)

where ||Sρ(x̄n)|| ≤ ϑ , γ = 0.5γ0 and γ0 > 0 is a designparameter.

The parameter updated law Ŵρ is designed as

˙̂Wρ = �ρ(l−1STρ (x̄n)s − σρ Ŵρ

)(61)

where �ρ = �Tρ > 0 and σρ > 0 are the design parameters.Consider the Lyapunov function candidate

Vn = 12gn

z2n +1

2W̃ Tn �

−1n W̃n +

1

2W̃ Tρ �

−1ρ W̃ρ +

1

2s2 + 1

2D̃2.

(62)

Invoking (48), (53), and (60), the time derivative of Vn is

V̇n = 1gn

zn żn − ġn2g2n

z2n + W̃ Tn �−1n ˙̃Wn+W̃ Tρ �−1ρ ˙̃Wρ + sṡ + D̃ ˙̃D

≤ zn( − knzn − W̃ Tn Sn(Zn) − D̂ + ε∗n) + g

dn

2g2n

z2n

+ zn Dgn

+ W̃ Tn �−1n ˙̃Wn + W̃ Tρ �−1ρ ˙̃Wρ− (c − 1.0)s2 − l−1sW̃ Tρ Sρ(x̄n) + 0.5l−2ε2ρ + 0.5θ20− (l − (1.0 + γϑ2))D̃2 + 1

γ||W̃ρ ||2

+ 0.5θ21 + 0.5ε2ρ. (63)Considering D̃ = D − D̂ and Assumption 3, we have

V̇n ≤ zn( − knzn − W̃ Tn Sn(Zn) + ε∗n) + g

dn

2g2n

z2n

+ zn D̃ + zn D(

1

gn− 1

)+ W̃ Tn �−1n ˙̃Wn + W̃ Tρ �−1ρ ˙̃Wρ

− (c − 1.0)s2 − l−1sW̃ Tρ Sρ(x̄n) + 0.5l−2ε2ρ + 0.5θ20− (l − (1.0 + γϑ2))D̃2 + 1

γ||W̃ρ ||2 + 0.5θ21 + 0.5ε2ρ

≤ −(

kn − 1.5 − gdn

2g2n

)z2n − zn W̃ Tn Sn(Zn)

+ W̃ Tn �−1n ˙̃Wn + W̃ Tρ �−1ρ ˙̃Wρ − l−1sW̃ Tρ Sρ(x̄n)− (c − 1.0)s2 − (l − (1.5 + γϑ2))D̃2

+ 1γ

||W̃ρ ||2 +(

0.5 + 0.5| 1g

n

− 1|2)

θ20

+ 0.5θ21 + 0.5ε∗2n + (0.5 + 0.5l−2)ε2ρ. (64)

Substituting (47) and (61) into (64), we obtain

V̇n ≤ −(

kn − 1.5 − gdn

2g2n

)z2n − (c − 1.0)s2

−(l − (1.5 + γϑ2))D̃2 − σnW̃ Tn Ŵn − σρ W̃ Tρ Ŵρ+ 1

γ||W̃ρ ||2 +

(0.5 + 0.5| 1

gn

− 1|2)

θ20

+ 0.5θ21 + 0.5ε∗2n + (0.5 + 0.5l−2)ε2ρ. (65)Considering the following facts:

2W̃ Tn Ŵn = ‖W̃n‖2 + ‖Ŵn‖2 − ‖W∗n ‖2≥ ‖W̃n‖2 − ‖W∗n ‖2 (66)

and

2W̃ Tρ Ŵρ = ‖W̃ρ‖2 + ‖Ŵρ‖2 − ‖W∗ρ ‖2

≥ ‖W̃ρ‖2 − ‖W∗ρ ‖2 (67)we have

V̇n ≤ −(

kn − 1.5 − gdn

2g2n

)z2n − (c − 1.0)s2

−(l − (1.5 + γϑ2))D̃2 − σn2

‖W̃n‖2

−(

σρ

2− 1

γ

)‖W̃ρ‖2 +

(0.5 + 0.5| 1

gn

− 1|2)

θ20

+ 0.5θ21 + 0.5ε∗2n + (0.5 + 0.5l−2)ε2ρ+ σn

2‖W∗n ‖2 +

σρ

2‖W∗ρ ‖2. (68)

The above DSC design procedure and stability analysis canbe summarized in the following theorem, which contains theresults for the uncertain strict-feedback nonlinear system (1)using the NDO and backstepping technique.

Theorem 1: Consider the uncertain strict-feedback nonlin-ear system (1) with the input saturation and the unknownexternal disturbance and suppose that full state information isavailable. The NDO is designed as (50), (51), (54), and (55).The updated laws of the NN weight values are chosen as (21),(34), (47), and (61). The nonlinear disturbance observer-basedDSC is proposed in (46). Given any positive number p, for allinitial conditions satisfying �n := {∑nj=1(z2j +(W̃ Tj � j W̃ j ))+∑n

j=2 η2j < 2 p}, the appropriate design parameters ki , l, c, τi ,�i , �ρ , σi , σρ , and γ can be chosen according to (72) suchthat all closed-loop signals are uniformly bounded convergenceunder the proposed dynamic surface control based on thenonlinear disturbance observer. Furthermore, the tracking errorz1 = x1−yd can be made small by proper choice of the designparameters ki , l, c, τi , �i , �ρ , σi , σρ , and γ.

Proof: For considering the convergence of disturbanceestimate error and closed-loop states, the Lyapunov func-tion candidate of the whole closed-loop control system isconsidered as

V =n∑

i=1Vi . (69)


Differentiating V and considering (28), (41), and (68), weobtain

V̇ ≤ −(

k1 − 1.5 − gd1

2g21

)z21 −

n∑i=2

(ki − 2.0 − g

di

2g2i

)z2i

−n∑

i=2

(1

τi− 1

)η2i −

n∑i=1

σi

2‖W̃i‖2

− (c − 1.0)s2 − (l − (1.5 + γϑ2))D̃2

−(

σρ

2− 1

γ

)‖W̃ρ‖2 + 0.5

n∑i=2

B2i

+(

0.5 + 0.5| 1g

n

− 1|2)

θ20 + 0.5θ21 + 0.5n∑

i=1ε∗2i

+ (0.5 + 0.5l−2)ε2ρ +n∑

i=1

σi

2‖W∗i ‖2 +

σρ

2‖W∗ρ ‖2

≤ −�V + C (70)where � and C are given by

� : = min

⎛⎜⎜⎜⎜⎜⎜⎝

(k1 − 1.5 − g

d1

2g21

),(

ki − 2.0 − gdi

2g2i

)

(1

τi− 1), (c − 1.0), (l − (1.5 + γϑ2))σi

λmax(�−1i )

,2( σρ2 − 1γ )λmax(�

−1ρ )

⎞⎟⎟⎟⎟⎟⎟⎠

C : = 0.5n∑

i=2B2i +

n∑i=1

σi

2‖W∗i ‖2 +

σρ

2‖W∗ρ ‖2

+(

0.5 + 0.5∣∣∣∣ 1g

n

− 1∣∣∣∣ 2

)θ20

+ 0.5θ21 + 0.5n∑

i=1ε∗2i + (0.5 + 0.5l−2)ε2ρ. (71)

To ensure the closed-loop system stability, the correspond-ing design parameters ki , l, c, τi , σρ , and γ should be chosento make the following inequalities hold:

k1 − 1.5 − gd1

2g21

> 0

ki − 2.0 − gdi

2g2i

> 0, i = 2, . . . , n

1

τi− 1 > 0, i = 2, . . . , n − 1

c − 1.0 > 0l − (1.5 + γϑ2) > 0σρ

2− 1

γ> 0. (72)

According to (70), we have

0 ≤ V ≤ C�

+[

V (0) − C�]e−�t. (73)

From (73), we can know that V is convergent, i.e.,limt−→∞ V = C/�. According to (73), it may directly show

that the signals e, zi , W̃i , W̃ρ , and D̃ are semiglobally uni-formly bounded when t → 0. Hence, the tracking error e, theapproximation errors W̃i , W̃ρ , and the disturbance estimationerror D̃ of the closed-loop system are bounded. This concludesthe proof. ♦

Remark 4: To enhance the closed-loop system robustnessof the uncertain nonlinear system, the nonlinear disturbanceobserver-based DSC scheme has been developed. For fullyutilizing the dynamic information of the external disturbance,the NDO is proposed to estimate the unknown disturbance ofthe uncertain nonlinear system and the output of the NDO isused to design the DSC law, as shown in (46). Due to theintroduction of the output of NDO, the control gain can beadjusted according to the variation of unknown disturbanceand the disturbance rejection ability of the closed-loop systemhas been improved.

Remark 5: In this paper, the NDO is developed to estimatethe unknown disturbance of the uncertain strict-feedback non-linear system. In the developed NDO, the known boundaryrequirement of the disturbance is canceled and the boundeddisturbance estimation error is guaranteed. At the same time,the slowly changeable assumption of the external disturbanceis eliminated.

Remark 6: In the developed DSC, the design parameters ki ,l, c, τi , �i , �ρ , σi , σρ , and γ need to be tuned to obtain a goodtransient performance and the closed-loop stable performance.If the tracking error is desired to be lower, we should increasek1. �i , �ρ , σi , and σρ are design parameters in adaptationlaw of NN weight value. Decreases in σi and σρ or increasesin the adaptive gain �i and �ρ will result in a better trackingperformance. Furthermore, the L∞ performance of systemtracking error can be guaranteed by choosing the properinitial conditions for all closed-loop system signals accordingto (73) [13], [32], [40], [42].

IV. SIMULATION STUDY

In this section, simulation results are presented to illus-trate the effectiveness of the developed nonlinear disturbanceobserver-based DSC and an example system is used in thesimulation study. In the simulation, the NDO is designedas (50), (51), (54), and (55). The updated laws of theNN weight values are chosen as (21), (34), (47), and (61).The nonlinear disturbance observer-based DSC is proposedin (46).

Let us consider the one-link manipulator with the inclusionof motor dynamics. The model of the one-link manipulator isgiven by [36]

D̄q̈ + Bq̇ + N sin(q) = τM τ̇ + H τ = u − Kmq̇ (74)

where q , q̇, and q̈ denote the link angular position, velocity,and acceleration, respectively. τ is the motor current. u is theinput control voltage. The parameter values with appropriateunits are given by D̄ = 1, M = 0.05, B = 1, Km = 10,H = 10, and N = 10.

Defining x1 = q , x2 = q̇, and x3 = τ , and considering theinput saturation, the above one-link manipulator model can be


Fig. 1. Output x1 (solid line) follows desired trajectory yd (dashed line) ofthe single-link robot system for case 1.

written as

ẋ1 = x2ẋ2 = x3

D̄− Bx2

D̄− N

D̄sin(x1)

ẋ3 = 1M

u(v(t)) − KmM

x2 − HM

x3y = x1.

When uncertainty and disturbance are involved, the dynam-ics of one-link manipulator can be expressed as the followingform: ⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ẋ1 = x2ẋ2 = x22 e−x

21 − Bx2

D̄− N

D̄sin(x1) + x3D̄

ẋ3 = − KmM x2 − HM x3 + 1M u(v(t)) + d(t)y = x1.

(75)

Define f1(x1) = 0, g1(x1) = 1, f2(x̄2) = x22e−x21 −

Bx2/D̄ − (N/D̄) sin(x1), g2(x̄2) = 1/D̄, f3(x̄3) =−(Km/M)x2 − (H/M)x3, and g3(x̄3) = 1/M . It is apparentthat the numerical example (75) is suitable for the case ofsystem (1). In the simulation, the external disturbance ischosen as d(t) = 0.4 cos(2t).

To proceed with the design of nonlinear disturbanceobserver-based DSC scheme, all design parameters are chosenas l = 200, c = 13, σi = 0.02, k1 = 20, k2 = 20, k3 = 10,and γ = 20. The initial state conditions are chosen as x0 = 0,x2 = 0, and x3 = 0. The input saturation value is given asuM = 80.

A. Case 1: For Constant Desired Trajectory

To illustrate the effectiveness of the developed nonlineardisturbance observer-based DSC design, the desired trajectoryis taken as yd = 1. Under the proposed nonlinear disturbanceobserver-based DSC scheme (46), the tracking control resultsare shown in Figs. 1–4. From Figs. 1 and 2, we note thatthe tracking performance is satisfactory and the tracking error

Fig. 2. Tracking error of the single-link robot system for case 1.

Fig. 3. Control input of the single-link robot system for case 1.

quickly converge to zero for the uncertain one-link manipula-tor system (75) in the presence of the time-varying externaldisturbance and input saturation. Although the better trackingerror is obtain without considering the input saturation, theaccepted tracking performance is still maintained for theuncertain one-link manipulator system (75) in the presenceof the time-varying external disturbance and input saturationunder our developed DSC scheme. Using the output of theNDO, the control input is bounded and convergent, as shownin Fig. 3. The plots of the NN weight values are shown inFig. 4, which are convergent.

B. Case 2: For Time-Varying Desired Trajectory

Here, the desired trajectory is taken as yd = sin(t) +cos(0.5t) to illustrate the effectiveness of the developednonlinear disturbance observer-based DSC design. Usingthe proposed nonlinear disturbance observer to design DSCscheme (46), all tracking control results are given inFigs. 5–8. According to Figs. 5 and 6, the satisfactory trackingperformance is obtained and the tracking error maintains in


Fig. 4. Norms of NN weight values for case 1.

Fig. 5. Output x1 (solid line) follows desired trajectory yd (dashed line) ofthe single-link robot system for case 2.

Fig. 6. Tracking error of the single-link robot system for case 2.

a small compact set for the uncertain one-link manipulatorsystem (75) under the integrated effects of the time-varyingexternal disturbance and input saturation. On the basis of the

Fig. 7. Control input of the single-link robot system for case 2.

Fig. 8. Norms of NN weight values for case 2.

output of the NDO, the control input command is bounded andconvergent, as shown in Fig. 7. From Fig. 8, the convergentplots of the NN weight values are noted.

Based on above simulation results, we can obtain that theproposed disturbance observer-based DSC scheme is valid forthe uncertain the one-link manipulator system with the time-varying unknown external disturbance and input saturation.

V. CONCLUSION

In this paper, the nonlinear disturbance observer-based NNDSC scheme has been developed for a class of uncertainnonlinear systems with input saturation. To improve the abil-ity of the disturbance attenuation and system performancerobustness, the NDO has been used to monitor the unknowncompounded disturbance, and its output signal is utilizedin the construction of nonlinear disturbance observer-basedNN DSC scheme. Closed-loop system stability and trackingperformance have been proved and analyzed using a rigorousLyapunov analysis. Finally, simulation results of a one-linkmanipulator control system have been presented to illus-trate the effectiveness of the proposed disturbance observer-based NN DSC scheme. In the future, the application of the


developed nonlinear disturbance observer-based DSC schemeshould be further studied. Furthermore, the DSC scheme canbe developed using L∞-type criteria to enhance the controlperformance when the control input saturation appear for thestudied uncertain nonlinear systems.

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Mou Chen (M’10) received the B.Sc. degree inmaterial science and engineering, and the M.Sc.degree and the Ph.D. degree in automatic controlengineering from the Nanjing University of Aero-nautics and Astronautics, Nanjing, China, in 1998and 2004, respectively.

He was an Academic Visitor with the Depart-ment of Aeronautical and Automotive Engineer-ing, Loughborough University, Leicester, U.K., from2007 to 2008, and a Research Fellow with theDepartment of Electrical and Computer Engineering,

National University of Singapore, Singapore, from 2008 to 2009. He iscurrently a Professor with the College of Automation Engineering, NanjingUniversity of Aeronautics and Astronautics. His current research interestsinclude 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 Universityof Virginia, Charlottesville, VA, USA. His researchhas been mainly in adaptive control, with particularinterests in adaptive control of systems with multiple

inputs and multiple outputs, with nonsmooth nonlinearities or uncertain faults,stability and robustness of adaptive control systems, and system passivitycharacterizations. He has authored or co-authored, and co-edited six books andover 350 papers on some related topics. His current research interests includeadaptive control of systems with actuator failures and structural damage,adaptive approximation-based control, and resilient aircraft and spacecraftcontrol applications.

Prof. Tao served as an Associate Editor of Automatica, the InternationalJournal of Adaptive Control and Signal Processing, and the IEEE TRANS-ACTIONS ON AUTOMATIC CONTROL, an Editorial Board Member of theInternational Journal of Control, Automation and Systems, and a Guest Editorof the Journal of Systems Engineering and Electronics.

Bin Jiang (SM’05) received the Ph.D. degreein automatic control from Northeastern University,Shenyang, China, in 1995.

He was a Post-Doctoral Fellow or Research Fellowin Singapore, France, and the U.S., respectively.He is currently a Professor and the Dean of theCollege of Automation Engineering with the NanjingUniversity of Aeronautics and Astronautics, Nan-jing, China. His current research interests includefault diagnosis and fault tolerant control, and theirapplications.

Dr. Jiang is a member of the IFAC Technical Committee on Fault Detection,Supervision, and Safety of Technical Processes. He currently serves as anAssociate Editor or Editorial Board Member for a number of journals,such as the IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY,the International Journal of Systems Science, the International Journal ofControl, Automation, and Systems, the International Journal of InnovativeComputing, Information and Control, the International Journal of AppliedMathematics and Computer Science, Acta Automatica Sinica, and the Journalof Astronautics.

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