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Automatica 47 (2011) 452–465

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Automatica

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Adaptive tracking control of uncertain MIMO nonlinear systems withinput constraints

Mou Chen a,c, Shuzhi Sam Ge b,c,∗, Beibei Ren c

a College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, Chinab Institute of Intelligent Systems and Information Technology & Robotics Institute, University of Electronic Science and Technology of China, 611731 Chengdu, Chinac Department of Electrical & Computer Engineering, National University of Singapore, 117576, Singapore

a r t i c l e i n f o

Article history:Available online 23 February 2011

Keywords:Nonlinear systemsInput constraintCommand filterAdaptive tracking controlBackstepping control

a b s t r a c t

In this paper, adaptive tracking control is proposed for a class of uncertain multi-input and multi-outputnonlinear systems with non-symmetric input constraints. The auxiliary design system is introduced toanalyze the effect of input constraints, and its states are used to adaptive tracking control design. Thespectral radius of the control coefficient matrix is used to relax the nonsingular assumption of the controlcoefficient matrix. Subsequently, the constrained adaptive control is presented, where command filtersare adopted to implement the emulate of actuator physical constraints on the control law and virtualcontrol laws and avoid the tedious analytic computations of time derivatives of virtual control laws in thebackstepping procedure. Under the proposed control techniques, the closed-loop semi-global uniformlyultimate bounded stability is achieved via Lyapunov synthesis. Finally, simulation studies are presentedto illustrate the effectiveness of the proposed adaptive tracking control.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

During the past several decades, adaptive control of nonlinearsystems has received much attention for establishing the globallyasymptotical stability of the closed-loop system (Ge, 1996a,b; Ge& Wang, 2003; Hung, Tuan, Narikiyo, & Apkarian, 2008; Krstić &Kokotović, 1995; Luo, Chu, & Ling, 2005; Makoudi & Radouane,2000; Mirkin & Gutman, 2005; Skjetnea, Fossen, & Kokotović,2000; Tang, Tao, & Joshi, 2007; Yao & Tomizuka, 2001; Yu & Sun,2001). In practice, most control plants are nonlinear, uncertain andmultivariable in character. It is important to investigate effectiveadaptive control techniques for uncertain multi-input and multi-output (MIMO) nonlinear systems. In Tang et al. (2007), directadaptive control was developed for a class of MIMO nonlinearsystems in the presence of uncertain failures of redundantactuators. In Yao and Tomizuka (2001), adaptive robust controlwas proposed for MIMO nonlinear systems in semi-strict feedbackforms. Robust adaptive tracking control was developed for the

This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Raul Ordóñezunder the direction of Editor Miroslav Krstic.∗ Corresponding author at: Institute of Intelligent Systems and Information

Technology & Robotics Institute, University of Electronic Science and Technologyof China, 611731 Chengdu, China. Tel.: +86 28 61830633; fax: +86 28 61831655.

E-mail addresses: [email protected] (M. Chen), [email protected](S.S. Ge), [email protected] (B. Ren).

0005-1098/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2011.01.025

time varying uncertain nonlinear systems with unknown controlcoefficients (Ge &Wang, 2003). As an effective control technology,adaptive control has been successively used in a variety ofpractical control systems. In Ge (1996a), adaptive control wasproposed for robots with both dynamic parameter uncertaintiesand unknown input scalings. Adaptive control for flexible jointrobots was presented based on singular perturbation theory andposition information in Ge (1996b). Adaptive recursive design wasdeveloped for a parametric uncertain nonlinear plant describingthe dynamics of a ship (Skjetnea et al., 2000). In Luo et al. (2005),inverse optimal adaptive control was presented for the attitudetracking of spacecraft. Adaptive controlwas studied for nonlinearlyparameterized uncertainties in robot manipulators (Hung et al.,2008). In the adaptive control of uncertain MIMO nonlinearsystems, one main challenge is the possible singularity of thecontrol coefficient matrix which makes the control design becomemore complicated. Existing research results of adaptive controltechniques for the MIMO nonlinear system mostly assume thatthe control coefficient matrix is known and nonsingular (Kwan& Lewis, 2000). In this paper, the spectral radius of the controlcoefficient matrix is introduced in the control design to relax thenonsingular assumption of the control coefficient matrix.

Since actuator physical constraints can severely degrade theclosed-loop system performance, control design for uncertainMIMO nonlinear systems with actuator constraints presents atremendous challenge. During the past decades, there has exten-sive research on the control of mechanical systems with various

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M. Chen et al. / Automatica 47 (2011) 452–465 453

constraints. Analysis and design of control systems with input sat-uration constraints have been studied in Cao and Lin (2003), Chen,Ge, and Choo (2009), Chen, Ge, and How (2010), Hu, Ma, and Xie(2008), Gao and Selmic (2006) and Zhong (2005). To handle thephysical limitation, constrained adaptive backstepping controlwasproposed in which command filters were used to implement theemulate of constraints on the control command and the virtualcontrol laws (Farrell, Polycarpou, & Sharma, 2003; Polycarpou, Far-rell, & Sharma, 2004, 2003; Sonneveldt, Chu, & Mulder, 2007).In Polycarpou et al. (2004), nonlinear approximation based back-stepping control was presented for nonlinear dynamical systemssubject to magnitude, rate, and bandwidth constraints. The con-trol input saturation was investigated via on-line approximationbased control for uncertain nonlinear systems (Polycarpou et al.,2003). Constrained adaptive backstepping control was presentedfor fighter aircraft in Sonneveldt et al. (2007). In the constrainedadaptive control, the key problem is how to analyze the constrainteffect of the actuator’s physical constraints. To this end, we intro-duce an auxiliary design system to analyze the constraint effect inthis paper. Based on the states of the auxiliary design system, con-strained adaptive control is investigated for a class of uncertainMIMO nonlinear systems with input constraints using backstep-ping technique.

Backstepping control has became one of the most popular ro-bust adaptive control design techniques for some special classesof nonlinear systems (Gong & Yao, 2001; Wang & Huang, 2005;Zhang, Ge, & Hang, 2000). In recent years, the universal approx-imation ability of neural network (NN) or fuzzy logical system(FLS) has been employed to design robust adaptive control comb-ing with backstepping technique for the uncertain MIMO nonlin-ear systems, and various robust adaptive control strategies havebeen proposed (Chang, 2000, 2001; Chang & Yen, 2005; Ge, 1998;Ge & Wang, 2004; Ge & Tee, 2007; Ge, Li, Zhang, & Lee, 2004;Ge, Zhang, & Lee, 2004; Lee & Lee, 2004; Zhang, Ge, & Lee, 2005).The proposed robust adaptive control based on NN or FLS is anefficient control approach of MIMO nonlinear systems, but themodel-based adaptive control should be widely developed due tothe relatively easy realization (Narendra & Annaswamy, 1989; Qu,Dorsey, & Dawson, 1994). Furthermore, the adaptive backsteppingcontrol of uncertainMIMOnonlinear systemswith non-symmetricinput constraints need to be further investigated.

In this paper, adaptive tracking control is proposed to handlethe input saturation and actuator physical constraints for uncertainMIMO nonlinear systems. The main contributions of the paper areas follows:

(i) To the best of our knowledge, it is the first time in the literaturethat the non-symmetric nonlinear input saturation constraintis considered for the adaptive tracking control of uncertainMIMO nonlinear systems.

(ii) The spectral radius of the control coefficient matrix is em-ployed in the control design to relax the nonsingular assump-tion of the control coefficient matrix.

(iii) To handle the non-symmetric input saturation constraint, theauxiliary design system is introduced to analyze the effect ofinput constraints, and the states of auxiliary design system areused to develop adaptive tracking control.

(iv) command filters are introduced to implement the emulate ofactuator physical constraints on the control command and vir-tual control laws, and avoid the tedious analytic computationsof time derivatives of virtual control laws in the backsteppingprocedure.

The rest of the paper is organized in the following manner.Section 2 presents the problem formulation and preliminaries.Adaptive tracking control is investigated for uncertain MIMOnonlinear systems with input saturation in Section 3, followed

Fig. 1. Non-symmetric input saturation constraint.

by the constrained adaptive control considering actuator physicalconstraints in Section 4. The simulation results are presented todemonstrate the effectiveness of proposed adaptive control inSection 5. Section 6 contains the conclusion.Notations: ‖ · ‖ denotes for Frobenius norm of matrices andEuclidean norm of vectors, i.e., given a matrix B and a vector ξ ,the Frobenius norm and Euclidean norm are given by ‖B‖2

=

tr(BTB) =∑

i,j b2ij and ‖ξ‖2

=∑

i ξ2i . xi = [x1, x2, . . . , xi]T ∈ Ri×m

stands the vector of partial state variables in the nonlinear system.For integer indices i and j, we define Tanh(zi) := diag

tanh

zijεij

,

εij > 0, Ψi = [kεi1, kεi2, . . . , kεim]T , k = 0.2758, ρi(xi) :=

diagρij(xi) and Θi = [Θi1,Θi2, . . . ,Θim]T . θi and Θi denote the

estimates of uncertain parameter vectors θi and Θi, respectively,and the estimate errors are defined as θi := θi − θi and Θi :=

Θi −Θi, i = 1, 2, . . . , n and j = 1, 2, . . . ,m.

2. Problem formulation

Consider a class of uncertain MIMO nonlinear systems in theform ofxi = Fi(xi)θi + (Gi(xi)+∆Gi(xi))xi+1

+Di(xi, t), i = 1, 2, . . . , n − 1. . .

xn = Fn(xn)θn + (Gn(xn)+∆Gn(xn))u + Dn(xn, t)y = x1 (1)

where xi ∈ Rm, i = 1, 2, . . . , n are the state vectors; θi ∈ Rqi , i =

1, 2, . . . , n are the uncertain parameter vectors; Fi ∈ Rm×qi ,i = 1, 2, . . . , n are known nonlinear functions; Gi ∈ Rm×m, i =

1, 2, . . . , n are known control coefficient matrices; Di ∈ Rm, i =

1, 2, . . . , n are unknown time-varying disturbances; u ∈ Rm is thecontrol input vector; y ∈ Rm is the system output vector; qi arepositive integers and ∆Gi ∈ Rm×m, i = 1, 2, . . . , n are unknownbounded perturbations of control coefficient matrices.

Considering actuator non-symmetric input constraints asshown in Fig. 1, the control input u = [u1, . . . , um]

T is defined by

ui =

urimax, if vi > vrimaxgri(vi), if 0 ≤ vi ≤ vrimaxgli(vi), if vlimax ≤ vi < 0ulimax, if vi < vlimax

(2)

where vi is ith element of the designed control law v = [v1,v2 . . . , vm]

T , vlimax < 0, and vrimax > 0 are known constants; andgri(vi) and gli(vi) are smooth continuous known nonlinear func-tions.

To facilitate control system design, the following assumptionsand lemmas are presented and will be used in the subsequentdevelopments.

Assumption 1 (Zhang & Ge, 2007, 2008). There exists positiveconstants kli0, kli1, kri0 and kri1 such that

454 M. Chen et al. / Automatica 47 (2011) 452–465

0 < kri0 ≤ g ′

ri(vi) ≤ kri1, vi ∈ [0, vrimax] (3)

0 < kli0 ≤ g ′

li(vi) ≤ kli1, vi ∈ [vlimax, 0). (4)

Assumption 2 (Tee & Ge, 2006). For the disturbance terms ∀(xi, t)∈ Ri×m

× R+, Dij(xi, t), i = 1, 2, . . . , n; j = 1, 2, . . . ,m, thereexist known smooth functions ρij(xi) ∈ R+,∀t > t0 and unknownbounded constantsΘij such that

|Dij(xi, t)| ≤ ρij(xi)Θij. (5)

Assumption 3 (Kim & Ha, 2000). For all known control coefficientmatrices Gi(xi), i = 1, 2, . . . , n of the uncertain nonlinear system(1), there exist known positive constants ζi > 0 such that‖Gi(xi)‖ ≤ ζi, ∀xi ∈ Ωi ⊂ Ri×m with compact subsetΩi containingthe origin.

Assumption 4. For 1 ≤ i ≤ n, there exist known constants ξi1 ≥ 0such that ‖∆Gi(xi)‖ ≤ ξi1.

Lemma 1 (Polycarpou & Ioannou, 1996). The following inequalityholds for any ε > 0 and for any η ∈ R

0 ≤ |η| − η tanhηε

≤ kpε (6)

where kp is a constant that satisfies kp = e−(kp+1), i.e, kp = 0.2758.

Lemma 2 (Usmani, 1987; Chen et al., 2010). No eigenvalue of amatrix A ∈ Rm×m exceeds any of its norm in absolute value, that is

|λi| ≤ ‖A‖, i = 1, 2, . . . ,m (7)

where λi is a eigenvalue of matrix A.

Lemma 3 (Chen et al., 2010). Considering a matrix B ∈ Rm×m withspectral radius ϱ(B), there exists a positive constant ∆ > 0 whichmakes matrix B + (ϱ(B)+∆)Im×m nonsingular.

The control objective is to make x1 follow a certain desired tra-jectory x1d to a compact set in the presence of system uncertaintiesand disturbances under the designed adaptive control law v.

Assumption 5. There exist ε0i such that for all t , ‖x(i)1d(t)‖ ≤ ε0i,i = 1, 2, . . . , n.

Remark 1. Assumption 1 means that the nonlinear functionsgri(vi) and gli(vi) of the non-symmetric input saturation arestrictly monotonous. Assumption 2 is reasonable since the time-dependent component of the disturbance with finite energy is al-ways bounded (Tee & Ge, 2006). Assumption 3 is similar to theAssumption A2 in Kim and Ha (2000). Assumption 4 means thatperturbations ∆Gi(xi) of control coefficient matrices Gi(xi), i =

1, 2, . . . , n are bounded. There are many practical systems can beexpressed as the nonlinear system form as shown in (1). For exam-ple, rigid robots and motors (Dawson, Carroll, & Schneider, 1994),ships (Tee &Ge, 2006) and aircraft (Tang et al., 2007; Tee, Ge, & Tay,2008).

Remark 2. In this paper, the matrix spectral radius is employed todesign adaptive control for uncertainMIMO nonlinear systems (1).We do not assume that all control coefficient matrices Gi(xi), i =

1, 2, . . . , n are invertible, but only require that the norm of controlcoefficient matrix is bounded. This point is always valid for apractical control plant. Considering Assumption 3 and Lemma 2,the spectral radius ϱ(Gi) of Gi(xi) satisfies ϱ(Gi) ≤ ζi (Chen et al.,2010). According to Lemma 3, we know that Gi(xi)+ (ζi + τi)Im×mare nonsingular with τi > 0, i = 1, 2, . . . , n.

3. Adaptive control design and stability analysis

In this section, adaptive control is proposed for the uncer-tain nonlinear system with control input saturation. The auxiliary

design system is adopted to analyze the input saturation con-straints. The spectral radius of the control coefficient matrix is in-troduced to design adaptive control and the bounded stability ofall signals in the closed-loop system is achieved.Step 1: Define error variables z1 = x1 − x1d, and z2 = x2 − α1,where α1 ∈ Rm will be defined. Considering (1) and differentiatingz1 with respect to time, we obtainz1 = F1(x1)θ1 + G1(x1)(z2 + α1)

+∆G1(x1)x2 + D1(x1, t)− x1d. (8)Consider the Lyapunov function candidate

V ∗

1 =12zT1 z1. (9)

Its derivative is given byV ∗

1 = zT1 F1(x1)θ1 + zT1G1(x1)(z2 + α1)+ zT1∆G1(x1)x2+ zT1D1(x1, t)− zT1 x1d

≤ zT1 F1(x1)θ1 + zT1G1(x1)(z2 + α1)+ ξ11‖z1‖‖x2‖

+

m−j=1

|z1j|ρ1j(x1)Θ1j − zT1 x1d. (10)

Invoking Lemma 1, we havem−j=1

|z1j|ρ1j(x1)Θ1j

≤

m−j=1

kε1j + z1j tanh

z1jε1j

ρ1j(x1)Θ1j

= Ψ T1 ρ1(x1)Θ1 + zT1 Tanh(z1)ρ1(x1)Θ1

≤ zT1 Tanh(z1)ρ1(x1)Θ1 +‖Ψ T

1 ρ1(x1)‖2

2+

‖Θ1‖2

2. (11)

Substituting (11) into (10), we haveV ∗

1 ≤ zT1 F1(x1)θ1 + zT1G1(x1)(z2 + α1)

+ ξ11‖z1‖‖x2‖ + zT1 Tanh(z1)ρ1(x1)Θ1

+‖Ψ T

1 ρ1(x1)‖2

2+

‖Θ1‖2

2− zT1 x1d. (12)

Invoking Lemma 3, choose the following virtual control law:

α1 = (G1(x1)+ γ1Im×m)−1(−K1z1 − F1(x1)θ1

− Tanh(z1)ρ1(x1)Θ1 + x1d) (13)where K1 = K T

1 > 0 and γ1 = ζ1 + τ1.Substituting (13) into (12) yields

V ∗

1 ≤ −zT1K1z1 + zT1G1(x1)z2 + zT1 F1(x1)θ1− zT1 F1(x1)θ1 + zT1 Tanh(z1)ρ1(x1)Θ1

− zT1 Tanh(z1)ρ1(x1)Θ1 + ξ11‖z1‖‖x2‖

− γ1zT1α1 +‖Ψ T

1 ρ1(x1)‖2

2+

‖Θ1‖2

2= −zT1K1z1 + zT1G1(x1)z2 − zT1 F1(x1)θ1

− zT1 Tanh(z1)ρ1(x1)Θ1 + ξ11‖z1‖‖x2‖


1 ρ1(x1)‖2

2+

‖Θ1‖2

2. (14)

Considering the error signals θ1 and Θ1, the augmented Lyapunovfunction candidate is written as

V1 = V ∗

1 +12θ T1Λ

−111 θ1 +

12ΘT

1Λ−112 Θ1 (15)

whereΛ11 = ΛT11 > 0 andΛ12 = ΛT

12 > 0.


The time derivative of V1 is given by

V1 ≤ −zT1K1z1 + zT1G1(x1)z2 − zT1 F1(x1)θ1− zT1 Tanh(z1)ρ1(x1)Θ1 + ξ11‖z1‖‖x2‖


1 ρ1(x1)‖2

2+

‖Θ1‖2

2

+ θ T1Λ−111

˙θ1 + ΘT

1Λ−112

˙Θ1. (16)

Consider the adaptive laws for θ1 and Θ1 as

˙θ1 = Λ11(F T

1 (x1)z1 − β11θ1) (17)

˙Θ1 = Λ12(ρ1(x1)Tanh(z1)z1 − β12Θ1) (18)

where β11 > 0 and β12 > 0.Substituting (17) and (18) into (16), and considering the follow-

ing facts by completion of squares:

−θ T1 θ1 ≤ −‖θ1‖

2

2+

‖θ1‖2

2(19)

−ΘT1 Θ1 ≤ −

‖Θ1‖2

2+

‖Θ1‖2

2(20)

we have

V1 ≤ −zT1K1z1 + zT1G1(x1)z2 + ξ11‖z1‖‖x2‖ − γ1zT1α1

+‖Ψ T

1 ρ1(x1)‖2

2−β11

2‖θ1‖

2+β11

2‖θ1‖

2

−β12

2‖Θ1‖

2+β12

2‖Θ1‖

2+

‖Θ1‖2

2. (21)

The first term on the right-hand side is negative, and the secondtermwill be canceled in the next step. The other termswill be con-sidered in stability analysis of the closed-loop system.

Step 2: Define the error variable z3 = x3 − α2. Considering (1) anddifferentiating z2 with respect to time, we obtain

z2 = F2(x2)θ2 + G2(x2)x3 +∆G2(x2)x3 + D2(x2, t)− α1. (22)

Consider the Lyapunov function candidate

V ∗

2 =12zT2 z2. (23)

Considering Lemma 1, the derivative of V ∗

2 is

V ∗

2 = zT2 F2(x2)θ2 + zT2G2(x2)(z3 + α2)

+ zT2∆G2(x2)x3 + zT2D2(x2, t)− zT2 α1

≤ zT2 F2(x2)θ2 + zT2G2(x2)(z3 + α2)

+ ξ21‖z2‖‖x3‖ +

m−j=1

|z2j|ρ2j(x2)Θ2j − zT2 α1

≤ zT2 F2(x2)θ2 + zT2G2(x2)(z3 + α2)

+ξ21‖z2‖‖x3‖ + zT2 Tanh(z2)ρ2(x2)Θ2

+‖Ψ T

2 ρ2(x2)‖2

2+

‖Θ2‖2

2− zT2 α1. (24)

Invoking Lemma 3, choose the virtual control law as

α2 = (G2(x2)+ γ2Im×m)−1(−GT

1(x1)z1 − K2z2

− F2(x2)θ2 − Tanh(z2)ρ2(x2)Θ2 + α1) (25)

where K2 = K T2 > 0 and γ2 = ζ2 + τ2.

Substituting (25) into (24), we obtain

V ∗

2 ≤ −zT2K2z2 + zT2G2(x2)z3 + zT2 F2(x2)θ2− zT2 F2(x2)θ2 + zT2 Tanh(z2)ρ2(x2)Θ2

− zT2 Tanh(z2)ρ2(x2)Θ2 − zT2GT1(x1)z1 − γ2zT2α2

+ ξ21‖z2‖‖x3‖ +‖Ψ T

2 ρ2(x2)‖2

2+

‖Θ2‖2

2= −zT2K2z2 + zT2G2(x2)z3 − zT2 F2(x2)θ2

− zT2 Tanh(z2)ρ2(x2)Θ2 − zT1G1(x1)z2 − γ2zT2α2

+ ξ21‖z2‖‖x3‖ +‖Ψ T

2 ρ2(x2)‖2

2+

‖Θ2‖2

2. (26)

Considering the error signal θ2 and Θ2, the augmented Lyapunovfunction candidate can be written as

V2 = V1 + V ∗

2 +12θ T2Λ

−121 θ2 +

12ΘT

2Λ−122 Θ2 (27)


22 > 0.Invoking (21) and (26), the time derivative of V2 is

V2 ≤ −

2−j=1

zTj Kjzj −2−

j=1

γjzTj αj +

2−j=1

ξj1‖zj‖‖xj+1‖

+ zT2G2(x2)z3 − zT2 F2(x2)θ2 − zT2 Tanh(z2)ρ2(x2)Θ2

+ θ T2Λ−121

˙θ2 + ΘT

2Λ−122

˙Θ2 +

2−j=1

‖Ψ Tj ρj(xj)‖

2

2

−β11

2‖θ1‖

2+β11

2‖θ1‖

2−β12

2‖Θ1‖

2

+β12

2‖Θ1‖

2+

2−j=1

‖Θj‖2

2. (28)


˙θ2 = Λ21(F T

2 (x2)z2 − β21θ2) (29)

˙Θ2 = Λ22(ρ2(x2)Tanh(z2)z2 − β22Θ2) (30)

where β21 > 0 and β22 > 0.Substituting (29) and (30) into (28), similar with (19) and (20)

we have

V2 ≤ −

2−j=1

zTj Kjzj + zT2G2(x2)z3 −

2−j=1

γjzTj αj

+

2−j=1

ξj1‖zj‖‖xj+1‖ +

2−j=1

‖Ψ Tj ρj(xj)‖

2

2

−

2−j=1

βj1

2‖θj‖

2+

2−j=1

βj1

2‖θj‖

2−

2−j=1

βj2

2‖Θj‖

2

+

2−j=1

βj2

2‖Θj‖

2+

2−j=1

‖Θj‖2

2. (31)

The first term on the right-hand side is negative, and the secondterm will be canceled in the next step. The other terms will beconsidered in stability analysis of the closed-loop system.Step i (1 ≤ i ≤ n − 1): Define the error variable zi+1 = xi+1 − αi.Considering (1) and differentiating zi with respect to time, we have

zi = Fi(xi)θi + Gi(xi)(zi+1 + αi)+∆Gi(xi)xi+1

+Di(xi, t)− αi−1. (32)



V ∗

i =12zTi zi. (33)

Invoking Lemma 1, the derivative of V ∗

i is

V ∗

i = zTi Fi(xi)θi + zTi Gi(xi)(zi+1 + αi)

+ zTi ∆Gi(xi)xi+1 + zTi Di(xi, t)− zTi αi−1

≤ zTi Fi(xi)θi + zTi Gi(xi)(zi+1 + αi)

+ ξi1‖zi‖‖xi+1‖ +

m−j=1

|zij|ρij(xi)Θij − zTi αi−1

≤ zTi Fi(xi)θi + zTi Gi(xi)(zi+1 + αi)

+ ξi1‖zi‖‖xi+1‖ + zTi Tanh(zi)ρi(xi)Θi

+‖Ψ T

i ρi(xi)‖2

2+

‖Θi‖2

2− zTi αi−1. (34)

Considering Lemma 3, we choose the following virtual control law:

αi = (Gi(xi)+ γiIm×m)−1(−GT

i−1(xi−1)zi−1 − Kizi

− Fi(xi)θi − Tanh(zi)ρi(xi)Θi + αi−1) (35)

where Ki = K Ti > 0 and γi = ζi + τi.

Substituting (35) into (34), we obtain

V ∗

i ≤ −zTi Kizi + zTi Gi(xi)zi+1 + zTi Fi(xi)θi− zTi Fi(xi)θi + zTi Tanh(zi)ρi(xi)Θ2

− zTi Tanh(zi)ρi(xi)Θi − zTi GTi−1(xi−1)zi−1

− γizTi αi + ξi1‖zi‖‖xi+1‖ +‖Ψ T

i ρi(xi)‖2

2+

‖Θi‖2

2= −zTi Kizi + zTi Gi(xi)zi+1 − zTi Fi(xi)θi

− zTi Tanh(zi)ρi(xi)Θi − zTi−1Gi−1(xi−1)zi − γizTi αi

+ ξi1‖zi‖‖xi+1‖ +‖Ψ T

i ρi(xi)‖2

2+

‖Θi‖2

2. (36)

Considering the error signals θi and Θi, the augmented Lyapunovfunction candidate can be written as

Vi = Vi−1 + V ∗

i +12θ Ti Λ

−1i1 θi +

12ΘT

i Λ−1i2 Θi (37)

whereΛi1 = ΛTi1 > 0 andΛi2 = ΛT

i2 > 0.Invoking (31) and (36), the time derivative of Vi is given by

Vi ≤ −

i−j=1

zTj Kjzj −i−

j=1

γjzTj αj +

i−j=1


+ zTi Gi(xi)zi+1 − zTi Fi(xi)θi − zTi Tanh(zi)ρi(xi)Θi

+ θ Ti Λ−1i1

˙θ i + ΘT

i Λ−1i2

˙Θ i +

i−j=1

‖Ψ Tj ρj(xj)‖

2

2

−

i−1−j=1

βj1

2‖θj‖

2+

i−1−j=1

βj1

2‖θj‖

2−

i−1−j=1

βj2

2‖Θj‖

2

+

i−1−j=1

βj2

2‖Θj‖

2+

i−j=1

‖Θj‖2

2. (38)

Consider the adaptive laws for θi and Θi as

˙θ i = Λi1(F T

i (xi)zi − βi1θi) (39)

˙Θ i = Λi2(ρi(xi)Tanh(zi)zi − βi2Θi) (40)

where βi1 > 0 and βi2 > 0.

Substituting (39) and (40) into (38), similar with (19) and (20)we have

Vi ≤ −

i−j=1

zTj Kjzj + zTi Gi(xi)zi+1 −

i−j=1

γjzTj αj

+

i−j=1


i−j=1

‖Ψ Tj ρj(xj)‖

2

2

−

i−j=1

βj1

2‖θj‖

2+

i−j=1

βj1

2‖θj‖

2−

i−j=1

βj2

2‖Θj‖

2

+

i−j=1

βj2

2‖Θj‖

2+

i−j=1

‖Θj‖2

2. (41)

The first term on the right-hand side is negative, and the secondterm will be canceled in the next step. The other terms will beconsidered in stability analysis of the closed-loop system.Step n: By differentiating zn = xn−αn−1 with respect to time yields

zn = Fn(xn)θn + Gn(xn)u +∆Gn(xn)u+Dn(xn, t)− αn−1. (42)


V ∗

n =12zTn zn. (43)

Note the fact ‖u‖ ≤ Umax with Umax > 0. Invoking Lemma 1, thederivative of V ∗

n is

V ∗

n = zTn Fn(xn)θn + zTnGn(xn)u + zTn∆Gnu

+ zTnDn(xn, t)− zTn αn−1

≤ zTn Fn(xn)θn + zTnGn(xn)u + ξn1‖zn‖‖u‖

+

m−j=1

|znj|ρnj(xn)Θnj − zTn αn−1

≤ zTn Fn(xn)θn + zTnGn(xn)u + ξn1‖zn‖‖u‖

+ zTn Tanh(zn)ρn(xn)Θn +‖Ψ T

n ρn(xn)‖2

2

+‖Θn‖

2

2− zTn αn−1. (44)

From (2), control inputs u have an upper limit and a lower limit. Forconvenience of input constraint effect analysis, the auxiliary designsystem is given by

e =

−Kn2e −

1‖e‖2

f (u,∆u, zn, xn)e

+ (Gn(xn)+ γnIm×m)(v − u), ‖e‖ ≥ σ0, ‖e‖ < σ

(45)

where f (u,∆u, zn, xn) = |zTnGn(xn)∆u| + 0.5(γn + ζn)2∆uT∆u +

|γnzTn u| + ξn1‖zn‖‖u‖, ∆u = u − v, Kn2 = K Tn2 > 0, γn = ζn + τn

and e ∈ Rm is the state of auxiliary design system. The designparameter σ is a positive constant which should be chosen asan appropriate value in accordance with the requirement of thetracking performance.

Define

h(Z) =12zTnK

Tn Knzn +

n−1−j=1

γj|zTj αj|

+

n−1−j=1


n−j=1

‖Ψ Tj ρj(xj)‖

2

2(46)

where Kn = K Tn > 0 and Z = [αj, zj, xj]T , j = 1, 2, . . . , n.


Invoking Lemma 3 and considering the input saturation effect,choose the following control law:

v = (Gn(xn)+ γnIm×m)−1v0

v0 = −GTn−1(xn−1)zn−1 − Kn(zn − e)− Fn(xn)θn

− Tanh(zn)ρn(xn)Θn + αn−1 −znh(Z)

ψ2 + ‖zn‖2

ψ =

−ψh(Z)

ψ2 + ‖zn‖2− kvψ, ‖zn‖ ≥ ℓ

0, ‖zn‖ < ℓ

(47)

where kv > 0 and ℓ > 0.The above design procedure can be summarized in the follow-

ing theorem, which contains the results of adaptive control for un-certain MIMO nonlinear systems (1).

Theorem 1. Considering the strict-feedback nonlinear system (1)with known coefficient matrices satisfies Assumptions 1–5, and giventhat the full state information is available. Under the control law (47),parameter updated laws (17), (18), (29), (30), (39), (40), (53), (54),and for any bounded initial condition, there exist design parametersσ > 0, Ki = K T

i > 0, Kn2 = K Tn2 > 0, βi1 > 0, βi2 > 0

and kv > 0 such that the overall closed-loop control system is semi-globally stable in the sense that all of the closed-loop signals e, zi,ψ , θiand Θi are bounded, where i = 1, 2, . . . , n. Furthermore, the trackingerror signals z1 remains within the compact setsΩz1 defined by

Ωz1 :=z1 ∈ Rm

| ‖z1‖ ≤√D

where D = 2(Vn(0)+Cκ) with C and κ as defined in (55).

Proof. When ‖e‖ ≥ σ , we consider the Lyapunov functioncandidate

Vn =Vn−1 + V ∗

n +12eT e +

12θ TnΛ

−1n1 θn +

12ΘT

nΛ−1n2 Θn +

12ψ2 (48)

whereΛn1 = ΛTn1 > 0 andΛn2 = ΛT

n2 > 0.Considering (41) and (44), the time derivative of Vn is

Vn ≤ −

n−1−j=1

zTj Kjzj + zTn−1Gn−1(xn−1)zn −

n−1−j=1

γjzTj αj

+

n−1−j=1

ξj1‖zj‖‖xj+1‖ + zTn Fn(xn)θn + ξn1‖zn‖‖u‖

+ zTnGn(xn)(v +∆u)+ zTn Tanh(zn)ρn(xn)Θn

− zTn αn−1 + eT e + θ TnΛ−1n1

˙θn + ΘT

nΛ−1n2

˙Θn

+

n−j=1

‖Ψ Tj ρj(xj)‖

2

2−

n−1−j=1

βj1

2‖θj‖

2+

n−1−j=1

βj1

2‖θj‖

2

−

n−1−j=1

βj2

2‖Θj‖

2+

n−1−j=1

βj2

2‖Θj‖

2

+

n−j=1

‖Θj‖2

2+ ψψ. (49)

Substituting (45)–(47) into (49), we obtain

Vn ≤ −

n−j=1

zTj Kjzj −n−1−j=1

γjzTj αj +

n−1−j=1


− γnzTn u + ξn1‖zn‖‖u‖ + zTnGn(xn)∆u + zTnKne

− zTn Fn(xn)θn − zTn Tanh(zn)ρn(xn)Θn + eT e

+ θ TnΛ−1n1

˙θn + ΘT

nΛ−1n2

˙Θn +

n−j=1

‖Ψ Tj ρj(xj)‖

2

2

−

n−1−j=1

βj1

2‖θj‖

2+

n−1−j=1

βj1

2‖θj‖

2−

n−1−j=1

βj2

2‖Θj‖

2

+

n−1−j=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2−

‖zn‖2h(Z)ψ2 + ‖zn‖2

+ ψψ

≤ −

n−j=1

zTj Kjzj − eT (Kn2 − Im×m) e − zTn Fn(xn)θn

− zTn Tanh(zn)ρn(xn)Θn + θ TnΛ−1n1

˙θn + ΘT

nΛ−1n2

˙Θn

−

n−1−j=1

βj1

2‖θj‖

2+

n−1−j=1

βj1

2‖θj‖

2−

n−1−j=1

βj2

2‖Θj‖

2

+

n−1−j=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2

+ψ2h(Z)

ψ2 + ‖zn‖2+ ψψ. (50)

Invoking the third equation of (47), we have

ψ2h(Z)ψ2 + ‖zn‖2

+ ψψ = −kvψ2. (51)

Substituting (51) into (50) yields

Vn ≤ −

n−j=1

zTj Kjzj − eT (Kn2 − Im×m) e − zTn Fn(xn)θn


˙θn + ΘT

nΛ−1n2

˙Θn

−

n−1−j=1

βj1

2‖θj‖

2+

n−1−j=1

βj1

2‖θj‖

2−

n−1−j=1

βj2

2‖Θj‖

2

+

n−1−j=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2− kvψ2. (52)

Consider the adaptive laws for θn and Θn as

˙θn = Λn1(F T

n (xn)zn − βn1θn) (53)

˙Θn = Λn2(ρn(xn)Tanh(zn)zn − βn2Θn) (54)

where βn1 > 0 and βn2 > 0.Substituting (53) and (54) into (52), similar with (19) and (20)

we have

Vn ≤ −

n−j=1

zTj Kjzj − eT (Kn2 − Im×m) e − kvψ2

−

n−j=1

βj1

2‖θj‖

2+

n−i=1

βj1

2‖θj‖

2−

n−j=1

βj2

2‖Θj‖

2

+

n−i=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2

≤ −κVn + C (55)

where


κ := min

2λmin

n−

j=1

Kj

, 2λmin(Kn2 − Im×m),

n−j=1

2βj1

λmax(Λ−1j1 )

,

n−j=1

2βj2

λmax(Λ−1j2 )

, kv

C :=

n−j=1

βj1

2‖θj‖

2+

n−j=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2.

(56)

To ensure that κ > 0, the design parameter Kn2 must makeKn2 − Im×m > 0.

From (55), if κ > 0, we can conclude that z1 converges to acompact set asymptotically, and therefore the control objectiveis reached when the input saturation constraint occurs, i.e., thedesired trajectory of MIMO nonlinear system is followed in thepresence of parametric uncertainties and disturbances under thesaturation constraint. On the other hand, we can conclude thatauxiliary design variables e and ψ , error signals zi, θi and Θiconverge to a compact set asymptotically.

It is worth pointing out that the above proof of Theorem 1only contains the result when the states of the auxiliary designsystem (45) satisfy the condition ‖e‖ ≥ σ , i.e., there exists inputsaturation. If ‖e‖ < σ means that there does not exist inputsaturation, we have ∆u = 0, i.e., u = v and the control input uis bounded. Thus, v is bounded. The stability proof of Theorem 1can be easily proved by considering Eqs. (48)–(55) when ‖e‖ < σ .The detailed proof is omitted. This concludes the proof.

Remark 3. In this section, the robust adaptive tracking control isproposed for a class of uncertain MIMO nonlinear systems withnon-symmetric input saturation constraints. To handle the non-symmetric input saturation, the auxiliary design system (45) isintroduced to analyze the effect of saturation constraint, and theauxiliary variable e is used to design the robust adaptive controllaw. It is apparent that the constrained control u produced bythe designed control command v can guarantee the closed-loopsystem stability. If e ≤ σ and e = 0, it means that there is nosaturation, i.e., there is u = v according to (45) (Polycarpou et al.,2003). It implies that vlimax ≤ vi ≤ vrimax and gri(vi) = gli(vi) = vi.

4. Constrained adaptive control design and stability analysis

Although the robust adaptive control for the uncertain MIMOnonlinear system (1) with non-symmetric input saturation con-straints has been successfully developed in Section 3, physics con-straints of virtual control laws have not been considered, and theanalytic computations of time derivatives of virtual control lawsαi (i = 1, . . . , n − 1) need to be done in the backstepping pro-cedure. In fact, the vast analytic calculation of the virtual controlderivatives is a drawback of backstepping control. Specially, theanalytic calculation of time derivatives of virtual control laws is te-dious for the MIMO nonlinear systems. In this section, we will in-vestigate the constrained robust adaptive control which considerthe mechanical or operating limitations of virtual control laws andcontrol command, and eliminate the analytic computations of thevirtual control law derivatives. Therefore, command filters are in-troduced to avoid the analytic calculation of the time derivativesof the virtual control laws.Step 1: Define error variables z1 = x1 − x1d and z2 = x2 − α1.Considering (1) and differentiating z1 with respect to time, weobtain

z1 = F1(x1)θ1 + (G1(x1)+∆G1(x1))(z2 + α1)

+D1(x1, t)− x1d (57)

where α1 is a virtual control lawwhich is produced by the nominalvirtual control law α10. The nominal virtual control law α10 is

Fig. 2. Configuration of the command filter, where i = 1, 2, . . . , n − 1, αn = v,αi0 are the nominal virtual control law or the nominal control law, αi are the virtualcontrol law or the control law, ξi and ωni are the bandwidth parameters.

filtered to provide the magnitude, rate and bandwidth limitedvirtual control law α1 and its derivatives α1 which are within theoperating envelope of the system. Such a command filter is shownin Fig. 2 to implement the emulate of any mechanical or operatingconstraints on virtual control law α10 (Polycarpou et al., 2004).

The nominal virtual control law α10 is given by

α10 = (G1(x1)+ γ1Im×m)−1(−K10(z1 − ϕ1)

− F1(x1)θ1 − Tanh(z1)ρ1(x1)Θ1 + x1d) (58)

where K10 = K T10 > 0, and ϕ1 ∈ Rm is the state vector of auxil-

iary design system which denotes the constraint effect due to themagnitude, rate and bandwidth limitation of the nominal virtualcontrol law. Note the following facts ‖α1‖ ≤ ε10 with ε10 > 0,where ε10 denotes the magnitude limit of α1 which is decided bythe command filter. Let δ1 = ζ1 +

√mγ1. For convenience of con-

straint effect analysis (Chen et al., 2010), the auxiliary design sys-tem is given by

ϕ1 =

−K11ϕ1 − f1(z1, ϕ1, θ1, Θ1)ϕ1+ (G1(x1)+ γ1Im×m)(α1 − α10), ‖ϕ1‖ ≥ σ1

0, ‖ϕ1‖ < σ1

(59)

where f1(z1, ϕ1, θ1, Θ1) =φ1(z1,θ1,Θ1)

‖ϕ1‖2, φ1(z1, θ1, Θ1) = a1‖K10‖

‖z1‖2+

12‖F1(x1)θ1‖

2+

δ1ε210

2 + ζ1ε10‖z1‖ + ‖z1‖‖F1(x1)θ1‖ +

12‖Tanh(z1)ρ1(x1)Θ1‖

2+ ‖x1d‖2

+ γ1zT1α1 + ‖z1‖‖Tanh(z1)ρ1(x1)Θ1‖, K11 = K T

11 > 0, a1 > 0 and σ1 is a positive design parameter.Consider the Lyapunov function candidate

V ∗

1 =12zT1 z1 +

12ϕT1ϕ1. (60)

Invoking (57)–(59), the time derivative of V1 is

V ∗

1 = −c1zT1K10z1 + zT1 F1(x1)θ1 + zT1G1(x1)(z2 + α1)

+ zT1∆G1(x1)x2 + zT1D1(x1, t)− zT1 x1d+ c1zT1K10z1 − ϕT

1K11ϕ1 − f1(z1, ϕ1, θ1, Θ1)‖ϕ1‖2

+ϕT1 (G1(x1)+ γ1Im×m)(α1 − α10)

≤ −c1zT1K10z1 + zT1 F1(x1)θ1 + zT1 Tanh(z1)ρ1(x1)Θ1

+ζ1

2‖z1‖2

+ζ1

2‖z2‖2

+ ζ1ε10‖z1‖ + ξ11‖z1‖‖x2‖

+12‖z1‖2

+‖x1d‖2

2+ c1‖K10‖‖z1‖2

− ϕT1K11ϕ1

− f1(z1, ϕ1, θ1,Θ1)‖ϕ1‖2+δ1

2‖ϕ1‖

2+δ1ε

210

2

+‖K10‖

2‖z1‖2

+3‖K10‖

2‖ϕ1‖

2+

32‖ϕ1‖

2

+12‖F1(x1)θ1‖2

+12‖Tanh(z1)ρ1(x1)Θ1‖

2+

‖x1d‖2

2+ zT1 F1(x1)θ1 + zT1 Tanh(z1)ρ1(x1)Θ1 − zT1 F1(x1)θ1

− zT1 Tanh(z1)ρ1(x1)Θ1 +‖Ψ T

1 ρ1(x1)‖2

2+

‖Θ1‖2

2


≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

+ζ1

2‖z2‖2

+ ξ11‖z1‖‖x2‖ +‖Ψ T

1 ρ1(x1)‖2

2− γ1zT1α1 − zT1 F1(x1)θ1 − zT1 Tanh(z1)ρ1(x1)Θ1

−(c1 + 0.5)‖K10‖‖z1‖2(a10 − 1)+‖Θ1‖

2

2(61)

where c1 > 0, a10 =a1

(c1+0.5) .

Considering the error signals θ1 and Θ1, the augmented Lya-punov function candidate can be written as

V1 = V ∗

1 +12θ T1Λ

−111 θ1 +

12ΘT

1Λ−112 Θ1 (62)


12 > 0.Obviously, we can choose a1 and c1 to render a10 − 1 > 0.

Invoking (61), the time derivative of V1 is

V1 ≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

+ζ1

2‖z2‖2

+ ξ11‖z1‖‖x2‖ +‖Ψ T

1 ρ1(x1)‖2

2− γ1zT1α1 − zT1 F1(x1)θ1 − zT1 Tanh(z1)ρ1(x1)Θ1

+ θ T1Λ−111

˙θ1 + ΘT

1Λ−112

˙Θ1. (63)


˙θ1 = Λ11(F T

1 (x1)z1 − β11θ1) (64)

˙Θ1 = Λ12(ρ1(x1)Tanh(z1)z1 − β12Θ1) (65)

where β11 > 0 and β12 > 0.Substituting (64) and (65) into (63), we obtain

V1 ≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

+ζ1

2‖z2‖2

+ ξ11‖z1‖‖x2‖ +‖Ψ T

1 ρ1(x1)‖2

2

− γ1zT1α1 −β11

2‖θ1‖

2+β11

2‖θ1‖

2

−β12

2‖Θ1‖

2+β12

2‖Θ1‖

2+

‖Θ1‖2

2. (66)

The first term and the second term on the right-hand sideare negative if c1K10 −

ζ12 +

12

Im×m > 0 and K11 −

3‖K10‖2 +

32 +

δ12

Im×m > 0. The other terms will be considered

in the next step or the stability analysis of the closed-loop system.Step i (2 ≤ i ≤ n−1): Define the error variables zi = xi −αi−1 andzi+1 = xi+1 −αi. Considering (1) and differentiating zi with respectto time, we obtain

zi = Fi(xi)θi + Gi(xi)(zi+1 + αi)+∆Gi(xi)xi+1

+Di(xi, t)− αi−1 (67)

where αi is a virtual control law which is produced by the nominalvirtual control law αi0. The nominal virtual control law αi0 arefiltered to provide the magnitude, rate and bandwidth limitedvirtual control law αi and its derivatives αi which are within theoperating envelope of the system. Such a command filter is similarto the first filter shown in Fig. 2 to implement any mechanical oroperating constraints on virtual control law αi0.

The nominal virtual control law αi0 is given by

αi0 = (Gi(xi)+ γiIm×m)−1πi

πi = −GTi−1(xi−1)zi−1 − Ki0(zi − ϕi)− Fi(xi)θi

− Tanh(zi)ρi(xi)Θi + αi−1

(68)

where Ki0 = K Ti0 > 0, and ϕi ∈ Rm is the state vector of the ith

auxiliary design systemwhich denotes the constraint effect due tothe magnitude, rate and bandwidth limitation of nominal virtualcontrol law αi0. In nominal virtual control law (68), αi−1 need notbe computed here which can be directly obtained from the firstcommand filter in Step i−1. Note the following facts ‖αi‖ ≤ εi0 and‖αi−1‖ ≤ εi1 with εi0 > 0 and εi1 > 0, where εi0 and εi1 denote themagnitude limit of αi and the rate limit of αi−1 which are decidedby the command filter. Let δi = ζi +

√mγi. For convenience of

constraint effect analysis, the auxiliary design system is given by

ϕi =

−Ki1ϕi − fi(zi, ϕi, θi, Θi)ϕi+ (Gi(xi)+ γiIm×m)(αi − αi0), ‖ϕi‖ ≥ σi

0, ‖ϕi‖ < σi

(69)

where fi(zi, ϕi, θi, Θi) =φi(zi,θi,Θi)

‖ϕi‖2, φi(zi, θi, Θi) = ai‖Ki0‖‖zi‖2

+

12‖Tanh(zi)ρi(xi)Θi‖

2+

12‖Fi(xi)θi‖

2+

ζi−12 ‖zi−1‖

2+ ζiεi0‖zi‖ +

‖zi‖‖Fi(xi)θi‖+ (0.5+0.5δi)ε2i1 +‖zi‖‖Tanh(zi)ρi(xi)Θi‖+γizTi αi,Ki1 = K T

i1 > 0, ai > 0 and σi is a positive design parameter.Consider the Lyapunov function candidate

V ∗

i =12zTi zi +

12ϕTi ϕi. (70)

Invoking (67)–(69), the time derivative of V ∗

i is

V ∗

i = −cizTi Ki0zi + zTi Fi(xi)θi + zTi Gi(xi)(zi+1 + αi)

+ zTi ∆Gi(xi)xi+1 + zTi Di(xi, t)− zTi αi−1

+ cizTi Ki0zi − ϕTi Ki1ϕi − fi(zi, ϕi, θi, Θi)‖ϕi‖

2

+ϕTi (Gi(xi)+ γiIm×m)(αi − αi0)

≤ −cizTi Ki0zi + zTi Fi(xi)θi + zTi Tanh(zi)ρi(xi)Θi

+ζi

2‖zi‖2

+ζi

2‖zi+1‖

2+ ζiεi0‖zi‖ + ξi1‖zi‖‖xi+1‖

+12‖zi‖2

+ε2i1

2+ ci‖Ki0‖‖zi‖2

− ϕTi Ki1ϕi

− fi(zi, ϕi, θi, Θi)‖ϕi‖2+δi

2‖ϕi‖

2+δiε

2i0

2

+ζi−1

2‖ϕi‖

2+ζi−1

2‖zi−1‖

2+

‖Ki0‖

2‖zi‖2

+3‖Ki0‖

2‖ϕi‖

2+

32‖ϕi‖

2+

12‖Fi(xi)θi‖2

+12‖Tanh(zi)ρi(xi)Θi‖

2+ε2i1

2+ zTi Fi(xi)θi

+‖Ψ T

i ρi(xi)‖2

2+ zTi Tanh(zi)ρi(xi)Θi − zTi Fi(xi)θi

− zTi Tanh(zi)ρi(xi)Θi +‖Ψ T

i ρi(xi)‖2

2+

‖Θi‖2

2


≤ −zTi

ciKi0 −

ζi

2+

12

Im×m

zi

−ϕTi

Ki1 −

3‖Ki0‖

2+ζi−1

2+

32

+δi

2

Im×m

ϕi

+ζi

2‖zi+1‖

2+ ξi1‖zi‖‖xi+1‖ − zTi Fi(xi)θi

− zTi Tanh(zi)ρi(xi)Θi − γizTi αi +‖Ψ T

i ρi(xi)‖2

2

−(ci + 0.5)‖Ki0‖‖zi‖2(ai0 − 1)+‖Θi‖

2

2(71)

where ci > 0, ai0 =ai

(ci+0.5) .

Considering the error signals θi and Θi, the augmented Lya-punov function candidate can be written as

Vi = Vi−1 + V ∗

i +12θ Ti Λ

−1i1 θi +

12ΘT

i Λ−1i2 Θi (72)

whereΛi1 = ΛTi1 > 0 andΛi2 = ΛT

i2 > 0.Similarly, we can choose ai and ci to render ai0−1 > 0. Invoking

(66) and (71), the time derivative of Vi is

Vi ≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−

i−j=2

zTj

cjKj0 −

ζj−1

2+ζj

2+

12

Im×m

zj

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

−

i−j=2

ϕTj

Kj1 − Kj1

ϕj +

ζi

2‖zi+1‖

2− zTi Fi(xi)θi

− zTi Tanh(zi)ρi(xi)Θi + θ Ti Λ−1i1

˙θ i + ΘT

i Λ−1i2

˙Θ i

−

i−1−j=1

βj1

2‖θj‖

2+

i−1−j=1

βj1

2‖θj‖

2−

i−1−j=1

βj2

2‖Θj‖

2

+

i−1−j=1

βj2

2‖Θj‖

2+

i−j=1

‖Θj‖2

2−

i−j=1

γjzTj αj

+

i−j=1


i−j=1

‖Ψ Tj ρj(xj)‖

2

2(73)

where Kj1 =

3‖Kj0‖

2 +ζj−12 +

32 +

δj2

Im×m.

Consider the adaptive laws for θi and Θi as

˙θ i = Λi1(F T

i (xi)zi − βi1θi) (74)

˙Θ i = Λi2(ρi(xi)Tanh(zi)zi − βi2Θi) (75)

where βi1 > 0 and βi2 > 0.Substituting (74) and (75) into (73), we obtain

Vi ≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−

i−j=2

zTj

cjKj0 −

ζj−1

2+ζj

2+

12

Im×m

zj

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

−

i−j=2

ϕTj

Kj1 − Kj1

ϕj +

ζi

2‖zi+1‖

2−

i−j=1

βj1

2‖θj‖

2

+

i−j=1

βj1

2‖θj‖

2−

i−j=1

βj2

2‖Θj‖

2+

i−j=1

βj2

2‖Θj‖

2

+

i−j=1

‖Θj‖2

2−

i−j=1

γjzTj αj

+

i−j=1


i−j=1

‖Ψ Tj ρj(xj)‖

2

2. (76)

The first four terms are negative if c1K10 −

ζ12 +

12

Im×m >

0, cjKj0 −

ζj−12 +

ζj2 +

12

Im×m > 0 (j = 2, . . . , i), K11 −

3‖K10‖2 +

32 +

δ12

Im×m and Kj1 −

3‖Kj0‖

2 +ζj−12 +

32 +

δj2

Im×m

(j = 2, . . . , i). The other terms will be considered in the next step.Step n: By differentiating zn = xn−αn−1 with respect to time yields

zn = Fn(xn)θn + Gn(xn)u +∆Gn(xn)u+Dn(xn, t)− αn−1 (77)

where u is a control law which is produced by the nominalcontrol law v. The nominal control law v is filtered to provide themagnitude, rate and bandwidth limited virtual control law u andits derivatives u which are within the operating envelope of thesystem. Such a command filter is similar to the first filter shownin Fig. 2 to implement the mechanical or operating constraints onvirtual control law v. Here, it is required that the command filtercan implement the same position constraints on adaptive controlv as shown in (2).

Define

h(Z) =12zTnK

Tn0Knzn0 +

n−1−j=1


+

n−1−j=1

γj|zTj αj| +

n−j=1

‖Ψ Tj ρj(xj)‖

2

2(78)

where Kn0 = K Tn0 > 0 and Z = [αj, zj, xj]T , j = 1, 2, . . . , n.

The nominal virtual control law v is given by

v = (Gn(xn)+ γnIm×m)−1v0

v0 = −GTn−1(xn−1)zn−1 − Kn0(zn − ϕn)− Fn(xn)θn

− Tanh(zn)ρn(xn)Θn + αn−1 −znh(Z)

ψ2 + ‖zn‖2(79)

ψ =

−ψh(Z)

ψ2 + ‖zn‖2− kvψ, ‖zn‖ ≥ ℓ

0, ‖zn‖ < ℓ

(80)

where ϕn ∈ Rm is the state vector of the auxiliary design systemwhich denotes the constraint effect due to themagnitude, rate andbandwidth limitation of nominal virtual control law. In nominalcontrol law (79), αn−1 need not be computed here which can bedirectly obtained from the command filter in Step n − 1. Note thefollowing facts ‖u‖ ≤ εn0 and ‖αn−1‖ ≤ εn1 with εn0 > 0 andεn1 > 0, where εn0 and εn1 denote the magnitude limit of u andthe rate limit of αn−1 which are decided by the command filter. Letδn = ζn +

√mγn. For convenience of constraint effect analysis, the

auxiliary design system is given by

ϕn =

−Kn1ϕn − fn(u, zn, ϕn, θn, Θn)ϕn+ (Gn(xn)+ γnIm×m)(u − v), ‖ϕn‖ ≥ σn

0, ‖ϕn‖ < σn

(81)

where fn(u, zn, ϕn, θn, Θn) =φn(u,zn,θn,Θn)

‖ϕn‖2, φn(u, zn, θn, Θn) =

an‖Kn0‖‖zn‖2+ζn2 ‖u‖2

+12‖Fn(xn)θn‖

2+

12‖Tanh(zn)ρn(xn)Θn‖

2+


ζn−12 ‖zn−1‖

2+‖zn‖‖Fn(xn)θn‖+‖zn‖‖Tanh(zn)ρn(xn)Θn‖+

δnε2n0

2 +

ε2n1+|γnzTn u|+ξn1‖zn‖‖u‖+12 z

TnK

Tn0Knzn0+

‖zn‖2h(Z)ψ2+‖zn‖2

,Kn1 = K Tn1 > 0,

an > 0 and σn is a positive design parameter.Consider the Lyapunov function candidate

V ∗

n =12zTn zn +

12ϕTnϕn. (82)

Invoking (77), (79) and (81), the time derivative of V ∗n is

V ∗

n = −cnzTnKn0zn + zTn Fn(xn)θn + zTnGn(xn)u

+ zTn∆Gn(xn)u + zTnDn(xn, t)− zTn αn−1

+ cnzTnKn0zn − ϕTnKn1ϕn − fn(zn, ϕn, θn, Θn)‖ϕn‖

2

+ϕTn (Gn(xn)+ γnIm×m)(u − v)

≤ −cnzTnKn0zn + zTn Fn(xn)θn +ζn

2‖zn‖2

+ zTn Tanh(zn)ρn(xn)Θn + |γnzTn u| +ζn

2‖u‖2

+ ξn1‖zn‖‖u‖ +12‖zn‖2

+ε2n1

2+ cn‖Kn0‖‖zn‖2

−ϕTnKn1ϕn − fn(zn, ϕn, θn, Θn)‖ϕn‖

2

+δn

2‖ϕn‖

2+δnε

2n0

2+ζn−1

2‖ϕn‖

2+ζn−1

2‖zn−1‖

2

+‖Kn0‖

2‖zn‖2

+3‖Kn0‖

2‖ϕn‖

2+

32‖ϕn‖

2

+12‖Fn(xn)θn‖2

+12‖Tanh(zn)ρn(xn)Θn‖

2

+ε2n1

2+ zTn Fn(xn)θn + zTn Tanh(zn)ρn(xn)Θn

− zTn Fn(xn)θn − zTn Tanh(zn)ρn(xn)Θn

+‖Ψ T

n ρn(xn)‖2

2+

‖Θn‖2

2−

12zTnK

Tn0Knzn0

≤ −zTn

cnKn0 −

ζn

2+

12

Im×m

zn

−ϕTn

Kn1 − Kn1

ϕn +

12zTnK

Tn0Knzn0

− zTn Fn(xn)θn − zTn Tanh(zn)ρn(xn)Θn

−(cn + 0.5)‖Kn0‖‖zn‖2(an0 − 1)

+‖Θn‖

2

2+

‖Ψ Tn ρn(xn)‖

2

2−

‖zn‖2h(Z)ψ2 + ‖zn‖2

(83)

where Kn1 =

3‖Kn0‖+ζn−1+δn+3

2

Im×m, cn > 0 and an0 =

an(cn+0.5) .

Considering the error signals θn and Θn, the augmented Lya-punov function candidate can be written as

Vn = Vn−1 + V ∗

n +12θ TnΛ

−1n1 θn +

12ΘT

nΛ−1n2 Θn +

12ψ2 (84)

whereΛn1 = ΛTn1 > 0 andΛn2 = ΛT

n2 > 0.Obviously, we can choose an and cn to render an0 − 1 > 0.

Invoking (76), (78), (79) and (83), the time derivative of Vn is

Vn ≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−

n−j=2

zTj

cjKj0 −

ζj−1

2+ζj

2+

12

Im×m

zj

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

−

n−j=2

ϕTj

Kj1 − Kj1

ϕj − zTn Fn(xn)θn


˙θn + ΘT

nΛ−1n2

˙Θn

−

n−1−j=1

βj1

2‖θj‖

2+

n−1−j=1

βj1

2‖θj‖

2−

n−1−j=1

βj2

2‖Θj‖

2

+

n−1−j=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2

+ψ2h(Z)

ψ2 + ‖zn‖2+ ψψ. (85)

Consider the adaptive laws for θn and Θn as

˙θn = Λn1(F T

n (xn)zn − βn1θn) (86)

˙Θn = Λn2(ρn(xn)Tanh(zn)zn − βn2Θn) (87)

where βn1 > 0 and βn2 > 0.Substituting (78), (86) and (87) into (85), we obtain

Vn ≤ −zT1

c1K10 −

ζ1

2+

12

Im×m

z1

−

n−j=2

zTj

cjKj0 −

ζj−1

2+ζj

2+

12

Im×m

zj

−ϕT1

K11 −

3‖K10‖

2+

32

+δ1

2

Im×m

ϕ1

−

n−j=2

ϕTj

Kj1 − Kj1

ϕj −

n−j=1

βj1

2‖θj‖

2

+

n−i=1

βj1

2‖θj‖

2−

n−j=1

βj2

2‖Θj‖

2+

n−i=1

βj2

2‖Θj‖

2

+

n−j=1

‖Θj‖2

2− kvψ2

≤ −κVn + C (88)

where

κ := min

2λmin(Q0), 2λmin(Q1),2λmin(Q2), 2λmin(Q3),

n−j=1

2βj1

λmax(Λ−1j1 )

,

n−j=1

2βj2

λmax(Λ−1j2 )

, kv

,C :=

n−j=1

βj1

2‖θj‖

2+

n−j=1

βj2

2‖Θj‖

2+

n−j=1

‖Θj‖2

2,

Q0 = c1K10 −

ζ1

2+

12

Im×m,

Q1 =

n−j=2

cjKj0 −

ζj−1

2+ζj

2+

12

Im×m

,

Q2 = K11 −

3‖K10‖

2+

32

+δ1

2

Im×m,

Q3 =

n−j=2

(Kj1 − Kj1), j = 2, . . . , n. (89)

To ensure the closed-loop stability, we can choose correspondingdesign parameters to make Q0 > 0, Q1 > 0, Q2 > 0 and Q3 > 0.The above design procedure can be summarized in the followingtheorem, which contains the results for the constrained adaptivecontrol of an uncertain nonlinear system.


Theorem 2. Considering the uncertain MIMO nonlinear system (1)satisfies Assumptions 1–5, and given that full state information isavailable. The control law is produced by nominal control law (79) us-ing the command filter. Under the parameter adaptation laws (64),(65), (74), (75), (86), (87) and for any bounded initial condition, theclosed-loop signals zi, ϕi, ψ , θi and Θi (i ≤ i ≤ n) are semi-globallystable in the sense that all of the closed-loop signals are bounded,where i = 1, 2, . . . , n. The tracking error z1 asymptotically convergesto a compact set Ωz1 defined by

Ωz1 :=

z1 ∈ Rm

| ‖z1‖ ≤√D

where D = 2(Vn(0)+Cκ) with C and κ as defined in (89).

It is apparent that the Theorem2 can be easily proved accordingto (84) and (88).

Remark 4. In the proposed constrained adaptive control, we cansee that the satisfactory closed-loop stability with suitable tran-sient performance can be achieved by properly adjusting designparameters Ki0, Ki1, βi1, βi2,Λi, and k, i = 1, 2, . . . , n. For example,the tracking error could be decreased by increasing the value of Ki0,but that increase would also increase the control signal, and couldexcite unmodeled dynamics. Therefore, caution must be exercisedin the choice of these parameters, due to the fact that there is sometrade-off between the control performance and other issues.

Remark 5. In the developed constrained adaptive control, if ϕi =

0 and ϕi = 0, there are αi0 = αi and u = v according to(59), (69) and (81), i.e., there are no constraints. At the same time,the nominal virtual control law (58), (67) and nominal controllaw (79) are the same as the virtual control law (13), (35) andcontrol law (47) of the proposed adaptive control in Section 3. Itshould be pointed out that we do not directly consider the inputsaturation constraint (2). However, the command filter can notonly implement the same position and also rate constraints can beconsidered on the adaptive control v by choosing the appropriatedesign parameter.

Remark 6. In practice, it is apparent that the magnitude of theactual/virtual control input, as well as their derivations shouldbe bounded due to the physical limitation. Thus, the commandfilter could be presented according to mechanical and operatingconstraints of actuator. Magnitude limit function and rate limitfunction can be chosen as conservative common saturation func-tion or other limit functions. If limit functions are chosen as conser-vative common saturation functions, the relationship between theinput and the output of the command filter can be found in Farrellet al. (2003).

5. Simulation results

Consider the uncertain MIMO nonlinear system with input sat-uration in the form of Chen et al. (2010)

x1 = F1(x1)θ1 + (G1(x1)+∆G1(x1))x2 + D1(x1, t)x2 = F2(x2)θ2 + (G2(x2)+∆G2(x2))u + D2(x2, t)y = x1 (90)

where

x1 = [x11, x12]T , x2 = [x21, x22]T ,

F1(x1) =

[0.2 sin(x11) cos(x12)

0.2x11x12

],

G1(x1) =

[g11(x) −2

5 g22(x)

],

g11(x) = 1.2 + cos(x11) sin(x12),g22(x) = 1.3 − cos(x12) sin(x11),

∆G1(x1) =

[0.2 sin(x11) 0

0 0.1 cos(x12)

],

F2(x2) =

[−x12x21 0

0 2x11x22

]G2(x2) =

[cos(x21) sin(x22) − sin(x22)

sin(x22) cos(x21)

],

∆G2(x2) =

[0.12 sin(x11x21) 0.11 cos(x11x21)0.15 cos(x11x21) 0.13 sin(x21x22)

],

D1(x1, t) =

[0.21(cos(x12))2 + 0.04 sin(0.3x12t)0.12(sin(x11))2 + 0.03 sin(0.2x11t)

],

D2(x2, t) =

[0.13(sin(x22))2 + 0.05 sin(0.2x22t)0.11(cos(x21))2 + 0.21 sin(0.3x21t)

].

For simulation purposes, parameter values are set to θ1 = −1,θ2 = 0.5, u1max = −u1min = 3.0, u2max = −u2min = 2.0,gri(vi) = gli(vi) = vi, γ1 = 3.0 and γ2 = 2.0.

Now, the control objective is to design adaptive control andconstraint adaptive control for system (90) such that the systemoutput y = x1 follows the desired trajectory x1d, where the desiredtrajectories are taken as x11d = 0.5[sin(1.5t) + sin(0.5t)] andx12d = 0.8 sin(t)+ 0.5 sin(0.5t).

The adaptive control is designed as follows:

α1 = (G1(x1)+ γ1Im×m)−1

−K1z1 − F1(x1)θ1 − Tanh(z1)ρ1(x1)Θ1 + x1d

e =

−K22e −

1‖e‖2

f (u,∆u, z2, x2)e

+ (G2(x2)+ γ2I2×2)(v − u), ‖e‖ ≥ σ0, ‖e‖ < σ

v = (G2(x2)+ γ2Im×m)−1v0

v0 = −GT1(x1)z1 − K2(z2 − e)− F2(x2)θ2

− Tanh(z2)ρ2(x2)Θ2 + α1 −z2h(Z)

ψ2 + ‖z2‖2

ψ =

−ψh(Z)

ψ2 + ‖z2‖2− kψ, ‖z2‖ ≥ ℓ

0, ‖z2‖ < ℓ

where k > 0, K1 = K T1 > 0, K2 = K T

2 > 0, K22 = K T22 > 0,

f (u,∆u, z2, x2) = |zT2G2(x2)∆u| + 0.5(γ2 + ζ2)2∆uT∆u and σ =

0.1. The adaptive laws for θ1 and Θ1 are chosen as (17) and (18). Theadaptive laws of θ2 and Θ2 are chosen as (53) and (54). The designparameters of the control are chosen as K1 = diag18.0, 18.0,K2 = diag120.0, 180.0, K22 = diag10.0, 10.0 and Λ11 =

Λ12 = Λ21 = Λ21 = diag0.01, 0.01.The simulation results of the tracking output are shown in

Figs. 3 and 4 with initial states x11 = 1.0 and x12 = 0.0. It canbe observed that the system output x11 and x12 follow the desiredtrajectory x11d and x12d well despite the unknown parameters, per-turbation of the control coefficient matrices and input saturation.From Figs. 5 and 6, we can see that the control inputs are satu-rated in the initialization transient phase. These simulation resultsshow that good tracking performance can be obtained under theproposed adaptive control.

To illustrate the effectiveness of the proposed constrainedadaptive control, the nominal virtual control law and the controlcommand are designed based on (58) and (79). Then, the nominalvirtual control law α10 and the nominal control command v areused to produce the virtual control law α1 and the system control


Fig. 3. Output x11 (solid line) follows desired trajectory x11d (dashed line).

Fig. 4. Output x12 (solid line) follows desired trajectory x12d (dashed line).

Fig. 5. Control signal u1 .

Fig. 6. Control signal u2 .

law u using the command filter as shown in Fig. 2. The designparameters of the filters are chosen as ω1n = 10, ξ1 = ξ2 = 0.707and ω2n = 100. To observer the variety of closed-loop systemcontrol performance for the different design parameters under theconstrained adaptive tracking control, the following two cases areconsidered:Case 1: K10 and K20 are chosen as K10 = diag18.0, 18.0 and K20 =

diag120.0, 180.0. Other design parameters are chosen as thesame design parameters as the corresponding design parametersin the adaptive tracking control.

Fig. 7. Output x11 (solid line) follows desired trajectory x11d (dashed line) for Case 1.

Fig. 8. Output x12 (solid line) follows desired trajectory x12d (dashed line) for Case 1.

Fig. 9. Control signal u1 for Case 1.


Case 2: The design parameters K10 and K20 are chosen as K10 =

diag10.0, 10.0 and K20 = diag120.0, 120.0. Other design pa-rameters are chosen as the same design parameters as the corre-sponding design parameters in the adaptive tracking control.

Under initial states are x11 = 1.0 and x12 = 0.0, the tracking re-sults of the Case 1 are shown in Figs. 7 and 8. It can be observed thatthe outputs x11 and x12 of Case 1 still follow the desired trajectoryx11d and x12d when the actuator constraints are considered. In ac-cordance with Figs. 9 and 10, it is observed that the control inputs


Fig. 11. Output x11 (solid line) follows desired trajectory x11d (dashed line) forCase 2.

Fig. 12. Output x12 (solid line) follows desired trajectory x12d (dashed line) forCase 2.



are saturated in the transient phase of Case 1. As a comparison, thecorresponding simulation results of Case 2 are shown in Figs. 11–14. From Figs. 11 and 12, we can observed that the differenttracking performance can be obtained by adjusting the design pa-rameters of the constrained adaptive tracking control. According toFigs. 7, 8, 11 and 12, we obtain that the tracking error could be de-creased by increasing the value of Ki0, but that increaseswould alsoincrease the control signal and could excite unmodeled dynamics.

6. Conclusion

Model-based adaptive control has been investigated for the un-certain MIMO nonlinear systems with input constraints in this pa-per. Considering actuator physical constraints, the adaptive con-trol and the constrained adaptive control in combination with thebackstepping technique and Lyapunov synthesis have been pro-posed. In the development of adaptive control, the auxiliary de-sign system has been introduced to analyze the effect of actuatorphysical constraint, and states of auxiliary design system are usedto develop adaptive control. The cascade property of the studiedsystems has been fully utilized in developing the control structureand parameter adaptive laws. It has proved that both the proposedadaptive control and the constrained adaptive control are able toguarantee the asymptotical stability of all signals in the closed-loop system. Finally, simulation studies have been presented toillustrate the effectiveness of the proposed adaptive and the con-strained adaptive control.

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MouChen receivedhis B.Sc. degree inmaterial science andengineering at Nanjing University of Aeronautics & Astro-nautics, Nanjing, China, in 1998, the M.Sc. and the Ph.D.degree in automatical control engineering at Nanjing Uni-versity of Aeronautics & Astronautics, Nanjing, China, in2004. He is currently an association professor in automa-tion college at Nanjing University of Aeronautics & Astro-nautics, China. From June 2008 to Step 2009, he was aresearch fellow in the Department of Electrical and Com-puter Engineering, the National University of Singapore.His research interests include nonlinear control, artificial

intelligence, imagine processing and pattern recognition, and flight control.

Shuzhi SamGe IEEE Fellow, IFAC Fellow, IET Fellow, P.Eng,is founding Director of Social Robotics Lab, Interactive Dig-ital Media Institute and Full Professor in the Departmentof Electrical and Computer Engineering, the National Uni-versity of Singapore and the Institute of Intelligent Sys-tems and Information Technology, University of ElectronicScience and Technology of China, Chengdu, China. He re-ceived his BSc degree from Beijing University of Aeronau-tics and Astronautics (BUAA), and the Ph.D. degree and theDiploma of Imperial College (DIC) from Imperial College ofScience, Technology and Medicine.

He has (co)-authored three books and over 300 international journal and con-ference papers. He has served/been serving as an Associate Editor for a number offlagship journals including IEEE Transactions on Automatic Control, IEEE Transac-tions on Control Systems Technology, IEEE Transactions on Neural Networks, andAutomatica. He also serves as an Editor of the Taylor & Francis Automation andControl Engineering Series. He is an elected member of Board of Governors, IEEEControl Systems Society. He provides technical consultancy to industrial and gov-ernment agencies. He is the Editor-in-Chief of the International Journal of SocialRobotics. His current research interests include social robotics, multimedia fusion,adaptive control, and intelligent systems.

Beibei Ren received the B.E. degree in the Mechanical &Electronic Engineering and the M.E. degree in Automationfrom Xidian University, Xi’an, China, in 2001 and in 2004,respectively, and the Ph.D. degree in the Electrical andComputer Engineering from the National University ofSingapore, Singapore, in 2010. Currently, she is workingas a postdoctoral scholar in the Department of Mechanicaland Aerospace Engineering, University of California, SanDiego. Her current research interests include nonlinearsystem control and its applications.

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