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IEEE/ASME TRANSACTIONS ON MECHATRONICS 1

Performance-Oriented Adaptive Robust Control of aClass of Nonlinear Systems Preceded by UnknownDead Zone With Comparative Experimental Results

Chuxiong Hu, Member, IEEE, Bin Yao, Senior Member, IEEE, and Qingfeng Wang, Member, IEEE

Abstract—This paper presents an integrated direct/indirectadaptive robust control scheme for a class of nonlinear dy-namic systems preceded by unknown nonsymmetric, nonequalslope dead-zone nonlinearity. Departing from existing approxi-mate adaptive dead-zone compensations, this paper uses indi-rect parameter estimation algorithms along with on-line condi-tion monitoring to obtain an accurate estimation of the unknowndead zone when certain relaxed persistent-excitation conditionsare satisfied—a theoretical result that cannot be achieved with theexisting methods. Such a result is obtained by making full use ofthe fact that though not being linearly parameterized globally, theunknown dead zone can still be linearly parameterized perfectlywithin certain known working ranges. With these accurate esti-mates of dead-zone parameters, perfect dead-zone compensation isthen constructed and utilized in the development of a performance-oriented adaptive robust control algorithm for the overall system.Consequently, asymptotic output tracking is achieved even in thepresence of unknown dead zone. In addition, the proposed algo-rithm achieves certain guaranteed robust transient performanceand final tracking accuracy even when the entire system may besubjected to other uncertain nonlinearities and time-varying dis-turbances. The proposed algorithm is also experimentally testedon a linear motor drive system preceded by a simulated unknownnonsymmetric dead zone. Comparative experimental results ob-tained validate the effectiveness of dead-zone compensation andthe high-performance nature of the proposed approach in practi-cal implementation.

Index Terms—Adaptive control, nonlinear systems, parameterestimation, robust control, uncertainty.

I. INTRODUCTION

NONSMOOTH nonlinearities such as dead zone, backlash,and hysteresis are ubiquitously present in hydraulic servo

Manuscript received February 12, 2011; revised May 19, 2011; accepted June28, 2011. Recommended by Technical Editor J. Xu. This paper was presentedin part at the 48th IEEE Conference on Decision and Control, Shanghai, China,December 16–18, 2009. The work was supported in part by the Ministry ofEducation of China through a Chang Jiang Chair Professorship, in part by theChina Postdoctoral Science Foundation under Grant 20110490378, and in partby the U.S. National Science Foundation under Grant CMMI-1052872.

C. Hu was with the State Key Laboratory of Fluid Power Transmis-sion and Control, Zhejiang University, Hangzhou 310027, China. He is nowwith the State Key Laboratory of Tribology, Department of Precision Instru-ments and Mechanology, Tsinghua University, Beijing 100084, China (e-mail:[email protected]).

B. Yao is with the State Key Laboratory of Fluid Power Transmission and Con-trol, Zhejiang University, Hangzhou 310027, China, and also with the Schoolof Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA(e-mail: [email protected]).

Q. Wang is with the State Key Laboratory of Fluid Power Transmis-sion and Control, Zhejiang University, Hangzhou 310027, China (e-mail:[email protected]).

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

Digital Object Identifier 10.1109/TMECH.2011.2162633

valves, dc servo motors, mechanical connections, and piezo-electric stages [1]–[4]. Among them, dead-zone nonlinearity isa relatively simple static input–output relationship which char-acterizes the insensitivity of the output-to-input values duringcertain working ranges. Should the parameters of the dead-zonenonlinearity be known, it is straightforward to construct a per-fect inversion of the dead-zone nonlinearity to compensate forits effect completely [5]. However, in reality, the dead-zone out-put is often difficult to measure and the dead-zone parametersare seldom known completely or remain the same values overthe entire lifespan of a machine. Thus, how to effectively com-pensate the effect of dead-zone nonlinearity in the presence ofparametric uncertainties without direct measurement of dead-zone output has always been a practically important problem.The problem is also difficult to solve as it is unclear if exactinversion of the dead zone for perfect dead-zone compensationis possible or not in such a scenario. It has been shown thatimperfect compensation of the dead zone in feedback controlsystems could severely deteriorate the achievable control per-formance, leading to undesirable control accuracy, limit cycles,and even instability [6].

During the past decades, extensive research has been de-voted to solving the aforementioned problem in one way oranother, with quite a large number of publications appeared [5],[7], [8], [9]. Specifically, an adaptive dead-zone inverse was firstproposed in [5] under the unrealistic assumption of set certaintyequivalence. Such an assumption was subsequently removedin [7] but only bounded output tracking errors are achieved. Thework in [10] continued the previous research and did achieveperfect asymptotical adaptive cancelation of an unknown deadzone. However, a much stronger assumption of both the inputand the output of the dead zone being measured was assumed,which cannot be met in most practical applications having deadzones. In [11]–[13], fuzzy-logic and neural network were used toprovide alternative interpretations to the basis functions neededfor adaptive dead-zone inversions [7]. However, no essential im-provement on theoretically achievable results are obtained, i.e.,only bounded output tracking errors are obtained. In [9], [14],and [15], the dead zone was modeled as a combination of a lin-ear input with either an unknown constant gain for symmetricdead zones [14] or time-varying unknown gain for nonsymmet-ric ones [9], [15] and a bounded disturbance-like term. Withthis formulation, traditional robust adaptive control techniquescan be conveniently applied to achieve bounded output trackingerrors. In [8], smooth basis functions as opposed to the discon-tinuous ones in [7] and [11] were explored to provide some

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2 IEEE/ASME TRANSACTIONS ON MECHATRONICS

approximate inversions of the dead-zone nonlinearity. Mode-less approaches such as the disturbance observer [16] are alsoemployed to compensate for nonsmooth nonlinearities [17].

It is noted that without assuming the dead-zone output beingmeasured, even under the assumption that certain persistent-excitation (PE) conditions are satisfied, none of the aforemen-tioned existing schemes can achieve perfect adaptive compen-sation of unknown dead zones for asymptotic output tracking,except resorting to the unrealistic chattering control inputs [18].The reason for this might come from the fact that the harddead-zone nonlinearity cannot be linearly parameterized glob-ally. As the existing schemes are based on direct adaptive controldesigns, they all need uncertain nonlinearities to be linearly pa-rameterized perfectly and globally for asymptotic output track-ing, which is not possible for unknown dead zones. Departingfrom these approximate dead-zone compensations, in this paper,we will make full use of the fact that, though not being linearlyparameterized globally, the unknown dead zone can still be lin-early parameterized when restricted to some known ranges. Assuch, when on-line parameter adaptation is turned ON only dur-ing those working ranges, accurate estimations of all unknowndead-zone parameters could be obtained. Consequently, perfectadaptive compensation of unknown dead zones for asymptoticoutput tracking can be achieved, provided that certain relaxedPE conditions are satisfied. This is made possible in this paperthrough the use of indirect parameter estimation algorithms withon-line condition monitoring.

In [19], an integrated direct/indirect adaptive robust control(DIARC) scheme was presented, in which the construction ofparameter adaptation law can be totally independent from thedesign of underline robust control law. This DIARC strategyallows various parameter adaptation algorithms having betterconvergence properties and practical modifications such as theexplicit on-line monitoring of signal excitation levels to be usedto improve the accuracy of parameter estimates in implementa-tion. Such a DIARC strategy has been successfully applied toseveral applications such as the precision motion control of lin-ear motors [20] and the coordinated contouring application [21].

This paper will make full use of the flexible parameter estima-tion process of the DIARC strategy and the dead-zone propertyof being linearly parameterized within certain known ranges todevelop performance-oriented control algorithms for a class ofuncertain nonlinear systems preceded by unknown dead zones.The DIARC schemes in [19] and [21] only consider systemswith smooth unknown nonlinearities that can be linearly pa-rameterized. As the dead-zone nonlinearity is nonsmooth andcannot be linearly parameterized globally, theoretically, the DI-ARC design in [19] cannot be straightforwardly applied to thetype of systems studied in this paper. Appropriate nonsmoothdead-zone inverse has to be developed and novel ideas areneeded to overcome the linear parametrization problem. It isshown that the proposed DIARC is able to achieve accurateestimates of all dead-zone parameters when certain relaxed PEconditions are satisfied. Consequently, perfect dead-zone com-pensation will be achieved and asymptotic output tracking with-out assuming the dead-zone output being measured is proventheoretically—a theoretical result that cannot be achieved by the

existing methods. Furthermore, even when the relaxed PE con-ditions are not met and the overall system may be subjected toother uncertain nonlinearities and time-varying disturbances,the proposed DIARC algorithm still achieves a guaranteed ro-bust transient performance and final tracking accuracy. Namely,the bounds of the output tracking errors are directly related to thecontroller parameters in a known form, not only in L2 norm as inthe traditional robust adaptive controls [8] but also in practicallymore meaningful L∞ norm.

The desired compensation idea in [22] has also been in-corporated into the proposed DIARC dead-zone compensationstrategy to synthesize high-performance motion controllers forelectrical drive systems in [23]. In this paper, comparative ex-periments will be carried out on a different linear motor-drivenstage preceded by a simulated unknown nonsymmetrical andnonequal slope dead-zone nonlinearity. In addition, the pro-posed algorithm will be compared with a typical example of theexisting robust adaptive dead-zone compensations—the controlalgorithm in [18]—along with control algorithms with no dead-zone compensation, dead-zone compensation based on nominalvalues, and dead-zone compensation based on actual values.Comparative experimental results show that the proposed DI-ARC dead-zone compensation outperforms the one in [18] sig-nificantly and achieves almost the same steady-state trackingperformance as that of the one with exact dead-zone compensa-tion (i.e., assuming that the dead-zone parameters are known).Overall, the much better tracking performance of the proposedalgorithm seen in the experiments validates the effectivenessof the proposed DIARC dead-zone compensation strategy inpractical applications.

II. PROBLEM STATEMENT

A. System Model

This paper considers the same class of nonlinear dynamicsystems preceded by unsymmetric dead zone as in [8], [14],and [15], which are described by

x(n) =p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t)) + Ew(t) + fu

y = x(t), w(t) = D(v(t)) (1)

where Yi, i = 1, 2, . . . , p, are some known continuous nonlin-ear functions, and parameters ai and E represent unknown con-stants. v(t) is the output from the controller, w(t) is the actualinput to the plant, and y(t) is the output from the plant. fu rep-resents the lumped uncertain nonlinearities including externaldisturbances. The actuator nonlinearity D(v(t)) is described asa dead-zone characteristic shown in Fig. 1(a).

With input v(t) and output w(t), the dead zone can be repre-sented as in [7] as follows:

w(t) = D(v(t)) =

⎧⎪⎨

⎪⎩

mrv(t) − mrbr , for v(t) ≥ br

0, for bl < v(t) < br

mlv(t) − mlbl , for v(t) ≤ bl(2)


HU et al.: PERFORMANCE-ORIENTED ADAPTIVE ROBUST CONTROL OF A CLASS OF NONLINEAR SYSTEMS 3

Fig. 1. (a) Dead-zone model. (b) Perfect dead-zone inverse.

where the parameters mr , ml , br , and bl are constants and standfor the right slope, left slope, right break-point, and left break-point of the dead zone, respectively.

Let E = 1 for that its effect can be considered in the un-known slope mr and ml . For notation simplicity, let θ ∈Rp+4 be the vector of all unknown constant parameters, i.e.,θ = [a1 , a2 , . . . , ap ,mr ,mrbr ,ml,mlbl ]T . The control objec-tive is to design a control law v(t) to ensure that all closed-loop signals are bounded and the plant state vector X =[x(t), x(t), . . . , x(n−1)(t)] tracks the specified desired trajec-tory Xd = [xd(t), xd(t), . . . , x

(n−1)d (t)] with certain guaranteed

transient responses. The following practical assumptions aremade.

Assumption 1: The dead-zone output w(t) is not measurableand the dead-zone parameters mr , ml , br , bl are unknown, buttheir signs are known as mr > 0, ml > 0, br > 0, bl < 0.

Assumption 2: The unknown parameter vector θ is within aknown bounded convex set Ωθ . Without loss of generality, itis assumed that ∀θ ∈ Ωθ , ai ∈ [ai min , ai max], i = 1, 2, . . . , p,and 0 < mr min ≤ mr ≤ mr max , 0 < mlmin ≤ ml ≤ mlmax ,0 < (mrbr )min ≤ mrbr ≤ (mrbr )max , (mlbl)min ≤ mlbl ≤(mlbl)max < 0, where ai min , ai max , mr min , mr max , mlmin ,mlmax , (mrbr )min , (mrbr )max , (mlbl)min , and (mlbl)max areall known constants.

Assumption 3: The uncertain nonlinearity fu can be boundedby

|fu | ≤ δ(X)fd(t) (3)

where δ(X) is a known positive function and fd(t) is an un-known but bounded positive time-varying function.

B. Dead-Zone Compensation

The essence of compensating dead-zone effect is to employ aperfect dead-zone inverse function v(t) = DI (w(t)) such thatD (DI (w(t))) = w(t),∀w(t). However, as seen from Fig. 1(a),the dead-zone function is not a one-one mapping during thezero-output zone of v(t) ∈ [bl , br ]. As such, there does not existan unique inverse and various choices of a particular dead-zoneinverse function in the controller design may have significantlydifferent results in implementation. This problem has largelybeen ignored in all the previous publications. Typically, only aparticular dead-zone inverse is provided without carefully ex-amining its potential implementation problems. For example,

in [7], the following underline perfect dead-zone inverse func-tion was used:

v(t) = DI (w(t)) =

⎧⎪⎪⎨

⎪⎪⎩

w + mrbr

mr, if w(t) ≥ 0

w + mlbl

ml, if w(t) < 0.

(4)

Such an inverse function may suffer from the following im-plementation problem. In addition to chattering of the controlinput signal v(t) between two distinct values of bl and br whenw(t) = 0 due to the unavoidable any extremely small calcula-tion errors of w(t) either by quantization or measurement noise,when the system comes to a stop in which w(t) continuouslyconverges to zero from some negative values, unnecessary jump-ing of v(t) from bl to br will occur as well. This tends to excitethe neglected high-frequency dynamics and prolong the settlingof the system response to the target value. To alleviate this im-plementation problem, in this paper, the following underlineperfect dead-zone inverse will be used:

v(t) = DI (w) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

w(t) + mrbr

mr, if w(t) > 0

argminc∈[bl ,br ]

|DI (w(t−)) − c|, if w(t) = 0

w(t) + mlbl

ml, if w(t) < 0

(5)where w(t−) represents the left limit of w(t). Such an inverse isgraphically shown in Fig. 1(b) for w(t) �= 0. Essentially, whenw(t) = 0, the proposed dead-zone inverse is to keep the previousvalue of v(t−) = DI (w(t−)) without jumping. Thus, in reality,for continuous w, when w(t) = 0, depending on past history,v(t) will be either bl or br only as v(t−) = bl when w reacheszero from w < 0 region and v(t−) = br when w reaches zerofrom w > 0 region. In implementation, to avoid the erroneousswitching due to any calculation error of w, a small toleranceband would be used when checking conditions like w = 0. It isnoted that the proposed dead-zone inverse has certain memoryeffect as opposed to the pure static nature of the dead-zoneinverse (4).

As the dead zone is assumed unknown, the aforementionedinverse can only be implemented with the estimates of the dead-zone parameters. Namely, let mr , (mrbr ), ml , (mlbl) be theestimates of mr , mrbr , ml , mlbl , respectively, and wd be thedesired control signal that would achieve the stated control ob-jective when there is no dead-zone effect. Then, viewing (5),a reasonable actual control input v(t) when the dead-zone (2)exists would be

v(t) = DI (wd)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

wd(t) + (mrbr )mr

, if wd(t) > 0

argminc∈[b l ,br ]

∣∣∣DI (wd(t−)) − c∣∣∣ , if wd(t) = 0

wd(t) + (mlbl)ml

, if wd(t) < 0(6)



Fig. 2. Designed parts and the nonlinear systems.

where br = (mr br )m r

and bl = (ml bl )m l

. With this dead-zone in-

verse, the resulting controlled system is shown in Fig. 2.For notational simplicity, in the following, (6) is rewritten as

v(t) = κ+(wd)wd(t) + (mrbr )

mr+ κ−(wd)

wd(t) + (mlbl)ml

(7)where κ+(wd) and κ−(wd) are defined by

κ+(wd)(t) =

⎧⎨

⎩

1, if (wd(t) > 0 or (wd(t) = 0 and|v(t−) − bl | ≥ |v(t−) − br |))

0, else

(8)

κ−(wd)(t) =

⎧⎪⎨

⎪⎩

1, if (wd(t) < 0 or (wd(t) = 0 and

|v(t−) − bl | < |v(t−) − br |))0, else.

(9)

C. Projection-Type Adaptation Law Structure

One of the key elements of the ARC design is to use thepractical available prior process information to construct theprojection type adaptation law for a controlled adaptation pro-cess. For this purpose, the widely used projection mapping Proj

θwill be used to keep the parameter estimates within the knownbounded set Ωθ , the closure of the set Ωθ as in [24]. The standardprojection mapping [25] is as follows:

Projθ(ζ) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ζ, if θ ∈ Ωθ or (θ ∈ ∂Ωθ

and nT

θζ ≤ 0)

(I − Γ

nθnT

θ

nT

θΓn

θ

)ζ, if θ ∈ ∂Ωθ and nT

θζ > 0

(10)where ζ ∈ Rp+4 is any function and Γ(t) ∈ R(p+4)×(p+4) canbe any time-varying positive-definite symmetric matrix. In (10),Ωθ and ∂Ωθ denote the interior and the boundary of Ωθ , re-spectively, and n

θrepresents the outward unit normal vector at

θ ∈ ∂Ωθ . The parameter estimate vector θ is updated using thefollowing projection-type adaptation law:

˙θ = Proj

θ(Γτ), θ(0) ∈ Ωθ (11)

where τ is any estimation function and Γ(t) > 0 is any contin-uously differentiable positive symmetric adaptation rate matrixwhich would be detailedly presented in Section III-C. With thisadaptation law structure, the following desirable properties hold.

(P1): The parameter estimates are always within theknown bounded set Ωθ , i.e., θ ∈ Ωθ ,∀t. Thus, from As-sumption 2, ∀t, ai min ≤ ai(t) ≤ ai max , i = 1, 2, . . . , p,0 < mr min ≤ mr (t) ≤ mr max , 0 < ml min ≤ ml(t) ≤

ml max , 0 < (mrbr )min ≤ (mrbr )(t) ≤ (mrbr )max , and

(mlbl)min ≤ (mlbl)(t) ≤ (mlbl)max < 0.(P2):

θT (Γ−1Projθ(Γτ)) − τ) ≤ 0, ∀τ. (12)

D. Dead-Zone Compensation Error Characterization

Noting property P1 for the dead-zone parameter estimates,the following lemma can be obtained as in [23]:

Lemma 1: With the dead-zone inverse (7), the error betweenthe actual dead-zone output w by (2) and the desired output wd

can be parameterized as

w − wd = ˜(mrbr )κ+(wd) + ˜(mlbl)κ−(wd)

− wd + (mrbr )mr

mrκ+(wd)

− wd + (mlbl)ml

mlκ−(wd) + d(t) (13)

where d(t) is a function defined as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if κ+(wd) = 1 and v(t) ≥ br

mrbr −wd + (mrbr )

mrmr , if κ+(wd) = 1 and 0 < v < br

mlbl −wd + (mlbl)

mlml, if κ−(wd) = 1 and bl < v < 0

0, if κ−(wd) = 1 and v(t) ≤ bl

(14)

and bounded above by

⎧⎪⎪⎨

⎪⎪⎩

∣∣∣∣(mrbr )max − mrmin(mrbr )min

mrmax

∣∣∣∣ , if κ+(wd) = 1∣∣∣∣(mlbl)min − mlmin(mlbl)max

mlmax

∣∣∣∣ , if κ−(wd) = 1.

(15)In addition, d(t) ∈ L2 [0,∞) when θ ∈ L2 [0,∞) and d(t) −→0 when θ(t) −→ 0 as t −→ ∞.

III. INTEGRATED DIARC SYNTHESIS

In this section, using the dead-zone inverse (7) and certainconditional parameter adaptation, the integrated DIARC strat-egy in [19] will be generalized to solve the control problemposted in section II for the system (1).

A. Integrated DIARC Law

With the use of the projection type adaptation law structure(11), the parameter estimates are bounded with known bounds,regardless of the estimation function τ to be used. In the fol-lowing, this property will be used to synthesize an integratedDIARC control law with the dead-zone inverse (7) for the sys-tem (1) which achieves a guaranteed transient and steady-state



output tracking accuracy in general. Define

s(t) =(

d

dt+ λ

)n−1

x(t) (16)

with λ > 0 which can be rewritten as

s(t) = ΛT X(t) (17)

where ΛT = [C0n−1λ

n−1 , C1n−1λ

n−2 , . . . , Cn−1n−1 ], X =

X − Xd , X = [x(t), x(t), . . . , x(n−1)(t)], and Xd =[xd(t), xd(t), . . . , x

(n−1)d (t)]. Noting (1) and (13) in Lemma 1,

the derivative of s(t) can be derived as

s(t) = ΛTv X(t) + xn (t)

= ΛTv X(t) − x

(n)d (t)+

p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

+ fu + w(t)

= ΛTv X(t) − x

(n)d (t)+

p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

+ fu + d(t) +wd(t) + ˜(mrbr )κ+(wd) + ˜(mlbl)κ−(wd)

− wd + (mrbr )mr

mrκ+(wd) −wd + (mlbl)

mlmlκ−(wd)

(18)

where ΛTv = [0, C0

n−1λn−1 , C1

n−1λn−2 , . . . , Cn−2

n−1 λ]. In the fol-lowing, a DIARC controller will be developed by treating wd

as the virtual control input to (18) as shown in Fig. 2, fromwhich the actual input v(t) to the dynamics system (1) will becalculated by the dead-zone inverse (7).

The following control law for wd is proposed:

wd = wda + wds, wda = wda1 + wda2

wds = wds1 + wds2,

wda1 = x(n)d (t) − ΛT

v X(t)

−p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

wda2 = −dc , wds1 = −ks1s. (19)

In (19), wda1 represents the usual model compensation withthe physical parameter estimates ai updated using an on-lineadaptation algorithm with condition monitoring to be detailedin Section III-B. wda2 is a model compensation term similar tothe fast dynamic compensation-type model compensation usedin the DARC designs [19], in which dc can be thought as theestimate of the low-frequency component of the lumped modeluncertainties defined later. wds represents the robust controlterm in which wds1 is a simple proportional feedback to stabilizethe nominal system with a constant gain ks1 , and wds2 is a robustfeedback term used to attenuate the effect of various modeluncertainties for a guaranteed robust control performance ingeneral.

From (18) and (19), the derivative of s(t) can be derived as

s(t) = fu + d(t) −p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

+[mr

mr(wds + wda2) −

mr

mrwda1 + ˜(mrbr )

− mr

(mrbr )mr

]κ+(wd) +

[ml

ml(wds + wda2)

− ml

mlwda1 + ˜(mlbl) − ml

(mlbl)ml

]κ−(wd). (20)

Define a constant dc and time-varying �(t) such that

dc + �(t) = fu + d(t) −p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

−[mr

mrκ+(wd) +

ml

mlκ−(wd)

]wda1

+[

˜(mrbr ) − mr

(mrbr )mr

]κ+(wd)

+[

˜(mlbl) − ml

(mlbl)ml

]κ−(wd)

= fu + d(t) + ϑψ(t) (21)

where

ϑ =[a1 , a2 , . . . , ap ,

mr

mrκ+(wd) +

ml

mlκ−(wd),

[˜(mrbr ) − mr

(mrbr )mr

]κ+(wd)

+[

˜(mlbl) − ml

(mlbl)ml

]κ−(wd)

],

ψ(t) = −[Y1 , Y2 , . . . , Yp , wda1 ,−1]T .

Conceptually, (21) lumps the original system uncertain nonlin-earity fu with the model uncertainties due to physical parameterestimation errors and divides it into the static component dc (orlow-frequency component in reality) and the high-frequencycomponents �(t). In the following, the low-frequency com-ponent dc will be compensated through fast adaptation similarto those in the direct ARC designs [24] as follows.

Let dcM be any preset bound and use this bound to constructthe following projection-type adaptation law for dc :

˙dc = Proj

dc(γs) =

{0, if |dc(t)| = dcM and dcs > 0γs, else

(22)with γ > 0 and dc(0) = 0. Such an adaptation law guarantees|dc(t)| ≤ dcM , ∀t. For notation simplicity, define

k� = κ+(wd)mr

mr+ κ−(wd)

ml

ml(23)

which has a value of 1 in the absence of dead-zone slopesestimation error. Substituting (21) into (20) and noting (19),



s(t) can be written as

s(t) = k�(wds + wda2) + dc + �(t)

= k�wds1 + k�wds2 − dc + (1 − k�)dc + �(t). (24)

Noting Assumption 2, Assumption 3, and property P1, thereexists a wds2 such that the following two conditions are satisfied:

1) swds2 ≤ 0

2) s[k�wds2 − dc + (1 − k�)dc + �(t)] ≤ ε + εdf2d (25)

where ε and εd are parameters which can be arbitrarily small.Essentially, condition 2 of (25) shows that wds2 is synthesized todominate the model uncertainties coming from both parametricuncertainties and uncertain nonlinearities, and condition 1 of(25) is to make sure that wds2 is dissipative in nature, so that itdoes not interfere with the functionality of the adaptive controlpart wda .

Remark 1: One example of wds2 satisfying (25) can be foundin the following way. Let h be any function satisfying

h ≥ |k�max‖wda2 | + ‖ϑmax‖‖ψ‖ + |d(t)|max (26)

where k�max = max{mr m a x

mr m in, m l m a x

ml m in}, ϑ1max = a1max − a1min ,

ϑ2max = a2max − a2min , . . . , ϑrmax = armax − armin ,(ϑr+1)max = max{mr m a x −mr m in

mr m in, m l m a x −ml m in

ml m in}, (ϑr+2)max =

max{(mrbr )max − (mrbr )min + (mrmax − mrmin) (mr br )m a xmr m in

,

(mlbl)max − (mlbl)min − (mlmax − mlmin) (ml bl )m inml m in

}(ϑr+2)max = max{(mrbr )max − (mrbr )min + (mrmax −mrmin) (mr br )m a x

mr m in, (mlbl)max − (mlbl)min − (mlmax −mlmin)

(ml bl )m inml m in

}, |d(t)|max = max{|(mrbr )max − mrmin(mr br )m in

mr m a x|,

|(mlbl)min − mlmin(ml bl )m a x

ml m a x|}. Then, wds2 can be chosen as

wds2 = −[

14εk�

minh2 +

14εdk�

minδ2(X)

]s (27)

where k�min = min{mr m in

mr m a x, m l m in

ml m a x}. Using the same techniques

as in [19], it is easy to show that the aforementioned choice ofwds2 does satisfy (25).

The following theorem summarizes the theoretically achiev-able guaranteed performance.

Theorem 1: When the DIARC control law (19) with the dead-zone inverse (7) and the projection type adaptation law (11) isapplied, regardless the estimation function τ to be used, gener-ally, all signals in the resulting closed-loop system are boundedand the output tracking is guaranteed to have a prescribed tran-sient performance and final tracking accuracy in the sense thatthe tracking error index s is bounded by

s(t)2 ≤ e−λv ts(0)2 +2(ε + εdf

2dmax)

λv[1 − e−λv t ] (28)

where λv = 2k�minks1 and fdmax represents the L∞ norm of the

bounded time function fd(t).Proof: See Appendix A. �

B. Parameter Estimation Algorithm With On-LineCondition Monitoring

Theorem 1 shows that, regardless of the specific estimationfunction τ to be used, with the proposed DIARC control law,all the physical signals have been guaranteed to be boundedand guaranteed output-tracking transient and steady-state per-formance have been achieved. Thus, this section focuses on de-termining the specific parameter adaptation algorithms so thatan improved steady-state tracking accuracy—asymptotic outputtracking—can be obtained in the absence of uncertain nonlinear-ities [i.e., assuming fu = 0 in (1)] with an emphasis on havinga good parameter estimation process as well. For this purpose,rather than using any transformed tracking error dynamics toconstruct parameter estimation model as in the direct adaptivecontrol designs, the actual plant model (1) will be directly usedto construct specific estimation functions as detailed below.

As seen from (14), the model error d(t) in (13) is nonzeroduring some working ranges, revealing the fact that the dead-zone nonlinearity (2) cannot be globally linearly parameterizedby its parameters mr , ml , br , and bl with known basis functions.As such, if the system dynamics (1) is directly used to constructparameter estimation algorithm, one will suffer from the sameproblem as in all previous research that, theoretically, due to theexistence of nonzero model error d(t), no asymptotic parameterestimation convergence can be achieved even when relevant PEconditions are satisfied. In the following, we will make full useof the fact that, though not being linearly parameterized globally,the unknown dead-zone nonlinearity can be perfectly linearlyparameterized during the working regions of v ≥ br or v ≤ bl

as d(t) = 0 in those cases. As such, if the parameter estimationis updated only when the control input v(t) inside the regions ofv ≥ br or v ≤ bl (i.e., the parameter estimation is stopped whenbl ≤ v ≤ br ), accurate estimations of all the unknown dead-zone parameters can be achieved provided that some relaxedPE conditions are satisfied. For this purpose, explicit on-linecondition monitoring will be used by noting the following result.

Lemma 2: [23]: Define a positive constant H as H =max{(mrbr )max − (mrbr )min , (mlbl)max − (mlbl)min}.Then, when |wd(t)| ≥ H , the dead-zone input v(t) by theproposed dead-zone inverse (7) would satisfy v(t) ≥ br orv(t) ≤ bl .

For notation simplicity, define

χ+(•) ={

1, if • ≥ 00, else

, χ−(•) ={

1, if • ≤ 00, else

(29)

and χ(wd(t)) = χ+ (wd(t) − H) + χ− (wd(t) + H) which is1 when |wd(t)| ≥ H and is 0 when |wd(t)| < H by notingLemma 2. Thus, multiplying both sides of the plant dynamics(1) by χ(wd(t)) leads to

χ (wd(t)) x(n) = χ (wd(t))p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

+ (mrv(t) − mrbr ) χ+ (wd(t) − H)

+ (mlv(t) − mlbl) χ− (wd(t) + H) (30)



which is linearly parameterized by the system parameters. Accu-rate parameter estimation can, thus, be obtained by conductingon-line parameter estimation only using the measurement datawhen (30) is true. Namely, first group the measurement data intothree different cases given next using the checkable conditionin Lemma 2, and then perform on-line parameter estimationfor relevant dead-zone parameters in each case separately asfollows.

Case I: The data in this group consist of all the measurementdata during the time intervals when wd(t) ≥ H only. For thisdataset, χ+ (wd(t) − H) = 1 and χ− (wd(t) + H) = 0. Thus,the plant dynamics (30) are in the following linear regressionmodel for parameter estimation

yr (t) = FTr (t)θr (31)

where

θr = [a1 , a2 , . . . , ap ,mr ,mrbr ]T (32)

is the system parameter vector to be identified, and yr (t) =x(n)(t) and FT

r (t) = [Y1 , Y2 , . . . , Yp , v(t),−1] represent thecorresponding model output and regressor vector, respectively.To avoid the potential noise problem associated with the calcu-lation of x(n) based on the state measurement data, appropriatefiltering should be used. For example, let Hf (s) be the transferfunction of any stable filter with a relative degree larger than orequal to 1 (e.g., Hf (s) = 1/(ωf s + 1)). Applying the filter toboth sides of (31), one obtains

yrf (t) = FTrf (t)θr (33)

where yrf = Hf [yr ] and Frf = Hf [Fr ]. Noting that (33) isin the standard linear regression form for parameter estimation,various estimation algorithms exist. Among them, the least-squares estimation algorithm is known to have the best param-eter estimation convergence property in general [26]. Thus, itis used to estimate the values of θr . With the covariance lim-iting modification, the resulting adaptation function τ for theestimation of the parameter vector θr using the projection typeadaptation law structure (11) is

τ = − 11 + υtr{FT

rf ΓFrf }Frf ς (34)

where ς and Γ are the prediction error and covariance matrixcalculated by

ς = FTrf θr − yrf

Γ =

⎧⎪⎨

⎪⎩αΓ −

ΓFrf FTrf Γ

1 + υtr{FTrf ΓFrf }

, if λmax (Γ(t)) ≤ ρM

0, otherwise

(35)

in which α ≥ 0 is the forgetting factor, ρM is the preset upperbound for the covariance matrix, and υ ≥ 0 with υ = 0 leadingto the unnormalized algorithm. It is known that such an estimatorleads to the convergence of θr to its true value θr so long as thefollowing PE condition is satisfied [19], [26]:

Δtt−T Frf F T

rf dτ ≥ βIp , for some β > 0 and T > 0. (36)

Remark 2: The parameter adaptation law in the existing ro-bust adaptive dead-zone compensations is an integral part ofthe overall control design for closed-loop system stability. Assuch, the parameter adaptation needs to be turned on all thetime to ensure the nominal stability of the closed-loop system.Subsequently, similar PE condition to (36) needs to hold truefor all time for the convergence of parameter estimates, whichis not realist at all in practical implementation. In contrast, thebaseline control law presented in this paper is a robust controllaw not only guaranteeing robust stability but also robust perfor-mance regardless the convergence of the parameter adaptationas stated in Theorem 1. As such, parameter adaptation doesnot have to be turned on all the time and conditional parameteradaptation with explicit on-line monitoring of the signal excita-tion level can be used to ensure quality of the on-line parameterestimates. For example, in the previous development, the param-eter estimation for θr is conducted only using the data during thetime intervals when wd(t) ≥ H . Subsequently, the PE condition(36) only needs to hold true during those time intervals for theconvergence of parameter estimates. In addition, one can evenexplicitly check if the PE condition (36) is satisfied or not andupdate the parameter estimates only when it is actually satisfied.

Case II: The data in this group consist of all the measurementdata during the time intervals when wd(t) ≤ −H only. For thisdataset, χ+ (wd(t) − H) = 0 and χ− (wd(t) + H) = 1, and theplant dynamics (30) are in the following linear regression modelfor parameter estimation:

yl(t) = FTl (t)θl (37)

where yl = x(n)(t), FTl (t) = [Y1 , Y2 , . . . , Yp , v(t),−1], and

θl = [a1 , a2 , . . . , ap ,ml,mlbl ]T . (38)

Thus, the same procedure as in Case I can be used to estimate thephysical parameter vector θl , and the estimates θl converge totheir true values when certain PE condition like (36) is satisfied.The details are omitted as they are identical to those in Case I.

Case III: This case consist of all the measurement data duringthe time intervals when |wd(t)| ≤ H . Since the unknown dead-zone nonlinearity cannot be linearly parameterized in this case,no on-line parameter estimation would be used in this case.

Overall, parameter estimations in Cases I and II enableall unknown physical parameters, i.e., θ = [a1 , a2 , . . . , ap ,mr ,mrbr ,ml,mlbl ]T , to be accurately estimated, provided thatPE conditions like (36) are satisfied.

C. Asymptotic Output Tracking

With the accurate parameter estimates obtained in Section III-B, in addition to the results stated in Theorem 1, an improvedsteady-state tracking performance is also obtained as summa-rized by the following theorem.

Theorem 2: In the presence of parametric uncertainties andunknown dead-zone nonlinearity only [i.e., assuming fu = 0 in(1)], when accurate parameter estimation is obtained using theconditional parameter adaptation with explicit on-line monitor-ing of signal excitation levels, the proposed control law (19) with



Fig. 3. A linear motor drive system.

the dead-zone inverse (7) achieves asymptotic output trackingas well, i.e., X(t) → 0 and s → 0 as t → ∞.

Proof: See Appendix B. �

IV. COMPARATIVE EXPERIMENTS

The proposed DIARC control algorithm is implemented ona precision HIWIN stage driven by linear motors shown inFig. 3 with additional 4.82-kg payload. Neglecting fast electricaldynamics and flexible modes of the mechanical systems, thedynamics of the linear motor preceded by simulated unknowndead zones can be described as follows:

x = Ew(t) − Bx − ASf (x) + fu

y = x(t), w(t) = D(v(t)) (39)

where x represents the position of the inertia load, v(t) is theinput to the unknown dead zone, w(t) represents the controlinput voltage to the motor, y(t) is the position of the motor,E is the motor input gain, B is the equivalent viscous frictioncoefficient, ASf (x) represents the nonlinear Coulomb friction,in which the amplitude A may be unknown but the continuousshape function Sf (x) is known, and fu represents the lumpeduncertain nonlinearities including various model approximationerrors and external disturbances. Noting that the aforementionedlinear motor dynamics is in the form of (1) with p = 2, a1 =B, a2 = A, and Y1 = −x, Y2 = −Sf (x).

The position sensor of the stage is a linear encoder with aresolution of 0.5μm after quadrature. The velocity signal isobtained by the difference of two consecutive position mea-surements. Standard off-line least-squares identification is per-formed to obtain the parameters of the linear motor, and the iden-tified values of the parameters are B = 2.2/s, A = 0.6/s, E =2.4 m/s2/V, fN = 0 m/s2 , where fN is the nominal value offu . All the control algorithms are implemented using a dSPACEDS1103 controller board. The controller executes programs ata sampling period of Ts = 0.2 ms, resulting in a velocity mea-surement resolution of 0.0025 m/s.

The control objective is to let the system state [x, x]T followtheir desired trajectory [xd, xd ]T . Assuming

xd = 0.21 + 0.08sin(πt − π/2)

+ 0.07sin(0.8πt − π/2) + 0.06sin(1.2πt − π/2). (40)

Initial values of plant states are set as X(0) = [0, 0]T .

A. Comparative Experiments I

The experiments are first conducted assuming that w(t) is theoutput of a simulated dead zone described by

w = D(v) =

⎧⎨

⎩

0.9(v − 1.2), for v ≥ 1.20, for − 1 < v < 1.21(v − (−1)), for v ≤ −1.

(41)Taking into account the motor gain of E = 2.4 m/s2/V,the parameter vector θ in Section II is then given by θ =[B, A, 0.9E, 0.9 × 1.2E, E, −E]T , which has a value of[2.2, 0.6, 2.16, 2.592, 2.4, −2.4]T with the aforementionedoff-line estimates of motor parameters. In the experiments, theactual dead-zone parameters are assumed to be unknown and aninitial value of θN = [1.5, 0.3, 1.5, 1.5, 1.5, −1.5]T is usedfor θ(0) to account for the unknown dead-zone parameters. Fur-thermore, the bounds of the parametric variations are assumedto be

θmin = [1.5, 0.3, 1.5, 1.5, 1.5, −3.5]T

θmax = [3, 1.3, 3.5, 3.5, 3.5, −1.5]T . (42)

To better illustrate the effectiveness of the proposed scheme,especially the dead-zone compensation, the following four con-trol algorithms for the system (39) are implemented and com-pared.C1: The proposed DIARC adapting the estimates of physical

parameters B, A, E, fN and ignoring the existence ofthe unknown dead zone (41). This case represents the sce-nario where the existence of dead zone is totally ignored.

C2: The proposed DIARC adapting the estimates of physicalparameters B, A, E, fN only and without updating thedead-zone parameter estimates. This case represents thescenario where the dead zone is compensated with theirnominal values and effects of the unknown dead zone areignored.

C3: The proposed DIARC presented in Sections II and III.C4: The proposed DIARC in which only B, A, E, fN are up-

dated based on the proposed on-line parameter adaptationalgorithm and the dead-zone (41) effect is compensatedby (7) with actual dead-zone parameter values assumingthe dead-zone parameters are known. Thus, this case rep-resents the ideal scenario where the dead-zone effect canbe completely compensated.

In C3, wds2 in (19) is given in Section III-B. Theoretically, weshould use the form of wds2 = −ks2s with ks2 being a nonlinearproportional feedback gain as given in (27) to satisfy the robustperformance requirement (25) globally. In implementation, alarge enough constant feedback gain ks2 is used instead to sim-plify the resulting control law. With such a simplification, thoughthe robust performance condition (25) may not be guaranteedglobally, the condition can still be satisfied in a large enoughworking range which might be acceptable to practical applica-tions as done in [27]. With this simplification, noting (19), wechoose wds = −kss in the experiments where ks represents the



TABLE IPERFORMANCE INDEXES OF COMPARATIVE EXPERIMENTS I AND II

Fig. 4. Output trajectory tracking of C1, C2, C3, and C4.

summation of ks1 and ks2 and is chosen as ks = 200. The con-stant λ in the filtered tracking error (16) is selected as λ = 200.The constant adaptation rate γ in (22) is set as (γ = 1000 andthe bound of dc is chosen to be dcM = 2. The initial values of allparameter estimates are θN = [1.5, 0.3, 1.5, 1.5, 1.5, −1.5]T

which are significantly different from their true values θ. Theinitial value of the adaptation matrix Γ in (34) is Γ(0) = 1000I6 ,and the forgetting factor is chosen as α = 0.1. The constant Hin Lemma 2 is set as H = 2. For a fair comparison, the con-troller parameters and the initial values in case C1, C2, and C4are chosen the same as in C3 when they have the same meaning(e.g., wds in C1, C2, and C4 are chosen the same as in C3). Theinitial values of the adaptation matrix Γ in (34) for C1, C2, andC4 is Γ(0) = 1000I4 for that the dead-zone parameters are notadapted; consequently, there are B, A, E, fN need to be esti-mated and their initial values are [1.5, 0.3, 1.5, 0]T . In C4, thedead zone (41) is known and can be completely compensatedby its inverse (7).

The experimental results in terms of all performance indexesafter running the gantry for one period are given in ExperimentI of Table I. Overall, C2, C3, and C4 achieve good steady-statetracking performance during movements that are better than C1.C3 and C4 achieve almost the same excellent tracking perfor-mances that are better than C2. The experimental results forall four controllers are shown in Figs. 4–8. Specifically, Fig. 4shows the output tracking performance of all four cases, andFig. 5 shows the corresponding tracking errors. From theseplots, it is observed that all four cases have almost the samegood initial output-tracking transient performance, illustratingthe strong robust transient performance of the proposed DIARCstrategy in general. It is also seen that steady-state output track-ing errors of C3 and C4 are much less than that of C2. This resultagrees with the prediction by theory in Section III that the pro-

Fig. 5. Output tracking errors of C1, C2, C3, and C4.

Fig. 6. Parameter estimates of C3.

posed DIARC in C3 is able to achieve asymptotic convergenceof dead-zone parameter estimates to their true values and theperfect dead-zone compensation of the proposed dead-zoneinverse (7) when the estimates converge. This is also re-vealed by history of the system parameter estimates inFig. 6. Furthermore, C2, C3, and C4 have much better out-put tracking performance than C1, which illustrates the ne-cessity of dead-zone compensation. The estimate values ofB = 2.25, A = 0.6, E = 2.65 shown in Fig. 7 are close totheir off-line identified values, and the estimate values ofθ = [2.15, 0.7, 2, 2.1, 2.2, −1.9]T shown in Fig. 6 near theapproximate accurate value of θ, which verify the asymptoticconvergence of system parameter estimates as well. The con-trol inputs of all four cases shown in Fig. 8 reveal that thecontrol input may exhibit some chattering phenomena duringthe periods when the motion is very small and the system isalmost operating within the dead-zone region due to the useof the nonsmooth direct dead-zone inverse (7). In summary, thegood output tracking performance of all four cases demonstratesthe robust performance of the proposed DIARC algorithm. Themuch improved output tracking performance of C2, C3, and C4over C1 illustrates the necessity of dead-zone compensation.



Fig. 7. Parameter estimates of C4.

Fig. 8. Control input history of C1, C2, C3, and C4.

The much improved steady-state tracking performance ofcase C3 over C2 validates the need for accurate on-linedead-zone parameter estimates. And the almost same asymp-totically converging steady-state output tracking performanceof cases C3 and C4 verifies the asymptotic convergence of thedead-zone parameter estimates to their true values, and furthervalidates the perfect compensation of unknown dead zones bythe proposed DIARC algorithm.

It is noted that the proposed strategy has also been combinedwith the desired compensation-type ARC designs [22], [28],[29] for the precision motion control of linear motors in [23]. Thecomparative experimental results obtained on a linear motor-driven industrial gantry by Anorad (HERC-510-510-AA1-B-CC2) showed the same trends as in the previous experiments,revealing the performance consistencies of the proposed strategyin actual applications.

B. Comparative Experiments II

To further illustrate the effectiveness and merits of the pro-posed scheme, the robust adaptive control algorithm proposedin [18], a typical example of the previously published algo-rithms for systems with nonsmooth nonlinearities such as dead

Fig. 9. Output tracking errors of C5 and C3.

zones, is also implemented and compared with the proposed one.As the control algorithm in [18] is designed for systems withsymmetric dead zones only, in the following, the simulated deadzone is set as equal slope and described by

w = D(v) =

⎧⎨

⎩

2.16v − 2.592, for v ≥ 1.20, for − 1 < v < 1.22.16v + 2.16, for v ≤ −1

(43)which leads to θ = [2.2, 0.6, 2.16, 2.592, 2.16, −2.16]T .The bounds of the parametric variations are assumed to be thesame as those in the previous section.

The proposed DIARC presented in Sections II and III andthe robust adaptive control algorithm presented in [18] are thenimplemented and compared. They are referred to as C3 and C5 inthe following, respectively. For the proposed DIARC algorithm(or C3), the control parameters are the same as those in theprevious section with the initial values of all parameter estimatesset as θN = [1.5, 0.3, 1.5, 1.5, 1.5, −1.5]T . For case C5, therobust adaptive control law is designed following exactly thesame procedure as that in [18]

v = ewd

wd = xd(t) − c1 z1 − ϕθp − c2z2 − z1 − sgn(z2)D

˙e = −βwdz2 ,˙θp = ΓϕT z2 ,

˙D = μ|z2 | (44)

where z1 = x(t) and z2 = z1 + c1z1 ; c1 = 100, c2 = 50, μ =1000, Γ = 10I2 , and β = 10. The initial values are e(0) =1/1.5, D(0) = 0, θp(0) = [1.5, 0.3]T , and v(0) = 0.

The experimental results in terms of all performance indexesafter running the gantry for one period are given in ExperimentII of Table I. It can be seen from these results that the proposedDIARC algorithm outperforms the robust adaptive control al-gorithm in [18] significantly, both in terms of the steady-statetracking performance and the transient tracking errors. The ex-perimental results for the two controllers are also shown in Fig. 9in terms of the output tracking performance and in Fig. 10 forthe control inputs. It is seen from Fig. 9 that the steady-stateoutput tracking error of the proposed DIARC is mostly less



Fig. 10. Control input history of C5 and C3.

than 20 μm and around 0, much less than that of C5. This resultverifies the asymptotical output tracking performance of theproposed DIARC in the presence of parametric uncertaintiesand unknown dead zones. The control inputs shown in Fig. 10reveal that the robust adaptive control algorithm in [18] exhibitsthe control input chattering phenomena during the entire motiondue to the use of the discontinuous control action [the last termin wd in (44)], while the proposed DIARC only has the chat-tering input during the periods when the motion is very smalland the system is essentially operating at the dead-zone regiononly. All these results further validate the effectiveness and thehigh-performance nature of the proposed DIARC algorithm inpractical applications.

V. CONCLUSION

In this paper, an integrated DIARC scheme has been devel-oped for a class of single-input-single-output uncertain nonlin-ear systems preceded by nonsymmetrical and nonequal slopedead-zone nonlinearity. By making full use of the dead-zonecharacteristics that it can be perfectly linearly parameterizedwithin certain known working ranges, the proposed controlleruses indirect parameter estimation algorithms with on-line con-dition monitoring for an accurate estimation of the unknowndead-zone nonlinearity. With these accurate estimates of dead-zone parameters, perfect asymptotic dead-zone compensation isthen constructed and employed in the development of an inte-grated DIARC algorithm for the overall system. Consequently,asymptotic output tracking has been achieved even in the pres-ence of unknown dead-zone nonlinearity. Furthermore, regard-less of the convergence of the on-line parameter estimates totheir true values, the proposed algorithm also achieves certainguaranteed robust transient performance and steady-state track-ing accuracy even when the overall system may be subjected toother uncertain nonlinearities and time-varying disturbances aswell. Comparative experimental results have been obtained ona linear motor drive system preceded by a simulated unknowndead-zone nonlinearity. The results validate the effectivenessof the proposed dead-zone compensation scheme. The excel-lent output tracking performances obtained in the experiments

also verify the high-performance nature of the proposed DIARCstrategy in actual applications.

APPENDIX A

Proof of Theorem 1: Noting (24) and (25), the derivative ofV = 1

2 s2 is

V = k�wds1s + s[k�wds2 − dc + (1 − k�)dc + �(t)]

≤ −k�ks1s2 + ε + εdf

2d

≤ −λv

2s2 + ε + εdf

2d . (45)

Thus

V ≤ −λvV + ε + εdf2d . (46)

Using the Comparison Lemma, (28) is true. The theorem can,then, be proved by noting that all on-line parameter estimates areguaranteed to be bounded regardless of the estimation functionto be used shown in P1. �

APPENDIX B

Proof of Theorem 2: When the PE conditions are satisfied,θ ∈ L2 and θ(t) → 0 as t → ∞. Thus, noting Lemma 1, wehave d ∈ L2 as well. Noting (19), (20), and (23), the derivativeof V = 1

2 s2 when fu = 0 is

V = k�wdss − dcs + d(t)s + ξs (47)

where

ξ=[(

mr

mrdc −

mr

mrwda1

)κ+(wd) +

(ml

mldc −

ml

mlwda1

)

κ−(wd) −p∑

i=1

aiYi(x(t), x(t), . . . , x(n−1)(t))

]

+(

˜(mrbr ) − mr

(mrbr )mr

)κ+(wd)

+(

˜(mlbl) − ml

(mlbl)ml

)κ−(wd).

(48)

Choose a positive-definite function as

Va = V +12γ

dc2. (49)

Then, the derivative of Va is

Va = V + γ−1 dc˙dc

= k�wdss + d(t)s + ξs + (γ−1 Projdc

(γs) − s)dc

≤ −k�ks1s2 + k�wds2s + d(t)s + ξs (50)

where P2 of the projection mapping [i.e., (12)] is used in derivingthe inequality of the last step. As ξ defined by (48) is linearw.r.t. to the parameter estimation error θ with all coefficientsbeing uniformly bounded by Theorem 1, the fact that θ ∈ L2



implies that ξ ∈ L2 . From condition 1 of (25), k�wds2s ≤ 0.Thus, (50) implies that s ∈ L2 as ξ ∈ L2 and d(t) ∈ L2 . It iseasy to verify that s is uniformly continuous. Thus, by Barbalat’slemma, s → 0 as t → ∞. Noting (17), X(t) → 0 as t → ∞,which completes the proof of the theorem. �

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[16] S. Shahruz, “Performance enhancement of a class of nonlinear systems bydisturbance observers,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 3,pp. 319–323, Sep. 2000.

[17] J. Yi, S. Chang, and Y. Shen, “Disturbance-observer-based hysteresis com-pensation for piezoelectric actuators,” IEEE/ASME Trans. Mechatronics,vol. 14, no. 4, pp. 456–464, Aug. 2009.

[18] J. Zhou, C. Wen, and Y. Zhang, “Adaptive backstepping control of a classof uncertain nonlinear systems with unknown backlash-like hysteresis,”IEEE Trans. Autom. Control, vol. 49, no. 10, pp. 1751–1757, Oct. 2004.

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[21] C. Hu, Bin Yao, and Q. Wang, “Integrated direct/indirect adaptive robustcontouring control of a biaxial gantry with accurate Parameter Estima-tions,” Automatica, vol. 46, no. 4, pp. 701–707, Apr. 2010.

[22] B. Yao, “Desired compensation adaptive robust control,” ASME J. Dyn.Syst., Meas., Control, vol. 131, no. 6, pp. 1–7, 2009.

[23] C. Hu, B. Yao, and Q. Wang, “Adaptive robust precision motion control ofsystems with unknown input dead-zones: A case study with comparativeexperiments,” IEEE Trans. Ind. Electron., vol. 58, no. 6, pp. 2454–2464,Jun. 2011.

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[27] C. Hu, Bin Yao, and Q. Wang, “Coordinated adaptive robust contouringcontroller design for an industrial biaxial precision gantry,” IEEE/ASMETrans. Mechatronics, vol. 15, no. 5, pp. 728–735, Oct. 2010.

[28] L. Lu, Z. Chen, B. Yao, and Q. Wang, “Desired compensation adaptiverobust control of a linear motor driven precision industrial gantry with im-proved cogging force compensation,” IEEE/ASME Trans. Mechatronics,vol. 13, no. 6, pp. 617–624, Dec. 2008.

[29] B. Yao, C. Hu, and Q. Wang, “Adaptive robust precision motion control ofa high-speed industrial gantry with cogging force compensations,” IEEETrans. Control Syst. Technol., 2011, to be published. [Online]. Available:http://ieeexplore.ieee.org, DOI: 10.1109/TCST.2010.2070504

Chuxiong Hu (S’09–M’11) received the B.Eng. andPh.D. degrees in mechatronic control engineeringfrom Zhejiang University, Hangzhou, China, in 2005and 2010, respectively.

He is currently with the State Key Laboratoryof Tribology in the Department of Precision Instru-ments and Mechanology, Tsinghua University, Bei-jing, China. His research interests include coordi-nated motion control, precision mechatronics, adap-tive control, robust control, and nonlinear systems.

Bin Yao (S’92–M’96–SM’09) received the B.Eng.degree in applied mechanics from the Beijing Univer-sity of Aeronautics and Astronautics, Beijing, China,in 1987, the M.Eng. degree in electrical engineeringfrom the Nanyang Technological University, Singa-pore, in 1992, and the Ph.D. degree in mechanical en-gineering from the University of California at Berke-ley, in 1996.

He has been with the School of Mechanical Engi-neering, Purdue University, West Lafayette, IN, since1996, where he became a Professor in 2007.

Dr. Yao was honored as a Kuang-piu Professor in 2005 and a ChangjiangChair Professor at Zhejiang University by the Ministry of Education of Chinain 2010 as well. He was the recipient of an NSF CAREER Award in 1998, anNSFC Joint Research Fund for Outstanding Overseas Chinese Young Scholarsin 2005, the O. Hugo Schuck Best Paper (Theory) Award from the AmericanAutomatic Control Council in 2004, and the Outstanding Young InvestigatorAward of the ASME Dynamic Systems and Control Division (DSCD) in 2007.He has chaired numerous sessions and served on a number of InternationalProgram Committees of various IEEE, ASME, and IFAC conferences includingthe General Chair of the 2010 IEEE/ASME International Conference on Ad-vanced Intelligent Mechatronics. He was a Technical Editor of the IEEE/ASMETRANSACTIONS ON MECHATRONICS from 2001 to 2005 and Associate Editor ofthe ASME Journal of Dynamic Systems, Measurement, and Control from 2006to 2009.

Qingfeng Wang (M’11) received the M.Eng. andPh.D. degrees in mechanical engineering fromZhejiang University, Hangzhou, China, in 1988 and1994, respectively.

He then joined the faculty of Zhejiang Univer-sity, where he became a Professor in 1999. He wasthe Director of the State Key Laboratory of FluidPower Transmission and Control at Zhejiang Uni-versity from 2001 to 2005 and currently serves asthe Director of the Institute of Mechatronic ControlEngineering. His research interests include electro-

hydraulic control components and systems, hybrid power systems and energy-saving techniques for construction machinery, and system synthesis for mecha-tronic equipment.

ieee/asme transactions on mechatronics 1 …byao/papers/tmech12_deadzone.pdfinversion of the dead...

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