682 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 4,...

682 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 4, AUGUST 2011

Extraction and Adaptation of Fuzzy Rules forFriction Modeling and Control Compensation

Yongfu Wang, Dianhui Wang, Senior Member, IEEE, and Tianyou Chai, Fellow, IEEE

Abstract—Modeling of friction forces has been a challenging taskin mechanical engineering. Parameterized approaches for model-ing friction find it difficult to achieve satisfactory performance dueto the presence of nonlinearity and uncertainties in dynamical sys-tems. This paper aims to develop adaptive fuzzy friction modelsby the use of data-mining techniques and system theory. Our maintechnical contributions are twofold: extraction of fuzzy rules andformulation of a static fuzzy friction model and adaptation of thefuzzy friction model by the use of the Lyapunov stability theory,which is associated with a control compensation of a typical motiondynamics. The proposed framework in this paper shows a success-ful application of adaptive data-mining techniques in engineering.A single-degree-of-freedom mechanical system is employed as anexperimental model in simulation studies. Results demonstrate thatour proposed fuzzy friction model has promise in the design of un-certain mechanical control systems.

Index Terms—Adaptation, control compensation, data mining,extraction, friction modeling, fuzzy rules.

I. INTRODUCTION

THE characteristics of friction force that exist in engineeringsystems depend on pressure, slip velocity, surface, and

material properties. Such a physical phenomenon may resultin some negative effects on system performance, for instance,self-excited vibration, steady-state limit cycle, poor trackingperformance, and friction-induced instabilities [1]. Over the pastfew decades, some parameterized friction models, which includeboth static and dynamic, have been proposed by researchers[2]–[8]. Although the existing friction models work well forsome cases, they have a limitation in that they can only be usedto characterize the friction force in a unified framework. This

Manuscript received June 8, 2010; revised November 6, 2010; acceptedJanuary 18, 2011. Date of publication April 5, 2011; date of current versionAugust 8, 2011. This work was supported by the National Natural Science Foun-dation of China under Grant 50875042, Grant 60821063, and Grant 50875039.

Y. F. Wang is with the School of Mechanical Engineering and Automa-tion and the State Key Laboratory of Integrated Automation of Process Indus-tries, Ministry of Education, Northeastern University, Shenyang 110004, China(e-mail: [email protected]).

D. H. Wang is with the Department of Computer Science and ComputerEngineering, La Trobe University, Melbourne, Victoria 3086, Australia, andalso with the State Key Laboratory of Integrated Automation of Process Indus-tries, Ministry of Education, Northeastern University, Shenyang 110004, China(e-mail: [email protected]).

T. Y. Chai is with the State Key Laboratory of Integrated Automation ofProcess Industry, Ministry of Education, Northeastern University, Shenyang110004, 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/TFUZZ.2011.2134104

drawback forces the control compensation to perform poorlybecause of the utilization of a mismatched friction model incontroller design [9].

In motion-control systems, complex interactions between thesurface and near-surface regions of the two interacting mate-rials result in friction, which can cause steady-state trackingerrors to increase or even result in oscillations. For the im-provement of the control performance, it is critical to add afriction-compensation term in controller design. Thus, effectivemodeling of the friction force becomes a key for the achievementof satisfactory performance in motion control. It has been exper-imentally verified that the friction force is a nonlinear functionof the velocity. From an analysis of these existing friction mod-els, we can see that the mathematical approach finds it difficultto deal with the problem of universal friction modeling due tothe nonlinearity, uncertainty, and time-varying nature of friction.Therefore, it is significant to explore data-driven approaches tomodel the friction force with an adaptation mechanism.

Recently, fuzzy modeling and control techniques have beensuccessfully applied to complex systems [10]–[18], where tra-ditional approaches can rarely achieve satisfactory results dueto the nonlinearity, uncertainty, and lack of sufficient domainknowledge. Adaptive fuzzy systems have attracted consider-able attention from the control-engineering community due totheir universal approximation power as regards nonlinear maps,learning capability, domain knowledge embedability, and resultinterpretation ability. The main merit of the use of adaptive fuzzysystems for modeling is that we can naturally integrate both nu-merical data and domain knowledge in a unified framework.The key step in fuzzy modeling is to generate an appropriatefuzzy rulebase by the use of the collected data, which can beconsidered as a data-mining task. These data-driven fuzzy rule-bases are not usually formed in an optimal fashion due to theuncertainties of fuzzy-subset assignment. Therefore, it is nec-essary to adjust the extracted fuzzy rules according to a controlperformance. This adaptation is important because we need theestimated friction model to adapt to changes in the environmentso that the tracking performance will be retained during thecontrol process.

Previous studies on extraction of fuzzy rules from numericaldata have offered solutions to build fuzzy systems for variousengineering applications [19], [20]. However, to date, there islimited work that addresses the adaptation issue of fuzzy rules.In [21], the author pointed out that the adaptation of fuzzy rulesextracted from raw data is needed, and this should be done byusing the system’s performance. In our paper, we improve adata-mining algorithm for extraction of fuzzy rules [22], whichforms the basis of modeling a static fuzzy friction model. An

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WANG et al.: EXTRACTION AND ADAPTATION OF FUZZY RULES FOR FRICTION MODELING AND CONTROL COMPENSATION 683

updating rule for the adjustment of the fuzzy friction modelis proposed by the use of the Lyapunov stability theory. Theadaptive fuzzy friction model obtained as a result is used as acompensation term in controller design, where the proportional–derivative (PD) type of controller is employed. It is shown thatsuch an adaptation of the fuzzy friction model can guarantee thecontroller performance.

The remainder of the paper is organized as follows.Section II reviews some classical friction models with discus-sions on soft-sensor-based modeling techniques. Section III de-scribes an improved data-mining algorithm for fuzzy-rule ex-traction and details the way to obtain the static fuzzy model forfriction force. Section IV proposes an updating rule that helpstune the parameters of the static fuzzy friction model accord-ing to the Lyapunov stability theory. The proposed method formodeling adaptive fuzzy friction models is associated with ahybrid control scheme, which is composed of the PD controllerand adaptive fuzzy friction compensation term. A theoreticalanalysis on the bound of tracking errors of the closed-loop sys-tem is given as well. Section V reports simulation results ona one-dimensional motion dynamics of a mass that moves ona surface with friction and a real-time motion control experi-ment to illustrate the effectiveness of our proposed modelingand control techniques. Section VI concludes this paper.

II. REVIEW OF FRICTION MODELS

In general, friction models can be divided into two regimes:static-friction models and dynamic-friction models. This sectiongives a brief review on these friction models and indicates someshortcomings of the existing models for control compensation.

A. Static-Friction Models

The classical friction models are static, which means thatthe friction is modeled as a static function of velocity. In thestatic models, three components are normally considered: staticfriction or stiction, viscous friction, and Coulomb friction. Thetotal friction force can be calculated as follows:

F (υ) = Fc + (Fs − Fc)e−|υ/υs |2 + Fυυ (1)

where Fc is the Coulomb-friction coefficient, Fs is the static-friction coefficient, v is the stem velocity, υs is called theStribeck velocity, and Fv is the viscous-friction coefficient.Some classical static-friction models are shown in Fig. 1

B. Dynamic-Friction Models

Recently, there has been a great deal of interest in the devel-opment of dynamic-friction models. Dynamic-friction modelsmainly include the LuGre model and the Bristle model.

The following dynamic LuGre friction-force model that de-scribes the friction effects more accurately and addresses lowvelocities and velocity reversals was proposed by Canudas andOlsson [2]:

⎧⎪⎨

⎪⎩

F = σ0z + σ1 z + σ2υ

z = υ − |υ|z/g(υ)

σ0g(υ) = Fc + (Fs − Fc)e−|υ/υs |2(2)

Fig. 1. Classical static-friction models.

where z is the average deflection of the bristles, υ is the relativevelocity between the two surfaces, g(υ) is the positive function,σ0 and σ2 are contact stiffness and viscous friction parameters,σ1 is the damping coefficient, and F is the friction force. TheLuGre model is widely applied and accepted for its simplicityand effectiveness. However, the LuGre model fails to character-ize the hysteresis nonlocal memory behavior of friction force inthe presliding regime.

The Bristle friction model attempted to capture the behaviorof the microscopical contact points between two surfaces [23].The force is given by

F =N∑

i=1

σ0(xi − bi) (3)

where N is the number of bristles, σ0 is the stiffness of thebristles, xi is the relative position of the bristles, and bi is the lo-cation where the bond is formed. However, this model is difficultto use in practice due to its complexity.

Several points related to the friction models discussed earlierare summarized as follows.

1) Many kinds of friction models are available; however,there is no unified framework. In practice, it is difficult todetermine which type of friction model is more suitablefor a specific task.

2) Although simplified friction models can be easily imple-mented for control compensation, it is hard to guaranteethe tracking accuracy of control systems. While more so-phisticated friction models can result in better control per-formances, they are more difficult to implement due to theoptimization of model parameters.

3) Theoretically, it is difficult to employ traditional frictionmodels in controller design and stability analysis of theclosed-loop systems.

From the viewpoint of intelligent modeling and control, soft-sensor-based techniques have good potential to compensate forthe effects of friction in motion-control systems. The soft sen-sors can be employed for the estimation of hard-to-measure


Fig. 2. Structure diagram of acquisition data.

system variables from the available data. Then, the estimatedvariables will be placed in the controller’s design. In this way,soft computing techniques such as neural networks and adaptivefuzzy systems have demonstrated strong potential for frictionmodeling and control compensation [9], [24]–[26].

III. STATIC FUZZY MODELING FOR FRICTION

Recently, fuzzy-modeling techniques have been successfullyapplied to modeling complex systems, where traditional ap-proaches rarely reach a satisfactory solution. The fuzzy systemproposed in [27] has attracted interest due to its universal ap-proximation capability and effective learning algorithm. Thekey step to successfully build a fuzzy system is to extract aset of fuzzy rules from expert knowledge or numerical data.This section proposes an improved data-mining algorithm toextract a fuzzy rulebase, which forms the basis of fuzzy frictionmodeling.

A. Acquisition of Friction Data

To model the friction force by the use of fuzzy systems, weneed to collect some experimental data and employ our im-proved data-mining algorithm to extract a set of fuzzy rules.Fig. 2 depicts the block diagram and the experimental compo-nents, which include a PC, linear servo motor, motor driver,position encoder, PC104+ card, and motion-control card. Therod-source device used as the attenuation scan is a component ofmedical equipment Positron Emission Tomography (PET) [28].

In this paper, we consider that the friction force is a functionof the velocity. Thus, some data pairs composed of the veloc-ity and the friction values for a moving object were generatedby the experimental system in Fig. 2. The velocity informationwas obtained by the use of the well-known M/T (where symbolM is the unit of length and symbol T is the time unit) methodbased on the encoder signals. However, the friction force valuethat corresponds to a specific velocity could not be evaluateddirectly. Thus, an indirect method was adopted to solve thisproblem. It is known that the control force is equal to the fric-tion force while the object moves at a constant rate. Therefore,we acquired the friction force information through the so-calledconstant velocity test [1]. Three experiments were run for eachparticular constant velocity in order to obtain an accurate fric-

Fig. 3. Friction force data measured.

tion measurement. The final friction force values took the meanvalues of the three experiments. The results are plotted in aforce–velocity graph (see Fig. 3), where the cross points rep-resent the average values of the friction force measure at thesampling velocity points.

B. Extraction of Fuzzy Rules for Friction Modeling

Given a set of velocity–force data pairs:

(x(p) , F (p)), p = 1, 2, . . . , N (4)

where x ∈ R is the velocity, and F ∈ R is the friction force.The task here is to generate a set of fuzzy rules from the giveninput–output pairs of (4) and use these fuzzy rules to determinea mapping f : x → F . Our proposed approach consists of thefollowing steps.

Step 1 (Divide the input and output spaces into fuzzy regions):Assume that the domain intervals of x and F are [x−, x+] and[F−, F+], respectively, where the domain interval of a variablemeans that, most probably, this variable will lie in this range.Divide each domain interval into ci regions (ci can be differentfor different variables, and the lengths of these regions canbe equal or unequal). The sets of linguistic labels are denotedby A(x) = {Ax

1 , . . . , Axc1} and B(F ) = {BF

1 , . . . , BFc0}, where

each linguistic label is associated with a fuzzy membershipfunction.

Step 2 (Convert ordinary records into fuzzy records): Theinput–output data pairs obtained are stored in relationaldatabases. The data in relational databases are stored in a ta-ble, where each row is a record and each column represents oneof the attributes of the records. Let L1 = {x, F} be a set ofattributes and tp be the pth record with certain attribute values.A set of records associated with attribute L1 is denoted by TL1 ,i.e.,

TL1 = {t1 , . . . , tp , . . . , tN }. (5)

The fuzzified tp is written as μ(tp), which is associated withmembership values of the set of attributes L2 = {A(x) ∪B(F )}. A set of fuzzified records associated with attribute L2is denoted by TL2 , i.e.,

TL2 = {μ(t1), . . . , μ(tp), . . . , μ(tN )}. (6)


TABLE IORDINARY RECORDS AND FUZZY RECORDS

An illustrative toy example is given in Table I.Step 3 (Calculate the degree of support): From a data-mining

perspective, the degree of support is the percentage of records,where the rule holds. If a fuzzy rule has practical meaning, itmust have a large-enough degree of support from sample data.Therefore, the degree of support for a specific fuzzy input spaceis a good indicator for extraction of fuzzy rules from numericaldata. The degree of support for a fuzzy rule is defined as follows:

Supp(x ⇒ F ) =

∑Np=1 μ(B F

l 0)p (F )μ(Ax

l 1)p (x)

∑Np=1 μ(Ax

l 1)p (x)

(7)

where μ(Axl 1

)p (x) and μ(B Fl 0

)p (F ) are the values of membership

functions for the pth record, respectively, and N is the total num-ber of records in TL2 , l0 ∈ {1, . . . , c0} and l1 ∈ {1, . . . , c1}.

For simplicity, we can use the following formula to calculatethe degree of support:

Supp(x ⇒ F ) =1N

N∑

p=1

μ(B Fl 0

)p (F )μ(Axl 1

)p (x). (8)

Step 4 (Generate a fuzzy rulebase by the use of the improveddata-mining algorithm): Details of the improved data-miningalgorithm with pseudocode can be found in Appendix B. We usea single-input–single-output example to show the process for afuzzy-rulebase generation. Suppose A = {Ax

1 , . . . , Ax5 } is a set

of linguistic labels for attribute x and that B = {BF1 , . . . , BF

5 }is another set of linguistic labels for attribute F . For {Ax

1 },we first calculate the respective degrees of support for pairs{Ax

1 , BF1 }, {Ax

1 , BF2 }, {Ax

1 , BF3 }, {Ax

1 , BF4 }, and {Ax

1 , BF5 }.

Then, we select a fuzzy subspace with the maximum degree ofsupport for this column. By the repetition of this process for{Ax

2 }, {Ax3 }, {Ax

4 }, and {Ax5 }, we obtain the following fuzzy

rules.1) IF x is Ax

1 , THEN F is BF1 .

2) IF x is Ax2 , THEN F is BF

2 .3) IF x is Ax

3 , THEN F is BF4 .


3 .5) IF x is Ax

5 , THEN F is BF5 .

Fig. 4 shows the generated fuzzy rulebase.Step 5 (Calculate the degree of confidence):

Conf(x ⇒ F ) =Supp(x ⇒ F )

1N

∑Np=1 μ(Ax

l 1)p (x)

. (9)

The degree of confidence will be used to remove some trivialrules from the rulebase. This should be done if the numberof input variables or the number of its fuzzy subsets is large.

Fig. 4. Complete fuzzy rulebase.

Although this reduction may result in a worse approximationperformance, it is very necessary to make such a tradeoff inpractice.

Remark 1: Fuzzy inference systems are composed of a setof fuzzy rules and an inference algorithm to compute systemoutputs for given inputs. Fuzzy inference algorithms are stan-dard and usually have little impact on the system’s performance.Therefore, the success of fuzzy modeling relies heavily on thequality of fuzzy rulebase. The generation of fuzzy rules gener-ally involves two steps: structure identification of fuzzy rulesand parameter optimization of membership functions. Structureidentification mainly aims to define appropriate fuzzy partitionsfor each input variable and determine a set of IF–THEN rules.Structure identification methods can be grouped into two cate-gories: a priori knowledge-based approach and data-driven ap-proaches. In general, the first approach employs domain expertsto estimate the numbers of fuzzy sets of input/output variablesand then summarizes the IF–THEN rules. Many successful ap-plications show that this approach works favorably, although it istime-consuming, and some empirical studies have to be carriedout before the finalization of the fuzzy rulebase. Data-drivenapproaches employ data mining or computational-intelligencetechniques, such as clustering algorithms, neural networks, andgenetic algorithms [29]–[32], to determine the numbers of fuzzysets for the input and output variables from numerical data.

Two popular and widely used fuzzy models for nonlinear re-gression are the Takagi–Sugeno fuzzy model [33] and the fuzzybasis function model [27]. It has been proved that these fuzzymodels have universal approximation power as regards non-linear maps defined over compact sets. To identify the fuzzybasis function model, Wang and Mendel proposed a simple andpractical algorithm, termed the WM algorithm, for the extrac-tion of fuzzy rules from numerical data [19]. Their approachemploys a priori knowledge or expert experience to partitionthe input/output variables and assign their fuzzy sets. Later onin [20], Wang improved the WM algorithm (termed the iWMalgorithm) by the use of data-mining concepts. These two algo-rithms for fuzzy modeling have been highly cited and widelyused by researchers and engineers in various domains due totheir simplicity and effectiveness. A further study of these al-gorithms revealed that there is further opportunity to improvethe completeness and robustness of the fuzzy rulebase. To make


the fuzzy system more robust against input noise or outliers inmodeling tasks, we improved the iWM algorithm in our recentwork [22], where the sampling output values were replaced byits membership function values in order to compute the aver-age output values for each fuzzy rule (details can be seen inAppendix A). Comparative studies demonstrated that such a re-placement with the complete rulebase extraction method (seeAppendix A) can improve the robustness of the resulting fuzzysystems with respect to the corrupted or noisy input data. Inthis paper, we adopt Wang’s fuzzy basis function model to ap-proximate the nonlinear friction. In the light of the approachesproposed in [19] and [20], the fuzzy rules were extracted bythe use of our proposed algorithm in [22], with five fuzzy setsfor the input and output variables. The numbers of fuzzy setswere set empirically and this assignment resulted in satisfac-tory performance. Note that the parameters of the membershipfunctions of the input variable were fixed, and the parametersof the membership functions of the output variable were ad-justed according to Lyapunov theory to ensure the stability andconvergence of the resulting closed-loop system.

C. Static Fuzzy Friction Models

The fuzzy rulebase consists of a collection of fuzzy IF–THENrules:

R(j ) : IF is Axl1

THEN F is BFl0

(10)

where x ∈ R and F ∈ R are the input and the output of thefuzzy system, respectively; Ax

l1and BF

l0are the fuzzy subsets

defined over the input and the output variables, respectively; andl0 ∈ {1, . . . , c0}, l1 ∈ {1, . . . , c1}, and j = 1, . . . ,M .

By the application of the fuzzy-inference method in [10], wecan establish static fuzzy friction models as follows.

Definition 1: Given the fuzzy rulebase (10), a fuzzy frictionmodel is defined as

F (x) =

∑Mj=1 F jμ(Ax

l 1)j (x)

∑Mj=1 μ(Ax

l 1)j (x)

(11)

where F j is the point at which the fuzzy membership functionμ(B F

l 0)j (F ) achieves its maximum value.

The fuzzy basis function is denoted by

ξj (x) =μ(Ax

l 1)j (x)

∑Mj=1 μ(Ax

l 1)j (x)

, j = 1, 2, . . . ,M. (12)

Then, the static fuzzy friction model (11) can be rewritten as

F (x) = θT ξ(x) (13)

where θ = [F 1 , F 2 , . . . , FM ]T is a parameter vector, andξ(x) = [ξ1(x), ξ2(x), . . . , ξM (x)]T is a regressive vector.

IV. FUZZY FRICTION MODELS WITH ADAPTION MECHANISM

Note that the parameter vector θ = [F 1 , F 2 , . . . , FM ]T is aconstant vector specified by the designer. Such a static parame-ter of the fuzzy friction model may not provide a good estimateof the true friction force. Therefore, it may directly degrade the

control performance. To overcome this shortcoming, an adapta-tion mechanism to tune the static fuzzy friction model is neces-sary. This adaptation will also improve controller performanceas the friction force changes during the control process.

It should be pointed out that the development of updatingrules for online tuning of the fuzzy friction model is challeng-ing. This is because the model adaptation must be associatedwith the control performance of a specific plant. To addressthe adaptation mechanism, we employ the following motiondynamics of a mass, moving on a surface with friction [5]:

mx = u − F (14)

where m, x, F, and u represent the mass, the acceleration ofthe mass, the friction force, and the control force applied to themass, respectively.

Suppose that the function F in (14) is known. Then, thecontrol law

u = kpe + kd e + F (15)

can be applied to the nonlinear system (14) to obtain the follow-ing error dynamics:

me + kd e + kpe = mxd (16)

where e = (xd − x), e = (xd − x), e = (xd − x), and xd is thereference input. The first two terms on the right-hand side of(15) represent a PD controller, while the remaining depend onthe system dynamics.

This controller is perhaps the simplest and most popular con-trol scheme used to solve the problem of point-to-point motioncontrol. However, this controller does not provide exact track-ing for time-varying position, although it guarantees a boundedtracking error.

The results given in (15) and (16) are possible only when F isknown. In practice, the friction term F is very difficult to model.It is impossible to express accurate friction forces by the use of amathematical formula. In this situation, an approximation of thefriction term can be done by adaptive fuzzy-inference systems.We replace the friction term F in (15) by a fuzzy-inferencesystem (13), i.e.

F (x) = θT ξ(x). (17)

By the use of this estimate, we have the following control law:

u = kpe + kd e + F . (18)

By the application of (18) to (14), after some manipulations, weobtain the tracking error dynamic equation as follows:

me + kd e + kpe = Fg − F (19)

where Fg = mxd + F is the general function of friction or,equivalently

e + k2 e + k1e = m−1 [Fg − F ] (20)

where k2 = m−1kd , k1 = m−1kp .First, let us define the optimal parameter θ∗ as follows:

θ∗ = arg minθ∈Ωθ

[supx∈Ω x‖F − Fg‖] (21)


where Ωθ and Ωx denote the sets of suitable bounds on θ and x,respectively. The minimum approximation error is then definedas

ε = Fg − θ∗T ξ(x). (22)

Then, a general function Fg given in (19) can be modeled by afuzzy-inference system as follows:

Fg = θ∗T ξ(x) + ε. (23)

From (17) and (23), the tracking error (20) can be rewritten as

e + k2 e + k1e = m−1 [θTξ(x) + ε] (24)

where θ = θ∗ − θ. The aforementioned equation is equivalentto the following:

e = Ae + Bu (25)

where

A =(

0 1−k1 −k2

)

, B =(

01

)

, u = m−1 [θTξ(x) + ε].

The adaptive fuzzy friction model is associated with a controlperformance and implemented through tune-in of the parametersin the output fuzzy subsets. An updating rule for adaptation offuzzy rules in terms of the consequent parameters is derived byusing the Lyapunov stability theory. The result is now stated asfollows.

Theorem 1: Let the following PD controller with the adap-tive control compensation term (i.e., a dynamic estimate of thefriction force) be applied to the nonlinear system (14):

u = kpe + kd e + F (26)

where

F = θT ξ(x). (27)

The following updating rule for the parameter vector θ ensuresthat the tracking error of the closed-loop system converges to abounded compact set:

θ =

⎧⎪⎨

⎪⎩

γξ(x)BT Pe, if(‖θ‖ < M) or

((‖θ‖ = M) and eT PBξT (x)θ ≤ 0)

P [·], if((‖θ‖ = M) and eT PBξT (x)θ > 0)

(28)

with

P [·] = γξ(x)BT Pe − γeT PBξT (x)θ

‖θ‖2 θ

where P [·] is the projection operator, and P = PT is the solutionof the following Lyapunov equation:

AT P + PA = −Q. (29)

Proof: Consider a Lyapunov function as follows:

V =12eT Pe +

m−1

2γθ

Tθ γ > 0. (30)

The derivative of V with respect to time is given by

V =12eT P e +

12eT Pe +

m−1

γθ

T ˙θ. (31)

Note that ˙θ = −θ, and by the use of (25), the aforementioned

equation becomes

V =12[eT (PA + AT P )e] + m−1(θξ(x) + ε)BT Pe

− m−1

γθ

Tθ. (32)

From the updating rule (28), we get

V = −12eT Qe + m−1εBT Pe. (33)

Therefore, we have

V ≤ −12‖e‖[λmin(Q)‖e‖ − 2m−1‖ε0‖λmax(P )]. (34)

V is negative as long as the following inequality holds:

‖e‖ >2‖ε0‖λmax(P )

mλmin(Q). (35)

According to the Lyapunov stability theory, the tracking errorwill continue to decrease if the norm of state error is bigger thanthe right-hand side of (35). This indeed gives a practical boundon the tracking errors, i.e.,

‖e‖ ≤ 2‖ε0‖λmax(P )mλmin(Q)

. (36)

Consequently, we conclude that all variables in the system arebounded and the tracking error e will be confined inside a com-pact set. This completes the proof.

Theorem 1 states that tracking error will eventually fall insidea compact set controlled by the modeling bound ε0 , parametermatrices P and Q, and the constant m. The zero tracking errorcan be achieved if the bound ε0 is equal to zero. Fig. 5 depictsthe flow chart of the overall control systems.

Remark 2: In this work, only a PD type of controller with acompensation term is considered. The reason behind this is thatour proposed modeling technique and controller design are ap-plied to a specific mechanical system with friction. Other typesof controllers may be employed to control more complicatedplants with friction. Further research in this direction includesthe incorporation of our proposed fuzzy friction model into var-ious controllers to improve control performance for dynamicsystems with friction. Also, it will be interesting to developmore advanced intelligent control systems by the combinationof our proposed techniques in this paper with a networked con-trol system design [18].

V. SIMULATION RESULTS

This section presents simulation results by the use of ourproposed adaptive data-mining approach for friction modelingand control compensation. The following motion-control systemis employed as a simulation plant:

mx = u − F (37)

where m, x, F, and u represent the mass, the acceleration ofthe mass, the friction force, and the control force applied to themass, respectively.


Fig. 5. Control process of the motion dynamics with friction.

TABLE IISAMPLE DATA FOR FRICTION MODELING

TABLE IIISAMPLE DATA STORED IN FUZZY RELATIONAL DATABASE

A. Parameter Setting

In this paper, F = dx is supposed to be the unknown frictionforce in our simulations. The parameters used in our simula-tions are specified as follows: m = 4/3 kg, x(0) = (0.0, 0.0)T ;reference signal xd = sin(t) and xd = cos(t); and kp = 200,kd = 20, k1 = 150, k2 = 15, and −40 < u < 40. For a givenparameter matrix

Q =(

6000 12001200 260

)

we solve (29) and obtain

P =(

600 2020 10

)

.

B. Extraction of Fuzzy Rules for Static Modeling

Step 1 (Definition of membership functions): We have oneinput linguistic variable x, which is the velocity of the mass, andone output variable F , which is the friction force. For the inputvariable x, five membership functions are assigned: μAx

1=

1/(1 + exp(2.5(x + 0.37))), μAx2

= exp(−5.0(x + 0.25)2),

μAx3

= exp(−5.0(x)2), μAx4

= exp(−5.0(x − 0.25)2), andμAx

5= 1/(1 + exp(−2.5(x − 0.37))). For the output variable

F , we use the triangle type of membership functions exceptfor the two end membership functions, which are in trapezoidforms (see Fig. 4). The following five sets of parameters specifythe five membership functions for the friction variable: [−20−10], [−20 −10 0], [−10 0 10], [0 10 20], and [10 20].

Step 2 (Sample data): First, we store the data available inSection III-A in ordinary relational databases. Let L1 = {x, F}be a set of attributes. For one record (row) tp , tp stands for thevalue of attribute (column) L1 . In order to present this methodin detail, we only use ten sample data to mine friction fuzzyrules in Table II.

Step 3. Extraction of fuzzy rules: In order to obtain fuzzyrules with a maximum degree of support, suppose A ={Ax

1 , . . . , Ax5 } is a set of linguistic labels for attribute x, and

B = {BF1 , . . . , BF

5 } is a set of linguistic labels for attribute F .We calculate all degrees of support by the use of (7) or (8), andthe fuzzified sample data in Table III. The results of the degreeof support are shown in Table IV. Based on the maximum degree


TABLE IVDEGREE OF SUPPORT

of support principle to assign the consequent of fuzzy rules, weobtain the following five fuzzy rules.


1 .2) IF x is Ax

2 , THEN F is BF1 .


2 or BF4 .


5 .5) IF x is Ax

5 , THEN F is BF5 .

Remark 3: In the case that there exist multiple largest degreesof support for the same fuzzy subspace (e.g., Ax

3 in Table IV), thefollowing strategy will be used to determine the correspondingconsequent part of the fuzzy rule: 1) Take the average valueof the centers of the fuzzy subsets with the largest degree ofsupport and 2) define a fuzzy subset as the consequent of thefuzzy rule if its center has minimum distance to the averagevalue of the centers. For instance, we will select BF

3 as theconsequent for the third rule, and in the static fuzzy frictionmodel (13), the parameter vector θ should be selected as θ =(−20,−20, 0, 20, 20).

C. Adaptation of Fuzzy Rules for Control Compensation

In simulations, we set γ = 500 and M = 25√

5. LetS =

∑5j=1 μAx

j(x) and ξ(x) = 1

S (μAx1(x), . . . , μAx

5(x))T . The

fuzzy friction model and the updating rule are given as follows:

F = θT ξ(x) (38)

θ =

⎧⎪⎨

⎪⎩

γξ(x)(20e + 10e), if (‖θ‖ < M) or

((‖θ‖ ≥ M) and eT PBξT (x)θ ≤ 0)

P [·], if ((‖θ‖ ≥ M) and eT PBξT (x)θ > 0)

(39)

with

P [·] = γξ(x)(20e + 10e) − γ(20e + 10e)ξT (x)θ‖θ‖2 θ.

Remark 4: It has been noted that the parameter setting in theupdating rule (39) is important, which can directly affect thecontrol performance. The empirical setting for the upper boundof ‖θ‖ is given by M = 1.25

√5θmax , where the θmax represents

the upper bound of sampling input data. Another parameter inthe updating rule (39) is the value of γ. It was observed fromour simulation studies that the control performance in terms oftracking errors can be better if a larger value of γ is used in theupdating rule (39). Furthermore, the updating rule described in(39) does not exactly match the original formula given in (28).Theoretically, the updating rule in (28) can ensure ‖θ‖ ≤ M .However, this inequality may not hold for some instants insimulations due to the sampling effect. Thus, we modified the

condition ‖θ‖ = M in (28) as ‖θ‖ ≥ M in (39). Simulationresults indicated that such an amendment works well.

To investigate the power of adaptation of the proposed fuzzyfriction model for control compensation, we intentionally setdifferent d values in the friction force model F = dx at threedifferent time slots, i.e., for 0 ≤ t ≤ 100, d = 40; for 100 < t ≤200, d = 35; and for 200 < t ≤ 300, d = 25.

For the purpose of performance comparison, the follow-ing PD controllers with compensation terms were used in oursimulations:

{u1 = kpe + kd e + F

u2 = kpe + kd e + F.(40)

To simulate the overall control system, we integrated the differ-ential equations of the closed-loop system (25) and the adap-tive law (39) by the use of the fourth/fifth-order Runge–Kuttamethod. The MATLAB ODE45 command with step size 0.005was used in our simulations.

To illustrate the effect of friction on control performance, aPD controller without a compensation term is applied to thesystem (37). Fig. 6(a) and (b) shows the simulation results ofthe tracking errors, which correspond to the system (37) withF = 0 and F = 40x, respectively. As can be seen, the effect onthe tracking performance is remarkable.

To see the power of adaptive control compensation and theeffectiveness of our proposed updating rule (39) for the improve-ment of the control performance, we applied the controllers u1and u2 to the dynamics (37) with an uncertain friction force.Fig. 7(a) and (b) depicts the simulation results of the trackingerrors e. The controller with the adaptive-control compensationterm outperforms the one with initial friction model compen-sation F = 40x. It can be seen that the proposed adaptationscheme of the fuzzy friction model works well, while some un-certainties on the friction force are present during the controlprocess. Fig. 8(a) and (b) shows the corresponding simulationresults of the tracking errors e.

From the trajectories depicted in Fig. 9, which correspondto the parameters θi, i = 1, 2, 3, 4, and 5, the adaptation ofthe parameters in the fuzzy friction model can be observed.In this simulation, the initial parameters were set as θ =(−20,−20, 0, 20, 20), and they were adjusted online by the useof(39).

D. Motion-Control Experiment

The system employed in this experimental study is a rodsource device by dc motors associated with the control devices.The rod source device and configuration of our experimentsare given in Fig. 10. The system components involved in theexperiment include a PC, a dc motor, a motor driver, a posi-tion encoder, and a multipoint control unit (MCU) based onHCTL1100. The dc motor has the encoder that transfers themoving position signal to the controller.

The HCTL1100 is the most important part of the whole sys-tem, which operates as the controller. The HCTL1100 serieshave high performance and are well suited to motion-controltasks. The HCTL1100 receives the position signal and passes it


Fig. 6. Trajectory curve of tracking error e(t) by the use of the same controller. (a) System without friction. (b) System with friction.

Fig. 7. Trajectory curve of tracking error e(t) by the use of different controllers. (a) u = kp e + kd e + F . (b) u = kp e + kd e + F .

Fig. 8. Trajectory curve of tracking error e(t) by the use of different controllers. (a) u = kp e + kd e + F . (b) u = kp e + kd e + F .

onto the host computer by the MCU. The HCTL1100 executesany one of three setup routines or four control modes selected byend users. The three setup routines include: reset, initialization/idle, and align. The four internal control modes available toend users include: position control, proportional velocity con-trol, trapezoidal profile control, and integral velocity control.It is noteworthy that in our experiments the PWM register inthe idle mode for HCTL1100 initialization was directly used in

external control modes. Our experimental results are reported inTable V. The returned encoder values produced by the PD con-troller without fuzzy compensation were regulated to be closeto the desired value, which resulted in some fluctuations. Thereturned encoder values produced by the PD controller withfuzzy compensation were regulated as well but did not havethe fluctuations. In this experiment, we only used the medianbyte of encoder while high and low bytes were ignored. If high


Fig. 9 Timeresponses of five adaptive parameters for d = 40, 35, and 25.

Fig. 10 Experimentalsetup and photograph of rod source device.

TABLE VCONTROL RESULTS

and low bytes are used, the experimental results can be furtherimproved.

VI. CONCLUSION

This paper develops a framework of modeling friction forceby the use of adaptive data-mining techniques. We first employan improved data-mining algorithm for the extraction of fuzzyrules, which sets up a static fuzzy friction model. Based on the

well-known Lyapunov stability theory, an updating law for fuzzyfriction model parameter adjustment is proposed taking into ac-count typical motion dynamics. Simulation results and real-timecontrol experiments demonstrate that our proposed data-drivenapproaches for modeling and adaptive compensation controltechnique are effective and useful in the improvement of controlperformance.

It is believed that there exists a specific adaptation law fora concrete mechanical system with friction. However, it seems


difficult to derive a common adaptation law that is well suited forgeneral mechanical dynamics with friction. From what we havedone in this work, the well-known Lyapunov theory will play akey role to derive the adaptation law to update the parametersin fuzzy systems.

APPENDIX A

ALGORITHMS FOR FUZZY RULE EXTRACTION

1. Main Idea of the iWM Data-Mining Algorithm

The iWM algorithm takes the following steps for the extrac-tion of fuzzy rules from numerical data.

Step 1: For each input–output pair (x(p) , y(p)), p =1, 2, . . . , N , compute

w(p) =m∏

j=1

μAi j(x(p)

ij ) (41)

where Aij are the fuzzy subsets of the input linguistic variable.If

∏Np=1 w(p) = 0, there will be no fuzzy rule generated; other-

wise, take the value of w(p) as a weight for y(p) and computethe weighted average

av =

∑Np=1 y(p)w(p)

∑Np=1 w(p)

. (42)

Step 2: An IF–THEN rule is generated with a consequent Bdetermined by the following methods.

M1: Among the K fuzzy subsets B1 , . . . , BK defined in theoutput space, find the Bj∗ such that

μB j ∗(av) ≥ μB j (av) (43)

for j = 1, 2, . . . ,K. Then, the consequent fuzzy subset B ischosen as Bj∗.

M2: Compute the weighted “variance”

σ =

∑Np=1 |y(p) − av|w(p)

∑Np=1 w(p)

. (44)

A fuzzy subset B is then generated with its membership functionas follows:

μB (y) = μ(y; av, σ). (45)

2. Improved iWM Data-Mining Algorithm

For real-world applications, the sample data may containsome outliers caused by faults in measuring instruments or thenoisy environment. We found that in the case of outliers, theiWM algorithm did not perform robustly for modeling tasks.With analytical and numerical studies, a solution to improvethe robustness of the modeling performance with respect to thenoisy data or outliers is given in [22]. The idea behind the pro-posed solution is simple, i.e., to weaken the effect of the outliersor noisy data on the weighted average value in (42) by the re-placement of the membership function values of the outputs.We use the following formula to compute the weighted average:

avB j =

∑Np=1 μB j (y(p))w(p)

∑Np=1 w(p)

. (46)

Among the K fuzzy subsets B1 , . . . , BK in the output space,find the Bj∗ such that

avB j ∗ ≥ avB j (47)

for j = 1, 2, . . . ,K. Then, the consequent fuzzy subset B ischosen as Bj∗.

By the combination of this formula with the algorithm de-scribed in Section III-B for fuzzy rule extraction, we can obtaina complete set of IF–THEN fuzzy rules with better robustness.

APPENDIX B

PSEUDOCODE TO GENERATE COMPLETE FUZZY RULEBASE

Input: The inputs comprise the following specifications givenby the user:

1) a set of sampling data;2) user-specified number of region c1 for attribute x;3) user-specified number of region c0 for attribute F ;4) input N .Output: Generate a fuzzy rulebase with completeness and

robustness.The detailed description of the algorithm is given as follows:Step 1: InitializationDefine global variables j, Max-sup and Max-num.Step 2: Obtain fuzzy rules based on maximum degree of

supportfor(j = 1; c1 ; j + +) {

Call Step 3;Obtain rule[j] based on Max-num:

}end

Step 3: Scan all fuzzy sets and calculate membership valuesfor x

for(l1 = 1; c1 ; l1 + +) {Calculate μAx

l 1(x);

Call Step 4;}return

Step 4: Scan all fuzzy sets for F and calculate degree ofsupport

for(l0 = 1; c0 ; l0 + +) {Calculate μB F

l 0(F );

Use (7) or (8) to calculate:E[l0 ] = Sup(x ⇒ F );If (E[l0 ] > E[0]){

Max-sup = E[l0 ];Max-num = l0 ;E[0] = E[l0 ];

}}return

The computational complexity in terms of the BigO notationdepends on the number of input variables and the number offuzzy sets for each variable. It is straightforward to calculatethis measure as O(c0c

21).


ACKNOWLEDGMENT

The authors would like to express their sincere appreciationto anonymous reviewers for their constructive comments on thiswork.

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Yongfu Wang received the Ph.D. degree in controltheory and engineering from Northeastern University,Shenyang, China, in 2005.

From 2005 to 2007, he worked as a PostdoctoralFellow with Neusoft Group, China. He is currently anAssociate Professor with the School of MechanicalEngineering and Automation, Northeastern Univer-sity, where he is also associated with the Key Labo-ratory of Integrated Automation of Process Industry,Ministry of Education. He has been a Visiting Scholarwith Case Western Reserve University, Cleveland,

OH, from 2010 to 2011. His current research interests include intelligent con-trol of mechanical systems, data mining, and signal processing.

Dianhui Wang (SM’03) received the Ph.D. degreefrom Northeastern University, Shenyang, China, in1995.

From 1995 to 2001, he worked as a Postdoc-toral Fellow with Nanyang Technological University,Singapore, and a Researcher with The Hong KongPolytechnic University, Hong Kong, China. He is cur-rently an Associate Professor with the Departmentof Computer Science and Computer Engineering, LaTrobe University, Melbourne, Victoria, Australia. Heis also associated with the Key Laboratory of Inte-

grated Automation of Process Industry, Ministry of Education, NortheasternUniversity. His current research interests include data mining and computa-tional intelligence systems for bioinformatics and engineering applications.

Tianyou Chai (M’90–SM’97–F’08) received thePh.D. degree in control theory and engineering fromNortheastern University, Shenyang, China, in 1985.

He is currently with the Key Laboratory of Inte-grated Automation of Process Industry, Ministry ofEducation, Northeastern University. His current re-search interests include adaptive control, intelligentdecoupling control, and integrated automation of in-dustrial process.

Dr. Chai was elected as a member of the ChineseAcademy of Engineering in 2003, an Academician of

the International Eurasian Academy of Sciences in 2007, and an InternationalFederation of Automatic Control Fellow in 2008.

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