Active control of friction self-excited vibration using neuro-fuzzy and data mining techniques

Expert Systems with Applications 40 (2013) 975–983

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Active control of friction self-excited vibration using neuro-fuzzy and data mining techniques Y.F. Wang a,c,⇑, D.H. Wang b,c, T.Y. Chai c a

School of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 11004, China Department of Computer Science and Computer Engineering, La Trobe University, Melbourne, VIC 3086, Australia c State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning 11004, China b

a r t i c l e

i n f o

Keywords: Data mining Neuro-fuzzy systems Active control Friction Self-excited vibration

a b s t r a c t Vibration caused by friction, termed as friction-induced self-excited vibration (FSV), is harmful to engineering systems. Understanding this physical phenomenon and developing some strategies to effectively control the vibration have both theoretical and practical significance. This paper proposes a self-tuning active control scheme for controlling FSV in a class of mechanical systems. Our main technical contributions include: setup of a data mining based neuro-fuzzy system for modeling friction; learning algorithm for tuning the neuro-fuzzy system friction model using Lyapunov stability theory, which is associated with a compensation control scheme and guaranteed closed-loop system performance. A typical mechanical system with friction is employed in simulation studies. Results show that our proposed modeling and control techniques are effective to eliminate both the limit cycle and the steady-state error. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Friction-induced self-excited vibration (FSV) is a complex and nonlinear physical phenomenon with some uncertainties. Friction and vibration are almost ubiquitous in real life. Sometimes they can be beneficial to us under special circumstances. Such as, friction can be utilized in automotive brakes and vibration can be applied in nuclear magnetic resonance. However, friction usually causes degradation of system performances in most of the mechanical systems. In the case that the friction term critically impacts on mechanical dynamics, its presence may induce limit cycles, steady-state errors and other undesirable effects. In general, vibration generates additional dynamic loads to degrade the system performances. Thus, it is significant to reduce or eliminate vibration caused by friction force for performance improvement. From engineering viewpoints, it is meaningful to understand the FSV mechanism and develop effective control algorithms (Chatterjee, 2007; Das & Mallik, 2006; Sinou & Dereure, 2006). Recently, active control techniques have received considerable attention from mechanical and control engineers. These active control schemes have been widely applied for precision instrumentation, aerospace, transportation systems and mechanical engineering. In vibration control, active control schemes use sensors to measure the feedback signals, and generate control actions using some special ⇑ Corresponding author at: School of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 11004, China. E-mail address: [email protected] (Y.F. Wang). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.08.005

control strategies for driving the actuator to reduce or eliminate vibration. To eliminate or inhibit the FSV, it is necessary to introduce a friction compensation term in controller design. Therefore, effective modeling of the friction force play a key role to control the FSV in mechanical systems. It has been experimentally verified that the friction force is a nonlinear function of both the velocity and the direction of rotation or motion. Readers may refer to empirical models reported in the literature (Armstrong & Canudas De Wit, 1994; Bender, Lampaert, & Swevers, 2005; Canudas De Wit, Ollson, Astrom, & Lischinsky, 1995; Dupont, Hayward, Armstrong, & Altpeter, 2004; Kim & Ha, 2004; Rizos & Fassois, 2009; Swevers, Al-Bender, Ganseman, & Prajogo, 2000). From an analysis of these exiting friction models, we can see that the mathematical approach has difficulty in dealing with the problem of universal friction modeling due to the nonlinearity, uncertainty and time-varying nature of friction. Thus, it is useful to explore data-driven approaches for modeling the friction force with an adaptation mechanism. Recently, fuzzy systems and neural network systems have been successfully applied to complex systems (Jiang, Zhang, & Zhang, 2011; Rana, 2011; Selmic & Lewis, 2002; Wang, 1993; Wang, Wang, & Chai, 2009; Wu, Lin, & Lee, 2011), where traditional approaches can rarely achieve satisfactory results due to the nonlinearity, uncertainty and lack of sufficient domain knowledge. Neuro-fuzzy systems have attracted considerable attention in the past due to their universal approximation power to nonlinear maps, learning capability, domain knowledge embedability and

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governed by the following differential equation (Hinrichs, Oestreich, & Popp, 1998; Zjinjade & Mallik, 2007):

mx00 ðsÞ þ kxðsÞ ¼ F f ðv Þ þ uc ;

ð1Þ

where m is the mass of the block, x is the displacement of the mass, uc is the control signal, Ff(v) is the friction force, v is the relative velocity. qffiffiffi Let t ¼ x0 s; x0 ¼ mk and



x_ ¼ dx=dt ¼ x0 =x0 ; €x ¼ dðxÞ=dt _ ¼ x00 =x20 :

ð2Þ

System (1) can be rewritten as: Fig. 1. Mass on a moving belt system.

result interpretation ability (Figueiredo & Gomide, 1999; Jang, 1992). The main merit of neuro-fuzzy systems for engineering modeling is that we can naturally integrate both numerical data and domain knowledge in a unified framework. The key step in building neuro-fuzzy system is to determine the architecture of a system, which can be done by data mining techniques. Notice that the data-mining-based neuro-fuzzy inference system (DNFIS) are not constructed in an optimal fashion in terms of parameter setting. Therefore, it is important to develop learning algorithms for tuning the parameters (weights) of neuro-fuzzy inference system (ANFIS). Traditional learning techniques for learner models, such as the well-known error back- propagation algorithm and its variations, are derived from various numerical optimization techniques. Although some theoretical results on adaptive neural control can be read in literature, it is rare to find reports that associate the learning algorithm with control system’s performances. In this paper, we try to make a link between the learning algorithm of neuro-fuzzy system and the stability performance of a closed-loop dynamical system. Concretely, we employ an improved data mining algorithm (Wang, Wang, & Chai, 2010) to extract a set of fuzzy rules. Based on these generated fuzzy rules, a neuro-fuzzy system is constructed for approximate the unknown friction force. Then, an active control scheme, the proportionalderivative (PD) controller with a friction compensation term, is applied to control the dynamical system. To eliminate the limit cycle and the steady-state error caused by frictions in the systems, a updating rule for the weights of the neuro-fuzzy system is derived from Lyapunov stability theory. It is shown that such a learning algorithm can guarantee the control performance. The remainder of the paper is organized as follows: Section 2 gives some information on description of mechanical systems used in this study and some observations on numerical analysis of the FSV. Section 3 mainly describes a data-driven approach for modeling the friction force using neuro-fuzzy systems. Section 4 proposes an updating rule for tuning the weights of the neuro-fuzzy system according to the Lyapunov stability theory, which is associated with a PD control scheme with a friction compensation term. Section 5 reports simulation results on a one-dimensional motion dynamics of a mass which moves on a surface with friction to illustrate the effectiveness of our proposed neuro-fuzzy system modeling and active control techniques. Section 6 concludes this work.

€xðtÞ þ xðtÞ ¼ Fðv Þ þ u;

ð3Þ

_ v 0 is the velocity of the belt, F(v) = Ff(v)/k, where v ¼ v 0  x0 x, u = uc/k. Friction-induced vibration, a type of self-excited vibration, is a serious problem in many engineering systems. The friction force acting on the system provides the energy needed to maintain these vibrations. The nature of the friction force, dependent on the relative (slip) velocity, time, temperature, material properties, geometry and roughness of sliding surfaces, normal load etc., is really complex. Modeling of friction force and friction vibration has attracted the attention of both physicists and engineers. To understand the physical phenomenon of the friction vibration, some numerical simulations were carried out using two typical friction models, i.e., the Coulomb friction model and Stribeck friction model. The Coulomb friction model can be expressed as:

Fðv Þ ¼ F c sgnðtÞ;

ð4Þ

where the friction force Fc is proportional to the normal load, i.e., Fc = lFN. Notice that the model (4) is an ideal relay model. The Coulomb friction model does not specify the friction force for zero velocity. The Stribeck friction model describes the steady-state friction behavior in sliding regime and hence are dependent on the sliding velocity v. This friction model incorporates Coulomb, viscous, and Stribeck friction: ds

FðtÞ ¼ F c þ ðF s  F c Þejt=ts j þ F t t;

ð5Þ

where ts is called the Stribeck velocity and Fv is the viscous friction coefficient. For u = 0 one has the case of pure self excitation in (3). In our simulations, Matlab command (ode45) was used to obtain numerical solutions corresponding to these friction models. The following parameters were used in the simulations: m = 0.6 [kg], x0 = (0.02, 0.02), FN = 15.0 [N], v0 = 0.3 [m/s], k = 763 [N/m]. Fig. 2 depicts the Coulomb and the Stribeck friction-induced limit cycles with various parameters in (3)–(5). These numerical results demonstrate the physical phenomena of the friction vibration. To eliminate or inhibit the frictioninduced self-excited vibration, it is important to add a compensation term based on friction model in PD controller design. But the existing friction models are parameterized and will not be able to characterize accurately all types of friction under an unified framework. Therefore, it is very necessary to make efforts on developing data-driven-based intelligent approaches for modeling friction force and controlling the friction vibration.

2. Friction-induced self-excited vibration

3. Modeling friction force using neuro-fuzzy systems

The free body diagram of a block of mass m, placed on a moving belt and constrained by a spring of stiffness k, is shown in Fig. 1. The non-dimensional equation of motion of a single-degreeof-freedom undamped oscillator with the proposed control is

Modeling friction force from a collection of sampling data can be implemented by various learner models. Usually, there are three key steps towards to a successful modeling: data collection and filtering; learner model identification and model parameter

Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983

977

Fig. 2. Friction-induced limit cycles. Coulomb model: (a) Fc = 15, (b) Fc = 7.5; Stribeck model: (c) Fc = 6, Fs = 9, Fv = 0.6; (d): Fc = 6, Fs = 7.5, Fv = 0.6.

optimization; and model verification. In this paper, we employ a neuro-fuzzy system as the learner model for modeling the friction force in the system (3). Also, we adopt the framework proposed in Wang and Mendel (1992) and Wang (2003) to extract the fuzzy rules of neuro-fuzzy system. Here, model parameter optimization and verification are associated with the closed-loop system performance. The rest of this section details data collection, the neurofuzzy system description and the model parameter initialization. In this paper, we consider the friction force as a function of the velocity. By employing an experimental system in Fig. 1, some data pairs composed of the velocity and the friction values for a motion object were generated. The velocity values were obtained by using the well-known M/T method based on the encoder signals. However, the friction force value corresponding to a specific velocity could not be evaluated directly. Therefore, an indirect method has to be adopted for problem solving. It is well known that a control force is equal to the friction force whilst the object moves at a constant rate. Hence, we acquired the friction force information through the so-called constant velocity testing method (Armstrong & Canudas De Wit, 1994). Multiple experiments were run with various velocity values in order to obtain an accurate friction measurement. We took the mean of the recorded friction values as the final friction force. Our neuro-fuzzy system is a simplified ANFIS system (Jang, 1993) which is comprised of three layers. Layer 1 accepts input variable, whose node represents input linguistic variable. Layer 2 is to calculate the membership values, whose nodes represent the terms of the respective linguistic variables. Layer 3 is the output layer (see Fig. 3-Loop with DNFIS). This neuro-fuzzy system realizes a fuzzy inference system with the following form of fuzzy rules: _ RðjÞ : if x_ is Axj then b F is wj ; j ¼ 1; 2; . . . ; N;

ð6Þ

_ wj are the consewhere Aj are the fuzzy sets of the input variable x, quent parameters of b F. Define ⁄(l) and ‘(l) as the output and input variables of a node in layer l, respectively. Layer 1 – Input layer: No computation is done in this layer. The _ single node in this layer, which corresponds to input variable x, only transmits input value to the next layer directly. That is

h

ð1Þ

_ ¼ ‘ð1Þ ¼ x:

ð7Þ

Layer 2 – Membership function layer: Each node in this layer is a membership function that corresponds one linguistic label (e.g., fast, slow, etc.) of input variable x_ in Layer 1. In other words, the membership value which specifies the degree to which an input value belongs to a fuzzy set is calculated in Layer 2: ð2Þ

hj

2 _ ðxb jÞ    2 ð2Þ ¼ e ðrj Þ ; ¼ lj ‘j

ð8Þ

where l is a membership function and the Gaussian function is adopted. Layer 3 – Output layer: The single node in this layer is labeled with R, which computes the overall output and can be computed as: ð3Þ b F ¼ h ¼

N X

ð3Þ

wj  ‘j ;

ð9Þ

j¼1

where the connecting weight wj is the output action strength of the Layer 3. Generation of the fuzzy rules mentioned above and the neurofuzzy system parameter initialization were done by using an improved data mining algorithm (Wang et al., 2010). The following gives some details on our fuzzy rule extraction method. Given a set of velocity-force data pairs:

ðx_ ðpÞ ; F ðpÞ Þ;

p ¼ 1; 2; . . . ; P;

ð10Þ

where x_ 2 R is the velocity and F 2 R is the friction force. Step 1: Divide the input and output spaces into fuzzy regions. Assume that the domain intervals of x_ and F are ½x_  ; x_ þ  and [F, F+], respectively, where the domain interval of a variable means that most probably, this variable will lie in this range.The sets of n o _ ¼ Ax1_ ; . . . ; AxN_ linguistic labels are denoted by AðxÞ and n o F F BðFÞ ¼ B1 ; . . . ; BM , where each linguistic label is associated with a fuzzy membership function, N and M can be equal or unequal. Step 2: Convert ordinary records into fuzzy records. The input–output data pairs obtained are stored in relational databases. The data in relational databases are stored in a table, where each row is a record and each column represents one of _ Fg be a set of attributes, the attributes of the records. Let L1 ¼ fx; tp be the p-th record with certain attribute values. A set of records associated to attribute L1 is denoted by T L1 , that is,

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Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983

Fig. 3. Active control process of the vibration system with friction.

T L1 ¼ ft1 ;    t p . . . ; t P g:

ð11Þ

The fuzzified tp is written as l(tp), associated with membership _ [ BðFÞg. A set of fuzzified values of the set of attributes L2 ¼ fAðxÞ records associated to attribute L2 is denoted by T L2 , that is,

T L2 ¼ flðt 1 Þ; . . . lðt p Þ . . . ; lðt P Þg:

ð12Þ

An illustrative toy example is given in Table 1. Step 3: Calculate the degree of support. From data mining perspectives, the degree of support is the percentage of records where the rule holds. If a fuzzy rule has practical meaning, it must have a large enough degree of support from sample data. Therefore, the degree of support for a specific fuzzy input space is a good indicator for extraction of fuzzy rules from numerical data. The degree of support for a fuzzy rule is defined as follows:

PP

p¼1

Supðx_ ) FÞ ¼

_ lðBFl Þp ðFÞlðAx_ Þp ðxÞ

PP

p¼1

j

_ lðAxj_ Þp ðxÞ

ð13Þ

;

_ and the lðBF Þp ðFÞ are values of membership funcwhere the lðAx_ Þp ðxÞ l j tions for the p-th record respectively, P is the total number of records in T L2 ; l 2 f1; . . . ; Mg and j 2 {1, . . . , N}. For simplicity, we can use the following formula to calculate the degree of support,

Supðx_ ) FÞ ¼

P X

1 _ l F p ðFÞlðAxj_ Þp ðxÞ: P p¼1 ðBl Þ

ð14Þ

Step 4: Generate a fuzzy rulesbase using the improved data mining algorithm. Details of the improved data mining algorithm with pseudo code can be found in Wang et al. (2010). We use a singleinput–single-output example to show the process for a fuzzy _

_

rulebase generation. Suppose A ¼ fAx1 ; . . . ; Ax5 g is a set of linguistic _ and B ¼ fBF1 ; . . . ; BF5 g is another set of linguistic labels for attribute x, _

labels for attribute F. For fAx1 g, we first calculate the respective n o n o n o n o _ _ _ _ degrees of support for pairs Ax1 ; BF1 ; Ax1 ; BF2 ; Ax1 ; BF3 ; Ax1 ; BF4 n o _ and Ax1 ; BF5 . Then, we select a fuzzy subspace with the maximum degree of support for this column. Repeating this process for n o n o n o n o _ _ _ _ Ax2 ; Ax3 ; Ax4 , and Ax5 , we obtain the following fuzzy rules: _

_

_ RðjÞ : if x_ is Axj then b F is Bl ; ðj ¼ 1; . . . ; N; l ¼ 1; . . . ; MÞ;

O:R

x_

F

F:R

Ax1

_

Ax2

_

Ax3

BF1

BF2

BF3

t1 t2 t3 t4 t5

2 5 3 7 3

8 3 5 8 3

l(t1) l(t2) l(t3) l(t4) l(t5)

0.3 0.4 0.3 0.4 0.6

0.7 0.6 0.7 0.6 0.4

0.4 0.3 0.8 0.2 0.1

0.6 0.7 0.2 0.8 0.9

0.9 0.5 0.7 0.3 0.8

0.1 0.5 0.3 0.7 0.2

)

_

ð15Þ

where an if–then rule is generated with a consequent B determined by the following method. Among the M fuzzy subsets B1, . . . , BM defined in the output space, find the Bl such that

  Sup x_ Aj ) F Bl P Supðx_ Aj ) F Bl Þ Table 1 Ordinary records and fuzzy records.

_

(1) If x_ is Ax1 , then F is BF1 ; (2) If x_ is Ax2 , then F is BF2 ; (3) If x_ is Ax3 , then _ _ F is BF4 ; (4) If x_ is Ax4 , then F is BF3 ; (5) If x_ is Ax5 , then F is BF5 . Fig. 4 shows the generated fuzzy rulebase. Step 5: Initialize the model parameters. In order to construct the initial weight of the neuro-fuzzy systems, firstly, the following fuzzy rulebase is extracted from a set of velocity-force data pairs (10) based on the maximum degree of support criterion, i.e.,

ð16Þ

for l = 1, 2, . . ., M. Then, the consequent fuzzy subset B is chosen as Bl . Next, we can obtain the following initial weight by comparing (6) and (15)

wj ¼ F Bl ; where F is the point at which fuzzy membership function achieves its maximum value. A basis function of neuro-fuzzy system is defined by,

ð17Þ

lBl ðFÞ

Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983

979

Fig. 4. A sample for a fuzzy rulebase generation.



_ ¼P uj ðxÞ

lj ‘jð2Þ

N j¼1





lj ‘jð2Þ

€e þ kd e_ þ kp e ¼ €xm ; :

ð18Þ

The normalized output can be written as

b _ ¼ wT uðxÞ; _ F ðxÞ T

where w = (w1, w2, . . . , wj) is a parameter vector, _ ¼ ðu1 ðxÞ; _ u2 ðxÞ; _ . . . ; uj ðxÞÞ _ T is a regressive vector. uðxÞ

_ €e ¼ ð€xm  €xÞ and xm represents the where e ¼ ðxm  xÞ; e_ ¼ ðx_ m  xÞ; desired position. Notice that, € xm ¼ 0 for the problem of point-to-point control, the above equation becomes:

ð19Þ

€e þ kd e_ þ kp e ¼ 0:

and

In practice, the position displacement term x can be measured by sensors. However, the friction term F is very difficult to be measured or modeled. Hence, it is hard to express accurate friction forces using mathematical formula. In this situation, an approximation of the friction term can be estimated using a neuro-fuzzy system:

Remark 1. The identification methods of the neuro-fuzzy system can be grouped into two categories: a priori knowledge based approach and data driven approaches. In general, the first approach employs domain experts to estimate the system structure and parameters, Many successful applications show that this approach works favorably although it is time consuming and some empirical studies have to be carried out. Data driven approaches employ data mining or computational intelligence techniques, such as clustering algorithms, neural networks and GA algorithms, to determine the system structure and parameters from numerical data. In the light of the approaches proposed in Linkens and Chen (1999), Wong and Chen (2000), Pedrycz (1998) and Liu and Li (2004), the numbers of fuzzy sets and the membership functions of input variable were extracted by using proposed algorithm. In this paper, we focus on proposing a new method how to establish the initial weight and to adjust weight on-line.

4. Self-tuning active controller design This section proposes an on-line learning algorithm to adjust the weights of the neuro-fuzzy friction model. Unlike the traditional approaches for training learner models, our proposed learning algorithm is based on Lyapunov stability theory which is associated with a performance analysis of the closed-loop system. In this paper, the parameters in fuzzy membership functions are fixed and only the weight vector w = (w1, w2, . . . , wj)T is updated according to the adaptation rule. Suppose that the function F in (3) is known. Then, the following PD-like control law,

u ¼ kp e þ kd e_  F þ x;

ð21Þ

ð20Þ

can be applied to the system (3) to and it results in an error dynamics, i.e.,

b _ _ ¼ wT uðxÞ: F ðxÞ

ð22Þ

ð23Þ

By applying the control law, u ¼ kp e þ kd e_  b F þ x, to the system (3), we obtain a tracking error equation of the closed-loop system as follows:

€e þ kd e_ þ kp e ¼ b F  F:

ð24Þ

Let w⁄ be the optimal weight vector, i.e.,

b w ¼ argminw2Xw ½supx2 _ Xx_ k F  Fk;

ð25Þ

_ where Xw and Xx_ denote the sets of suitable bounds on w and x, respectively. Using these notations, a minimum approximation error function can be defined as:

_ e ¼ F  wT uðxÞ:

ð26Þ

Hence, the real friction force F can be modeled by:

_ þ e: F ¼ wT uðxÞ

ð27Þ

From (19) and (27), the error Eq. (24) can be rewritten as:

~ T uðxÞ _  e; €e þ kd e_ þ kp e ¼ w

ð28Þ

~ ¼ w  w . where w The above Eq. (28) is equivalent to the following state-space form:

~; e_ ¼ Ae þ Bu where

ð29Þ

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Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983

 A¼

0 kp

 1 ; kd

B¼

  0 1

;

~ T uðxÞ ~¼w _  e; u

kmin ðQÞ  1 2 1 T 2 1 V_ 6  jej þ jB Pej  ðjej2 þ 2eBT Pe 2 2 2 þ jBT Pej2 Þ

_ T: e ¼ ðe; eÞ

6 Theorem 1. The closed-loop system is stable in a sense of limt?1je(t)j = 0, if the PD-like controller u ¼ kp e þ kd e_  b F þ x is _ and the applied to the system (3)(see Fig. 3), where b F ¼ wT uðxÞ weight vector w is updated on-line according to the following rule:

( _ ¼ w

t

0

_ P½; if ðkwk ¼ R and eT PBuT ðxÞw < 0Þ; ð30Þ T

T

_ _ T Pe þ c e PBu 2ðxÞw where P½ ¼ cuðxÞB w is a Projection operator, R is a kwk norm bound of the weight vector and P = PT P 0 is the solution of the following Lyapunov equation:

AT P þ PA ¼ Q :

1 T 1 ~ T wÞ; ~ e Pe þ ðw 2 2c

jej2 dt 6

2 1 ðjVð0Þ þ VðtÞjÞ þ jBT Pj2 kmin ðQ Þ  1 kmin ðQ Þ  1 Z t  jej2 dt

ð37Þ

0

If e 2 L2, we have e 2 L2. Since all variables in the right-hand side of _ _ (29) is bounded, eðtÞ is bounded, i.e., eðtÞ 2 L1 . Using the Barbalat lemmas (Sastry & Bodson, 1989), we have limt?1je(t)j = 0. This completes the proof. h

ð31Þ 5. Simulation results

Proof. Consider a Lyapunov candidate as follows:

V¼

ð36Þ

where kmin(Q) is the minimum eigenvalue of Q. By integrating both sides of (36) and assuming that kmin(Q) > 1 (since Q is chosen by the designer), after some simple manipulation, we can obtain

Z _ T Pe; if ðkwk < RÞ or ðkwk ¼ R and eT PBuT ðxÞw _ cuðxÞB P 0Þ;

kmin ðQÞ  1 2 1 T 2 jej þ jB Pej ; 2 2

c > 0:

ð32Þ

The derivative of V with respect to time is given by,

This section presents simulation results using our proposed modeling and control techniques. The following motion control system is employed as a simulation plant:

€xðtÞ þ xðtÞ ¼ Fðv Þ þ u;

1 1 1 T _ ~ ~ w: V_ ¼ eT Pe_ þ e_ T Pe þ w 2 2 c

ð33Þ

ð38Þ

where F is the friction force and u is the control force applied to the mass.

~_ ¼ w _ and using Eq. (29), the above equation becomes: Notice that w 5.1. Parameter setting

1 1 T ~ T uðxÞ ~ w: _ _  eÞBT Pe þ w V_ ¼ ½eT ðPA þ AT PÞe þ ðw 2 c

ð34Þ

From the adaptive law (30) and Lyapunov Eq. (31), we get:

1 V_ ¼  eT Qe  eBT Pe: 2

ð35Þ

Therefore, we have,

In this paper, the Coulomb and Stribeck friction models are used in the simulations. The parameters used in the system design are specified as follows: m = 0.6 [kg], FN = 15 [N], x(0) = (0.02, 0.02)T, xm = (0, 0)T, kp = 150, kd = 15, 40 < u < 40. For a given parameter  6000 1200 matrix Q ¼ , we solve the Eq. (31) and obtain 1200 260  600 20 P¼ . 20 10

0.01 0.00 −0.01 −0.02

0

50

100

150 Time(s)

200

250

300

50

100

150 Time(s)

200

250

300

−0.015

−0.01

−0.005

0

0.005

0.01

0.02 0.01 0.00 −0.01 0

0.02 0.01 0.00 −0.01 −0.02

Fig. 5. Time responses of the controlled system with Coulomb friction (displacement, velocity, limit cycle), when u ¼ kp e þ kd e_  F þ x and k ¼ 763 [N/m].

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Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983 Table 2 Sample data for friction modeling.

Table 3 Sample data stored in fuzzy relational database.

T L1

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

T L2

Ax1

Ax2

Ax3

Ax4

Ax5

BF1

BF2

BF3

BF4

BF5

x_ F

0.5 20

0.4 18

0.3 17

0.2 16

0.1 14

+0.1 +14

+0.2 +16

+0. 3 +17

+0.4 +18

+0.5 +20

l(t1) l(t2) l(t3) l(t4) l(t5) l(t6) l(t7) l(t8) l(t9) l(t10)

1.00 0.61 0.14 0.01 0.00 0.00 0.00 0.00 0.00 0.00

0.04 0.32 0.88 0.88 0.32 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.01 0.14 0.61 0.61 0.14 0.01 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.32 0.88 0.88 0.32 0.04

0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.14 0.61 1.00

1.00 0.80 0.70 0.60 0.40 0.00 0.00 0.00 0.00 0.30

0.00 0.20 0.30 0.40 0.60 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.60 0.40 0.30 0.20 0.00

0.30 0.00 0.00 0.00 0.00 0.40 0.60 0.70 0.80 1.00

Table 4 Degree of support. BF1

BF2

BF3

BF4

BF5

Ax1

_

0.16

0.02

0.00

0.00

0.03

Ax2

_

0.16

0.09

0.00

0.00

0.00

Ax3

_

0.03

0.04

0.00

0.04

0.03

Ax4

_

0.00

0.00

0.00

0.09

0.16

_ Ax5

0.03

0.00

0.00

0.02

0.16

Sup

5.2. Neuro-fuzzy system initialization Step 1: Definition of membership functions. _ the velocity of the mass, We have one input linguistic variable x, and one output variable F, the friction force. For the input variable _ x, five membership functions are assigned: lAx_ ¼ exp 1

ð5:0ðx_ þ 0:50Þ2 Þ;

lAx2_ ¼ expð5:0ðx_ þ 0:25Þ2 Þ; lAx3_ ¼ expð5:0 2 2 _ Þ; lAx_ ¼ expð5:0ðx0:25Þ _ _ ðxÞ Þ; lAx_ ¼ expð5:0 ðx0:50Þ Þ. For 5 4 2

the output variable F, we use triangle type of membership functions except for the two end membership functions, which are in trapezoid forms (see Fig. 4). The following five sets of parameters specify the 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 in Section 3 in ordinary rela_ Fg be a set of attributes. For one retional databases. Let L1 ¼ fx; cord (row) tp, tp stands for the value of attribute (column) L1. In order to present this method in detail, we only use ten sample data to mine friction fuzzy rules in Table 2. Step 3: Extraction of fuzzy rules. In order to obtain fuzzy rules with a maximum degree of supn o _ _ port, suppose A ¼ Ax1 ; . . . ; Ax5 is a set of linguistic labels for attrin o _ and B ¼ BF1 ; . . . ; BF5 is a set of linguistic labels for attribute bute x,

_

_

_

_

_

F. We calculate all degrees of support using (13) or (14), and the fuzzified sample data in Table 3. The results of the degree of support are shown in Table 4. Based on the maximum degree of support principle for assigning the consequent of fuzzy rules, we _ obtain the following five fuzzy rules: (1) If x_ is Ax1 Then F is BF1 ; _ _ (2) If x_ is Ax2 Then F is BF1 ; (3) If x_ is Ax3 Then F is BF2 or BF4 ; (4) If x_ x_ x_ F is A4 Then F is B5 ; (5) If x_ is A5 Then F is BF5 .

Remark 2. In the case that there exist multiple largest degree of _ support for the same fuzzy subspace (e.g., Ax3 in Table 4), the following strategy will be used to determine the corresponding consequent part of the fuzzy rule: (i) take the average value of the centers of the fuzzy subsets with the largest degree of support; (ii) define a fuzzy subset as the consequent of the fuzzy rule if its center has minimum distance to the average value of the centers. For instance, we will select BF3 as the consequent for the third rule, and in the data-driven-based neuro-fuzzy friction model (19) the parameter vector w should be selected as w = (20,  20, 0, 20, 20). 5.3. Adaptation of neuro-fuzzy friction model for control compensation pffiffiffi In our simulations, we set c ¼ 2000; R ¼ 25 5. Let   T P _ and uðxÞ _ ¼ 1S lAx_ ðxÞ; _ . . . ; lAx_ ðxÞ _ S ¼ 5j¼1 lAx_ ðxÞ . The neuro-fuzzy 1

j

5

friction model and the updating weights are given as follows:

b _ F ¼ wT uðxÞ;

ð39Þ

0.02 0.00 −0.02

0

50

100

0

50

100

150 Time(s)

200

250

300

150

200

250

300

0.05 0.00 −0.05

Time(s)

0.05 0.00 −0.05 −0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Fig. 6. Time responses of the controlled system with Coulomb friction (displacement, velocity, limit cycle), when u ¼ kp e þ kd e_  b F þ x and k = 763 [N/m].

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Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983

0.02 0.00 −0.02

0

50

100

150 Time(s)

200

250

300

0

50

100

150 Time(s)

200

250

300

0.05 0.00 −0.05 0.05 0.00 −0.05 −0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

F þ x and k = 2000 [N/m]. Fig. 7. Time responses of the controlled system with Coulomb friction (displacement, velocity, limit cycle), when u ¼ kp e þ kd e_  b

_ ¼ w



_ _ cuðxÞð20e þ 10e_ Þ; if ðkwk < RÞ or ððkwk P RÞ and eT PBuT ðxÞw 6 0Þ _ > 0Þ; P½; if ððkwk P RÞ and eT PBuT ðxÞw ð40Þ

with

_ uT ðxÞw _ _  cð20e þ 10eÞ _ P½ ¼ cuðxÞð20e þ 10eÞ w: 2 kwk

Remark 3. It has been noticed that the parameter setting in the updating rule (40) is important, which can directly affect the control performance. The empirical pffiffiffi setting for the upper bound of the kwk is given by R ¼ 1:25 5wmax , where the wmax represents the upper bound of sampling input data. Another parameter in the updating rule (40) is the value of c. It was observed from our simulation studies that the control performance in terms of tracking errors can be better if a larger value of c is used in the updating weights (40). Furthermore, the updating weights described in (40) does not exactly match the original formula given in (30). Theoretically, the updating weights in (30) can ensure kwk 6 R. However, this inequality may not hold for some instants in simulations due to the sampling effect. Thus, we modified the condition kwk ¼ R in (30) as kwk P R in (40). Simulation results indicated that such an amendment works well. 5.4. Active control for Coulomb-induced limit cycle This simulation aims to explore the power of adaptive neurofuzzy friction compensation used in the control law (29) on the system performance as the parameter of the friction model changes. To do this, we set the Coulomb friction force F as:

F ¼ F c sgnðtÞ;

ð41Þ

where Fc = lFN, and k = 763 [N/m] in (3) for this simulation. Usually, the friction force in mechanical systems will be reduced for time being. To simulate the process, this subsection gives a scenario study to see the power of adaptation of the proposed neuro-fuzzy friction model for control compensation. We intentionally set different l values in the Coulomb friction force model at three time slots, that is, for 0 6 t 6 100, l = 0.70; for 100 < t 6 200, l = 0.60 and for 200 < t 6 300, l = 0.50. For the purpose of performance comparison, the following PD controllers with compensation terms were used in our simulations:

(

u1 ¼ kp e þ kd e_  F þ x; F þ x: u2 ¼ kp e þ kd e_  b

ð42Þ

To see the effectiveness of our proposed updating weights (40) for improving the control performance, we applied the controllers u1 and u2 to the motion dynamics (3) with uncertain friction force, respectively. Fig. 5 shows the simulation results that correspond to the varying friction force in (3) and a fixed compensation term F = 0.70FNsgn(t) in u1. Fig. 6 depicts the simulation results with the adaptive neuro-fuzzy friction compensation term in u2. As can be seen that our proposed adaptation scheme of the neuro-fuzzy friction model works well whilst some uncertainties on the friction force present during the control process. It has been observed that the controller with the adaptive control compensation term can eliminate both limit cycle and steady-state error. It is interesting to notice that the controller with fixed friction model compensation has no power to remove the limit cycle and the steady-state error due to the lack of adaptation mechanism.

5.5. Active control for Stribeck-induced limit cycle To further investigate the adaptive power of the neuro-fuzzy friction compensation, this section reports some simulation results, where the Coulomb friction used in the above simulations was replaced by the Stribeck friction. Here, the friction force is specified as: ds

FðtÞ ¼ F c þ ðF s  F c Þejt=ts j þ F t t;

ð43Þ

where Fc = 0.40FN, Fs = 0.60FN, Fv = 0.04, FN = 15 [N], ts = 1, ds = 1. Moreover, the structural parameter k = 763 [N/m] was changed to k = 2000 [N/m] in (3) for this simulation. Fig. 7 shows that the active controller with the adaptive neurofuzzy friction compensation, i.e., u ¼ kp e þ kd e_  b F þ x, still eliminate the Stribeck-induced limit cycle and the steady-state error simultaneously. From the simulation results, we can conclude that the proposed adaptive neuro-fuzzy friction compensation used in the PD controller is robust to changes in friction, and even structural parameters. The adaptation mechanism plays a key role to eliminate the limit cycle and the steady-state error.

Y.F. Wang et al. / Expert Systems with Applications 40 (2013) 975–983

6. Conclusions To eliminate or inhibit the friction-induced self-excited vibration, this paper develops a framework of modeling friction force and control compensation using neuro-fuzzy system and data mining techniques. An improved data mining algorithm is employed to extract a complete and robust fuzzy rulebase, which forms a basis of a data-driven neuro-fuzzy friction model. Based on the well known Lyapunov stability theory, the parameters of the neurofuzzy friction model are on-line adjusted to ensure the desired performances of the closed-loop system. A typical motion dynamics with friction is used in our simulation studies. Results demonstrate that our proposed approaches for friction modeling and adaptive compensation control technique are effective for controlling friction self-excited vibration. Acknowledgements This work was supported in part by the Program for New Century Excellent Talents in University under Grant NCET-09-0273, the Natural Science Foundation of China under Grant 51275085, 61020106003 and 51135003, the Science and Technology Foundation of Shenyang City under Grant F10-205-1-40, the National Basic Research Program of China under Grant 2009CB320601, the Fundamental Research Fund of Central Universities under Grant N110503001, and the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT0816, China. References Armstrong, B., & Canudas De Wit, P. C. D. (1994). A survey of models, analysis tools and compensation methods for control of machines with friction. Automatica, 30(7), 1083–1138. Bender, F. A., Lampaert, V., & Swevers, J. (2005). The generalized Maxwell slip model: a novel model for friction simulation and compensation. IEEE Transactions on Automatic Control, 50(11), 1883–1887. Canudas De Wit, P. C. D., Ollson, H., Astrom, K. J., & Lischinsky, P. (1995). A new model for control of systems with friction. IEEE Transactions on Automatic Control, 40(3), 419–425. Chatterjee, S. (2007). Non-linear control of friction-induced self-excited vibration. International Journal of Non-linear Mechanics, 42(3), 459–469. Das, J., & Mallik, A. K. (2006). Control of friction driven oscillation by time-delayed state feedback. Journal of Sound and Vibration, 297(3), 578–594. Dupont, P., Hayward, V., Armstrong, B., & Altpeter, F. (2004). Single state elastoplastic friction models. IEEE Transactions on Automatic Control, 47(5), 787–792. Figueiredo, M., & Gomide, F. (1999). Design of fuzzy systems using neuro-fuzzy networks. IEEE Transactions on Neural Networks, 10(4), 815–827.

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