Backstepping Sliding Mode Control with FWNN for ... - Springer Link

International Journal of Control, Automation, and Systems (2013) 11(2):398-409 DOI 10.1007/s12555-012-9115-3

ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

Backstepping Sliding Mode Control with FWNN for Strict Output Feedback Non-Smooth Nonlinear Dynamic System Seong-Ik Han and Jang-Myung Lee* Abstract: An output feedback backstepping sliding mode control scheme was developed for precision positioning of a strict single-input and single-output (SISO) non-smooth nonlinear dynamic system that could compensate for deadzone, dynamic friction, uncertainty and estimations of immeasurable states. An adaptive fuzzy wavelet neural networks (FWNNs) technique was used to provide improved approximation ability to the system uncertainty. The adaptive laws were derived for application to estimate the deadzone and friction parameters using recursive backstepping controller design procedures. In addition, the sliding mode control method was also combined to enforce the robustness of the output feedback backstepping controller against disturbance. The Lyapunov stability theorem was used to prove stability of the proposed control system. The usefulness of the proposed control system was verified by simulations and experiments on a robot manipulator in the presence of a deadzone and friction in the actuator. Keywords: Backstepping sliding mode control, fuzzy wavelet neural networks, non-smooth nonlinearity, robot manipulator, strict output feedback SISO dynamic system.

1. INTRODUCTION Several adaptive and robust control approaches have recently been developed to examine the control problem of complex nonlinear systems. The backstepping technique is very popular control method because it provides a systematic procedure to design stabilizing controllers, according to a step-by-step algorithm [1,2]. Another advantage of backstepping control is that it guarantees global or regional regulation and tracking properties, and avoids the cancellation of useful nonlinearities. More efficient controllers have been developed by combining the backstepping method with other intelligent techniques, such as fuzzy [3,4] and neural networks [5,6]. Fuzzy technology is an efficient tool for treating complex nonlinear process based on human experiences or experts. Adaptive fuzzy control has been applied successfully in many nonlinear control systems, and guarantees improved performance and system stability in a Lyapunov sense [7,8]. On the other hand, for some complicated processes, this knowledge __________ Manuscript received March 17, 2012; revised December 25, 2012; accepted February 3, 2013. Recommended by Editorial Board member Won-jong Kim under the direction of Editor Poogyeon Park. This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the Human Resources Development Program for Specialized Environment Navigation /Localization Technology Research Center support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2012-H1502-12-1002). Seong-Ik Han and Jang-Myung Lee are with the Department of Electronic Engineering, Pusan National University, Jangjeondong, Geumjeong-gu, Busan 609-735, Korea (e-mails: {skhan, jmlee}@pusan.ac.kr). * Corresponding author. © ICROS, KIEE and Springer 2013

might not be sufficient. Neural networks (NNs) are important tools, and improved control performance has been obtained from fuzzy neural networks (FNNs) by taking advantage of the two techniques [9,10]. Nevertheless, to solve the majority of approximation problems, NNs require a large number of neurons and the network convergence is generally slow. Wavelet function (WFs) approaches have been used to overcome the disadvantages of NNs [11,12]. WFs use localized basis functions to achieve the desired input-output mapping. The FWNNs combine wavelet theory with adaptive control and fuzzy logic [13,14]. The synthesis of a fuzzy wavelet neural inference system provides fast learning capability to approximate the nonlinearities that contain uncertainties. The sliding mode control (SMC) technique is robust to system uncertainty due to the use of a sliding surface [15]. Scarrat et al. designed a hybrid backstepping controller with first and second order sliding mode control [16]. Lin et al. proposed the adaptive backstepping sliding mode control for the second order LIM system [17]. On the other hand, they did not consider both the output feedback control and the nonsmooth nonlinearity at the same time. Non-smooth nonlinearities, such as backlash, deadzone, hysteresis, saturation and friction, are common in many industrial plants. A deadzone exits in mechanical systems, such as ball-screw, servo motor and hydraulic system. Friction always exits in mechanical systems with linear and rotary relative contact motion. The characteristics of these nonlinearities are poorly known, time varying and usually limit the system performance. Most backstepping based adaptive control schemes dealt mainly with linear systems. A desirable control design for real systems, where non-smooth nonlinearities play

Backstepping Sliding Mode Control with FWNN for Strict Output Feedback Non-Smooth Nonlinear Dynamic System

an important role, requires the accommodation of nonsmooth nonlinearities. A range of design methods for deadzone [18-22] and friction [23-25] compensation have been proposed, and satisfactory performance has been accomplished. On the other hand, these approaches have been rarely reported to compensate for both deadzone and friction effects of a dynamic system with two nonlinearities in the same actuators. In this paper, the elasto-plastic dynamic friction model [24] was used as a dynamic friction model combined with an inverse deadzone model [18]. Many cases exist where the information of nonlinear system states cannot be completely available by measurements in most robotic systems, hydraulic systems and other servo dynamic systems. For these systems, state observer based or output feedback adaptive intelligent control schemes have been developed [26-28]. On the other hand, most of these methods did not address the non-smooth nonlinearities. The present paper addresses the adaptive fuzzy wavelet output feedback control problem for a SISO non-smooth nonlinear system based on backstepping control. Backstepping control was combined with the sliding mode control technique to enhance the robustness to compensate for uncertainty. The FWNNs could approximate the unknown nonlinear function, and an adaptive observer was also introduced to estimate the immeasurable states. A backstepping output feedback controller with the SMC and adaptive laws for the deadzone and friction parameters was designed recursively using the Lyapunov functions. The proposed control approach was applied successfully to solve the problem of both a reducing non-smooth nonlinear effect and state estimation of an uncertain nonlinear robot system by simulations and experiments. 2. PROBLEM FORMULATION

x1 = x2 + f1 ( x1 ) − F f 1 + d1 , xn −1 = xn + f n −1 ( xn −1 ) − F fn −1 + d n −1 ,

(1)

xn = D (u ) + f n ( xn , u ) − F fn + d n ,

y = x1 , T

n

2.2. Deadzone nonlinearity The mathematical model for deadzone nonlinearity, w, is described as ⎧mr (u − d r ) ⎪ D(u ) = ⎨0 ⎪m (u − d ) l ⎩ l

for for for

u ≥ dr

⎫ ⎪ dl < u < d r ⎬ , ⎪ u ≤ dl ⎭

(2)

where mr and ml are the slopes of the deadzone, and dr and, dl are the deadzone width parameters. The practical assumptions of the deadzone are given for the control problem: Assumption 1: The deadzone output D(u) is unavailable for measurement. The deadzone parameters mr, ml, dr and dl are unknown but their signs are known, mr > 0, ml > 0, d r ≥ 0 and dl ≤ 0. Assumption 2: The deadzone slopes are bounded by known constants mr min , mr max , ml min and mr max , such that 0 < mr min ≤ mr ≤ mr max , and 0 < ml min ≤ ml ≤ ml max . The deadzone inverse technique is a useful compensating method for examining the deadzone effect [18]. Setting ud as the control signal from the controller to achieve the control object for the plant without a deadzone, the following control signal, u, can be generated according to the certainty equivalence deadzone inverse u = D −1 (ud ) =

ud + dˆmr u + dˆml (1 − υ ), υ+ d mˆ l mˆ r

(3)

where mˆ r , mˆ l , dˆmr and dˆml are the estimates of mr , ml , mr d r and ml dl , respectively and ⎧ 1 if ud ≥ 0 υ=⎨ ⎩ 0 if ud < 0.

(4)

The resulting error between u and ud is given by

2.1. Strict output feedback dynamic system A SISO strict output feedback nonlinear dynamic system [26] was considered as follows: x2 = x3 + f 2 ( x2 ) − F f 2 + d 2 ,

399

where x = [ x1 ,… , xn ] ∈ R are the state vector, u ∈ R is the control input, y ∈ R is the output of system, f i ( xi ), (1 ≤ i ≤ n − 1) and f n ( x , u ) are unknown smooth nonlinear function, di , (1 ≤ i ≤ n) are the bounded disturbance uncertainties, F fi , (1 ≤ i ≤ n) are nonlinear dynamic frictions and the actuator nonlinearity D(u) is described as a deadzone characteristic. x1 was assumed to be the only state available for measurement; the other states were not unavailable.

⎛ ⎞ u + dˆmr (5) D(u ) − ud = ⎜ dmr − d m r ⎟υ ⎜ ⎟ mˆ r ⎝ ⎠ ⎛ ⎞ u + dˆml + ⎜ dml − d m l ⎟ (1 − υ ) + ε d , ⎜ ⎟ mˆ l ⎝ ⎠ where m r = mr − mˆ r , m l = ml − mˆ l , dmr = d mr − dˆmr , d = d − dˆ , and εd is the bounded function for all u ml

ml

ml

values [12]. 2.3. Dynamic friction The following elasto-plastic friction model was considered [24]: zi = vi − α i ( zi , vi )σ 0i hi (vi ) zi ,

(6)

where zi is the mean defection vector of the elastic bristle; hi (vi ) = vi / Fci + ( Fsi − Fci )exp(−(vi / vsi )2 ) ; vi is the relative velocity; σ 0i > 0 denotes the stiffness of elastic bristle; Fci is the Coulomb friction; Fsi is the stiction level, and vsi is the Stribeck angular velocity.

(

)

Seong-Ik Han and Jang-Myung Lee

400

α i ( zi , vi ) presents an adhesion map that controls the rate of change in zi, particularly, when the bristle displacement, zi, is smaller than a given breakaway displacement zbai , and α i ( zi , vi ) was set to zero, and consequently (6) reduces to zi = vi . This means that in the range zi ≤ zbai , the model shows a purely elastic pre-sliding regime, and does not exhibit drift [24]. The dynamic friction is described as follows: F fi = σ 0i zi + σ1i zi + σ 2i vi = σ 0i zi − α iσ 3i hi (vi ) zi + σ 4i vi ,

(1 ≤ i ≤ n),

(7)

where σ1i > 0 is the damping coefficient matrix in the elastic range, σ 2i > 0 is the viscous friction coefficient matrix, σ 3i = σ 0iσ1i and σ 4i = σ1i + σ 2i . In the elastoplastic model, α i ( z , v) is an important factor, but is difficult to determine experimentally. Therefore, a procedure to estimate this and other friction parameters should be considered. Because, the state variable, z, cannot be measured directly, the friction state observer was suggested to estimate zi as follows: zˆ = v − aˆ σˆ h (v ) zˆ , (1 ≤ i ≤ n), (8) i

i

i

0i i

i

i

where zî is an estimation of zi, αî is an estimation of α i and σˆ 0i is an estimation of σ 0i . The estimation error of the friction state variable can be obtained from (6) and (8) as follows: z = −α σ h z − α h zˆ σ − σˆ h zˆ α , (9) i

i

0i i i

i i i

0i

0i i i i

where zi = zi − zî , σ 0i = σ 0i − σˆ 0i , and αi = α i − αî . Thus, the estimation of friction, F fi , can be given as

Fˆ fi = σˆ 0i zî − αîσˆ 3i hi zî + σˆ 4i vi ,

(10)

where σˆ 3i and σˆ 4i are estimations of σ 3i and σ 4i , respectively, σ 3i = σ 3i − σˆ 3i , and σ 4i = σ 4i − σˆ 4i . From (7) and (10), it follows that F fi = F fi − Fˆ fi = (σ 0i − α iσ 3i hi ) zi + zîσ 0i − α i zî hiσ 3i − σˆ 3i zî hiα i + viσ 4i + ε fi ,

(11)

(i = 1,… , n, j = 1,… , M ),

xi − bij aij

, aij and bij

are the dilation and translation parameters, µ F l ( xi ) is i

fuzzy wavelet membership function, and M is the number of rules in the fuzzy rule base. 3) Fuzzy rule layer: Multiplication is used as the following AND operator in this layer: n

OlIII = ψ l ( xi ) = ∏ i =1 μ F l ( xi ), (l = 1,… , M ) . i

(13)

4) Output layer: The output of the fuzzy wavelet systems is expressed as M

O fw ( x) = ∑ l =1 wlψ l ( xi ) = φ Tψ ( x),

(14)

where wl is the weight between the rule layer and output layer and φ = [ w1 , , wM ]T is the weight vector and ψ ( x) = [ψ 1 ( x1 ),… ,ψ l ( xn )]T . The output of the feedforward path from the input nodes can be expressed as: m

(15)

i =1

2.4. Adaptive fuzzy wavelet neural networks system The proposed FWNNs combine a fuzzy system with wavelet neural networks as shown in Fig. 1. 1) Input layer: The fuzzy inference engine performs mapping from an input linguistic vector x = [ x1 ,… , xn ]T ∈ R n to an output linguistic scalar variable y ∈ R . 2) Membership layer: In this layer, each neuron represents the membership function of a linguistic variable. In this paper, the Mexican Hat wavelet basis function was adopted as a membership function, which is given as i

where ω is the frequency, zij =

O f = ∑ w fiϕi ( xi ) = wTf ϕ ( x),

where εfi is a bounded estimation error of friction.

O IIj = μ F l ( xi ) = cos(ω zij ) exp(− zij2 ),

Fig. 1. Structure of the FWNNs system.

(12)

where wf = [ w f 1 ,… , w fn ]T are the weight vector between the input nodes and output node and ϕi ( xi ) = 1/(1 + exp(− xi ) is the connecting function. The final output of the FWNNs system can be described as OoIV ( x ) = φ Tψ ( x) + wTf ϕ ( x) = θ T ξ ( x),

where θ = [φ

(16)

w f ]T and ξ ( x) = [ψ ( x) ϕ ( x)]T .

2.4. State observer The FWNNs system in (16) was used to approximate the unknown nonlinear functions in (1) as [27,28] fi ( xi ,θi ) = θiT ξi ( xi ), fî ( xî ,θi ) = θiT ξi ( xî ) (1 ≤ i ≤ n),

(17)


where xî is an estimation of xi . The weight θi was assumed to vary in the bounded area M θ f which is defined as M θi = {θi ∈ R n : θi ≤ mθi },

(18)

where mθi > 0 is a pre-specified parameter, whereas xî and xi remain in the boundary area Ux and U xˆ , respectively. The value of θf for the optimal approximation is

{

}

θi* = arg min sup x∈U x , xˆ∈U xˆ fî ( xî ,θi ) − fi ( xi ) . (19) θi ∈Mθi

The approximation errors were defined as ε ai = fi ( xi ) − fî ( xî ,θi* ) , ε bi = f i ( xi ) − fî ( xî ,θi ) . (20)

Denote wai = ε ai + di and wbi = ε bi + di , (i = 1,… , n) . Because the states, xi (2 ≤ i ≤ n), were not measured, state observers should be provided to estimate the unmeasured states. The FWNNs state observers similar to that proposed by Deng [26] can be written as

xî = xî +1 + ki ( y − xˆ1 ) + fî ( xî , θi ) − Fˆ fi (i = 1,… , n − 1), xˆn = τ + kn ( y − xˆ1 ) + fˆn ( xˆn , θ n ) − Fˆ fn .

(21)

Define the observer error, x = x − xˆ, where x = [ x1 , ..., xn ]T . The equation of the observation errors is expressed as xi = xi +1 − ki ( x1 − x1 ) + [ fi ( xi ) − fî ( xî ,θi ) + di ] − F + ε (i = 1,… , n − 1), fi

(22)

fi

Equation (22) can be rewritten in the following compact form: n

(23)

i =1

yd(i − 2) ,

Fˆ f 1 ,… , Fˆ fi −1 ) − yd(i −1) , i = 2,..., n − 1, ϖ n = xˆn − λn −1 ( xˆ1 ,… , xˆn −1 ,θ1 ,… ,θ n −1 , y, yd ,… ,

(26) yd( n − 2) ,

Fˆ f 1 ,… , Fˆ fn ) − yd( n −1) .

(27)

The time derivative of ϖ 1 and ϖ i with x2 = xˆ2 + x2 for i = 2,..., n, gives ϖ1 = xˆ2 + f1 ( x 1 ) − F f 1 + d1 − y d + x2

= ϖ 2 + λ1 ( xˆ1 ,θ1 , y, yd ) + θ1T ξ1 ( xˆ 1 ) − Fˆ f 1 − y d + x2

+θ1T ξ1 ( xˆ 1 ) − F f 1 + wa1 + ε f 1 ,

(28)

∂λi −1 i −1 ∂λi −1 ∂λi −1 xˆ − ∑ θ − y ˆk k k =1 ∂θ k k ∂y k = 2 ∂x

−∑ i −1

∂λi −1

( k −1) k = 2 ∂yd

i −1

∂λi −1 ˆ F fk − yd(i ) ˆ k =1 ∂F

yd( k ) − ∑

fk

− Fˆ fi − F fi + ε fi

where wb = [ wb1 ,… , wbn ]T , and ε f = [ε f 1 ,… , ε fn ]T ,

= ϖ i +1 + λi ( xˆ1 ,..., xî ,θ1 ,...,θi −1 , y, yd ,..., yd(i −1) ) T

Bi = ⎡⎣ 0 1 0 ⎤⎦ ,

and ki (1 ≤ i ≤ n) are positive constants chosen so that A is Hurwitz. This study assumed that wbi ≤ βi , (i = 1, … , n) and ε f ≤ ε fm , where β i > 0 and ε fm > 0 are unknown constants. From (23), x = e At x (0) + ∫ e A(t −τ ) ( wb + ε f )dτ . 0

(25)

ϖ i = xî − λi −1 ( xî ,… , xî −1 ,θ1 ,… ,θi −1 , y, yd ,… ,

−∑

= Ax + wb + ε f ,

t

ϖ1 = e ,

i −1

i =1 n

1 0 0⎤ ⎥ 0 1 0⎥ , ⎥ ⎥ 0 0 0 ⎦⎥

3.1. The recursive backstepping procedures for derivation of the controller In this section, the controller and adaptive laws will be designed using recursive procedures of backstepping technique with AFWNNs and sliding mode control. By defining the tracking error as e = x1 − yd , the error surfaces are defined as follows:

ϖ i = xî +1 + ki x1 + θiT ξi ( xî ) + θiT ξi ( xî ) + wai − wbi

x = Ax + ∑ Bi ⎡⎣ f i ( xi ) − fî ( xî ,θi ) + di ⎤⎦ + ε f

⎡ −k1 ⎢ −k A=⎢ 2 ⎢ ⎢ ⎣⎢ −kn

3. DESIGN OF CONTROLER AND DERIVATION OF ADAPTIVE LAWS

− F f 1 + wa1 + ε f 1

fn

= Ax + ∑ Bi [ε bi + di ] + ε f

Because A is Hurwitz and if wb and ε f are small and bounded, the estimation error x tends to zero asymptotically as t → ∞. On the other hand, asymptotic stability cannot be guaranteed if wb and ε f are not small. Therefore, a robust control and friction estimator should be considered to attenuate the effect of the nonlinear friction, deadzone and uncertainty.

= xˆ2 + θ1T ξ1 ( xˆ 1 ) − Fˆ f 1 − y d + x2 + θ1T ξ1 ( xˆ 1 )

xn = − kn ( x1 − x1 ) + [ f n ( xn ) − fˆn ( xˆn , θ n ) + d n ] − F + ε . fn

401

(24)

+ ki x1 + θiT ξi ( xî ) + θiT ξi ( xî ) i −1

∂λi −1 xˆk +1 + θ kT ξ k ( xˆk ) + kk x1 ˆ x ∂ k k =2

−∑

(

i −1

)

∂λi −1 ∂λi −1 θk − xˆ2 + θ1T ξ1 ( x1 ) + x2 + wb1 θ y ∂ ∂ k k =2

(

−∑ i −1

−∑

∂λi −1

( k −1) k = 2 ∂yd

i −1

∂λi −1 ˆ F fk − yd(i ) ˆ ∂ F k =2

yd( k ) − ∑

fk

+ wai − wbi − Fˆ fi − F fi + ε fi

)


402

= ϖ i +1 + λi ( xˆ1 ,..., xî ,θ1 ,...,θi −1 , y, yd ,..., yd(i −1) ) +Ωi

− Fˆ fi −

+θiT ξi ( xî )

yd(i )

+ wai − wbi − F fi + ε fi , ϖ n =

∂λ ∂λ − i −1 x2 − i −1 wb1 ∂y ∂y i = 2,..., n − 1 ,

(29)

∑

∑

∑

∂λ = ud + Ω n − Fˆ fn − yd( n ) + θnT ξ n ( xˆn ) − n −1 x2 ∂y −

(30)

Ωi = ki x1 + θiT ξi ( xî ) i −1

∂λi −1 xˆk +1 + θ kT ξ k ( xˆk ) + kk x1 ˆ ∂ x k k =2

(

i −1

)

∂λi −1 ∂λi −1 θk − xˆ2 + θ1T ξ1 ( x1 ) θ y ∂ ∂ k k =2

(

−∑ i −1

−∑

∂λi −1

( k −1) k = 2 ∂yd

)

∂λi −1 ˆ F fk , i = 2,..., n − 1, ˆ k = 2 ∂F fk

n −1

∂λn −1 n −1 ∂λn −1 xˆ −∑ θ î i i =1 ∂θi i i =1 ∂x

Ω n = kn x1 + θ nT ξ n ( xˆn ) −∑

∂λn −1 ˆ F fi . ˆ i =1 ∂F fi

The following Lyapunov function candidate was considered as follows: n

VI = ∑ Vi ,

(31)

i =1

where 1 Vi = ϖ i2 , i = 1,..., n . 2

Assuming that

wa1 + ε f 1 ≤ ρ1 ,

(

)

1 1 2 ϖ 1 x2 ≤ ϖ 12 + x2 , 2 2 ∂λi −1 ∂λ −ϖ i x2 − ϖ i i −1 wb1 ∂y ∂y

(35)

1 x2 2

2

(36)

2

⎛ ∂λ ⎞ 1 + ⎜ i −1 ⎟ ϖ i2 + wb21 , i = 2,..., n, (37) ∂ y 2 ⎝ ⎠

and the following hyperbolic tangent function relation can be introduced as ⎛ϖ ρ ⎞ 0 < ϖ i ρi − ϖ i ρi tanh ⎜ i i ⎟ ≤ 0.2785κ i = κ i′ , ⎝ κi ⎠ i = 1,..., n .

⎛ϖ ρ ⎞ 1 λ1 = −c1ϖ 1 − ϖ 1 − θ1T ξ1 ( xˆ 1 ) − ρ1 tanh ⎜ 1 1 ⎟ 2 ⎝ κ1 ⎠

n −1

−∑

∂λi −1 wb1 + ϖ i ρi + ϖ iθiT ξi ( xî ) − ϖ i F fi , ∂y i = 2,..., n − 1 , (34) ∂λ ∂λ Vn ≤ ϖ n ud + Ω n − Fˆ fn − yd( n) − n −1 x2 − n −1 wb1 ∂y ∂y T + ϖ n ρ n +ϖ nθn ξ n ( xˆn ) − ϖ n F fn

(38)

The virtual control functions, λi, was specified as follows:

n −1 ∂λn −1 ∂λ ( xˆ2 + θ1T ξ1 ( x1 )) − ∑ (ni −−1)1 yd(i ) ∂y i =1 ∂yd

−

)

−ϖ i

≤

i −1

yd( k ) − ∑

(

From the inequality ab ≤ ( a 2 + b 2 ) / 2 , it follows that

where θi = θi* − θi ,

−∑

(33)

∂λ Vi ≤ ϖ i ϖ i +1 + λi + Ωi − Fˆ fi − yd(i ) − ϖ i i −1 x2 ∂y

⎛ ⎞ u + dˆmr + ⎜ dmr − d m r ⎟υ ⎜ ⎟ mˆ r ⎝ ⎠ ⎛ ⎞ u + dˆml + ⎜ dml − d m l ⎟ (1 − υ ) . ⎜ ⎟ mˆ l ⎝ ⎠

∂λn −1 wb1 + wan − wbn − F fn + ε fn ∂y

⎛ ⎞ u + dˆmr + ⎜ dmr − d m r ⎟υ ⎜ ⎟ mˆ r ⎝ ⎠ ˆ ⎛ ⎞ u + d ml + ⎜ dml − d m l ⎟ (1 − υ ) + ε d , ⎜ ⎟ mˆ l ⎝ ⎠

)

+ x2ϖ 1 + ϖ 1 ρ1 +ϖ 1θ1T ξ1 ( xˆ1 ) − ϖ 1 F f 1 ,

D(u ) + kn x1 + θ nT ξ n ( xˆn ) + θnT ξ n ( xˆn ) + ε an − ε bn n −1 ∂λn −1 n −1 ∂λn −1 − Fˆ fn − F fn + ε fn − xˆ − θ î i i =1 ∂θi i i =1 ∂x n −1 ∂λ ∂λn −1 (i ) n −1 ∂λn −1 ˆ − n −1 y − y − F fi − yd( n) ( i −1) d ˆ ∂y ∂ F ∂ y i =1 i =1 fi d

∑

(

V1 ≤ ϖ 1 ϖ 2 + λ1 + θ1T ξ1 ( xˆ 1 ) − Fˆ f 1 − y d

(32) wai + wbi + ε fi ≤ ρi ,

wan + wbn + ε fn + ε d ≤ ρn , and ρi > 0 are constants for i = 1,..., n, the time derivative of (31) considering (28)(30) becomes

(39)

+ Fˆ f 1 + y d ,

2

⎛ ∂λ ⎞ λi = −ciϖ i − ϖ i −1 − Ωi − ⎜ i −1 ⎟ ϖ i ⎝ ∂y ⎠ ⎛ϖ ρ ⎞ − ρi tanh ⎜ i i ⎟ + Fˆ fi + yd(i ) , i = 2,..., n − 1, ⎝ κi ⎠

(40)

where ci and κ i are positive design parameters. (39) and (40) were substituted into (33) and (34), respectively, to give V1 ≤ −c1ϖ 12 + ϖ 1ϖ 2 +ϖ 1θ1T ξ1 ( xˆ1 ) − ϖ 1 F f 1 + κ1′ +

1 x2 2

2

,

(41)


Vi ≤ −ciϖ i2 + ϖ iϖ i +1 − ϖ iϖ i −1 + ϖ iθiT ξi ( xî ) − ϖ i F fi + κ i′ +

1 x2 2

2

(42)

1 + wb21. 2

s = γ 1ϖ 1 + γ 2ϖ 2 + + γ n −1ϖ n −1 + ϖ n ,

(43)

where γ i > 0, i = 1,… , n − 1, are constants. Define the following Lyapunov function candidate n 1 1 T 1 1 VII = VI + s 2 + ∑ θi θi + m r2 + m l2 2 2η mr 2η ml i =1 2ηθ i

+

θn = ηθ nξ n ( xˆn ) s, σˆ 0 n = −η0 n zˆn s , σˆ = η zˆ h s , 3n

3.2. The derivation of the controller and adaptive laws with sliding mode control The adaptive output feedback backstepping control was modified to enforce robustness by adding sliding mode control. The modification begins by defining the following sliding surface in terms of the error coordinates

n ⎛ 1 2 1 2 1 1 d mr + d ml + ∑ ⎜ zi2 + σ 02i 2ηdr 2η dl 2η0i i =1 ⎝ 2

⎞ 1 1 1 + α iσ 32i + σ 42i + αi2 ⎟ , 2η3i 2η4i 2ηα i ⎠

+

+

i =1

σˆ 3i = η3i zî hi (γ i s + ϖ i ), (i = 1, …, n − 1) , σˆ 4i = −η4i vi (γ i s + ϖ i ), (i = 1,…, n − 1) , αˆ i = ηα iσˆ 3i zî hi (γ i s + ϖ i ), (i = 1,… , n − 1) ,

(

(

+ −α nσ 0 n hn zn2 − α n hn zˆnσ 0 n zn − σˆ 0 n hn zˆnα n zn n −1

n −1

i =1

i =1

2

) ))

(

n

(46)

−α i hi zîσ 0i zi −σˆ 0i hi zîαi zi ) + ∑ κ i′ + i =1

n x2 2

2

n

1 (n − 1) 2 + wa21 + wb1 + ∑ γ iκ i′ 2 2 i =1

≤ − K1s 2 + sζ + Δ f ,

(62)

where 1 1 n −1 ζ = γ 1 x2 + γ 1wa21 + ∑ γ i x2 2 2 i =2

2

i

(61)

+ ∑ϖ i (σ 0i − α iσ 3i hi ) zi + ∑ϖ i −α iσ 0i hi zi2

⎛ sρ ⎞ ⎛ ∂λ ⎞ n −1 + ∑ γ i (1 − ϖ i ) ⎜ i −1 ⎟ −∑ γ i ρi tanh ⎜ i ⎟ y ∂ ⎝ ⎠ ⎝ κi ⎠ i=2 i =1

i

(60)

+ ∑ γ i −α iσ 0i hi zi2 − α i hi zîσ 0i zi − σˆ 0i hi zîα i zi

2

0i i

(59)

n −1 1 n −1 2 γ w + ∑ i b1 ∑ γ iε fi 2 i=2 i =1

i =1 n −1

(45)

⎛ ∂λ ⎞ ud = −ϖ n −1 − Ω n − ⎜ n −1 ⎟ ϖ n + Fˆ fn − yd( n ) ⎝ ∂y ⎠ n −1 n −1 n −1 1 −∑ γ iϖ i +1 + ∑ γ i ciϖ i + ∑ γ iϖ i −1 + γ 1ϖ 1 2 i =1 i =1 i=2

0i

(58)

n −1

The control input and the adaptive laws were chosen:

⎛ sρ ⎞ − ρ n tanh ⎜ n ⎟ − K1s − K 2 sgn( s ) , ⎝ κn ⎠ θi = ηθ i (γ i s + ϖ i )ξi ( xî ), (i = 1,… , n − 1) , σˆ = −η zˆ (γ s + ϖ ), (i = 1,…, n − 1) ,

(57)

+ ∑ γ i (σ 0i − α iσ 3i hi ) zi + (σ 0 n − α nσ 3n hn ) zn

From (43),

n −1

(56)

1 1 n −1 ⎛ − K 2 s − K1s 2 + s ⎜ γ 1 x2 + γ 2 wa21 + ∑ γ i x2 2 2 i=2 ⎝

(44)

⎛ 1 + ∑ ⎜ zi zi + σ 0iσ 0i η 0i i =1 ⎝

ϖ n = −γ 1ϖ 1 − γ 2ϖ 2 − − γ n −1ϖ n −1 + s .

(55)

3n n n

T VII ≤ − [ϖ 1 ϖ 2 ϖ n −1 ] Θ [ϖ 1 ϖ 2 ϖ n −1 ]

n

⎞ 1 1 1 α iσ 3iσ 3i + σ 4iσ 4i + αiαi ⎟ . η3i η 4i ηα i ⎠

(54)

where K1 > 0 and K 2 ≥ 0 are positive constants. Substituting (47)-(61) into (45) and considering the previous derived results, gives

n 1 T 1 1 VII = VI + ss + ∑ θi θ i + m r m r + m l m l η η η mr ml i =1 θ i

1 1 d mr d mr + d ml d ml ηdr ηdl

(53)

σˆ 4 n = −η4 n vn s , αˆ n = ηα nσˆ 3n zˆn hn s , u + dˆmr υs , mˆ r = −ηmr d mˆ r u + dˆml mˆ l = −ηml d (1 − υ ) s , mˆ l dˆmr = η drυ s , dˆml = ηdl (1 − υ ) s .

where η( i) > 0 are design constants. The time derivative of VII can be expressed as

+

403

2

+

1 n −1 γ i wb21 ∑ 2 i=2

n −1

(47)

+ ∑ γ i σ 0i − α iσ 3i hi zi + σ 0n − α nσ 3n hn zn

(48)

+ ∑ γ i α iσ 0i hi zi

(49)

+σˆ 0i hi zî αi zi

(50)

2

(51) (52)

i =1 n −1 i =1

(

2

+ α i hi zî σ 0i zi

)

+ ⎛⎜ α nσ 0n hn zn + α n hn zˆn σ 0 n zn ⎝ +σˆ 0 n hn zˆn α n zn ) ,


404

From (16), (48) and (53), the adaptive laws of the k th connecting weight vector and the weight of the feedforward path are determined by

n −1

Δ f = ∑ ϖ i σ 0i − α iσ 3i hi zi i =1 n −1

(

+ ∑ ϖ i α iσ 0i hi zi i =1

+σˆ 0i hi zî αi zi +

2

+ α i hi zî σ 0i zi n

) +∑ κ i′ + n2 i =1 n

2

x2

(n − 1) 2 1 2 wb1 + wa1 +∑ γ iκ i′ , 2 2 i =1

and the positive definite matrix Θ can be described as 0 ⎡ c1 0 ⎤ ⎢0 c ⎥ 0 2 ⎥. Θ=⎢ ⎢ ⎥ ⎢ ⎥ ⎣⎢γ 1 γ 2 γ n −1 + cn −1 ⎦⎥

(63)

where Q = (2 K1 − 1/ χ 2 ), K1 > 1/ 2 χ 2 and χ is a positive constant. Integrating both sides of (63) from t = 0 to t = ∞, yields T 1 2 VII (T ) ≤ VII (0) − λmin (Q) ∫ s dt 0 2 T 1 2 + χ 2 ∫ ζ dt + Δ f 0 2

2

s dt ≤ 2VII (0) + χ 2 ∫

T 0

2

ζ dt + 2Δ f , (65)

where λmin (Q) denotes the minimum eigenvalue of Q. If the system begins with the initial conditions, (65) can be expressed as λmin (Q ) ∫

T 0

2

s dt ≤ χ 2 ∫

T 0

2

(68)

w fi = η fiϕi ( xî )ϖ i , (1 ≤ i ≤ n − 1) ,

(69)

w fn = η fnϕ n ( xˆn ) s,

(70)

where ηψ i and η fi are design constants. In general, the adaptive laws in (48) and (53) need to be modified to guarantee that θi ∈ M θi using the projection algorithm [29]. The modified projection adaptive law can be expressed as

(71) where the projection operator is defined as Proj [ηiξi ( xî )ϖ i ] = ηi ξi ( xî )ϖ i − ηiϖ i

ζ dt + 2Δ f .

(66)

Consider the uncertainty, ζ ∈ L∞ [0, ∞), i.e., there is an 2 ε d > 0 such that ζ ≤ ε d . Then VII ≤ −λmin (Q ) s . 2 2 2 If λmin (Q) > ( χ ε d + 2Δ f ) / δ is chosen, there is a 2 positive constant β and δ such that Vn ≤ − β s < 0 for all s > δ . This suggests that T > 0 such that s ≤ δ for all t ≥ T . This means that all variables are bounded. Finally, suppose ζ ∈ L2 [0, ∞) ∩ L∞ [0, ∞) and Δ f ∈ L2 [0, ∞) ∩ L∞ [0, ∞). Then, ϖ i → 0, ( i = 1,… , n), s → 0, θi → 0, σ i → 0 (i = 0,3, 4), αi → 0, m r → 0, m l → 0, B mr → 0 and B ml → 0 as t → ∞ according to Barbalat’s Lemma [15].

θiT ξi ( xî ) θi

2

θi . (72)

Similar relationships can be applied to other adaptive laws. 4. SIMULATED AND EXPERIMENTAL VERIFICATION

(64)

for all 0 ≤ T < ∞. This suggests that all the states and signals are bounded. Because VII (t ) ≥ 0, the inequality in (64) indicates the following inequality T

φn = ηψ nψ n ( xˆn ) s,

⎧ηiξi ( xî )ϖ i if θi < mθ i or θ i = mθ i ⎪ and ηi ξi ( xî )ϖ iθ i ≥ 0 ⎨ ⎪ ⎩ Proj [ηiξi ( xî )ϖ i ] if θ i = mθ i and ηiξi ( xî )ϖ iθi < 0,

2

⎞ 1⎛ 1 ⎞ 1⎛ 1 VII = − ⎜ 2 K1 − 2 ⎟ s 2 − ⎜ s − χζ ⎟ 2⎝ 2⎝ χ χ ⎠ ⎠ 1 1 1 + χ 2ζ 2 + Δ f ≤ − Qs 2 + χ 2ζ 2 + Δ f , 2 2 2

0

(67)

θi =

Then, (61) can be described as

λmin (Q ) ∫

φi = ηψ iψ i ( xî )ϖ i , (1 ≤ i ≤ n − 1) ,

This section describes the simulated and experimental evaluation of the proposed control scheme is described. As a design example, the single manipulator of the Scorbot robot system in the presence of a deadzone and friction in the joints was chosen. Fig. 2 shows a photograph of the Scorbot robot control system. Only the second upper arm was considered the control application among the four links of the Scorbot robot manipulator. In Fig. 2, the deadzone nonlinearities were generated in the timing-belt and harmonic driver, which are connected between DC motors and each joint. The dynamic equation for the second single link of the Scorbot robot manipulator can be expressed as Jq + G (q ) + F f = D (u ) ,

u=

nkt V, Rm

(73) (74)

where J = mL2 / 3, q is the angular position of link, G (q ) = mLgcosq, g is the acceleration due to gravity and F f = σ 0 z + σ 1 z + σ 2 q. In (74), V = ud is the control input voltage, n = 65.5 is the gear ratio of the harmonic drive gear, Rm = 0.8294Ω is the motor resistance, kt = 1.82 × 10−2 Nm / A is the torque constant, and u is the control input torque. The chosen values of the link parameters were m = 3.59 kg, L = 0.41m and g = 9.806


(a) Manipulator.

405

(b) Timing belt in the actuator.

Fig. 2. Photograph of the Scrobot robot system. m/s2. The friction parameters determined experimentally were selected as Fs = 0.06 Nm, Fc = 0.054 Nm, σ 0 = 2300 Nm/rad, σ1 = 5.4 Nmsec/rad, σ 2 = 2.3 Nmsec/rad, respectively. The initial range of transient function values of the elasto-plastic model was considered approximately α = 0.8 ~ 1. The initial values of the deadzone parameters are selected as mr = 1, ml = 1, d r = 0.3 and dl = −0.32. The desired trajectory of the manipulator was yd (t ) = 0.005sin(1.2566 × t ) (rad). The state variables were defined as x1 = q and x2 = q. The state equation of (73) can be expressed as x1 = x2 , x2 = f 2 ( x1 ) − bF f 2 + bD (u ),

(75)

where f 2 ( x1 ) = −bmLg cos( x1 ), F f 2 = F f , and b = 1/ J . The sliding surface was defined as s = γ 1ϖ 1 + ϖ 2 . 4.1. Simulation The fuzzy wavelet membership functions were chosen as follows: µ F1 = i

1 1 + exp ⎡⎣( xi − 5 × 10−3 ) 2 ⎤⎦

,

µ F 2 = cos ⎡⎣0.1( xi − 10−3 ) ⎤⎦ exp ⎡⎣ −( xi − 10−3 ) 2 ⎤⎦ , i

µ µ

Fi3

= cos ⎡⎣0.1( xi − 5 × 10−4 ) ⎤⎦ exp ⎡⎣ −( xi − 5 × 10−4 )2 ⎤⎦ ,

Fi4

= cos [ 0.1( xi − 0)] exp ⎡⎣ −( xi − 0)2 ⎤⎦ ,

µ F 5 = cos ⎡⎣0.1( xi + 5 ×10−4 ) ⎤⎦ exp ⎡⎣ −( xi + 5 × 10−4 ) 2 ⎤⎦ , i

µ

Fi6

= cos ⎡⎣0.1( xi + 10−3 ) ⎤⎦ exp ⎡⎣ −( xi + 10−3 )2 ⎤⎦ ,

µF 7 = i

(a) Tracking errors of the BC and BSC systems.

1 , (i = 2) . ⎡ 1 + exp ⎣ ( xi + 5 × 10−3 ) 2 ⎤⎦

The following design parameters of the controller and state observer were chosen as c1 = 25, K1 = 5, K 2 = 0.25, χ = 0.25, γ 1 = 0.5, ρ2 = 0.5, κ 2 = 0.01, k1 = 100, and k2 = 1200 . In the simulation, the following five controllers were designed to evaluate the performance of the proposed control system: Backstepping controller (BC), Backstepping sliding mode-based controller (BSC), Backstepping sliding mode-based controller with the deadzone observer (BSDC), Backstepping based controller with the deadzone and friction observer (BDFC), and backstepping sliding mode-based controller with the deadzone and friction observer (BSDFC). All simulated tests were carried using the Matlab package.

(b) x1 (solid), xˆ1 (dashed), x2 (dotted) and xˆ2 (dash_ dotted) of the BSC system. Fig. 3. Simulated results of the BC and BSC systems. Fig. 3 presents the simulated results of the BC and BSC systems, where the joint position tracking error of the BSC system was improved by virtue of adding robustness effect of the sliding mode control. Fig. 3(b) shows the estimated states by the state observer. As shown in Fig. 3(a), however, satisfactory tracking performance for the BC and BSC systems could not be obtained due to both the deadzone and friction. Fig. 4 shows the simulated results using the deadzone and friction estimators. In Fig. 4(a), the BSDC system provides an improved tracking performance than that of the BSC by deadzone compensation. On the other hand, the size of the tracking errors of the BSDC system was still high due to friction. The magnitude of the tracking errors of the BSDFC system was reduced significantly by virtue of the proposed friction and deadzone compensation. The friction torque, Ff, of the joint was also estimated well, as shown in Figs. 4(b). In Fig. 4(c), the controller output approached the deadzone output. The estimation results of the other parameters of the friction and deadzone were omitted due to a limitation in the paper length. The disturbances composed of the 0.5V pulse with a 0.5sec width and a 1.5 times higher mass of the nominal link were applied to the control input and plant at 5sec, 10sec and 15sec, to examine the robustness of the proposed controller. In Fig. 5, for the given disturbance, the tracking performance of the proposed BSDFC system was more robust than that of the BDFC system due to the robustness of the SMC system.

406


Therefore, the simulated results revealed the robustness of the proposed BSDFC system to the uncertainty of the robot manipulator as well as the effects of both the deadzone and friction.

(a) Tracking errors of the BSC, BSDC and BSDFC systems.

(b) F f and Fˆ f of the BSDFC system.

(c) ud and D(u) of the BSDFC system. Fig. 4. Simulated results of the BSC, BSDC and BSDFC systems.

Fig. 5. Simulate disturbance responses of the BDFC and BSDFC systems.

4.2. Experiment An experiment of the proposed control scheme was performed to confirm the simulated results. The fuzzy wavelet membership functions were chosen as the same functions of simulation. The design parameters of the controller and state observer were chosen as c1 = 60, K1 = 70, K 2 = 6, χ = 0.1, γ 1 = 10, ρ2 = 2, κ 2 = 0.1, k1 = 100 and k2 = 800. In the experiment, five control systems, similar to those in the simulation, were also designed to examine the usefulness of the proposed control system. The designed controllers generated in the computer were implemented by the Matlab RTI system using the MF624 board of the Humusoft Company [30]. The control signals were transferred to the DC servo motor of the Scorbot robot through the servo drive. The sample time was selected as 500 Hz. Fig. 6 presents the experimental results of the BSDFC system with the tracking errors of the BSC and BSDFC control system. Fig. 6(a) and Table 1 show that the magnitude of the position tracking error of the BSDFC system is reduced considerably compared to those of the other two control systems due to compensation for both deadzone and friction. The estimated states are represented in Fig. 6(b) and (c), where the filtered state for x2 is included to evaluate the performance of the state observer. Fig. 6(d) and (e) show the estimated friction and control input. Finally, to examine the robustness of the proposed control scheme experimentally, a 1.8kg increase in mass was attached in the link as a disturbance, as shown in Fig. 7, and a 1V pulse with a 0.5sec width was applied to the in DC motor drive at 5sec and 15sec, respectively, during 25sec. For these disturbances, Fig. 8 shows the tracking errors of the BC, BDFC and BSDFC systems, where the proposed BSDFC control is the most robust to these disturbances. From these experimental results, the proposed control scheme compensates effectively for non-smooth deadzones and friction nonlinearities and uncertainties. On the other hand, for a high order nonlinear dynamic system, the proposed BSDFC system may have the problems associated with the complexity of the controller and over-parametrization due to the inherent problem of explosion of terms of the backstepping control scheme. For these problems, an estimation method for both the differential term of the virtual control and uncertainty needs to be considered, and the tuning functions to simplify the estimation law can be introduced if the structure of the uncertainty has a linear-in-parameter, which cannot be surely guaranteed in a real nonlinear system. Therefore, to enhance the implementation capacity for many complex industrial plants, a technique for simplifying the proposed controller scheme will be considered in a future study.


(a) Position tracking errors of the BSC, BSDC and BSDFC systems.

407

Fig. 7. Photograph of the manipulator with added mass.

(b) x1 (solid) and xˆ1 (dash). Fig. 8. Experimental disturbance responses of the BC, BDFC and BSDFC systems. 5. CONCLUSIONS

(c) Filtered x2 (solid) after differentiation of x1 and xˆ2 (dash).

(d) Fˆ f .

In this paper, the output feedback backstepping sliding mode control scheme was developed to provide enhanced position tracking performance of the strict feedback SISO nonlinear dynamic system in the presence of both a deadzone and friction. Initially, the output feedback backstepping controller was designed using recursive procedures with the friction and deadzone estimators of the elasto-plastic model based friction and unknown deadzone parameters. The sliding mode control was added to enforce the robustness of the backstepping controller. The uncertainties in each recursive step of the backstepping design were approximated by the FWNNs. The adaptive laws for the controller, friction and deadzone observers were derived from the Lyapunov stability theorem. As a design example, a Scorbot robot manipulator in the presence of joint friction and deadzone was chosen. The favorable position tracking performance of the proposed control scheme was validated by some simulations and experiments.

[1]

[2]

(e) ud. Fig. 6. Experimental results of the BSDFC system.

[3]

REFERENCES M. Kristic, L. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. Y. Zhang, B. Fidan, and P. A. Ioannou, “Backstepping control of linear time-varying systems with known and unknown parameters,” IEEE Trans. Autom. Contr., 2003, vol. 48, no. 11, pp. 1908-1925, 2003. S. Jagannathan and F. L. Lewis, “Robust backstepping control of a class of nonlinear systems using

408

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14] [15] [16]

[17]

[18]


fuzzy logic,” Information Science, vol. 123, no. 3, pp. 223-240, 2000. R. J. Wai, F. J. Lin, and S. O. Hsu, “Intelligent backstepping control for linear induction motor drive,” IET Control Theory Appl., vol. 148, no. 3, pp. 193-202, 2001. Y. Li, S. Qiang, X. Zhuang, and O. Kaynak, “Robust and adaptive backstepping control for nonlinear systems using RBF neural network,” IEEE Trans. Neural Networks, vol. 15, no. 3, pp. 693-701, 2004. C. Kwan and F. L. Lewis, “Robust backstepping control of nonlinear systems using neural networks,” IEEE Sys. Man Cyber. Part A: Sys. and Humans, vol. 30, no. 6, pp. 753-766, 2000. C. H. Wang, H. L. Liu, and T. C. Lin, “Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems,” IEEE Trans. Fuzzy Sys., vol. 10, no. 1, pp. 39-49, 2002. H. X. Li and S. C. Tong, “A hybrid adaptive fuzzy control for a class of nonlinear MIMO systems,” IEEE, Trans. Fuzzy Sys., vol. 11, no. 1, pp. 24-34, 2003. W. Y. Wang, T. T. Lee, C. L. Liu, and C. H. Wang, “Function approximation using fuzzy neural networks with robust learning algorithm,” IEEE Trans. Syst. Man B Cybern., vol. 27, no. 4, pp. 740747, 1997. F. J. Lin, W. J. Hwang, and R. J. Wai, “A supervisory fuzzy neural network control system for tracking periodic inputs,” IEEE Trans. Fuzzy Sys., vol. 7, no. 1, pp. 41-52, 1999. C. K. Lin, “Adaptive tracking controller design for robotic systems using Gaussian wavelet networks,” IET Control Theory Appl., vol. 149, no. 4, pp. 316322, 2002. R. J. Wai and H. H. Chang, “Backstepping wavelet neural network control for indirect field-oriented induction motor drive,” IEEE Neural Net., vol. 15, no. 2, 367-382, 2004. C. J. Lin and C. C. Chin, “Prediction and identification using wavelet-based recurrent fuzzy neural networks,” IEEE Trans. Syst. Man B Cybern., vol. 34, no. 5, pp. 2144-2154, 2004. H. Adeli and X. Jiang, “Dynamic fuzzy wavelet neural network model for structural system identification,” ASCE, vol. 132, pp. 102-112, 2006. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, 1991. J. C. Scarrat, A. S. I. Zinober, R. E. Mills, M. RiosBolivar, A. Ferrara, and L. Giacomini, “Dynamic adaptive first and second-order sliding mode backstepping control of nonlinear nontriangular uncertain system,” Trans. ASME, vol. 122, pp. 746752, 2000. F. J. Lin, P. H. Shen, and S. P. Hsu, “Adaptive backstepping sliding mode control for linear induction motor drive,” IET Electr. Power Appl., vol. 149, no. 3, pp. 184-194, 2002. G. Tao and G. P. V. Kokotovic, “Adaptive control

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26] [27] [28]

[29] [30]

of plants with unknown dead-zones,” IEEE Trans. Autom., Contr., vol. 39, no. 1, pp. 59-68, 1994. F. L. Lewis, W. K. Tim, L. Z. Wang, and Z. X. Li, “Deadzone compensation in motion control systems using adaptive fuzzy logic control,” IEEE Trans. Contr. Sys. Tech., vol. 7, no. 6, pp. 731-742, 1999. X. S. Wang, C. Y. Su, and H. Hong, “Robust adaptive control of a class of nonlinear systems with unknown dead-zone,” Automatica, vol. 40, no. 3, pp. 407-413, 2004. Z. Wang, Y. Zhang, and H. Fang, “Neural adaptive control for a class of nonlinear systems with unknown deadzone,” Neural Comput. & Appplic., vol. 17, pp. 339-345, 2008. N. Zhou, Y. J. Liu, and S. C. Tong, “Adaptive fuzzy output feedback control of uncertain nonlinear systems with nonsymmetric dead-zone input,” Nonlinear Dyn., vol. 63, pp. 771-778, 2011. C. Canudas de Wit, H. Olsson, and K. J. Astrom, “A new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, no. 3, pp. 419-425, 1995. P. Dupong, V. Hayward, B. S. R. Armstrong, and J. Altpeter, “Single state elasto-plastic friction models,” IEEE Trans Automat. Contr., vol. 47, no. 5, pp. 787-792, 2002. F. Al-Bender, V. Lampaert, and J. Swevers, “The generalized Maxwell-slip model: a novel model for friction simulation and compensation,” IEEE Trans. Automat. Contr., vol. 50, no. 11, pp. 1883-1887, 2005. H. Deng and M. Krstic, “Output-feedback stochastic nonlinear stabilization,” IEEE Trans. Autom., Contr., vol. 44, no. 2, pp. 328-333, 1999. G. Rigatos, “Adaptive fuzzy control of DC motors using state and output feedback,” Electric Power Sys. Research, vol. 79, pp. 1579-1592, 2009. S. C. Tong and Y. Li, “Observer-based fuzzy adaptive control for strict-feedback nonlinear systems,” Fuzzy Sets and Sys., vol. 160, pp. 17491764, 2009. L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cliffs, New York, 1995. Humusoft Comp., MF 624 Multifunction I/O Card Manual, Czech Republic, 2006.

Seong-Ik Han received his B.S. and M.S. degrees in Mechanical Engineering from Pusan National University, Korea, in 1987 and 1989, respectively, and his Ph.D. degree in Mechanical Design Engineering from Pusan National University in 1995. From 1995 to 2009, he was an associate professor of Electrical Automation of Suncheon First College, Korea. Now he is with the Electronic Engineering, Pusan National University, Korea. His research interests include intelligent control, nonlinear control, robotic control, hydraulic servo system control, vehicle system control and steel process control.


Jang-Myung Lee received his B.S. and M.S. degrees in Electronic Engineering from Seoul National University, Seoul, Korea, in 1980 and 1982, respectively, and his Ph.D. degree in Computer Engineering from the University of Southern California (USC), Los Angeles, in 1990. Since 1992, he has been a professor with the Intelligent Robot Laboratory, Pusan National University, Busan, Korea. His current research interests include intelligent robotic systems, ubiquitous ports, and intelligent sensors. Dr. Lee is a past president of the Korean Robotics Society, and a vice president of ICROS. He is also the head of National Robotics Research Center, SPENALO.

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