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Direct Adaptive Fuzzy Sliding Mode Control for Under-actuated Uncertain Systems

Shun-Feng Su1, Yao-Chu Hsueh1, Cio-Ping Tseng1, Song-Shyong Chen2, and Yu-San Lin3

1Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, 2Department of Information and Networking Technology, Hsiuping University of Science and Technology, Taichung, Taiwan, 3National Chung-Shan Institute of Science and Technology, Taoyuan, Taiwan
Correspondence to: Shun-Feng Su (sfsu@mail.ntust.edu.tw)
Received December 1, 2015; Revised December 24, 2015; Accepted December 25, 2015.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

The development of the control algorithms for under-actuated systems is important. Decoupled sliding mode control has been successfully employed to control under-actuated systems in a decoupling manner with the use of sliding mode control. However, in such a control scheme, the system functions must be known. If there are uncertainties in those functions, the control performance may not be satisfactory. In this paper, the direct adaptive fuzzy sliding mode control is employed to control a class of under-actuated uncertain systems which can be regarded as a combination of several subsystems with one same control input. By using the hierarchical sliding control approach, a sliding control law is derived so as to make every subsystem stabilized at the same time. But, since the system considered is assumed to be uncertain, the sliding control law cannot be readily facilitated. Therefore, in the study, based on Lyapunov stable theory a fuzzy compensator is proposed to approximate the uncertain part of the sliding control law. From those simulations, it can be concluded that the proposed compensator can indeed cope with system uncertainties. Besides, it can be found that the proposed compensator also provide good robustness properties.

Keywords : Adaptive fuzzy control, Under-actuated systems, Sliding mode control
1. Introduction

Most mechanical systems, such as individual links of robotic manipulators, are assumed to be directly controllable; that is, all components of those systems are wished to be individually actuated. However, it is not always possible because there may exist failure or attrition in those components and actuators. In such cases, the system becomes under-actuated. An under-actuated system is a system having a fewer number of control inputs than the degree of freedom of the system to be controlled [1]. In fact, there also exist many under-actuated systems in real applications such as mentioned in [1, 2], free-flying space robots, underwater robots, walking robots, mobile robots, flexible-link robots, surface vessels, helicopters, etc. The investigations of under-actuated mechanical systems are valuable in many applications. For instance, if under-actuated control can work well, the number of actuators can be reduced so that the weight or size of the system can become more economical. Such advantages of considering under-actuated mechanical systems can also be found for walking robots, aircrafts, spacecrafts, etc. Sometimes, the control algorithms for under-actuated systems can be utilized to partially retrieve the functions of a malfunction-plant. By a proper under-actuated control algorithm, as presented in [3, 4], the robot arm still can provide partial functions. Therefore, the development of the control algorithms for under-actuated systems is important.

In [2], a way of transforming an under-actuated system in a Lagrange equation form into a decoupled representation is proposed. Then, the control methods about decoupled systems have been discussed for years and can be used to design controllers for under-actuated systems. The idea is to partition the system into several subsystems, and each subsystem can be treated as a second order canonical form system so that the control of the whole system can easily be designed. Sliding mode control is widely used to cope with many nonlinear systems. As mentioned in [5?12], major advantages of sliding mode control include insensitive to external disturbances, robustness against system uncertainty, stability guaranteed of the system, fast dynamic response, and simple to implement. In [13], a multi-level (hierarchical) sliding model based decoupling control was proposed to successfully stabilize a class of under-actuated systems. In the approach, a sliding surface is defined for each subsystem respectively and they are linearly combined to form a so-called decoupling sliding surface. Then, the control law can be derived based on the decoupling sliding surface. With different decoupling sliding surface designs, the obtained control law is adaptive as presented in [14, 15]. However, there are several parameters required in the decoupling sliding surface. Since these parameters are not easily identified, several estimators for those parameters have been developed in [10, 11, 16]. Instead of estimating those parameters, in this study, we intent to directly employ adaptive fuzzy control schemes for the design of the decoupling sliding surface [13]. From our simulation, it can be found that such an approach is indeed effective.

Fuzzy control has been wildly used in many applications [8, 10]. Based on the property that a fuzzy system can be an universal function approximator, as proved in [17, 18], the so-called adaptive fuzzy controls have been developed by using the feedback linearization or the backstepping approach as discussed in [19]. This kind of approaches is an important methodology for dealing with system uncertainties. In those approaches, the consequent of a considered fuzzy system (approximator) is updated by a specified adaptive law, which is often derived based on the Lyapunov stable theory. If such a fuzzy system is utilized to directly approximate the designated reference control law, it is often referred to as the direct adaptive fuzzy control [20]. In this paper, the direct adaptive fuzzy sliding mode control will be considered in defining the decoupling sliding surface for a class of under-actuated systems.

As mentioned, a direct adaptive fuzzy sliding mode control system is proposed based on the control design discussed in [13]. According to the proposed decoupling sliding surface in [13] and the feedback linearization control technique, a sliding control law is proposed in our study. This control law consists of an equivalent control law and a hitting control law. However, since the under-actuated system considered in this study is assumed to be uncertain, the equivalent control part of the sliding control law is not available because the system functions are unknown. Thus, the direct adaptive fuzzy control is employed to approximate the uncertain equivalent control part, where the input of the fuzzy approximator is defined as the decoupling sliding surface. The detailed proposed control scheme will be introduced in the following sections.

The remainder of this paper is organized as follows: in Section 2, the considered under-actuated system is described and the model based sliding control law is derived. In Section 3, the design of the fuzzy approximator is briefly introduced. In Section 4, the proposed direct adaptive fuzzy sliding mode control system is presented. In Section 5, the commonly-used doubled-inverted pendulum system is considered as the test platform to demonstrate the effectiveness of the proposed designs. Finally, the conclusions are given in Section 6.

2. System Description

In this study, a single-input-multiple-output under-actuated system is considered and can be described as:

${x˙1(t)=x2(t)x˙2(t)=f1(x)+b1(x)u(t)x˙3(t)=x4(t)x˙4(t)=f2(x)+b2(x)u(t)?x˙n-1(t)=xn(t)x˙n(t)=fm(x)+bm(x)u(t)$

where f1(x), f2(x), . . . , and fm(x) are system functions with uncertainties, b1(x), b2(x), . . . and bm(x) are system gain functions and are supposed to be known, u(t) is the control input of the system, m is the number of the sub-systems, each of which is a second order system, and n is the number of states. Then, it is easy to verify that n = 2m. The state vector can be written by x = [x1, x2, ..., xn]T. The system can be regarded as m sub-systems (?1, ?2, . . . , ?m) with second-order canonical forms, as

$?:{?1:{x˙1(t)=x2(t)x˙2(t)=f1(x)+b1(x)u?2:{x˙3(t)=x4(t)x˙4(t)=f2(x)+b2(x)u??m:{x˙n-1(t)=xn(t)x˙n(t)=fm(x)+bm(X)u$

As mentioned, those system functions are subject to uncertainties. Thus, we assume that they can be written as

${f1?(x)=f^1?(x)+Δf1?(x)f2?(x)=f^2?(x)+Δf2?(x)?fm?(x)=f^m?(x)+Δfm?(x)$

where f?i(x) |i=1,2,···,m represents the nominal function of fi(x) and Δfi(x) |i=1,2,···,m represents the uncertain term of fi(x). Hence, system (1) is rewritten as

${x˙1(t)=x2(t)x˙2(t)=f^1(x)+Δf1(x)+b1(x)u(t)x˙3(t)=x4(t)x˙4(t)=f^2(x)+Δf2(x)+b2(x)u(t)?x˙n-1(t)=xn(t)x˙n(t)=f^m(x)+Δfm(x)+bm(x)u(t)$

As mentioned, the sliding mode control in considered in this study. Thus, the system is divided into m subsystems that are in canonical forms and each subsystem is controlled by its corresponding sliding surface. The sliding surfaces of those subsystems are defined as:

${s1=c1x1+x2s2=c2x3+x4?sm=cmxn-1+xn$

where ci |i=1,2,···,m> 0 are bounded sliding coefficients and usually are selected in a trial and error manner. The basic idea is to design a controller based on those sliding surface so that the system can have required control performance.

Now, consider the first system, (?1). Choose a Lyapunov function as

$V1=12s12$

It is desired that the controller can make V?1 negative to guarantee the first subsystem stable. In other words, we hope

$V˙1=s1s˙1=s1?(c1x2+f1+b1u)<0$

In the sliding mode control design, the controller can be selected as

$u=1b1[-f1-c1x2-k1sgn(s1)]=ueq1+uhit1$

where k1 is a positive bounded constant. Here,

$ueq1=1b1[-f1-c1x2]$

is referred to as the equivalent control input and

$uhit1=1b1[-k1sgn(s1)]$

is the hitting control input. With the same idea, the control inputs of the other subsystems can also be obtained.

In above, all subsystems are ensured to be stable by their individual controllers. However, since the system considered is under-actuated and there is only one control input. Thus, a way of combining all controllers into one must be proposed. In [13], a hierarchical sliding surface is proposed and is written as

$sc=s1-kd2s2+kd3s3-?±kdmsm$

where kd2, kd3, . . . and kdm are positive constants and are called decoupling constants, and ± is a sign symbol for the m-th term. When m is odd, ± is +; otherwise, ± is ?. Decoupling constants are determined in a trial and error manner.

Similarly, a Lyapunov function is defined as

$V=12sc2.$

Then, we hope to have

$V˙=scs˙c=sc{c1x2+f1-kd2c2x4-kd2f2?±kdmcmxn±kdmfm+(b1-kd2b2+kd3b3?±kdmbm)?u}<0$

Thus, the controller can be selected as

$u*=1b1-kd2b2+kd3b3?±kdmbm×{-c1x2-f1+kd2c2x4+kd2f2??kdmcmxn?kdmfm-kcsgn(sc)}=1b1-kd2b2+kd3b3?±kdmbm×{-c1x2-f1+kd2c2x4+kd2f2??kdmcmxn?kdmfm}+{-kcsgn(sc)}b1-kd2b2+kd3b3?±kdmbm=ueq+uhit$

where kc is a positive bounded constant. Again, the equivalent control (ueq) and the hitting control (uhit) can be defined accordingly. By substituting Eq. (12) into Eq. (11), we have V? < 0 |sc≠0 and V? = 0 |sc=0. Hence, system (1) is asymptotically stable.

However, it should be noted that the control input u* need to have knowledge about the system functions, fi(x) |i=1,2,···,m. But, as mentioned earlier, those system functions are subject to uncertainties or even unknown. In the literature, there is an approach called adaptive fuzzy control that has been proposed to resolve this kind of problems [13, 20?24]. The idea of direct adaptive fuzzy control is to employ a fuzzy approximator with a finite number of fuzzy rules to online approximate (estimate) the desired controller. In our study, we also employ this kind of approach to approximate the equivalent control. This approach is called the decoupling sliding direct adaptive fuzzy control.

3. Fuzzy Approximator

As mentioned, the direct adaptive fuzzy control is to use a fuzzy approximator (system) to approximate an uncertain control law directly. In the process, an adaptive law must be obtained to update the fuzzy approximator. In this section, the fuzzy approximator is briefly introduced.

The fuzzy approximator used consists of a set of fuzzy IF-THEN rules as:

$Rule-l:IF?sc?is?Al?THEN?yF?is?θdl?for?l=1,2,3,…,M$

where sc is the input, yF is the output, Al is the l-th rule, $θdl$ is the fuzzy singleton corresponding to fuzzy sets, and M is the number of fuzzy rules. Then, the overall output of the fuzzy approximator is

$yf?(sc?θdl)=∑l=1MθdlμAl∑l=1MμAl,$

where μAl (sc) is the membership function of fuzzy set Al. Eq. (14) can be written in a linear-in-parameter (LIP) form as:

$yf?(sc?θdl)=θdTωd$

where $θdT=[θd1,θd2,θd3,…,θdM]$ and is called the adjustable parameter vector, and $ωd=[ωd1,ωd2,ωd3,…,ωdM]T$ and is a regressive vector, where

$ωdl=μAl∑l=1MμAl,l=1,2,3,…,M$

As mentioned, this approximator is directly employed to model the control input.

Since the number of fuzzy rules is finite, approximation errors must be taken into consideration as in [19]. A robust control law denoted as ur is considered into the control law to cope with possible approximator error and/or external disturbance, and then the control input becomes

$u=ud+ur=θdTωd+ur.$

In fact, in Eq. (12), there is a hitting control term, which can also be viewed as a robust control. Thus, in our approach, the fuzzy approximator will be employed to estimate ueq in Eq. (12) and the hitting control uhit will be considered as ur.

By the knowledge in [19], Eq. (14) is a universal approximator. In [13], the optimal fuzzy approximator can be denoted as

$ud*(sc?θd*)=ud*=θd*Tωd$

where $θd*T=[θd1*,θd2*,?,θdM*]$ is an optimal parameter vector and is defined as:

$θd*≡argθd∈Ωθdmin?{supsc∈Ωs?u*-ud*(sc?θd)?}$

where Ωθd and Ωs are the constant sets of θd and sc, respectively. In the following, the adaptive control law of the fuzzy approximator will be derived.

4. Adaptive Fuzzy Sliding Mode Control

In the section, the adaptive law for the fuzzy approximator as (15) that can online approximate the uncertain equivalent control law in (12) is proposed. The whole system stability is also shown in the section.

By adding and subtracting u* into Eq. (10), we have

$s˙c=c1x2+f1-kd2c2x4-kd2f2?±kdmcmxn±kdmfm+(b1-kd2b2+kd3b3?±kdmbm)?(u+u*-u*)=(b1-kd2b2+kd3b3?±kdmbm)?(u-u*)-kcsgn(sc)$

Then, let u = ud and we have

$s˙c=-(b1-kd2b2+kd3b3?±kdmbm)(u*-ud*+ud*-ud)-kcsgn(sc)=-(b1-kd2b2+kd3b3?±kdmbm)(?+θ?dTωd)-kcsgn(sc)$

where $?d=u*-ud*$ is the approximation error and $θ?dTωd=ud*-ud$ is the learning error with $θ?dT=(θd*-θd)T$. With the property of universal approximation theory for fuzzy approximators as stated in [18], ?dL is valid.

Now, define a Lyapunov function as:

$V=12sc2+12αθ?dTθ?d$

where α > 0 is the learning rate of the fuzzy compensator. To guarantee the system stable, it is desired that V? must less than 0. It can be obtained that

$V˙=scs˙c-1αθ?dTθ˙d=sc?(b1-kd2b2+kd3b3?±kdmbm)?(?d+θ?dTωd)p-kc?sc?-1αθ?dTθ˙d$

$θ˙d=αsc?(b1-kd2b2+kd3b3?±kdmbm)?ωd,$

We can have

$V˙=-sc?(b1-kd2b2+kd3b3?±kdmbm)?(?d-ur)-kc?sc?$

Since ?dL, if a suitable kc can be selected to suppress the approximation error, V? < 0 |sc≠0 and V? = 0 |sc=0 can be achieved. In fact, similar to traditional robust control schemes, kc can be selected to be a bound of the related function as discussed in [25]. Therefore, it is evident that the system can be stabilized by the adaptive fuzzy control law as (25).

5. Simulations

In our study, in order to demonstrate the effects of the proposed approach, a physical system, a doubled-inverted pendulum system is considered as the platform for control. The doubled-inverted pendulum system is often used for verifying theoretical conjecture such as in [9?11, 13]. It is a 6-order system and an under-actuated SIMO system with three degrees of freedom. The mathematical model and parameters of the considered doubled-inverted pendulum system are referred to [13]. The system is shown in Figure 1 and is described in a decoupled form as

${x˙1=x2x˙2=f1+b1ux˙3=x4x˙4=f2+b2ux˙5=x6x˙6=f3+b3u$

where

$f1=A21l1m1sin(θ2-θ1)+1l1g?sin?θ1-A11l1mccos?θ1?sin?θ1,b1=A22l1m1sin(θ2-θ1)-1l1mccos?θ1-A12l1mccos?θ1?sin?θ1,f2=A11l2m1sin(θ2-θ1),b2=A12l2m1sin(θ2-θ1),f3=A11mcsin?θ1,$

and

$b3=1mc+A12mcsin?θ1$

with

$A11=a22(l1θ˙12-g?cos?θ1)-a12l2θ˙22Δ,A12=-a22?sin?θ1Δ·mc,A21=-a12(l1θ˙12-g?cos?θ1)+a11l2θ˙22Δ,A22=a12?sin?θ1Δ·mc,a11=1m1+1mcsin2?θ1,a12=-cos(θ2-θ1)m1,a22=1m1+1m2,$

and

$Δ=a11a22-a122.$

Here, x1 = θ1 is the angle of pole 1 with respect to the vertical axis, x2 = θ?1 is the angle velocity of pole 1 with respect to the vertical axis, x3 = θ2 is the angle of pole 2 with respect to the vertical axis, x4 = θ?2 is the angle velocity of pole 2 with respect to the vertical axis, x5 is the position of the cart, x6 is the velocity of the cart, l1 is the length of pole 1, l2 is the length of pole 2, g is the acceleration of gravity, m1 is the mass of the ball at the top of pole 1, m2 is the mass of the ball at the top of pole 2, and mc is the mass of the cart. Because of temperature, mechanical attritions, or others factors, the parameters of the system may change. As shown in Eq. (4), there will be Δfi in the system equation. In order to check whether the proposed approach can cope with this situation, a constant Cu is introduced into the system function in our simulation. Thus, fi in Eq. (26) becomes Cufi for i = 1, 2 and 3 (or compared to Eq. (5)ufi = fi + Δfi).

The parameters used in our simulation are: for the hitting (robust) control, kc = 15, for the sliding surfaces of those subsystems; c1 = 5, c2 = 3, and c3 = 0.5; for the hierarchical sliding surface, kd2 = 0.05 and kd3 = 0.5, and for the fuzzy compensator, the learning constant α = 300. The initial states are x1(0) = 30°, x2(0) = 0, x3(0) = ?30°, x4(0) = 0, x5(0) = 1m, and x6(0) = 0. Figures 2, C?4 show the response of all six states for Cu = 1, 0.8, and 0.4, respectively. It can be observed the control performance is severely degraded when Cu = 0.4. Thus, in this study, the fuzzy compensator is implemented for the system with Cu = 0.4. The result is presented in Figure 5. It can be found that the performance is significantly improved.

In order to have a clear comparison, the following indices are considered for those experiments.

1. (i) Settling time: It is the time that the considered variable starts to stay in the range of ±δ. For each variable in a system, the δ is chosen independently.

2. (ii) Average steady state error: It denoted the average error when the system enters the settling stage and for state x, it can be calculated as $Average?steady?state?error=1b-a+1∑k=ab?x?(k)?$

where t is the total simulation time, $a=settlingtime0.01$ and $b=t0.01$.

The control performances in terms of those indices are tabulated in Tables 1 and 2. It is clearly evident that the proposed fuzzy compensator can have much better control performance.

In order to show the robust capability of the proposed approach, a noise as cos(2πt+5)+2 cos(3πt+1)+5 sin(2πt)+ 2 sin(3πt?1)+2 sin(2πt?2)+3 sin(πt?3) is added into the system, where Cu = 1 is considered. The results are tabulated in Tables 3 and 4. Again, it is clearly evident that the proposed fuzzy compensator can have much better control performance.

6. Conclusions

A direct adaptive fuzzy sliding mode control has been proposed for under-actuated systems based on a hierarchical sliding control. Presently, the adaptive law design is not further modified to avoid the parameter drifting problem. But, it is known that the so-called robust modification techniques as discussed in [1, 21] can easily be applied to modify the proposed adaptive law for avoiding the parameter drifting problem. In our simulations, the doubled-inverted pendulum system is considered. From those simulations, it can be concluded that the proposed compensator can overcome those system uncertainties. Besides, it can be found that the compensator also provide good robustness properties.

Conflict of Interest

Figures
Fig. 1.

The considered doubled-inverted pendulum system.

Fig. 2.

System responses without compensator for Cu = 1. (a) and (b) are the responses of all states, (c) the sliding surface sc.

Fig. 3.

System responses without compensator for Cu = 0.8. (a) and (b) are the responses of all states, (c) the sliding surface sc.

Fig. 4.

System responses without compensator for Cu = 0.4. (a) and (b) are the responses for all states, (c) the sliding surfaces s1, s2, s3 and sc, (d) the control input u.

Fig. 5.

System responses with compensator for Cu = 0.4. (a) and (b) are the responses for all states, (c) the sliding surfaces s1, s2, s3 and sc, (d) the control input u and the compensator input.

TABLES

### Table 1

Settling times under various system uncertainties and with the use of the proposed fuzzy compensator

Cu variableδ10.80.40.4+fuzzy compensator
x10.056.246.65-4.63
x20.17.939.08-6.26
x30.54.284.64-2.57
x40.57.639.30-5.85
x50.18.028.31-7.56
x60.28.319.999.996.44
s10.18.259.98-7.14
s20.57.809.47-5.98
s30.56.218.48-5.72
sc0.050.777.60-3.74

The notation (?) represents variable does not converge into the desired ±δ region until the end of our simulation time.

### Table 2

Average steady state errors under various system uncertainties and with the use of the proposed fuzzy compensator

Cu variable10.80.40.4+fuzzy compensator
x10.01390.0212-0.0115
x20.04130.0587-0.0285
x30.11970.1747-0.0986
x40.21340.3111-0.1310
x50.03480.0500-0.0411
x60.10750.29450.22270.0728
s10.05110.1297-0.0298
s20.23930.3009-0.1420
s30.18600.2566-0.0974
sc0.00280.0236-0.0133

The notation (?) represents variable does not converge into the desired ±δ region and average steady state error is not defined.

### Table 3

Settling times for systems with disturbance

Control input variableδWithout compensatorWith fuzzy compensator
x10.059.675.89
x20.19.979.63
x30.52.742.50
x40.58.865.83
x50.19.537.70
x60.29.939.19
s10.1-9.81
s20.58.505.95
s30.59.815.77
sc0.05-9.94

### Table 4

Average steady state errors for systems with disturbance

Control input variableWithout compensatorWith fuzzy compensator
x10.02340.0131
x20.08660.0474
x30.13100.1092
x40.16900.1907
x50.05590.0426
x60.11050.1086
s1-0.0555
s20.25180.2154
s30.31920.1506
sc-0.0483

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23. Cao, J, Ji, X, Li, P, and Liu, H (2011). Design of adaptive interval type-2 fuzzy control system and its stability analysis. International Journal of Fuzzy Systems. 13, 334-343.
24. Lee, CH, and Hsueh, HY (2013). Observer-based adaptive control for a class of nonlinear non-affine systems using recurrent-type fuzzy logic systems. International Journal of Fuzzy Systems. 15, 55-65.
25. Hsueh, YC, and Shun-Feng, S 2011. Direct adaptive fuzzy decoupled sliding control for a class of under-actuated uncertain systems., Proceedings of 2011 International Conference on Machine Learning and Cybernetics, Guilin, China, Array, pp.226-231.
Biographies

Shun-Feng Su received the B.S. degree in electrical engineering, in 1983, from National Taiwan University, Taiwan, R.O.C., and the M.S. and Ph.D. degrees in electrical engineering, in 1989 and 1991, respectively, from Purdue University, West Lafayette, IN. He is now a Chair Professor of the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taiwan, R.O.C. He is an IEEE Fellow and CACS fellow. He has published more than 200 refereed journal and conference papers in the areas of robotics, intelligent control, fuzzy systems, neural networks, and non-derivative optimization. His current research interests include computational intelligence, machine learning, virtual reality simulation, intelligent transportation systems, smart home, robotics, and intelligent control. Dr. Su is very active in various international/domestic professional societies. He is now the president of International Fuzzy Systems Association. He also serves as Associate editors of IEEE Transactions on Cybernetics, IEEE Transactions on Fuzzy Systems, and IEEE Access, and the Editor-in-Chief of International Journal of Fuzzy Systems.

Yao-Chu Hsueh received the A.S. degree in electrical engineering from Oriental Institute of Technology, Taipei, Taiwan, in 2000, the B.S. degree in electrical engineering from Ta Hwa Institute of Technology, Hsinchu, Taiwan, in 2002, the M.S. degree in electronic engineering from China Institute of Technology, Taiwan, in 2004, and the Ph.D. degree in electrical engineering from National Taiwan University of Science and Technology, Taipei, in 2009.

Cio-Ping Tseng received the M.S. degrees from National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. in 2011. She current research interests include nonlinear control, fuzzy modeling and control, and robust adaptive control.

Song-Shyong Chen received the B.S. and M.S. degrees from the National Taiwan Institute of Technology, Taipei, Taiwan, R.O.C., and the Ph.D. degree in electrical engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., in 1992, 1994, and 2003, respectively. He is currently an Associate Professor with the Department of Electrical Engineering, Cheng Shiu University, Taiwan. His current research interests include nonlinear control, fuzzy modeling and control, and robust adaptive control.

Yu-San Lin is working in the Chung-Shan institute of Science and Technology, Taoyuan City, Taiwan, R.O.C. His current research interests include image processing and intelligent transportation systems.

July 2017, 17 (2)