Adaptive Fuzzy Tracking Control of Nonlinear Systems ... - IEEE Xplore

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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 46, NO. 6, JUNE 2016

Adaptive Fuzzy Tracking Control of Nonlinear Systems With Asymmetric Actuator Backlash Based on a New Smooth Inverse Guanyu Lai, Zhi Liu, Yun Zhang, and C. L. Philip Chen, Fellow, IEEE

Abstract—This paper is concentrated on the problem of adaptive fuzzy tracking control for an uncertain nonlinear system whose actuator is encountered by the asymmetric backlash behavior. First, we propose a new smooth inverse model which can approximate the asymmetric actuator backlash arbitrarily. By applying it, two adaptive fuzzy control scenarios, namely, the compensation-based control scheme and nonlinear decomposition-based control scheme, are then developed successively. It is worth noticing that the first fuzzy controller exhibits a better tracking control performance, although it recourses to a known slope ratio of backlash nonlinearity. The second one further removes the restriction, and also gets a desirable control performance. By the strict Lyapunov argument, both adaptive fuzzy controllers guarantee that the output tracking error is convergent to an adjustable region of zero asymptotically, while all the signals remain semiglobally uniformly ultimately bounded. Lastly, two comparative simulations are conducted to verify the effectiveness of the proposed fuzzy controllers. Index Terms—Actuator backlash, adaptive fuzzy control, compensation, smooth inverse, uncertain nonlinear systems.

I. I NTRODUCTION DAPTIVE control of the uncertain nonlinear systems using fuzzy logic systems (FLSs) and neural networks (NNs) has attracted tremendous attention over these latest decades, and many important works have been reported [1]–[29], [71]–[76]. Chen et al. [1] originally proposed an adaptive fuzzy controller that incorporated a minimal learning parameter (MLP) mechanism. Motivated by it, a series of state-feedback adaptive control scenarios were constructed in [2]–[7] successively. It is well known that the measure of system states in practice is usually costly

A

Manuscript received January 7, 2015; revised April 20, 2015; accepted June 8, 2015. Date of publication June 29, 2015; date of current version May 13, 2016. This work was supported in part by the National Natural Science Foundation of China under Project U1134004, in part by the Natural Science Foundation of Guangdong Province for Distinguished Young Scholars under Grant S20120011437, in part by the Ministry of Education of New Century Excellent Talent under Grant NCET-12-0637, in part by the 973 Program of China under Grant 2011CB013104, and in part by the Doctoral Fund of Ministry of Education of China under Grant 20124420130001. This paper was recommended by Associate Editor H. Zhang. G. Lai, Z. Liu, and Y. Zhang are with the School of Automation, Guangdong University of Technology, Guangzhou 510006, China (e-mail: [email protected]; [email protected]; [email protected]). C. L. P. Chen is with the School of Science and Technology, University of Macau, Macau 999078, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2443877

and tedious, so Tong et al. [8], [9] proposed a class of fuzzy filter observer-based controllers, and Boulkroune et al. [10] presented a novel strictly positive real (SPR) filter approach. Next, Boulkroune et al. [11] further designed a unified adaptive fuzzy observer without leaning on the SPR condition or the special observation error dynamics. Note that, the universal approximation abilities of the FLSs and NNs are only valid on a compact set [14]. To overcome the difficulty, Wu et al. [13] and Chen and Jiao [14] have proposed some globally adaptive neural or fuzzy control protocols. Furthermore, the pioneering work [16] was the first to consider the adaptive dynamic programming (ADP)-based optimal tracking control of an unknown general nonlinear system using the recurrent NN (RNN), and in [17], they originally proposed a new heuristic dynamic programming algorithm for the optimal tracking control of time-delayed nonlinear system. In [18], a new-type of stochastic Lyapunov–Krasovskii functional was initially proposed such that an efficient guaranteed cost controller was established for the time-delayed stochastic fuzzy systems. In addition, the works in [19]–[21], respectively, investigated the integral Lyapunov functional for multi-input and multioutput nonlinear systems, the Lyapunov–Krasovskii functional for time-delayed nonlinear systems, and the barrier Lyapunov functional for output-constrained nonlinear systems. More importantly, some successful applications of adaptive neural or adaptive fuzzy control in industrial systems have been shown in [25]–[29] and [78]. It must be stressed that the works mentioned above were commonly based on the ideal actuator whose input is always equal to its output. However, in practical actuator, there frequently exist four classes of hard (nondifferentiable) nonlinearities, namely the dead zone [55]–[59], the saturation [60]–[62], [77], the backlash [30]–[39], and the hysteresis [40]–[54]. Certainly, each of them will severely limit the system performance and even lead to the instability. But in contrast to the dead zone and saturation, the backlash nonlinearity imposes more challenging difficulties on the adaptive controller design due to the multivalued mapping and the rate-dependent mathematical description. So the suppression of the actuator backlash in control design has become one active researched topic in cybernetics. In [33], a decoupling control approach was designed to compensate for the sandwiched backlash encountered by the multivariable nonlinear systems. In [36], a fuzzy control mechanism without depending on a special backlash compensator

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LAI et al.: ADAPTIVE FUZZY TRACKING CONTROL OF NONLINEAR SYSTEMS WITH ASYMMETRIC ACTUATOR BACKLASH

was developed to compensate for the delay effect caused by the backlash. Zhou and Zhang [35] had originally proposed an efficient smooth inverse for the hard backlash characteristic, and a compensation control scheme based on it was then constructed. Instead of using the compensation strategy, the Liu et al. [31] directly regarded the backlash effect as a bounded disturbance-like term in nonlinear systems, and then a neural adaptive controller was constructed. Recently, Liu and Tong [30] have considered a class of discrete-time nonlinear systems with the nonsmooth backlash behavior, and a fuzzy approximation-based adaptive output tracking controller was developed to guarantee a good system performance. It is found that above results were established on a symmetric or at least semi-symmetric backlash model. Actually, the completely asymmetric backlash has ever been considered by Vörös [37], [38], but the works mainly concerned with its modeling and identification. To date, there have been few completed results available for the fuzzy adaptive control of uncertain nonlinear systems with the asymmetric actuator backlash. Inspired by above observations, this paper explores the adaptive fuzzy control of uncertain nonlinear system whose actuator is subjected to the asymmetric backlash effect. Essentially, one of the most difficult problems in the development of our control schemes is to seek an appropriate inverse model for the asymmetric actuator backlash, which is also the first step for the construction of the feedforward compensator. As the result of solving this challenge, two adaptive fuzzy controllers (compensation-based and nonlinear decompositionbased controllers) are developed to guarantee the asymptotic convergence of the tracking error to a small region of zero, and the semiglobal uniform ultimate boundedness of all signals in the closed-loop system. More contributions of the present work are epitomized as follows. 1) It is worth mentioning that an efficient smooth inverse for the symmetric backlash has been raised in [35], but it is not suitable to the asymmetric one. As an extension to it, a new smooth inverse model for the asymmetric actuator backlash is initially proposed in this paper. 2) The newly proposed smooth inverse model can effectively be linearly parameterized, and combining with a key result derived in Proposition 1, an adaptive feedforward compensator is constructed to compensate for the asymmetric backlash effect in actuator. With the fuzzy backstepping adaptive control design of compensator input, a new compensation-based controller (i.e., the compensator output) is developed in Theorem 1. 3) It is found that the compensation-based controller recourses to a known slope ratio of backlash model. To remove the restriction, this paper further proposes a nonlinear decomposition-based controller. First, with the new smooth inverse, we decompose the nonsmooth backlash model into a proper form. Then both significant properties on this decomposition are further derived in Propositions 2 and 3 so that it can be fused with available nonlinear control techniques (e.g., Nussbaum functional approach and robust mechanism). Consequently, a more

Fig. 1.

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Visualization of asymmetric actuator backlash.

universal fuzzy adaptive controller is constructed in Theorem 2. 4) Both of the proposed fuzzy adaptive controllers incorporate the FLSs as well as the MLP mechanism so that they are more efficient in practical systems. II. P RELIMINARIES AND P ROBLEM S TATEMENT A. System Model and Asymmetric Backlash Nonlinearity This paper considers a class of uncertain nonlinear systems preceded by the asymmetric backlash as follows: x˙ i (t) = fi (¯xi ) + xi+1 ,

i = 1, . . . , n − 1

x˙ n (t) = fn (¯xn ) + u u = B(v) y(t) = x1

(1)

where x¯ i = (x1 , . . . , xi )T ∈ Ri and x¯ n = (x1 , . . . , xn )T ∈ Rn are the state-space vectors. fi (¯xi ) : Ri → R and fn (¯xn ) : Rn → R are unknown continuous functions. In (1), y(t) denotes the system output. The real control signal to be designed is v(t), and control to the nonlinear plant is u(t) = B(v). Actually, the asymmetric backlash is the usual phenomenon in practical application and its mathematical description is given as u = B(v) ⎧ ⎪ ⎨ml (v − Bl ), if v˙ < 0 and u(t) = ml (v(t) − Bl ) = mr (v − Br ), if v˙ > 0 and u(t) = mr (v(t) − Br ) ⎪ ⎩ other cases u(t− ),

(2)

where ml and mr represent the left and right slopes, respectively. Bl < 0 and Br > 0 are two constants which collectively determine the width of nonsmooth backlash nonlinearity. All these parameters are not assumed to be measurable. In addition, the notation u(t− ) indicates that the actuator’s output u(t) does not vary at this instant. A graphical result of asymmetric actuator backlash is shown in Fig. 1. Remark 1: Many physical systems can be modeled by the form of (1), e.g., the one-link robot manipulator [31], the turret-barrel system [33], the chemical stirred tank reactor system [30], the inverted pendulum system [27], [43], [46],

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the mass-springer-damper system [11], [63], the Duffing oscillator [10], the Brusselator system [64], etc. Remark 2: Most of previous works dealing with the actuator backlash were based on the symmetric model (ml = mr and Br = |Bl |) or semi-symmetric model (ml = mr and Br = |Bl |) for the facilitation of the control design. In physical systems and devices, however, the backlash behavior is not always captured perfectly by these idealized model. The truth motivates us to consider the asymmetrical actuator backlash encountered by the uncertain nonlinear systems. The difficulties of handling the asymmetric actuator backlash lie in two aspects. First, it is not clear how to construct an asymmetric smooth inverse model which can effectively be fused with available adaptive techniques. Second, even if such inverse is possible to be found, it is still a challenging difficulty to establish an asymmetric backlash compensator due to the fact that all model parameters (ml , mr , Bl , and Br ) are unknown. This paper is intensively focused on these problems, and will solve them successively. Different from the symmetric backlash, it is found from Fig. 1 that the input v(t) of the asymmetric backlash actuator will be naturally saturated, i.e., the two unparallel lines will finally develop an intersection point as shown in Fig. 1. When the abscissa value of the intersection point is positive, we denote the value by M ∈ R+ . Otherwise, we denote it by −M. Assumption 1: It is assumed that the actual control signal v(t) does not reach the saturated point, i.e., it satisfies the constraint |v| ≤ M. Remark 3: Such condition is not restrictive. In the engineering control systems with backlash, it is right that the two slopes mr and ml are not equal, but they would be close up to a scale so that the value M or −M of the intersection point is often sufficiently large to cover the control input. Therefore, the constraint |v| ≤ M is appropriate in practical application. Also, it guarantees the controllability of the system (1). The desired trajectory is denoted by yr (t) whose time derivatives up to nth order are assumed to be continuous and bounded. The control goal is to design a proper control input v(t) such that the tracking error z1 (t) = y(t) − yr (t) is convergent to an adjustable neighborhood of zero, while all closed-loop signals remain semiglobally uniformly ultimately bounded (SGUUB). B. Fuzzy Logic Systems A completed FLS is mainly composed of a singleton fuzzifier, a product inference, a center-average defuzzifier, and a collection of fuzzy rules R(l) : IF x1 is F1l and · · · and xn is Fnl , THEN yF is Gl where x = [x1 , x2 , . . . , xn ]T ∈ Rn and yF are the input and output signals of the FLS, respectively. μFl (xi ) and μGl (yF ) are i the membership functions (MFs) of the fuzzy sets Fil and Gl , respectively. N is the number of the used fuzzy rules.


Then an FLS is expressed as N yF (x) =

n l l=1 θ i=1 μFil (xi ) N n l=1 i=1 μFil (xi )

(3)

where θ l is the point at which μGl (θ l ) = 1. The fuzzy MFs are commonly selected as Gaussian functions with the following exponential form: (xi − ali )2 (4) μFl (xi ) = exp − i bli where ali and bli are, respectively, the centers and widths of fuzzy MFs. Denote the fuzzy basis functions by n i=1 μF l (xi ) . (5) φi (xi ) = N n i l=1 i=1 μF l (xi ) i

Then the FLS in (3) can be re-expressed as yF (x) = θ T φ(x), where φ = [φ1 , . . . , φN ]T and θ = [θ1 , θ2 , . . . , θN ]T . If all the fuzzy MFs are selected as the Gaussian exponential form, then we have the following lemma. Lemma 1 [8]–[12], [74]: Let f (x) be a continuous function defined on a compact set 0 . Then for any positive constant ε > 0, there exists an FLS like (3) such that sup | f (x) − yF | ≤ ε.

(6)

x∈0

It should be pointed out that above universal approximation property only holds on a compact set, and that is why the results obtained are SGUUB [14]. C. Nussbaum-Type Function A function N(η) is defined as the Nussbaum-type function

k if it satisfies the conditions limk→±∞ sup 1/k 0 N(s)ds = ∞

k and limk→±∞ inf 1/k 0 N(s)ds = −∞. Actually, many functions satisfy these conditions, e.g., exp(η2 ) cos η, η2 cos η, etc. Lemma 2 [46]–[48], [59]: Let V(t) and η(t) be smooth functions defined on [t0 , tf ) with V(t) ≥ 0, ∀t ∈ [t0 , tf ). N(·) is an even smooth Nussbaum-type function. If the following inequality: t ˙ )e−h2 (t−τ ) dτ (7) V(t) ≤ h1 + (g(τ )N(η(τ )) + 1)η(τ t0

holds for ∀t ∈ [t0 , tf ), where h1 ∈ R and h2 ∈ R+ are some scalars, and g(t) is a bounded function whose value is within / l . Then it is concluded that V(t), l : [l− , l + ] for 0 ∈ t ˙ must be bounded on [t0 , tf ). ϑ(t), and t0 (g(t)N(ϑ) + 1)ϑdτ Moreover, tf can be extended to the infinity if the solution of the resulting closed-loop system is bounded [59]. III. T WO A DAPTIVE F UZZY C ONTROLLERS BASED ON N EW S MOOTH I NVERSE A. Novel Smooth Inverse of Asymmetric Actuator Backlash Essentially, the compensation for the actuator backlash depends on the construction of its inverse model. For the symmetric case, an efficient inverse model has been presented


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The adaptive compensator input is denoted by −1 (v) = ϑˆ T ωd (t) ud (t) = BI

(15)

where ωd (t) is given by ωd (t) =

1 (v, Xr (˙ud ), Xl (˙ud ))T . Xr (˙ud ) + rXl (˙ud )

(16)

Then the adaptive compensation error is derived as d(t) = B(v) − BI

Fig. 2. Visualization of the new smooth inverse model and its approximation.

in [35]. Motivated by it, this paper further proposes a new smooth inverse model for asymmetric backlash as follows: v = BI(us )

1 1 us + Br Xr (˙us ) + us + Bl Xl (˙us ) = mr ml

(8)

where Xr (˙us ) and Xl (˙us ) are two positive adjustable functions defined as 1 + tanh(κ u˙ s ) Xr (˙us ) = (9) 2 1 − tanh(κ u˙ s ) . (10) Xl (˙us ) = 2 κ is a positive adjustable parameter. Moreover, a graphic result of such inverse model is depicted in Fig. 2. Note that, the larger the adjustable parameter κ is, the smaller the approximation error will be.

us = ϑ T ωs (t)

(11)

where ϑ and ωs (t) are

1 (Xr (˙ud ) + rXl (˙ud ))ud − ϑˆ rlT Xrl (˙ud ) m ˆ T = − mBr , − mBl , Xrl (˙ud ) = [Xr (˙ud ), Xl (˙ud )]T 1 zn T ˆ = −kn zn − 2 zn Wn φ (Zn )φ(Zn ) − ˆn tanh ςn 2an 1 z ˆ i φ T (Zi )φ(Zi ) − î tanh i = −ki zi − 2 zi W ςi 2ai = xi − αi−1 i = 1, . . . , n and α0 = yr (t)

v= ϑˆ rl ud αi zi

(17)

(18) (19) (20) (21) (22)

and construct the following adaptation rules: ˙ˆ = ri z2 φ T (Z )φ (Z ) − σ W W i i i i 0i ˆ i i i 2a2i ˆ˙i = zi i tanh(zi /ςi ) − k0i î ϑ˙ˆ = γ z ω − σ ϑˆ n d

(23) (24) (25)

where ai , ki , ri , i , ςi , and γ are the positive constants. −1 ∈ R3×3 is a symmetric positive-definite matrix. σ0i , k0i , and σ are the positive modification parameters. Z1 = [y, yr , y˙ r ]T ∈ R3 ˆ 1, . . . , W ˆ i−1 , yr , . . . , yr(i) ]T ∈ and Zi = [¯xiT , ˆ1 , . . . , î−1 , W 4i−1 for i = 2, . . . , n. Then the tracking error is asympR totically convergent to an adjustable region of zero, while all the signals in the closed-loop system remain SGUUB. Proof: Consider a quadratic Lyapunov function as V=

ϑ = (m, −mBr , −mBl ) (12) 1 ωs (t) = (13) (v, Xr (˙us ), Xl (˙us ))T . Xr (˙us ) + rXl (˙us ) Due to that the backlash parameters ϑ are unknown in practice, we employ the adaptive technique to estimate them, that is T ˆ ϑ(t) = m, − mBr , − mBl . (14)

(v) = ec + ϑ˜ T ωd

where ϑ˜ T ωd is the linearly parameterizable term, and ec = B(v) − ϑ T ωd is called as the unparameterizable error. Proposition 1: There exists a sufficiently large adjustable parameter κ in (8) such that the unparameterizable error ec (t) = B(v) − ϑ T ωd remains bounded. The detailed proof is given in the Appendix. Theorem 1: Consider the closed-loop system consisting of nonlinear plant (1). Suppose that Assumptions 1 and 2 hold. If we design the control law as

B. Scheme I: Compensation-Based Controller It is noticeable that the parameters of the backlash model are unknown so that the newly proposed inverse model v = BI(us ) in (8) cannot directly serve as a compensator. Then it needs to construct an adaptive mechanism to estimate the unknown backlash parameters online. Assumption 2: It is assumed that the slopes ratio r = mr /ml of the asymmetric backlash model is a known constant. The known fixed relation is important to reparameterize the smooth inverse model v = BI(us ) so that an adaptive mechanism can be established. For the simplicity, denote a reference slope by m = mr . Then (8) is linearly parameterized as

−1

T

Va =

n 1 i=1 n i=1

2

z2i + Va

(26)

1 2 1 2 1 T −1 ˜i + ϑ˜ ϑ˜ ˜i + W 2ri 2i 2γ n

(27)

i=1

˜ i = Wi − W î where ri , γ , and i are the positive scalars. W and ˜i = i − î . Wi and i are the unknown constants given latter. For the consistence, we define u = αn = B(v) and

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z0 = zn+1 = 0. Then the time derivative of V(t) is yielded V˙ =

n

zi (xi+1 + fi (¯xi ) − α˙ i−1 ) + V˙ a

i=1

= z1 (z2 + α1 + f1 (¯x1 ) − y˙ r ) ⎛ n i−1 ∂αi−1 + xj+1 + fj x¯ j zi ⎝zi+1 + αi + fi (¯xi ) − ∂xj i=2

j=1

i−1 i−1 ∂αi−1 ˙ ∂αi−1 ˙ ˆj− ˆj W ˆ ∂ ˆj ∂ Wj j=1 j=1 ⎞ i−1 ∂αi−1 ( j+1) ⎠ − y + V˙ a . ( j) r j=0 ∂yr

−

(28)

It is observed that ϕi (Zi )(i = 1, . . . , n) are unknown due to the existence of the system uncertainties fi (¯xi ). We employ the FLSs to approximate them, i.e., ϕi (Zi ) = ψiT φi (Zi ) + δi (Zi ), where ψi is the idealized fuzzy weight and φi (Zi ) ∈ RNi is the known function vector. δi (Zi ) is the approximation error. To optimize the adaptation mechanism, we define Wi = ||ψi ||2 , where || · || is the Euclidean norm. Invoking Lemma 1 and Proposition 1, we know |δi (Zi )| ≤ i (i = 1, . . . , n − 1) and |δn (Zn ) + ec (t)| ≤ n . Then (28) is derived as zi (zi+1 − zi−1 + αi + ϕi (Zi )) + V˙ a

i=1 n 1 2 T ≤ zi (αi + δi (Zi )) + z ψ ψ φ T (Zi )φi (Zi ) 2 i i i i 2a i i=1 i=1

i=1

≤

n−1

2

a2i + V˙ a

zi (αi + i tanh(zi /ςi )) +

i=1

n z2i i=1

2a2i

Wi φiT (Zi )φi (Zi )

n n 1 2 a + zn ϑ˜ T ωd + (|zi |i − i zi tanh(zi /ςi )) + 2 i i=1

i=1

n 1 ˙ˆ ˜ iW + zn (ud + n tanh(zn /ςn )) − W i ri i=1

n 1 ˙ 1 ˙ˆ − ˜i î − ϑ˜ T −1 ϑ. i γ i=1

By using the adaptation laws (23)–(25), we have n k0i 2 σ ϑ˜ T ϑ˜ ˜i − −1 γ 2i 2λ max i=1 i=1 n n σ0i 2 σ0i 2 k0i 2 σ T 1 ˜i + W + + W ϑ ϑ + a2i − 2ri 2ri i 2i i 2γ 2 i=1 i=1

+ 0.2785i ςi ≤ −K V + R (33)

V˙ ≤ −

n

ki z2i −

where K and R are indirect designed parameters given by K = min 2ki , k0i , σ0i , σ/λmax −1 (34)

n σ0i k0i 2 σ T 1 R= ϑ ϑ + a2i + 0.2785i ςi . Wi2 + i + 2ri 2i 2γ 2 i=1

(35) By implementing the integral operation of the differentiation inequality (33), we have V(t) ≤ V(t0 )e−K (t−t0 ) +

lim z2 t→∞ 1

n

+

ki z2i

R 1 − e−K (t−t0 ) K

(36)

where t0 is an initial moment. Invoking (26), we conclude the following result:

i=1 n = zi αi + ψiT φi (Zi ) + δi (Zi ) + V˙ a

n 1

n

(32)

(29) ϕ1 (Z1 ) = f1 (¯x1 ) − y˙ r i−1 fi (¯xi ) ∂αi−1 ϕi (Zi ) = − fj x¯ j i−1 ∂xj j=1 ∂αi−1 ∂αi−1 ˙ ∂α ˆ j + i−1 ˙ˆj − xj+1 + W ˆ ∂xj ∂ ˆj ∂ Wj ∂αi−1 + ( j) yr( j+1) (30) + zi−1 . ∂yr

V˙ =

n 1 − ˜i zi i tanh(zi /ςi ) − ˙î i i=1 i=1 n 1 ri 2 T ˙ î ˜i zi φi (Zi )φi (Zi ) − W W − ri 2a2i i=1

n 1 1 2 + ϑ˜ T γ zn ωd − −1 ϑ˙ˆ + ai + 0.2785i ςi . γ 2

V˙ ≤ −

i=1

Denote the combined nonlinear functions by

n

Using the inequality |zi | − zi tanh(zi /ςi ) ≤ 0.2785ςi and the controller in (18)–(22), the inequality (31) is formulated as

(31)

≤

2R . K

(37)

Therefore, the tracking error is convergent to an adjustable ˜ i , ˜i , and region of zero. From (36), it is concluded that zi , W ˆ and the virtual controllers αi ˆ i , î , ϑ, ϑ˜ are bounded so that W are also bounded. Then the boundedness of the system states x1 , . . . , xn are recursively obtained. Remark 4: In practical implementation of the compensation control scheme, it is worth mentioning that some algorithms (see [37], [38]) for identifying the slope ratio of backlash model should be included due to Assumption 2. Note that, these identification methods have to depend on both actuator input and output. In some physical backlash actuators, however, the actuator output is usually difficult to be measured, which also results in the unavailability of slope ratio. To remove the restriction in Assumption 2, another fuzzy adaptive controller will be proposed in next section.


C. Scheme II: Nonlinear Decomposition-Based Controller We will establish a nonlinear decomposition-based fuzzy controller which is still on the basis of the newly proposed smooth inverse model of (8). Note that, the inverse v = BI(us ) can nonlinearly be decomposed into the following form: us =

ml mr (Br Xr (˙us ) + Bl Xl (˙us )) ml mr v− . ml Xr (˙us ) + mr Xl (˙us ) ml Xr (˙us ) + mr Xl (˙us ) (38)

For the simplicity, we denote ml mr c1 (t) = ml Xr (˙us ) + mr Xl (˙us ) ml mr (Br Xr (˙us ) + Bl Xl (˙us )) c2 (t) = − . ml Xr (˙us ) + mr Xl (˙us )

n 1 i=1 n i=1

2

z2i + Vb

(48)

1 2 1 2 ˜ + ˜ . W 2ri i 2i i n

(49)

i=1

By differentiating V(t), and repeating (28)–(31), we have the following result: V˙ ≤

n

zi (αi + δi (Zi )) +

i=1

(41)

We give two propositions on above nonlinear decomposition. Proposition 2: In (41), c1 (t) is a positive function and there exists a sufficiently large adjustable parameter κ such that both c1 (t) and c2 (t) are bounded. The proof is presented in the Appendix. This proposition reveals that the uncertain control input gain c1 (t) is bounded and nonzero so that the Nussbaum functional approach in Lemma 1 can be employed in controller design. Proposition 3: The approximation error es = u − us of the smooth inverse model to the asymmetric backlash nonlinearity is bounded. The proof of this Proposition 3 is similar to that in Proposition 1. So it is omitted here due to the limit space. Remark 5: With the established nonlinear decomposition in (41) and Propositions 2 and 3, we further propose a new fuzzy adaptive control scheme without using Assumption 2. Theorem 2: Consider the closed-loop system consisting of the nonlinear plant (1). Suppose that Assumption 1 holds. If we design the adaptive fuzzy controller as ˆn T zn W zn φ (Zn )φ(Zn ) + ˆn tanh (42) v = N(η) kn zn + 2 ςn 2an 1 zi T ˆ (43) αi = −ki zi − 2 zi Wi φ (Zi )φ(Zi ) − î tanh ς 2ai i zi = xi − αi−1 i = 1, . . . , n and α0 = yr (t) (44) and construct the adaptive mechanism as ˆn T zn W zn η˙ = γ zn kn zn + φ (Zn )φ(Zn ) + ˆn tanh ςn 2a2n ˙ˆ = ri z2 φ T (Z )φ (Z ) − σ W W i i i i 0i ˆ i i i 2a2i ˙î = zi i tanh(zi /ςi ) − k0i î

V=

(39)

So (38) can be written as us (t) = c1 (t)v + c2 (t).

ˆ i and î are the online estimaIn Theorem 2, W tions of Wi and i , respectively. Wi (i = 1, . . . , n) and i (i = 1, . . . , n − 1) are consistent with those in scheme I. In particular, n is denoted by the upper bound of es (t) + c2 (t) + δn (Zn ). Proof: A quadratic Lyapunov function is selected as

Vb =

(40)

1255

+

n 1

2

i=1

≤

n−1

a2i + V˙ b

zi (αi + δi (Zi )) +

i=1

+

n 1

2

i=1

n 1 2 T z ψ ψ φ T (Zi )φi (Zi ) 2 i i i i 2a i i=1

n 1 2 T z ψ ψ φ T (Zi )φi (Zi ) 2 i i i i 2a i i=1

a2i + zn (es (t) + c1 (t)v + c2 (t) + δn (Zn )) + V˙ b

n 1 ˜i zi i tanh(zi /ςi ) − ˙î i i=1 i=1 n 1 ri 2 T ˙ î ˜i z φ (Zi )φi (Zi ) − W − W 2 i i ri 2a i i=1

n 1 1 2 + (c1 (t)N(η) + 1)η˙ + a + 0.2785i ςi . γ 2 i

≤−

n

ki z2i −

(50)

i=1

Using the adaptation laws in (45)–(47), we have V˙ ≤ −

n

ki z2i −

i=1

+

n σ0i i=1

+

2ri

n n k0i 2 σ0i 2 ˜ ˜i − W 2i 2ri i i=1

Wi2

i=1

k0i 2 1 2 + + ai + 0.2785i ςi 2i i 2

1 (c1 (t)N(η) + 1)η˙ γ

(51)

where K and R are given by (45) (46) (47)

where N(η) is the Nussbaum-type function which is selected 2 as eη cos η or η2 cos η. The FLS inputs Zi (i = 1, . . . , n) are defined the same as those in Theorem 1. Then the tracking error is convergent to a small neighborhood of zero asymptotically, and all closed-loop signals remain SGUUB.

(52) K = min{2ki , k0i , σ0i }

n σ0i 2 k0i 2 1 2 R= W + + ai + 0.2785i ςi . (53) 2ri i 2i i 2 i=1

Implementing the integral operation of (51), one further has R V(t) ≤ V(t0 )e−K (t−t0 ) + 1 − e−K (t−t0 ) K 1 t + (c1 (τ )N(η(τ )) + 1)η(τ ˙ )e−K (t−τ ) dτ. (54) γ t0 Invoking Lemma 2, it is concluded that V(t) is bounded at [t0 , t) and no finite-time escape phenomenon occurs,

1256


TABLE I D ESIGNED PARAMETERS OF C ONTROL S CHEMES I AND II (E XAMPLE 1)

which implies that the time can be extended to the infinity, t → ∞. Let P be the upper bound of the term

i.e., t (c ˙ )e−K (t−τ ) dτ . Then we obtain t0 1 (τ )N(η(τ )) + 1)η(τ lim z21 ≤

t→∞

2P 2R + . K γ

(55)

Therefore, the asymptotic convergence of the tracking error to an adjustable neighborhood of zero is guaranteed. Remark 6: It is observed from (34)–(37) and (52)–(55) that the size of the tracking error definitely depends on the designed parameters. By increasing the parameters ki , k0i , σ0i , σ , ri , γ , and i , while reducing ai and ςi , a smaller tracking error is guaranteed by the proposed fuzzy controllers. Remark 7: It is worth mentioning that the proposed two fuzzy controllers have the theoretical differences. In the first scheme, with the linearly parameterizable property of the inverse model and the key result of Proposition 1, an adaptive feedforward compensator is developed to compensate for the backlash effect in actuator. By contrast, the second scheme directly decomposes the nonsmooth backlash into a proper form which has the properties of Propositions 2 and 3, and then a robust control mechanism is established to suppress the actuator backlash without depending on a certain compensator. In addition, in comparison with the first control scheme, the second one further removes the restriction in Assumption 2. Remark 8: This paper differs from [39] in the following three aspects. First, [39] considered the Brunovsky nonlinear system which is a special case of the one considered in this paper. Second, unlike [39], this paper proposes a new smooth backlash inverse model. Third, the proposed controllers does not assume the system uncertainties to be linearly parameterizable. Remark 9: It is worth mentioning that the tracking control performance of the proposed controllers largely depends on the selection of fuzzy MFs. One way for their selection is based on the expert experiences from practical engineering systems, as considered in this paper. Another solution, which may be more promising and innovative, is to fuse some optimization algorithms, e.g., genetic algorithm, particle swarm optimization, and ant colony optimization to search the optimal values for fuzzy MFs (exactly for their centers and widths) so that the quality of the proposed fuzzy controllers is further improved.

IV. S IMULATION S TUDY AND P ERFORMANCE R ESULTS Two simulations will be conducted to testify the effectiveness of the proposed fuzzy controllers in Section III.

Example 1: Consider an uncertain nonlinear systems preceded by the asymmetric backlash characteristic as x13 2 x˙ 1 = x2 + 0.1x1 e−0.5x1 cos(x1 ) + 1 + x12 x14 sin(x2 ) 0.2 x˙ 2 = B(v) + x2 sin + 1 + x12 1 + x12 y = x1

(56)

where u = B(·) is the asymmetric backlash model with the parameters Bl = −3, Br = 4, and r = 1.05. The reference signal is given by yr (t) = sin(t). Subsequently, the proposed two control schemes in Section III are constructed and applied to the nonlinear system in (56). Note that, these two control scenarios are commonly based on the FLSs. Then the fuzzy MFs for Z1 ∈ R3 and Z2 ∈ R7 are given as μFp1 (x1 ) = exp [(x1 − 4 + 2p1 )2 /2] 1

μFp2 (x2 ) = exp [−(x2 − 4 + 2p2 )2 /4] 2

μFp3 (yr ) = exp [−(yr − 4 + 2p3 )2 /2] 3

μFp4 (˙yr ) = exp [−(˙yr − 2 + p4 )2 /4] 4

ˆ 1 ) = exp [−(W ˆ 1 + 4.5 − 1.5p5 )2 /2] μFp5 (W 5

μFp6 (ˆ1 ) = exp [−(ˆ1 + 4.5 − 1.5p6 )2 /1.5] 6

(57)

where pi = 1, . . . , 3 for i = 1, . . . , 6. Then the fuzzy basis function vectors φ1 (Z1 ) and φ2 (Z2 ) are constructed according to (5). It is found that φ1 (Z1 ) and φ2 (Z2 ) are involved in 27 and 729 fuzzy rules, respectively. If we directly use traditional learning mechanism to estimate the fuzzy weights, there will be 756 differentiation equations need to be computed, which may triggers a large online learning time. To resolve this problem, an MLP mechanism is incorporated in both fuzzy controllers so that there are only two update laws ˙ˆ ) left regardless of the increase of fuzzy rules. The ˙ˆ and W (W 1 2 nonlinear system in (56) is initialized as y(t0 ) = x1 (t0 ) = 0.5, x2 (t0 ) = 0.4, v(t0 ) = 0, and u(t0 ) = 0. A. Control Scheme I Based on Theorem 1 and Table I, a compensation-based fuzzy adaptive controller is constructed with the following ˆ 2 (t0 ) = 0.5, ˆ 1 (t0 ) = 0.4, W initial adaptive parameters W ˆ 0 ) = [1, −4, 3]T . ˆ1 (t0 ) = 0.5, ˆ2 (t0 ) = 0.8, and ϑ(t Applying it to the simulated nonlinear system, the performance results are obtained in Fig. 3(a)–(f). In Fig. 3(a), it is clear that the system output can well track the desired trajectory, while the corresponding tracking error in Fig. 3(b) is convergent to a small region (about [−0.02, 0.02]) of the zero after 10 s. Fig. 3(c) presents the profiles of the actuator input (control signal) and actuator output, respectively.


(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

(e)

(e)

(f)

(f)

1257

Fig. 3. Performance results of the nonlinear system under the control of proposed scheme I (compensation-based fuzzy adaptive controller). (a) Tracking trajectories. (b) Tracking error. (c) Actuator input and output. (d) System state. (e) Adaptive parameters related to neural networks. (f) Adaptive parameters related to robust mechanisms.

Fig. 4. Performance results of the nonlinear system under the control of the proposed scheme II (nonlinear decomposition-based controller). (a) Tracking trajectories. (b) Tracking error. (c) Actuator input and output. (d) Nussbaumtype functional. (e) System state. (f) Adaptive parameters related to robust mechanisms.

It is worth mentioning that a minor chattering exists in the control signal, which is caused by the actuator backlash. However, it does not affect the tracking performance as shown in Fig. 3(a) and (b). Fig. 3(d) shows the trajectory of the asymmetric actuator backlash, and such result is corresponding to Fig. 3(c). Moreover, it is observed from Fig. 3(c) that the control input v(t) does not reach the saturated point, which numerically proves the reasonability of Assumption 1. From the simulated result in Fig. 3(d), it is validated that the system state x2 remains bounded. In Fig. 3(e) and (f), the adaptive ˆ i and î are, respectively, searching for the ideparameters W alized fuzzy weights and robust upper bounds online so that a well tracking control performance is guaranteed.

B. Control Scheme II According to Theorem 2 and Table I, a nonlinear decomposition-based controller is established with the ˆ 1 (t0 ) = 0.4, following initial adaptive parameters W ˆ 2 (t0 ) = 12, ˆ1 (t0 ) = 0.1, ˆ2 (t0 ) = 0.3, and η(t0 ) = 0.1. The W initial system states and FLSs chosen are the same as those used in scheme I. By applying the developed controller to the nonlinear system in (56), some simulated results are obtained and recorded in Fig. 4(a)–(f). Fig. 4(a) and (b) shows that the output tracking error is convergent to a small neighborhood of the origin asymptotically, which verifies the conclusion in Theorem 2. In Fig. 4(c), it is found that the control signal has a somewhat large amplitude, and this is caused by the use of

1258


TABLE II D ESIGNED PARAMETERS OF C ONTROL S CHEMES I AND II (E XAMPLE 2)

Fig. 5.

Single-link robot arm with a gearing connection.

(a)

the Nussbaum-type functional which can accelerate the convergence of tracking error. As shown in Fig. 4(d), the Nussbaum functional will converge to a proper constant eventually, and always remains bounded. Fig. 4(e) and (f) depicts the profiles of the realtime system state and the robust adaptive parameters, respectively. Remark 10: The comparative simulations conducted above prove that both of the proposed fuzzy controllers in Theorems 1 and 2 can suppress the asymmetric actuator backlash effect, and guarantee the good tracking performance. In comparison with the first scheme, the second one does not require a known slope ratio of the backlash model, but it needs a larger control amplitude. Note that, both controllers consist of an approaching phase appearing at the startup of the system. How to eliminate it relies on the design of a more proper error surface, and we keep this problem as the work to be investigated in future. Note that, the backlash characteristic exists in a wide range of the mechanical connections, e.g., the gearing box. Subsequently, we further consider a practical engineering system. Example 2: Consider a robot manipulator in the following Fig. 5. Its gearing connection is described by an asymmetric backlash model, and it is well known that its dynamics has the following Lagrangian equation form [24]:

(b)

(c)

(58)

Fig. 6. Tracking control performance of the single-link robot manipulator. (a) Position tracking of the robot arm system. Actuator input and output of control (b) scheme I and (c) scheme II.

where q and q˙ represent the position and velocity of the link, respectively. J is the rotational inertia of the servo motor, and D is the damping coefficient. M is the mass of the object. l represents the distance from the axis of joint to the center of the mass. Their simulated values are specified as J = 1, D = 2, M = 2, and l = 1. The backlash parameters are ml = 1, mr = 1.1, Bl = −3, and Br = 4. In (58), g denotes the gravitational acceleration. In the simulation, the control task is to drive the servo motor using the proposed two fuzzy adaptive controllers such that the output position q(t) tracks a desired curve yr (t) = sin(t). The robot arm system is initialized as q(t0 ) = 0.3, q˙ (t0 ) = 0.1, v(t0 ) = 0, and u(t0 ) = 0. To construct the proposed adaptive controllers, we give the designed

parameters in Table II, and select the initial adaptive parameˆ 2 (t0 ) = 0.5, ˆ1 (t0 ) = 0.5, ˆ2 (t0 ) = 0.8, ˆ 1 (t0 ) = 0.4, W ters as W ˆ 0 ) = [1, −4, 3]T . Some comparative η(t0 ) = 0.1, and ϑ(t performance results are, respectively, recorded in Fig. 6(a)–(c). It is observed that both fuzzy adaptive controllers guarantee a good position tracking performance. Remark 11: It is noted that a minor chattering exists in the trajectory of the control input v(t), which is mainly produced to suppress the nonsmooth backlash effect in actuator, and as the result of suppression, the actuator output u(t) is basically smooth so that the position tracking control performance is well guaranteed.

J q¨ + D˙q + Mgl sin(q) = B(v),

u = B(v)


V. C ONCLUSION This paper presents a new smooth inverse for asymmetric actuator backlash model, and based on it, two fuzzy adaptive controllers are developed successively. It has been proven from the Lyapunov theory and the comparative simulations that both control schemes guarantee a good tracking performance. However, there are still many challenges to be considered in next step. 1) It is not clear how to construct an MLP mechanism under the output-feedback control framework instead of the statefeedback one considered in this paper. 2) Note that, the results obtained in this paper are SGUUB. It would be more valuable to extend them to be GUUB. A PPENDIX In this Appendix, we prove the Propositions 1 and 2 in closer detail. A. Proposition 1 Proof: To prove Proposition 1, the first step is to reparameterize the asymmetric backlash model u = B(v) in (2) as the following form: B(v) = mr (v − Br )ξr (t) + ml (v − Bl )ξl (t) + ξp (t)up

(59)

where the switch functions ξl (t), ξr (t), and ξp (t) are defined as 1, if u˙ > 0 1, if u˙ < 0 ξl (t) = ξr (t) = 0, other cases 0, other cases 1, if u˙ = 0 ξp (t) = (60) 0, other cases. In addition, invoking Assumption 1, we conclude r=

M − sgn(r − 1)Bl M − sgn(r − 1)Br

(61)

where sgn(·) is the sign function. Case 1: If u˙ > 0 (i.e., ξl = 0, ξr = 1, and ξp = 0), then we obtain ec (t) = B(v) − ϑ T ωd = mr (v − Br ) − ϑ T ωd

1 mv + = 1− Xr (˙ud ) + rXl (˙ud )

Xl (˙ud ) Xr (˙ud ) mBl + − 1 mBr . X (˙u ) + rXl (˙ud ) Xr (˙ud ) + rXl (˙ud ) r d ! "

(62)

B1

Clearly, there exists a sufficiently large parameter κ such that the term B1 is bounded and ⎧ ⎪ if u˙ d > 0 ⎨1 + ε1 (t) 1 = 1/r + ε2 (t) (63) if u˙ d < 0 ⎪ Xr (˙ud ) + rXl (˙ud ) ⎩ 2/(r + 1) if u˙ d = 0 where ε1 (t) and ε2 (t) can approach to zero arbitrarily by increasing the adjustable parameter κ. Denote the upper bound of B1 by B¯ 1 . Subsequently, we prove the boundedness of the first term in (62).

1259

1) When u˙ d > 0, by combing with the result |v| ≤ M in Assumption 1, it is derived from (62) and (63) that |ec (t)| ≤ |ε1 mM| + B¯ 1 .

(64)

Thus, ec (t) is bounded under the case u˙ d > 0. 2) When u˙ d < 0, the unparameterizable error is deduced as # #

# # 1 |ec (t)| ≤ ## 1 − − ε2 mM ## + B¯ 1 r # # # (Br − Bl )mM # # + |ε2 mM| + B¯ 1 . (65) # ≤# M − sgn(r − 1)B # l

Thus, ec (t) is bounded under the case u˙ d < 0. 3) When u˙ d = 0, it derives # #

# # 2 mM ## + B¯ 1 |ec (t)| ≤ ## 1 − r+1 # # # # (Br − Bl )mM # + B¯ 1 . # ≤# 2M − sgn(r − 1)(Br + Bl ) #

(66)

Under the case u˙ d = 0, ec is also bounded. Case 2: If u˙ < 0 (ξl = 1, ξr = 0, and ξp = 0), by following the similar process of case 1, it is not hard to prove the boundedness of the unparameterizable error ec (t). Case 3: If u˙ = 0 (ξl = 0, ξr = 0, and ξp = 1), then

mv ec (t) = up (t) − Xr (˙ud ) + rXl (˙ud ) Xr (˙ud ) Xl (˙ud ) mBl + mBr . + X (˙u ) + rXl (˙ud ) X (˙u ) + rXl (˙ud ) r d ! r d " B0

(67) For a sufficiently large κ, it is clear that B0 is bounded, and denote its upper bound by B¯ 0 . Similar to the case 1, it is also required to prove the boundedness of the first term in (67). 1) When u˙ d > 0, it is obtained from (63) and (67) that # # |ec (t)| ≤ #up − (1 + ε1 )mv# + B¯ 0 ≤ |up − mv| + |ε1 mM| + B¯ 0 .

(68)

It is found that mr (v − Br ) ≤ up ≤ ml (v − Bl ). Invoking the slope ratio r in (61), we further have # # $ % # (Br − Bl )mM # 1 # + |mBl | |ec (t)| ≤ max |mBr |, ## M − sgn(r − 1)Bl # r + |ε1 mM| + B¯ 0 . (69) Then it is concluded that ec (t) is bounded when u˙ d > 0. 2) When u˙ d < 0, from (67), it is derived # # $ % # (Br − Bl )mM # 1 # # + |mBr | |ec (t)| ≤ max |mBl |, # r M − sgn(r − 1)B # l

+ |ε2 mM| + B¯ 0 .

(70)

Then it knows that ec (t) is also bounded when u˙ d < 0. 3) When u˙ d = 0, we have # # # (Br − Bl )mM # # # |ec (t)| ≤ # M − sgn(r − 1)Bl # $ % 1 + max |mBr |, |mBl | + B¯ 0 . (71) r Then ec (t) is bounded when u˙ d = 0.

1260


B. Proposition 2 Proof: Three cases should be considered. Case 1: If u˙ s = 0, then c1 (t) and c2 (t) are specified as 2ml mr ml + mr ml mr (Bl + Br ) c2 (t) = − . ml + mr Clearly, the Proposition 2 holds under this case. Case 2: If u˙ s > 0, the functions c1 (t) and c2 (t) are c1 (t) =

mr mr Xl (˙us ) − c1 (t) Xr (˙us ) ml Xr (˙us )

mr Bl Xl (˙us ) c2 (t)mr Xl (˙us ) . + c2 (t) = −mr Br − Xr (˙us ) ml Xr (˙us )

c1 (t) =

(72) (73)

(74) (75)

From the definitions of Xr (˙us ) and Xl (˙us ), it knows that Xr (˙us ) → 1 and Xl (˙us ) → 0 as κ → +∞ when u˙ s > 0. For any > 0, there exists a sufficiently large κ such that 0 < mr − ≤ c1 (t) ≤ 2mr +

(76)

−mr Br − ≤ c2 (t) ≤ −mr Br + .

(77)

Case 3: If u˙ s < 0, it is observed that Xr (˙us ) → 0 and Xl (˙us ) → 1 as κ → +∞. Then c1 (t) and c2 (t) are expressed ml ml Xr (˙us ) − c1 (t) Xl (˙us ) mr Xl (˙us )

ml Br Xr (˙us ) c2 (t)ml Xr (˙us ) + . c2 (t) = −ml Bl − Xl (˙us ) mr Xl (˙us )

c1 (t) =

(78) (79)

Thus, for any > 0, there also exists a sufficiently large parameter κ such that 0 < ml − ≤ c1 (t) ≤ 2ml +

(80)

−ml Bl − ≤ c2 (t) ≤ −ml Bl + .

(81)

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Guanyu Lai received the B.S. degree in electrical engineering and automation and the M.S. degree in control science and engineering from the Guangdong University of Technology, Guangdong, China, in 2012 and 2014, respectively, where he is currently pursuing the Ph.D. degree with the Department of Automation. His current research interests include neural networks control, adaptive fuzzy control, image-based visual servoing, and mobile robotics.

Zhi Liu received the B.S. degree from the Huazhong University of Science and Technology, Wuhan, China, in 1997, the M.S. degree from Hunan University, Changsha, China, in 2000, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2004, all in electrical engineering. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou, China. His current research interests include fuzzy logic systems, neural networks, robotics, and robust control.

Yun Zhang received the B.S. and M.S. degrees in automatic engineering from Hunan University, Changsha, China, in 1982 and 1986, respectively, and the Ph.D. degree in automatic engineering from the South China University of Science and Technology, Guangzhou, China, in 1998. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou. His current research interests include intelligent control systems, network systems, and signal processing.

C. L. Philip Chen (S’88–M’88–SM’94–F’07) received the M.S. degree from the University of Michigan, Ann Arbor, MI, USA, in 1985, and the Ph.D. degree from Purdue University, West Lafayette, IN, USA, in 1988, both in electrical engineering. He was a tenured professor, the Department Head, and an Associate Dean in two different universities at U.S. for 23 years. He is currently the Dean with the Faculty of Science and Technology and the Chair Professor with the Department of Computer and Information Science, University of Macau, Macau, China. His current research interests include systems, cybernetics, and computational intelligence. Dr. Chen has been the Editor-in-Chief of the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART A: S YSTEMS AND H UMANS since 2014 and an Associate Editor of several IEEE T RANSACTIONS. He is the Chair of TC 9.1 Economic and Business Systems of International Federation of Automatic Control. He is a fellow of the American Association for the Advancement of Science.