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Neural Controller Design-Based Adaptive Control for Nonlinear MIMO Systems With Unknown Hysteresis Inputs Yan-Jun Liu, Shaocheng Tong, C. L. Philip Chen, Fellow, IEEE, and Dong-Juan Li

Abstract—This paper studies an adaptive neural control for nonlinear multiple-input multiple-output systems in interconnected form. The studied systems are composed of N subsystems in pure feedback structure and the interconnection terms are contained in every equation of each subsystem. Moreover, the studied systems consider the effects of Prandtl–Ishlinskii (PI) hysteresis model. It is for the first time to study the control problem for such a class of systems. In addition, the proposed scheme removes an important assumption imposed on the previous works that the bounds of the parameters in PI hysteresis are known. The radial basis functions neural networks are employed to approximate unknown functions. The adaptation laws and the controllers are designed by employing the backstepping technique. The closed-loop system can be proven to be stable by using Lyapunov theorem. A simulation example is studied to validate the effectiveness of the scheme. Index Terms—Adaptive control, intelligent control, neural networks (NNs), nonlinear control theory, Prandtl–Ishlinskii (PI) hysteresis inputs.

I. I NTRODUCTION HE NEURAL networks (NNs) and the fuzzy logic systems (FLSs) as the intelligent methods have been widely applied in many fields and received much attention. For instance, in [1]–[3], the authors proposed an efficient attraction domain segmentation method to analyze the multistability of recurrent NNs with time-varying delays. A novel stabilization control problem of Markovian stochastic jump systems against sensor fault, actuator fault, and input disturbances was studied in [4] by designing a sliding mode observer-based control. A self-optimizing NN was proposed in [5] for extended Hammerstein non-Gaussian noises systems. Recently, the adaptive technique of nonlinear systems with unknown functions has drawn considerable interest because the NNs and the FLSs have excellent function approximation ability [6], [7].

T

Manuscript received June 6, 2014; revised September 26, 2014 and December 21, 2014; accepted December 22, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61374113, and Grant 61473139, and in part by the Program for Liaoning Excellent Talents in University under Grant LR2014016. This paper was recommended by Associate Editor Z.-G. Hou. Y.-J. Liu and S. Tong are with the College of Science, Liaoning University of Technology, Jinzhou 121001, China (e-mail: [email protected]). C. L. P. Chen is with the Faculty of Science and Technology, University of Macau, Macau, China. D.-J. Li is with the School of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou 121001, China. 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.2015.2388582

Based on the neural or fuzzy approximation, many important adaptive control approaches were studied in [8]–[16] for several different classes of continuous unknown dynamic nonlinear SISO systems. Some results on SISO cases were extended in [17]–[23] to control uncertain continuous nonlinear multiple-input multiple-output (MIMO) systems. Recently, the stability problem of uncertain discrete-time nonlinear systems was studied in [24]–[28] by designing adaptive neural controller. These results in [24]–[28] guarantee that all the signals of the systems are semi-global uniformly ultimately bounded (SGUUB) and the output tracks a specified bounded signal. In [29], a near-optimal control for discrete nonlinear systems with control constraints was solved by adaptive NN iterative dynamic programming. A neural control internal model was developed in [30] for unknown discrete nonlinear nonaffine processes. By using the universal approximators, some applications were given for the robots [31]–[35], the servo mechanisms [36], a two continuous stirred tank reactor process [37], the Amira’s ball-and-beam system [38], and the half-car active suspension system [39]. In [40], an excellent adaptive robust approach was developed for solving the consensus problem of multiagent systems and a more practical application in real-world was considered because the agent’s dynamics includes the uncertainties and external disturbances. A novel neural-based adaptive control was designed to solve the tracking problem of manipulators with uncertain kinematics, dynamics, and actuator model and an outstanding Jacobian adaptive scheme is used to estimate the unknown kinematics parameters. In [41] and [42], an important adaptive neural approach was proposed for the leader-following control of multiagent systems. In this result, it is first to take into account the uncertainty in the agent’s dynamics with timevarying and the proposed algorithm for each following agent is only dependent on the information of its neighbor agents. The survey and summarization of recent development can be found in [7], [43], and [44]. It can be found from the above results that the effects of nonsmooth input nonlinearities are neglected. The deadzone, the saturation, and the hysteresis are very important input nonlinearity structure and they are frequently involved in various engineering plants, such as electric servomotors, mechanical connections, piezoceramics, and shape memory alloys, etc. [45]. Recently, some researchers studied adaptive control of nonlinear systems with input nonlinearities. Dead-zone compensation was constructed in [46] and [47] for

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motion systems by using adaptive fuzzy or neural control. The adaptive decentralized stabilization was given in [48] for nonlinear systems with interconnected terms and nonsymmetric dead-zones. The stability problem of nonlinear MIMO systems with the nonsymmetric input saturation constraint was considered in [49] by designing the adaptive control. The adaptive neural control was proposed for a general class of nonlinear MIMO systems with saturation and dead zone [50]. Some elegant approaches referred to as truncated predictor feedback were recently established in [51]–[53] for control systems with both input saturation nonlinearities and time delays. The hysteresis is a more general class of input nonlinearities. To overcome the effects of the hysteresis on the control performance, Tao and Kokotovic [54] presented a hysteresis inverse control for the plants with unknown parameter and backlash hysteresis. In [55], an adaptive control approach was proposed for nonlinear systems with backlash-like hysteresis without using the hysteresis inverse. An adaptive robust control algorithm was developed in [56] for the systems in [55]. In [45], an adaptive wavelet NN controller with hysteresis estimation was presented to betterment the control performance of a piezo-positioning mechanism. The increasing attention has received for the control of nonlinear systems with PI hysteresis owing to growing industrial demands. Two adaptive control approaches without using a hysteresis inverse were designed in [57] and [58] for nonlinear systems with the PI hysteresis model. In [59], based on an implicit inversion scheme, an adaptive control was constructed for continuous-time linear systems with PI hysteresis. The common property of the adaptive control algorithms proposed in [57]–[59] is that the considered nonlinear systems with PI hysteresis are required to satisfy the restrictive condition that the unknown parameters appear linearly in the systems on known functions. If the completely unknown functions appear in the systems, the above mentioned approaches are not ensured to stabilize the corresponding closed-loop system. At the same time, this restrictive condition is not also to satisfy development requirement in the industrial applications. At present, only few results are designed to cope with nonlinear systems with both completely unknown function and PI hysteresis. An adaptive neural control was proposed in [60] for unknown SISO pure-feedback systems with PI hysteresis. In practice, many real systems are multivariable. Therefore, Ren et al. [61] studied the adaptive neural variable structure of uncertain nonlinear MIMO systems with both unknown state delays and PI hysteresis [61]. However, the major structural limitations imposed on the systems in [61] are that: 1) the matching condition is required, i.e., the uncertainties appear only in the last equation of each subsystem and 2) the output of the PI hysteresis for each subsystem appear linearly in unknown functions of the corresponding subsystem, i.e., the uncertain systems are in affine form. This paper will try to study the control design for the nonlinear systems without these limitations. It is well known that a large number of real systems can be expressed in MIMO, nonaffine, and interconnected forms, and the PI type hysteresis is often included in some real systems.

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This paper addresses the stability and control issues of uncertain nonlinear interconnected MIMO systems in pure feedback nonaffine form. However, owing to nonaffine form for the control inputs and the couplings existing between the subsystems, the controller design, and stability analysis for such a class of systems are more difficult and complicated. The main contributions are summarized as follows. 1) When the PI hysteresis and unknown functions are simultaneity involved in the controlled systems, a more general class of MIMO systems is considered in this paper. The main characteristics are that the systems are composed of N interconnected subsystems and interconnection terms appear in each equation of subsystems; and the output of the PI hysteresis is nonlinearly appeared in unknown functions. It is the first time to control this class of systems. 2) It is mentioned that the control approaches in [55] and [56] have a restrictive assumption that the density function in PI hysteresis and its integral are bounded by known constants, which are difficult to be determined in practice. The approach proposed in this paper removes this assumption, that is, it does not need to assume that bounds are known. The bounds are compensated by designing adaptive tuning algorithms. 3) The existence of PI hysteresis leads to producing different form’s Lyapunov functions and special design procedure to solve the difficulties in constructing the controllers and adaptation laws. An adaptive compensation structure is employed to conquer the obstacles in the design, which are from unknown bounds for both density function and its integral in PI hysteresis. Finally, all the signals in the closed-loop system are certified to be SGUUB and the outputs follow the desired signals to a small compact set. The effectiveness is illustrated by using the simulation example.

II. P ROBLEM F ORMULATION AND P RELIMINARIES A. System Formulation Consider nonlinear interconnected systems ⎧ ⎪ x ˙ x ¯ , . . . , x ¯ , . . . , x ¯ = ⎪ j,r j,r 1, r −l j, r +1−l N, r −l j j ( j j1 ) (j ( j jN ) jj ) ⎪ ⎨ rj = 1, . . . , lj − 1 (1) ⎪ ⎪ x˙ j,lj = j,lj X, u1 , . . . , uj−1 , uj ⎪ ⎩ yj = xj,1 , j = 1, . . . , N where xj,rj , rj = 1, . . . , lj are the states of the jth subsystem, lj is the order of the jth subsystem, yj ∈ R is the output of the jth subsystem, j,rj (·), rj = 1, . . . , lj , j = 1, . . . , N

T ∈ Rrj , are unknown functions, x¯ j,rj = xj,1 , . . . , xj,rj rj = 1, . . . , lj are the vector of partial state variables in

T the jth subsystem, X = x¯ 1,l1 , . . . , x¯ N,lN is the state variable of the whole system, j, rj , lj , N are positive integers. lji = lj − li denotes the order difference between the jth and ith subsystems; uj is the hysteresis nonlinearity as

uj (t) = Pj τj (t) = pj0 τj (t) − dj τj (t)

(2)

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both the PI hysteresis model and unknown functions. The approaches are obtained under the condition that p¯ j0 and p¯ jr are required to be bounded by known constants. These known constants are also utilized in the controllers and adaptation laws directly. But it is difficult to determine the prior knowledge of known bounds in practice. In this paper, p¯ j0 and p¯ jr are assumed to be unknown constants and it does not need to determine the prior knowledge of bounds. In the following, an adaptive compensation algorithm is implemented to estimate the unknown constants. B. System Transformation and Basic Assumptions

Fig. 1.

Hysteresis curves given by (2).

D where τj (t) is the input of the hysteresis, pj0 = 0 pj (r)dr and D dj [τj ](t) = 0 pj (r)Fjr [τj ](t)dr with ⎧ Fjr τj (0) = fjr τj (0), 0 ⎪ ⎪ ⎨ Fjr τj (t) = fjr τj (t), Fjr τj (tk ) (3) tk < t ≤ tk+1 , 0 ≤ k ≤ N − 1 ⎪ ⎪ ⎩ fjr (τ, w) = max(τ − r, min(τ + r, w)) . In (3), pj (r) is a given continuous density function satisfying ∞ pj (r) > 0 and 0 rpj (r)dr < ∞; D is a constant so that the density function pj (r) vanishes for large values of D, in general, D = ∞ is chosen as the upper limit of integration in the literature; 0 = t0 < t1 < · · · tN = tE is a partition of [0, tE ] such that the function τj is monotone on each of the subintervals (tk , tk+1 ]. The PI hysteresis model generated by (2) 2 with pj (r) = e−0.07(r−1) and the input τj (t) = 8 sin(3t)/(1+t) is illustrated in Fig. 1. The control objective of this paper is to design an adaptive neural control approach for (1) and (2) so that all the signals in the closed-loop system are SGUUB, and the tracking error between yj and the reference signal yr,j (t) converges to a neighborhood around zero, where yr,j (t) and its time (l ) derivatives up to the lj th order yr,jj (t) are continuous and bounded. Assumption 1: pj0 and pj (r) are assumed to satisfy p j0 < pj0 and pj (r) < p¯ jr for all r ∈ [0, D] where p j0 and p¯ jr are unknown constants. Remark 1: When the PI hysteresis model is considered, an adaptive variable structure NN scheme was investigated in [61] for uncertain MIMO nonlinear systems, which are required to satisfy the matching condition, i.e., the system uncertainties appear only in the last equation of each subsystem. From the system (1), it can be seen that unknown functions appear not only in the last equation of each subsystem, but also in other equations. In addition, the output of PI hysteresis model is nonlinearly given in (1). The output of PI hysteresis model is linearly contained in the systems considered in [61]. Thus, the systems studied in this paper are more general compared with the previous works. Remark 2: In [60] and [61], two adaptive neural control approaches were developed to stabilize nonlinear systems with

Define bj,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,(rj +1) , . . . , x¯ N,(rj −ljN ) ∂j,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,(rj +1) , . . . , x¯ N,(rj −ljN ) = (4) ∂xj,(rj +1) bj,lj X, u1 , . . . , uj−1 , uj ∂j,lj X, u1 , . . . , uj−1 , uj = . (5) ∂uj Assumption 2: The functions bj,rj (·), rj = 1, . . . , lj , j = 1, . . . , N are bounded, i.e., there exist the constants b¯ j,rj ≥ b j,rj > 0 such that b j,rj ≤ bj,rj (·) ≤ b¯ j,rj for j = 1, . . . , N. Without losing generality, this paper assumes that b j,rj ≤ bj,rj (·) ≤ b¯ j,rj . Lemma 1: Consider χ˙ (t) = −aχ (t) + cq(t), where a > 0 and c > 0 are constants and q(t) > 0 is a function. Then, given initial bounded condition χ (t0 ) ≥ 0, it has χ (t) ≥ 0, ∀t ≥ t0 . θj,r According to the mean value theorem, there exist xj,rjj+1 θ

and uj j such that j,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,(rj +1) , . . . , x¯ N,(rj −ljN ) 0 , . . . , x ¯ = j,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,rj , xj,r N, r −l ( j jN ) j +1 θj,rj + bj,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,rj , xj,rj +1 , . . . , x¯ N,(rj −ljN ) 0 xj,rj +1 − xj,r j +1 j,lj X, u1 , .. . , uj−1 , uj = j,lj X, u1 , . . . , uj−1 , u0j θ + bj,lj X, u1 , . . . , uj−1 , uj j uj − u0j 0 θj,r , 0 < θj,rj < 1 where xj,rjj+1 = θj,rj xj,rj +1 + 1 − θj,rj xj,r j +1 0 θj and uj = θj uj + 1 − θj uj , 0 < θj < 1. θj,r

θ

0 = 0 and u0j = 0. Then, xj,rjj+1 = θj,rj xj,rj +1 and Let xj,r j +1

uj j = θj uj . Accordingly, one has j,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,(rj +1) , . . . , x¯ N,(rj −ljN ) = j,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,rj , 0, . . . , x¯ N,(rj −ljN ) + bj,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,rj , θj,rj xj,rj +1 , . . . , x¯ N,(rj −ljN ) xj,rj +1 (6)

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j,lj X, u1 , . . . , uj−1 , uj = j,lj X, u1 , . . . , uj−1 , 0 θ + bj,lj X, u1 , . . . , uj−1 , uj j uj . (7) In the following design procedure, we let: Aj,rj (·) = j,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,rj , 0, . . . , x¯ N,(rj −ljN ) θj,r Bj,rj (·) = bj,rj x¯ 1,(rj −lj1 ) , . . . , x¯ j,rj , xj,rjj+1 , . . . , x¯ N,(rj −ljN ) Aj,lj (·) = j,lj X, u1 , . . . , uj−1 , 0 θ Bj,lj (·) = bj,lj X, u1 , . . . , uj−1 , uj j . Substituting (2), (6), and (7) into (1), it follows that: ⎧ ⎨ x˙ j,rj = Aj,rj (·) + Bj,rj (·) xj,(rj +1) , rj = 1, . . . , lj − 1, (8) x˙ = Aj,lj (·) + Bj,lj (·) pj0 τj (t) − Bj,lj (·) dj τj (t) ⎩ j,lj yj = xj,1 , j = 1, . . . , N. θj,r

θ

Remark 3: It is apparent that xj,rjj+1 and uj j are included in Bj,rj (·) and Bj,lj (·), respectively. Therefore, Bj,rj (·) and Bj,lj (·) are still nonaffine functions in character. Due to this property, it results in the difficulties for designing the controllers and adaptation laws. C. NN Approximation Property In [7], it is shown that for any given continuous function q(x) : Rn → R on a compact set U ∈ Rn and an arbitrary ε > 0, there ST (x)W such that

exist the RBFNN

T l

sup f (X) − S (x)W < ε, where W ∈ R is the weight vector,

Step 1: Define the tracking error as ej,1 = xj,1 − yr,j and e˙ j,1 is e˙ j,1 = x˙ j,1 − y˙ r,j = Aj,1 (·) + Bj,1 (·) xj,2 − y˙ r,j . Based on the NN approximator, we obtain qj,1 Xj,1 = Aj,1 (·) − y˙ r,j ∗ T ∗ Xj,1 Wj,1 Xj,1 = Sj,1 + εj,1

where πi = [πi1 , . . . , πin ]T and ωi are the center and the width of the Gaussian function, respectively. Based on the neural approximation, the continuous function q(x) is approximated as q(x) = ST (x)W ∗ + ε∗ (x) W∗

(10) ε∗

where denotes the optimal weight vector and is called the approximation error. In general, W ∗ and ε∗ are bounded. In the controller design, the unknown functions of the systems and the uncertain parameters of PI hysteresis model can not be used. In the following section, we will use the NNs excellent approximation ability to approximate these functions and the uncertain parameters of PI hysteresis model will be estimated by using the adaptive compensation algorithm.

(12)

T where Xj,1 = x¯ 1,(1−lj1 ) , . . . , xj,1 , . . . , x¯ N,(1−ljN ) , y˙ r,j ∈ j,1 ∗ is the optimal parameter vector, ε ∗ X and Wj,1 j,1 is the j,1 approximation error to be bounded by the constant ε¯ j,1 . Substituting (12) into (11) yields ∗ T ∗ Xj,1 Wj,1 Xj,1 + Bj,1 (·) xj,2 . + εj,1 (13) e˙ j,1 = Sj,1 ∗ 2 ˆ Let j,1 = b−1 W j,1 j,1 and j,1 ≥ 0 is employed to denote the estimation of j,1 . The adaptation law is designed as 1 1 2 T ˙ˆ = −η e S Xj,1 Sj,1 Xj,1 j,1 j,1 ˆ j,1 + 2 j,1 j,1 2 σj,1

νj,1 = −kj,1 ej,1 −

1 1 T ˆ j,1 Sj,1 Xj,1 Sj,1 Xj,1 (15) ej,1 2 2 σj,1

where kj,1 > 0 is the design parameter to be specified later. Consider the Lyapunov function Vj,1 =

1 2 1 ˜ 2j,1 ej,1 + b j,1 2 2

(16)

ˆ j,1 − j,1 . ˜ j,1 = where ˙ˆ . ˜ j,1 The time derivative for Vj,1 is V˙ j,1 = ej,1 e˙ j,1 + b j,1 j,1 By substituting (13) and (15) into V˙ j,1 , one has ∗ T ∗ Xj,1 Wj,1 Xj,1 + ej,1 εj,1 V˙ j,1 = ej,1 Sj,1 ˙ˆ ˜ + B (·) e e − k B (·) e2 + b j,1

j,1 j,2

j,1 j,1

j,1

j,1

j,1

1 1 T ˆ j,1 Sj,1 Xj,1 Sj,1 Xj,1 . − Bj,1 (·) e2j,1 2 2 σj,1

j,1

(17)

Based on the following facts:

∗ ∗ T Xj,1 Wj,1 ≤ ej,1 Sj,1 Xj,1 Wj,1 ej,1 Sj,1 ≤

b j,1 e2j,1 2 2σj,1

1 2 T Xj,1 Sj,1 Xj,1 + σj,1 j,1 Sj,1 2 (18)

III. A DAPTIVE NN C ONTROLLER D ESIGN In this section, we will incorporate the backstepping technique into the adaptive NN control for the systems described in (1) or (8). The detailed design procedures are shown in the following procedures.

(14)

where ηj,1 > 0 and σj,1 > 0 are the design parameters. Introduce ej,2 = xj,2 − νj,1 where νj,1 denotes the virtual controller, which is chosen as

X∈U

l > 1 is the NN node number, X ∈ Rn is the input vector, and S(x) = [s1 (x), . . . , sl (x)]T is Gaussian basis function vector. Gaussian basis function can be expressed as x − πi 2 , i = 1, . . . , l (9) si (x) = exp − ωi2

(11)

and 1 1 1 ∗ Xj,1 ≤ b j,1 cj,1 e2j,1 + ε¯ 2 . ej,1 εj,1 2 2 b j,1 cj,1 j,1

(19)

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∗ is the optimal parameter vector and ε ∗ X where Wj,r j,rj is j,rj j the approximation error to be bounded by the constant ε¯ j,rj . Equation (23) can be rewritten as ∗ T ∗ e˙ j,rj = Sj,r Xj,rj Wj,r Xj,rj + εj,r j j j

Equation (17) becomes 1 V˙ j,1 ≤ Bj,1 (·) ej,1 ej,2 − b j,1 2kj,1 − cj,1 e2j,1 2 1 1 1 2 ˙ˆ 2 ˜ j,1 + ε¯ + σ + b j,1 j,1 2 b j,1 cj,1 j,1 2 j,1 1 1 T ˜ j,1 Sj,1 Xj,1 Sj,1 Xj,1 − b j,1 e2j,1 2 2 σj,1

− Bj,rj −1 (·) ej,rj −1 + Bj,rj (·) xj,rj +1 . (20)

c where kj,1 is required to satisfy kj,1 = 2kj,1 − cj,1 > 0 with cj,1 > 0. From (14), we have

˙ˆ − 1 1 b e2 T ˜ j,1 ˜ b j,1 j,1 j,1 j,1 j,1 Sj,1 Xj,1 Sj,1 Xj,1 2 2 σj,1

(21)

ν˙ j,rj −1 =

k=1

m=1 rj −1

∂vj,(rj −1) ∂xk,m

(28)

vj,rj = −kj,rj ej,rj −

1 1 T ˆ j,rj Sj,r ej,rj Xj,rj Sj,rj Xj,rj s j 2 2 σj,r (29)

(23)

where kj,rj > 0 is the design parameter to be specified later. Consider the Lyapunov function 1 1 ˜ 2j,r Vj,rj = Vj,rj −1 + e2j,rj + b j,rj j 2 2

(30)

ˆ j,rj − j,rj . The time derivative for Vj,rj is ˜ j,rj = where ˙ˆ ˜ j,rj ˙ ˙ Vj,rj = Vj,rj −1 + ej,rj e˙ j,rj + b j,rj j,rj and it can be obtained from (27) and (29) that ∗ T ∗ Xj,rj Wj,r Xj,rj + ej,rj εj,r V˙ j,rj = V˙ j,rj −1 + ej,rj Sj,r j j j − Bj,rj −1 (·) ej,rj −1 ej,rj + Bj,rj (·) ej,rj ej,rj +1 ˙ˆ ˜ − k B (·) e2 + b j,rj j,rj

j,rj

j,rj

j,rj

j,rj

1 1 T ˆ j,rj Sj,r Xj,rj Sj,rj Xj,rj . − Bj,rj (·) e2j,rj j 2 2 σj,r

(31)

j

x˙ k,m

rj ∂vj,rj −1 ˙ ∂vj,rj −1 (m) ˆ j,m + + y . (m−1) r,j ˆ ∂ j,m m=1 m=1 ∂yr,j

Define the following unknown function: qj,rj zj,rj = Aj,rj (.) + Bj,rj −1 (·) ej,rj −1 − v˙ j,rj −1 .

Similar to the derivation in (18), the following inequalities can be obtained: (24)

(25)

According to the definition of Aj,rj (.) and from (24), we know that qj,rj (Xj,rj ) is a function of Xj,rj = [¯x1,(rj −lj1 ) , . . . , x¯ j,rj , . . . , x¯ N,(rj −ljN ) , ϕj,rj −1 , ∂vj,(rj −1) / ∂xk,1 , . . . , ∂vj,(rj −1) / ∂xk,rj −1−ljl , k = 1, . . . , N] ∈ j,rj with rj −1

ϕj,rj −1

1 1 2 T ˙ˆ ˆ j,rj + ej,rj Sj,rj Xj,rj Sj,rj Xj,rj j,rj = −ηj,rj 2 2 σj,r

j

where νj,rj −1 has been obtained in step rj − 1 and it is a function of x¯ 1,(rj −1−lj1 ) , . . . , x¯ j,(rj −1) , . . ., x¯ N,(rj −1−ljN ) , (r −1) ˆ j,1 , . . . , ˆ j, r −1 . Then, ν˙ j, r −1 and yr,j , y˙ r,j , . . . , yr,jj (j ) (j ) can be expressed by N rj −1−l jl

∗ 2 ˆ Define j,rj = b−1 j,rj Wj,rj and let j,rj ≥ 0 be the ˆ j,rj is designed as estimation of j,rj . The adaptation law for

where σj,rj > 0 is the design parameter. Define ej,rj +1 = xj,rj +1 − vj,rj where vj,rj denotes the virtual controller which is designed as

1 1 c 2 ˜ 2j,1 + Bj,1 (·) ej,1 ej,2 ej,1 − ηj,1 b j,1 V˙ j,1 ≤ − b j,1 kj,1 2 2 1 1 1 1 + ε¯ 2 + σ 2 + ηj,1 b j,1 2j,1 . (22) 2 b j,1 cj,1 j,1 2 j,1 2 Step rj rj = 2, . . . , lj − 1 : In this step, the virtual control input νj,rj will be designed. Define ej,rj = xj,rj − νj,rj −1 and differentiating it yields e˙ j,rj = Aj,rj (·) + Bj,rj (·) xj,(rj +1) − ν˙ j,rj −1

(27)

j

˜ j,1 ˆ j,1 ≤ −ηj,1 b j,1 ˜ 2j,1 − ηj,1 b j,1 ˜ j,1 j,1 = −ηj,1 b j,1 1 1 ˜ 2j,1 + ηj,1 b j,1 2j,1 . ≤ − ηj,1 b j,1 2 2 By using (21), (20) can be further written as

5

rj ∂vj,rj −1 ˙ ∂vj,rj −1 (m) ˆ j,m + = y (m−1) r,j ˆ ∂ j,m m=1 m=1 ∂yr,j

being a computable variable. Based on the NN approximator, it follows that: ∗ T ∗ Xj,rj Wj,r Xj,rj + εj,r qj,rj Xj,rj = Sj,r j j j

∗ 1 2 T Xj,rj Wj,r ≤ σj,r ej,rj Sj,r j j 2 j 2 2 1 b j,rj ej,rj + j,rj Sj,rj Xj,rj 2 2 σj,r j

(32)

and 1 1 1 ∗ ε¯ 2 Xj,rj ≤ b j,rj cj,rj e2j,rj + ej,rj εj,r j 2 2 b j,rj cj,rj j,rj

(33)

where cj,rj > 0 is the design parameter. According to (28) and by using the similar derivations in (21), we have ˙ˆ − ˜ j,rj b j,rj j,rj

1 1 T ˜ j,rj Sj,r b j,rj e2j,rj Xj,rj Sj,rj Xj,rj j 2 2 σj,r j

(26)

1 1 ˜ 2j,r + ηj,rj b j,r 2j,r . ≤ − ηj,rj b j,rj j j j 2 2

(34)

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Substituting (32)–(34) into (31) yields 1 1 c 2 ˜ 2j,r e − ηj,rj b j,rj V˙ j,rj ≤ V˙ j,rj −1 − b j,rj kj,r j j j,rj 2 2 − Bj,rj −1 (·) ej,rj −1 ej,rj + Bj,rj (·) ej,rj ej,rj +1 1 1 2 1 2 + ε¯ j,r + σj,r + ηj,rj b j,rj 2j,rj (35) j j b j,rj cj,rj 2 2 c = 2k where kj,rj is required to satisfy kj,r j,rj − cj,rj > 0. j In step rj − 1, we obtain

V˙ j,rj −1 ≤ −

rj −1 rj −1 1 1 c 2 ˜ 2j,k b j,k kj,k ej,k − ηj,k b j,k 2 2 k=1

k=1

rj −1 1 ηj,k b j,k 2j,k 2 k=1

+ Bj,rj −1 (·) ej,rj −1 ej,rj + +

rj −1 1

2

k=1

1 2 ε¯ 2 + σj,k . b j,k cj,k j,k

V˙ j,rj ≤ −

2

c 2 b j,k kj,k ej,k −

k=1 rj

rj 1

2

1 1 Vj,lj = Vj,lj −1 + e2j,lj + b p d˜ 2 2 2γj j,lj j0 j0 D 1 1 ˜ 2j,l + + b j,lj b p˜ 2 (t, r) dr (43) j 2 2δj j,lj 0 j

k=1

(37)

k=1

Remark 4: In the previous affine systems, in general, Bj,rj (·) is included in the approximated function. However, in this paper, because Bj,rj (·) is a nonaffine function which contains the variable xj,rj +1 , Bj,rj (·) cannot be approximated in step rj . Therefore, we set the unknown function Bj,rj −1 (·) to be in the approximated function qj,rj Xj,rj and included qj,rj Xj,rj does not contain the variable xj,rj +1 . In addition, the coupling term Bj,rj (·) ej,rj ej,rj +1 can be canceled in step rj + 1. Step lj : In step lj − 1, ej,lj = xj,lj − vj,lj −1 has been defined. By differentiating ej,lj , we get e˙ j,lj = Aj,lj (·) + Bj,lj (·) pj0 τj (t)

− Bj,lj (·) dj τj (t) − v˙ j,lj −1

˜ j,lj = ˆ j,lj − j,lj , p˜ j (t, r) = pˆ j (t, r) − pj (r), and where ˜dj0 = dˆ j0 − dj0 ; ˆ j,ρj ≥ 0, pˆ j (t, r) ≥ 0, and dˆ j0 ≥ 0 are ∗ 2 W utilized to denote the estimations of j,lj = b−1 j,lj , pj (r) j,lj

and dj0 = p−1 , respectively, and δj > 0, γj > 0 are the design j0 parameters. Differentiating Vj,lj along with (42) yields D ∂ pˆ j (t, r) 1 dr p˜ j (t, r) V˙ j,lj = V˙ j,lj −1 + b j,lj τj ∂t 0 1 ˙ˆ + 1 b p d˜ dˆ˙ ˜ j,lj + b j,lj j,lj j0 j0 2 γj j,lj j0 ∗ T ∗ Xj,lj Wj,l Xj,lj + ej,lj Sj,l + ej,lj εj,l j j j − Bj,lj −1 (·) ej,lj −1 ej,lj + Bj,lj (·) ej,lj p j0 τj (t) D

(44) − ej,lj Bj,lj (·) pj (r)Fjr τj (t)dr. 0

(38)

The adaptation laws are designed as ˙ˆ = −η j,lj j,lj ˆ j,lj 1 1 2 T + e S Xj,lj Sj,lj Xj,lj 2 j,lj j,lj 2 σj,l j

∂ pˆ j (t, r) = −δj pˆ j (t, r) + δj ej,lj Fjr τj (t) ∂t ˙ dˆ j0 = −γj dˆ j0 − γj ej,lj τj0

(47)

1 1 T ˆ j,lj Sj,l Xj,lj Sj,lj Xj,lj ej,lj j 2 2 σj,l j

D − kj,lj ej,lj − sgn ej,lj pˆ j (t, r) Fjr τj (t) dr

(48)

where v˙ j,lj −1 is a function of x¯ 1,(lj −1−lj1 ) , . . . , x¯ j,(lj −1) , . . . , (l −1) ˆ j,1 , . . . , ˆ j,lj −1 , and x¯ N,(lj −1−ljN ) , yr,j , y˙ r,j , . . . , yr,jj , and it can be expressed as v˙ j,lj −1 =

N lj −1−l jl ∂vj,(lj −1) k=1 m=1 lj −1

+

∂xk,m

x˙ k,m lj

∂vj,lj −1 ˙ ∂vj,lj −1 (m) ˆ j,m + y . (39) (m−1) r,j ˆ j,m ∂ m=1 m=1 ∂yr,j

Define the following unknown function: qj,lj zj,lj = Aj,lj (·) + Bj,lj −1 (·) ej,lj −1 − v˙ j,lj −1 .

(42)

0

Consider the Lyapunov function

˜ 2j,k ηj,k b j,k

1 + ηj,k b j,k 2j,k + Bj,rj (·) ej,rj ej,rj +1 2 k=1 rj 1 1 2 2 + ε¯ + σj,k . 2 b j,k cj,k j,k

− Bj,lj −1 (·) ej,lj −1 + Bj,lj (·) pj0 τj (t) D

pj (r)Fjr τj (t)dr. − Bj,lj (·)

(36)

Further, (35) is expressed as rj 1

It can be determined from Aj,lj (·), Bj,lj −1 (·) and (39) that qj,lj (Xj,lj ) is a function of Xj,lj = [¯x1,(lj −lj1 ) , . . . , x¯ j,lj , . . . , x¯ N,(lj −ljN ) , ϕj,lj −1 , ∂vj,(lj −1) /∂xk,1 , . . . , ∂vj,(lj −1) /∂xk,lj −1−ljl , lj −1 k = 1, . . . , N] ∈ j,lj with ϕj,lj −1 = m=1 (∂vj,lj −1 / (m−1) (m) ˙ˆ + lj (∂v ˆ j,m ) /∂y )y being a computable ∂ j,m j,lj −1 r,j r,j m=1 variable. qj,lj Xj,lj can be approximated by the NN ∗ T ∗ qj,lj Xj,lj = Sj,l Xj,lj Wj,l Xj,lj (41) + εj,l j j j ∗ is the optimal parameter vector and ε ∗ X where Wj,l j,lj is j,lj j the approximation error which is bounded by the constant ε¯ j,lj . Based on (40) and (41), (38) can be transformed into ∗ T ∗ Xj,lj Wj,l Xj,lj + εj,l e˙ j,lj = Sj,l j j j

(40)

(45) (46)

where τj0 = −

0

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with σj,lj > 0 and kj,lj > 0 being the design parameter, and pˆ j (0, r) ≥ 0. Remark 5: It can be observed that (46) satisfies the conditions of Lemma 1. When the initial condition for pˆ j (t, r) is chosen as pˆ j (0, r) ≥ 0, it can be known that pˆ j (t, r) ≥ 0 for ∀t ≥ 0. Because pˆ j (t, r) is the estimation of the unknown constant pj (r) and its initial value is an artificial quantity, it is reasonable that we can assume that pˆ j (0, r) ≥ 0. The actual controller is constructed as τj = dˆ j0 τj0 .

≤

2 1 2 1 b j,lj j,lj 2 ej,lj Sj,lj Xj,lj + σj,l 2 2 σj,l 2 j

c = 2k where kj,lj is required to satisfy kj,l j,lj − cj,lj > 0. j According to (47), one has ˜dj0 1 b j,l p d˙ˆ j0 + b j,l p ej,lj τj0 (t) j j0 γj j j0 = −b j,l p d˜ j0 dˆ j0 2 = −b j,lj p j0 d˜ j0 − b j,lj p j0 d˜ j0 dj0

1 1 2 2 ≤ − b j,lj p j0 d˜ j0 + b j,lj p j0 dj0 . 2 2

1 b δj j,lj

0

(50)

(51)

(52)

1 b δj j,lj

D

p˜ j (t, r)

0

D ∂ pˆ j (t, r) 1 dr ≤ − b j,lj p˜ 2j (t, r) dr ∂t 2 0 D 1 + b j,lj p¯ 2jr dr. (58) 2 0

By taking rj = lj − 1 in (37) and substituting (56)–(58) into (55), it yields lj

lj

1 1 c 2 ˜ 2j,k b j,k kj,k ej,k − ηj,k b j,k 2 2 k=1 k=1 D 1 1 2 − b j,lj pj0 d˜ j0 − b j,lj p˜ 2j (t, r) dr 2 2 0 lj D 1 1 ηj,k b j,k 2j,k + b j,lj p¯ 2jr dr + 2 2 0 k=1 lj 1 1 1 2 2 2 + b j,lj pj0 dj0 + ε¯ + σj,k . (59) 2 2 b j,k cj,k j,k

V˙ j,lj ≤ − (53)

It can be obtained from (49) that ej,lj Bj,lj (·) pj0 τj (t) = ej,lj Bj,lj (·) pj0 dˆ j0 τj0 (t). It is obviously seen from (48) that ej,lj τj0 (t) ≤ 0. Then, according to Assumption 2, the above equation can become

k=1

(54)

Using (16), (30), and (43), it obtains

By substituting (48), (49), (54) into (44) and employing (50)–(53), we have 1 1 c 2 ˜ 2j,l ≤ V˙ j,lj −1 − b j,lj kj,l e − ηj,lj b j,lj j j j,lj 2 2 1 1 1 2 2 − Bj,lj −1 (·) ej,lj −1 ej,lj + ε¯ j,l + σj,l j 2 b j,lj cj,lj 2 j 1 ˙ + d˜ j0 b p dˆ j0 + b j,lj p j0 ej,lj τj0 (t) γj j,lj j0 D ∂ pˆ j (t, r) 1 1 2 dr p˜ j (t, r) + ηj,lj b j,lj j,lj + b j,lj 2 δj ∂t 0 D

(55) − b j,lj ej,lj p˜ j (t, r) Fjr τj (t) dr 0

(57)

By completion of squares, (57) becomes

0

V˙ j,lj

p˜ j (t, r)

0

where cj,lj > 0 is the design parameter. According to (45) and similar to (21), we have 1 1 T ˜ j,lj ˜ j,lj Sj,l ˆ˙ j,lj − Xj,lj Sj,lj Xj,lj b j,lj e2j,lj b j,lj j 2 2 σj,l j

ej,lj Bj,lj (·) pj0 τj (t) ≤ ej,lj b j,lj p j0 dˆ j0 τj0 (t) ≤ ej,lj b j,lj p j0 d˜ j0 + 1 τj0 (t).

∂ pˆ j (t, r) dr − b j,lj ej,lj ∂t D

× p˜ j (t, r) Fjr τj (t) dr 0 D = −b j,lj p˜ j (t, r) pˆ j (t, r) dr.

D

and

1 1 ˜ 2j,l + ηj,lj b j,l 2j,l ≤ − ηj,lj b j,lj j j j 2 2 and it follows from Assumption 2 that: D

pj (r)Fjr τj (t)dr −ej,lj Bj,lj (·) 0

D

≤ b j,lj ej,lj pj (r) Fjr τj (t) dr.

(56)

According to (46), we obtain

j

1 1 1 ∗ Xj,lj ≤ b j,lj cj,lj e2j,lj + ε¯ 2 ej,lj εj,l j 2 2 b j,lj cj,lj j,lj

j0

j

(49)

By using the similar derivations in (18), the following inequalities can be obtained: ∗ T Xj,lj Wj,l ej,lj Sj,l j j

7

Vj,lj

lj

lj

k=1

k=1

1 2 1 ˜ 2j,k = ej,k + b j,k 2 2 1 1 + b p d˜ 2 + b 2γj j,lj j0 j0 2δj j,lj

D 0

p˜ 2j (t, r) dr. (60)

According to (60), (59) can be rewritten as V˙ j,lj ≤ −βj,lj Vj,lj + μj,lj

(61)

where c c , . . . , b j,lj kj,l , η , . . . , η , γ , δ βj,lj = min b j,1 kj,1 j,1 j,l j j j j (62)

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and

and by taking

lj

μj,lj

D 1 1 = ηj,k b j,k 2j,k + b j,lj p¯ 2jr dr 2 2 0 k=1 lj 1 1 1 2 2 2 + b j,lj p j0 dj0 + ε¯ + σj,k . (63) 2 2 b j,k cj,k j,k k=1

Theorem 1: Consider the systems (1). On the compact sets j,rj , and under Assumptions 1 and 2, by constructing the virtual controllers vj,ri in (15), (29) and the actual ˆ j,rj controller τj in (49), and choosing the adaptation laws in (14), (28), and (45), pˆ j (t, r) in (46), dˆ j0 in (47) for rj = 1, . . . , lj , j = 1, . . . , N, if the design parameters are chosen as cj,rj > 0, γj > 0, δj > 0, σj,rj > 0, ηj,rj > 0, kj,rj ≥ cj,rj /2, the obtained scheme can guarantees that: 1) all the signals in the closed-loop system are SGUUB and 2) yj = xj,1 follows yr,j to a compact set defined by:

(64)

j = xj,1

xj,1 − yr,j ≤ μj,lj /βj,lj . β

t

Proof: Multiplying both sides in (61) by e j,lj gets d β t β t Vj,lj e j,lj ≤ μj,lj e j,lj . dt Integrating the above inequality over [0, t] leads to μj,lj −βj,l t μj,lj j + 0 ≤ Vj,lj (t) ≤ Vj,lj (0) − . e βj,lj βj,lj

(65)

Since μj,lj and βj,lj are positive constants, (65) implies that μj,lj −β t 0 ≤ Vj,lj (t) ≤ Vj,lj (0)e j,lj + . (66) βj,lj ˜ j,k , d˜ j0 , p˜ j (t, r) From (60) and (66), it can be seen that ej,k , ˆ j,k , are bounded. Since j,k dj0 and pj (r) are bounded, ˆdj0 , and pˆ j (t, r) are bounded. yr,j is assumed to be bounded and ej,1 = xj,1 − yr,j , then, xj,1 is bounded. Because ˆ j,1 , xj,1 , y˙ r,j are bounded, from (15), α j,1 is bounded. ej,1 , Thus, xj,2 = ej,2 + vj,1 must be bounded. Similarly, it can be proven in turn that vj,rj and xj,rj are bounded. According to the actual controller τj in (49), it can be known that τj is bounded. Therefore, it is concluded that all signals in the resulting closed-loop system are bounded. From (60) and (65), it is easy to obtain

! μ 2μj,lj

xj,1 − yr,j = ej,1 ≤ 2 Vj,l (0) − j,lj e−βj,lj t + . j βj,lj βj,lj

(67)

" If Vj,lj (0) = μj,lj /βj,lj , then, it holds xj,1 − yr,j ≤ 2μj,lj /βj,lj . If Vj,lj (0) = μj,l"j /βj,lj , from (67), it can be concluded that given any z > 2μj,lj /β j,lj , there exists Tz such that for any t > Tz , it has xj,1 − yr,j ≤ z . Specially, given any μj,lj −βj,l Tz 2μj,lj ! j + , e z = 2 Vj,lj (0) − βj,lj βj,lj μj,lj Vj,lj (0) = βj,lj

# μj,lj μj,lj 1 2 2 Vj,lj (0) − Tz = − ln z − 2 μj,lj βj,lj βj,lj

" % $ it has j = xj,1

xj,1 − yr,j ≤ μj,lj /βj,lj . IV. S IMULATION E XAMPLE

In this section, a simulation will be provided to demonstrate the feasibility of the approach. Consider the systems, which are described by ⎧ x˙ 1,1 = 1,1 x1,1 , x2,1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 1,2 = 1,2 x1,1 , x1,2 , x2,1 , x2,2 , u1 x˙ 2,1 = 2,1 x1,1 , x2,1 , x2,2 (68) ⎪ ⎪ x ˙ x = , x , x , x , u , u ⎪ 2,2 1,1 1,2 2,1 2,2 1 2 ⎪ ⎩ 2,2 yj = xj,1 , j = 1, 2 T

where x¯ j,2 = xj,1 , xj,2 and yj are the state and output of the jth subsystem, respectively, j = 1, 2; uj is the input of the jth subsystem and the output PI hysteresis as shown in (2): uj (t) = D

p τ (t) − dj τj (t) with pj0 = 0 pj (r)dr and dj τj (t) = j0D j

0 pj (r)Fjr τj (t)dr. In this simulation 2 2 1,1 (.) = 0.5 x1,1 + x2,1 + 1 + 0.1e−x1,1 x2,1 x1,2

1,2 (.) = x1,1 x1,2 + x2,1 x2,2 + 2 + cos x1,1 x2,1 u1

2,1 (.) = x1,1 x2,1 + 2 + sin x1,1 x2,1 x2,2 2 2,2 (.) = x1,1 x1,2 + x2,1 x2,2 + u1 + e−x2,1 (u2 + 0.5 sin (u2 )) 2

and pj (r) is chosen as pj (r) = 0.02e−0.0015(r−1) , r ∈ [0, 100]. The initial conditions are x1,1 (0) = 0.2, x1,2 (0) = −0.5, x2,1 (0) = 0.5, x2,2 (0) = 1. The control objective is to apply the adaptive neural control approach to the system (68) such that: 1) the outputs y1 and y2 can track the given signals yr,1 = 0.2 cos (0.5t) + 0.2 sin (2t) and yr,2 = 0.5 cos(t) + 0.3 sin (0.5t) to a small bounded compact set and 2) the boundedness of all the closed-loop signals is guaranteed. T X The NN S1,1 1,1 W1,1 contains ten nodes (i.e., l1,1 = 10), with centers πl evenly spaced in [−2, 2] × [−2.5, 2.5] × [−3, 3], and widths ωl = 3, l T= 1, . . . , l1,1 X1,2 W1,2 where X1,1 = x1,1 , x2,1 , y˙ r,1 . The NN S1,2 contains 25 nodes (i.e., l1,2 = 25), with centers πl evenly spaced in [−2, 2] × [−1.5, 1.5] × [−2.5, 2.5] × [−2.5, 2.5] × [−2, 2] × [−1.5, 1.5] × [−2.5, 2.5], = 5, l = 1, . . . , l1,2 , where and widths ω l

X1,2 = x1,1 , x1,2 , x2,1 , x2,2 , ϕ1,1 , ∂v1,1 /∂x1,1 , ∂v1,1 /∂x2,1 . T X NN S2,1 (i.e., l2,1 = 27), 2,1 W2,1 contains 27 nodes with centers πl l = 1, . . . , l2,1 evenly spaced in [−2.5, 2.5] × [−1.5, 1.5] × [−2.5,

2.5], and widths ωl = 3, l = 1, . . . , l2,1 , where X2,1 = x1,1 , x2,1 , y˙ r,2 . The NN T X S2,2 2,2 W2,2 contains 30 nodes (i.e., l2,2 = 30), with centers πl evenly spaced in [−2.5, 2.5]×[−1.5, 1.5]×[−2.5, 2.5] × [−2, 2] × [−2.5, 2.5] × [−3, 3] × [−3, 3], = 1.5, l = 1, . . . , l2,2 , where and widths ωl

X2,2 = x1,1 , x1,2 , x2,1 , x2,2 , ϕ2,1 , ∂v2,1 /∂x1,1 , ∂v2,1 /∂x2,1 . 2

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Fig. 2. (a) y1 (solid line) and yr,1 (dashed line). (b) y2 (solid line) and yr,2 (dashed line) in this paper.

9

ˆ 1,1 (solid line), ˆ 2,1 (dashed line), ˆ 1,2 Fig. 4. Adaptation laws ˆ 2,2 (dotted line). (dotted-dashed line), and

The design parameters are selected as k1,1 = k1,2 = 15, k2,1 = k2,2 = 6, δ1 = δ2 = 8, γ1 = γ2 = 5, and η1,1 = 6, η1,2 = 8, η2,1 = η2,2 = 10, σ1,1 = σ1,2 = 8, σ2,1 = σ2,2 = 12. Figs. 2–4 show the simulation results which are obtained by applying the control controller (69) to the system (68). Fig. 2 shows the tracking trajectories and it can be seen that the good tracking performances are obtained. Fig. 3 illustrates the trajectories of the control signals. The boundedness of adapˆ 2,1 , ˆ 1,2 , and ˆ 2,2 are given in Fig. 4. It ˆ 1,1 , tation laws is obvious that they are bounded. Thus, we can conclude that all the signals in the closed-loop system are bounded from these figures.

V. C ONCLUSION Fig. 3. (a) Control signals τ1 (solid line) and u1 (dashed line). (b) Control signals τ2 (solid line) and u2 (dashed line).

According to the above design procedure, the actual controllers τj , j = 1, 2 are constructed as τj = dˆ j0 τj0

(69) 2 )e ˆ ST X S X where τj0 = (−1/2)(1/σj,2 −

j,2

j,2 j,2 j,2 j,2 j,2 D

kj,2 ej,2 −sgn ej,2 0 pˆ j (t, r) Fjr τj (t) dr, ej,2 = xj,2 − vj,1 , 2 )e T vj,1 = −kj,1 ej,1 (−1/2)(1/σj,1 j,1 ˆ j,1 Sj,1 Xj,1 Sj,1 Xj,1 , and ej,1 = xj,1 − yr,j . ˆ j,rj , rj = 1, 2, j = 1, 2, pˆ j (t, r) The adaptation laws for and dˆ j0 are given as 1 1 2 T ˆ j,rj + ˆ˙ j,rj = −ηj,rj ej,rj Sj,rj Xj,rj Sj,rj Xj,rj 2 2 σj,r j

∂ pˆ j (t, r) = −δj pˆ j (t, r) + δj ej,lj Fjr τj (t) ∂t ˙ dˆ j0 = −γj dˆ j0 − γj ej,lj τj0 ˆ 1,1 (0) = 0.3, where the initial values are shown as ˆ ˆ ˆ 2,1 (0) = 0.3, 1,2 (0) = 0.5, 2,2 (0) = 0.5, pˆ j (0, r) = 0.2, dˆ j0 (0) = 0.2.

An adaptive neural control approach has been proposed for uncertain nonlinear interconnected MIMO systems in pure feedback nonaffine form with the PI hysteresis model. The considered systems are composed of N interconnected subsystems. The interconnection terms appear in each equation of each subsystem, and the output of the PI hysteresis is nonlinearly appeared in unknown functions. The present consideration renders the existing results as particular cases of this paper. Moreover, a restrictive assumption imposed on the previous works has been removed. The proposed approach can guarantee that all the closed-loop signals are SGUUB and the outputs follow the desired signals to a small compact set. The simulation results have showed that the scheme can realize the desired performance.

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Yan-Jun Liu received the B.S. and M.S. degrees in applied mathematics and control theory and control engineering from the Shenyang University of Technology, Shenyang, China, in 2001 and 2004, respectively, and the Ph.D. degree in control theory and control engineering from the Dalian University of Technology, Dalian, China, in 2007. He is currently a Professor with the College of Science, Liaoning University of Technology, Jinzhou, China. His current research interests include adaptive fuzzy control, nonlinear control, neural network control, reinforcement learning, and optimal control.

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Shaocheng Tong received the B.S. degree from Jinzhou Normal College, Jinzhou, China, the M.S. degree from Dalian Marine University, Dalian, China, in 1982 and 1988, respectively, both in mathematics, and the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 1997. He is currently a Professor with the College of Science, Liaoning University of Technology, Jinzhou. His current research interests include fuzzy and neural networks control theory and nonlinear control, adaptive control, and system identification.

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 is currently a Chair Professor with the Department of Computer and Information Science and the Dean of the Faculty of Science and Technology, University of Macau, Macau, China. His current research interests include systems, cybernetics, and computational intelligence. Dr. Chen is a fellow of American Association for the Advancement of Science. He is currently an Editor-in-Chief of the IEEE Transactions on Systems, Man, and Cybernetics: Systems.

Dong-Juan Li received the B.S. degree in applied chemistry from the Shenyang University of Technology, Shenyang, China, in 2003, and the M.S. degree in chemical engineering technology from Dalian Polytechnic University, Dalian, China, in 2007. She is currently a Lecturer with the Department of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou, China. Her current research interests include process control, adaptive control, and neural network control.