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Adaptive NN Controller Design for a Class of Nonlinear MIMO Discrete-Time Systems Yan-Jun Liu, Li Tang, Shaocheng Tong, and C. L. Philip Chen, Fellow, IEEE

Abstract— An adaptive neural network tracking control is studied for a class of multiple-input multiple-output (MIMO) nonlinear systems. The studied systems are in discrete-time form and the discretized dead-zone inputs are considered. In addition, the studied MIMO systems are composed of N subsystems, and each subsystem contains unknown functions and external disturbance. Due to the complicated framework of the discretetime systems, the existence of the dead zone and the noncausal problem in discrete-time, it brings about difficulties for controlling such a class of systems. To overcome the noncausal problem, by defining the coordinate transformations, the studied systems are transformed into a special form, which is suitable for the backstepping design. The radial basis functions NNs are utilized to approximate the unknown functions of the systems. The adaptation laws and the controllers are designed based on the transformed systems. By using the Lyapunov method, it is proved that the closed-loop system is stable in the sense that the semiglobally uniformly ultimately bounded of all the signals and the tracking errors converge to a bounded compact set. The simulation examples and the comparisons with previous approaches are provided to illustrate the effectiveness of the proposed control algorithm. Index Terms— Adaptive control, discrete-time, input nonlinearity, neural networks, uncertain nonlinear systems.

I. I NTRODUCTION

I

N the past two decades, the adaptive control technique for nonlinear systems has attracted increasing attention. Specially, because of the inherent approximation property of the neural networks (NNs) and the fuzzy logic systems [1]–[3], the intelligent control and design had been a very important research domain. For example, the adaptive fuzzy tracking controls were addressed in [4] and [5] for uncertain nonlinear single-input single-output (SISO) systems. Subsequently, some result focused on the adaptive fuzzy or neural control for uncertain nonlinear SISO systems with stochastic disturbances [6]–[10]. A novel adaptive control design was presented in [11] for nonlinear pure-feedback systems with the Manuscript received April 6, 2013; revised March 2, 2014 and May 31, 2014; accepted June 3, 2014. Date of publication July 21, 2014; date of current version April 15, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61104017 and Grant 61374113 and in part by the Program for Liaoning Innovative Research Team in University under Grant LT2012013 and Program for Liaoning Excellent Talents in University under Grant LR2014016. Y. J. Liu, L. Tang, and S. Tong are with the College of Science, Liaoning University of Technology, Liaoning 121001, China (e-mail: [email protected]; [email protected]; [email protected]). C. L. P. Chen is with the Faculty of Science and Technology, University of Macau, Macau, 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/TNNLS.2014.2330336

unmeasured states and without using backstepping. In [12], a novel learning algorithm with modeling and identification was proposed for nonlinear multiple-input single-output systems using the radial basis functions NNs (RBFNNs). The proposed method simplifies NN training through using an adaptive computation algorithm. Chen et al. [13] developed originally a general framework of the so-called distributed cooperative learning where the learned RBFNNs are proved to have the better generalization capability. Li et al. [14] first time proposed a novel adaptive sliding-mode control method for the nonlinear uncertain vehicle active suspension systems via the FLS approach and handled the actuator nonlinearity and parameter uncertainties in the vehicle suspension model. A consensus control was designed in [15] for nonlinear multiagent systems with state time-delay based on the RBFNNs. The adaptive control approaches were studied in [16]–[20] for some real systems, which are important for the practical applications. In the above results, the approaches were obtained for nonlinear systems in continuous time form. To this end, some researchers had devoted many efforts for studying the control problem of nonlinear discrete-time systems. In [21], the output feedback adaptive control was investigated for a class of nonlinear systems in discrete-time form with unknown control gains. Using the NN, the elegant adaptive control schemes were developed in [22] and [23] for nonlinear discrete-time SISO systems with completely unknown functions. Two important discrete-time controllers using the NN and the FLS were proposed in [24] and [25] for nonlinear dynamical multiple-input multiple-output (MIMO) systems. Some significant works were made in [26]–[31] for nonlinear discrete-time MIMO systems in the triangular structure. Two excellent real-time discrete neural controllers were designed in [32] and [33] for electric induction motors. Some excellent and important works were made in [34]–[40] to solve the optimal control problems for nonlinear discrete-time systems based on the NNs. However, the above control approaches do not consider the effect of the input nonlinearities, such as the dead zone, the backlash, the hysteresis, and so on. Their existence may be an unstable source of the systems. Recently, the adaptive control design was studied for continuous-time nonlinear dynamic systems preceded by unknown symmetric [41] or nonsymmetric [42], [43] dead zone. The adaptive tracking control was addressed in [44] for a class of uncertain MIMO nonlinear systems with nonsymmetric input constraints. The auxiliary design system is introduced to analyze the effect of the input

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 5, MAY 2015

constraints, and its states are used in adaptive tracking control design. The linear parameterized condition was sufficed in these works. Two earlier dead-zone compensation methods were developed in [45] and [46] for motion control systems using the adaptive fuzzy and NN control theories. An adaptive neural controller was proposed for nonlinear systems with nonlinear dead-zone and multiple time-delays [47]. The effectiveness of the approach was verified by using simulations and experiments for a turntable servo system with permanent magnet synchronous motor. Two adaptive NN schemes were provided in [48] and [49] for uncertain nonlinear systems with input nonlinearities. An adaptive NN control was developed in [50] for a class of continuous stirred tank reactors with dead-zone inputs. An output feedback control using the adaptive fuzzy was studied in [51] for a class of nonlinear systems with dead-zone inputs. All the results focused on nonlinear continuous time systems with input nonlinearities. When the considered plants are nonlinear discrete-time systems, these results cannot be directly used in the practical applications. For discrete-time linear plants with unknown dead zones, Tao and Kokotovic [52] proposed a controller structure with an adaptive dead-zone inverse, and the dead zone is allowed to be with unequal slopes. A fuzzy logic compensator was constructed in [53] for controlling uncertain nonlinear discrete-time systems with dead-zone input. In [54], an adaptive critic NN controller based on reinforcement learning was framed to stabilize a class of nonlinear systems with both unknown functions and nonsymmetric dead-zone inputs. These results assumed that the construction of the systems satisfies the matching condition. In [55], a backstepping control approach was studied for the class of discrete time nonlinear system in the presence of input nonlinearities like saturation and dead zone. Unfortunately, the existing approaches were designed for simple SISO systems [52]–[55], or MIMO systems satisfying the matching condition [43]–[54]. In comparison with those methods in continuous time systems, little results are available for discrete-time systems. In particular, for those MIMO discrete-time nonlinear systems in the triangular structure with input nonlinearities, the controller design and the stability analysis become more complicated and very difficult. Thus, the adaptive control of nonlinear MIMO discrete-time triangular systems with input nonlinearities needs to be further investigated. Motivated by the aforementioned works, we study an adaptive neural tracking control for a class of nonlinear MIMO nonlinear discrete-time triangular systems with completely unknown function. Because the dead zone is an important input nonlinearity phenomenon and it is frequently encountered in various engineering systems, this paper will take into account the effect of the dead-zone input. By using the difference Lyapunov method, it is proved that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB) and the track errors converge to a bounded compact set. The effectiveness of the proposed algorithm is verified by using two simulation examples. The main contributions of this paper are summarized as follows. In contrast to a great number of works on nonlinear continuous-time systems, there are a few results being

available for discrete-time systems. Specifically, no effective results are made for controlling MIMO nonlinear discrete-time systems in the triangular structure with unknown dead-zone input. The proposed NN control algorithm in this paper can solve this control problem. The considered systems are composed of N subsystems, and each subsystem is in strict feedback form and contains unknown dead-zone input. The structure property of the systems is inevitable to bring about a completed design procedure. In addition, due to the existence of the dead zone and noncausal problem of the discrete-time systems, there are some difficulties in the design. Many approaches have been obtained for nonlinear MIMO continuous-time systems in a similar form of this paper. However, there are the major differences in the controller design and the concepts for the discrete-time systems and the continuous-time systems. Owing to the fact that the dead zone is in the feedforward loop of the systems, it will result in additional complexities and difficulties in the controller design and the stability analysis. In order to control this class of systems, a systematic design procedure and new controllers are provided constructively. To handle the dead-zone input, an adaptation compensatory term is introduced to analyze the effect of the input constraints. II. S YSTEM D ESCRIPTION AND P RELIMINARIES Consider nonlinear discrete-time MIMO systems as follows: ⎧ ξ j,i j (k + 1) ⎪ ⎪ ⎪ ⎪ ⎪ = f j,i j (ξ¯ j,i j (k)) + g j,i j (ξ¯ j,i j (k))ξ j,i j +1 (k) ⎪ ⎪ ⎪ ⎪ ⎪ i j = 1, . . . , n j − 1 ⎨ (1) ξ j,n j (k + 1) ⎪ ⎪ ⎪ = f j,n j (ξ(k), u¯ j −1 (k)) + g j,n j (ξ(k))u j (k) ⎪ ⎪ ⎪ ⎪ ⎪ +d j (k)y j (k) ⎪ ⎪ ⎩ j = 1, . . . , N = ξ j,1 (k), where ξ j (k) = [ξ j,1 (k), . . . , ξ j,n j (k)]T is the state of the j th subsystem, ξ(k) = [ξ1T (k), . . . , ξ NT (k)]T is the state of the whole system, and ξ¯ j,i j (k) = [ξ j,1 (k), . . . , ξ j,i j (k)]T ∈ R i j is the i j th state of the j th subsystem; y j (k) is the output of the j th subsystem and y(k) = [y1 (k), . . . , y N (k)]T ∈ R N is the output vector of the systems; f j,i j (·) and g j,i j (·) are unknown smooth nonlinear functions; d j (k) is the external disturbance to be bounded by |d j (k)| ≤ d¯ j ; j , i j , and n j are positive integers; u¯ j −1 (k) = [u 1 (k), . . . , u j −1 (k)] and u j (k) = D[v j (k)] are taken as the input of the j th subsystem and the output of the dead zone, where v j (k) is the input of the dead zone. The dead zone u j (k) can be defined as ⎧ ⎪ ⎨m r j (v j (k) − br j ), if v j (k) ≥ br j u j (k) = D(v j (k)) = 0, if − bl j < v j (k) < br j (2) ⎪ ⎩ m l j (v j (k) + bl j ), if v j (k) ≤ −bl j where m r j and m l j are the right and left slopes of the dead zone, respectively; br j and bl j stand for the breakpoints; m r j , m l j , br j , and bl j are the positive constants. The dead zone can

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be expressed as u j (k) = m j (k)v j (k) + b j (k) where

(3)



m r j , if v j (k) > 0 m l j , if v j (k) ≤ 0 ⎧ ⎪ if v j (k) ≥ br j ⎨−m r j br j , b j (k) = −m j (k)v j (k), if − bl j < v j (k) < br j ⎪ ⎩ m l j bl j , if v j (k) ≤ −bl j .

m j (k) =

If we choose the following ideal virtual control for the first equation in (7) with i 1 = 1:

∗ (k) = − f 1,1 (ξ¯1,1 (k)) − yd1 (k + 1) /g1,1 (ξ¯1,1 (k)) ξ1,2

We know min(m r j , m l j ) = m j ≤ |m j (k)| ≤ m¯ j = max(m r j , m l j ) and |b j (k)| ≤ b¯ j = max(m r j br j , m l j bl j ). Similar to the results in [42], m j and m¯ j are known. The control objective of this paper is to design the adaptive controllers v j (k), j = 1, . . . , N for the system (1) so that all the signals in the closed-loop system are SGUUB, and the system outputs y j (k) follow the desired signals yd, j (k) as close as possible, where yd, j (k) are known smooth and bounded. Assumption 1: The signs of the smooth functions g j,i j (·) are known and there are positive constants g j,i j and g¯ j,i j such that g j,i j ≤ |g j,i j (·)| ≤ g¯ j,i j . We assume that g j,i j (·) are positive in this paper, i.e., g j,i j ≤ g j,i j (·) ≤ g¯ j,i j . Remark 1: It should be mentioned that several adaptive control approaches have been obtained in [26]–[28], [30], and [31] for MIMO nonlinear systems in a similar form of this paper. However, the dead-zone input is ignored in these results. When the dead-zone input is taken as a crucial component of the systems, these approaches cannot guarantee the stability of the closed-loop system. Furthermore, due to the fact that the discrete-time dead zone appears in the feedforward loop of the systems, the corresponding control problem becomes more challenging. The following designs are given to control the MIMO system (1). In this paper, the RBFNNs are employed to approximate the continuous function f (y) f NN (y) = σ T S(y)

(4)

where y ∈ is the input variable of the NNs, σ = [σ1 , . . . , σl ]T is the weight vector with l being the NN node number, S(y) is the smooth basis function vector to be S(y) = [s1 (y), . . . , sl (y)]T , and si (y) is chosen as the commonly used Gaussion functions   −(y − μi )T (y − μi ) , i = 1, . . . , l (5) si (y) = exp υi2 Rq

where μi = [μi1 , . . . , μiq ]T and υi are the center and the width of the Gaussian functions, respectively. From (4), it yields that f (y) = σ

∗T

∗

S(y) + ε (y)

Consider the first subsystem in the system (1) ⎧ ⎪ ⎨ξ1,i1 (k + 1) = f 1,i1 (ξ¯1,i1 (k)) + g1,i1 (ξ¯1,i1 (k))ξ1,i1 +1 (k) (7) i 1 = 1, . . . , n 1 − 1 ⎪ ⎩ ξ1,n1 (k + 1) = f 1,n1 (ξ(k)) + g1,n1 (ξ(k))u 1 (k).

(6)

where σ ∗ and ε∗ (y) are the ideal constant weight and the optimal approximation error, respectively. Assumption 2: There are positive constants σ¯ and ε¯ so that σ ∗  ≤ σ¯ and |ε∗ (y)| ≤ ε¯ where y ∈  y , and  y is a compact set.

then, the first equation in (7) will be stabilized. Similarly, the ideal virtual control for stabilizing the second equation in (7) can be selected as

∗ ∗ (k) = − f1,2 (ξ¯1,2 (k)) − ξ1,2 (k + 1) /g1,2 (ξ¯1,2 (k)) ξ1,3 ∗ (k + 1) is a future virtual control. It leads to that the where ξ1,2 ∗ (k) is infeasible. If the above process ideal virtual control ξ1,3 is repeated to construct the actual control u ∗1 (k). It will lead to that u ∗1 (k) is infeasible because of the unavailable future information. This is a major problem in discrete-time domain by using the backstepping design. Based on the above reasons, the system equation must be transformed into a special form. The j th subsystem of the system (1) can be rewritten as ⎧ ⎪ ⎪ ⎪ξ j,i j (k + n j − i j + 1) ⎪ ⎪ ⎪ =  j,i j (k) +  j,i j (k) × ξ j,i j +1 (k + n j − i j ) ⎨ (8) i j = 1, . . . , n j − 1 ⎪ ⎪ ⎪ξ j,n j (k + 1) = f j,n j (k) + g j,n j (k)u j (k) + d j (k) ⎪ ⎪ ⎪ ⎩ y (k) = ξ (k) j j,1

where  j,n j −1 (k) =  j,n j −1 (ξ¯ j,n j (k))  j,n j −1 (k) =  j,n j −1 (ξ¯ j,n j (k)) f j,n j (k) = f j,n j (ξ(k)), g j,n j (k) = g j,n j (ξ(k)). Note that it can be known from the definition of  j,i j (k) that  j,i j (k) is still satisfying Assumption 1, that is, g j,i j ≤  j,i j (k) ≤ g¯ j,i j . The transformation basic idea is described in [21]–[26] to be omitted in this paper. Remark 2: It can be easily seen that if the exact system model is known and the disturbance converges to zero, then the desired virtual controls and the ideal practical controls will drive the output to track the desired output exactly. But, these conditions usually cannot be satisfied. In this paper, the NNs will be used to approximate the unknown functions. In the following, the controllers and the adaptation laws are designed based on (8). III. C ONTROLLER D ESIGN AND S TABILITY A NALYSIS A. Controller Design In this section, we will incorporate the backstepping design procedure into the adaptive NN control for the j th subsystem described in (1) or (8). The design procedure contains n j steps for the j th subsystem. From Step 1 to Step n j − 1, the virtual controllers β j,i j , i j = 1, . . . , n j − 1 are designed. The

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controller v j is constructed in Step n j . The detailed design procedures are described in the following steps. Step 1: Consider the tracking error z j,1 (k) = ξ j,1 (k) − yd, j (k), and its n j th difference from the first equation in (8) can be expressed as z j,1 (k + n j ) = ξ j,1 (k + n j ) − yd, j (k + n j ) =  j,1 (k) − yd, j (k + n j ) +  j,1 (k)ξ j,2 (k + n j − 1). (9) Because the unknown terms  j,1 (k) and  j,1 (k) appear in (9), in general, the ideal controller cannot be used. Then, we define the unknown function U j,1 (k) = −[ j,1 (k) − yd, j (k + n j )]/ j,1 (k).

(10)

(11)

∗ ∈ R l j,1 , S (Z (k)), and ε ∗ (Z (k)) are the where σ j,1 j,1 j,1 j,1 j,1 ideal constant weight, the basis function vector, and the optimal approximation error, respectively, and they satisfy ∗  ≤ σ ¯ j,1 , S j,1 (Z j,1 (k))2 ≤ l j,1 , and |ε∗j,1(Z j,1 (k))| ≤ σ j,1 ε¯ j,1 ; Z j,1 (k) = [ξ¯ Tj,n j (k), yd, j (k + n j )]T ∈  j,1 is the input ∗ and let vector of the NNs; σˆ j,1 (k) is the estimation of σ j,1 ∗ σ˜ j,1 (k) = σˆ j,1 (k) − σ j,1 . The virtual controller β j,1 (k) is designed as T β j,1 (k) = σˆ j,1 (k)S j,1 (Z j,1 (k))

(12)

and the adaptation law for the NN weight is chosen as σˆ j,1 (k + 1)

= σˆ j,1 (k1 ) − j,1 S j,1 (Z j,1 (k1 ))z j,1 (k +1)+τ j,1σˆ j,1 (k1 ) (13) where j,1 = Tj,1 > 0 and τ j,1 > 0 are the design parameters, and k1 = k − n j + 1. Subtracting σ j,1 on both sides of (13), it has σ˜ j,1 (k + 1) = σ˜ j,1 (k1 ) − j,1



S j,1 (Z j,1 (k1 ))z j,1 (k + 1)+τ j,1 σˆ j,1 (k1 ) . (14)

By introducing z j,2 (k + n j − 1) = ξ j,2 (k + n j − 1) − β j,1 (k) and substituting (10)–(12) into (9), it follows that: z j,1 (k + n j )

T =  j,1 (k) σ˜ j,1 (k)S j,1 (Z j,1 (k)) − ε∗j,1 (Z j,1 (k))

+z j,2 (k +n j −1) .

(15)

Define the following Lyapunov function: n j−1

v j,1 (k) =

z 2j,1 (k)/g¯ j,1 +



T σ˜ j,1 (k1 +q) −1 ˜ j,1 (k1 +q). (16) j,1 σ

The first difference of (16) is v j,1 = v j,1 (k + 1) − v j,1 (k)

T = z 2j,1 (k + 1) − z 2j,1 (k) /g¯ j,1 + σ˜ j,1 (k + 1) + 1)

−2ε∗j,1(Z j,1 (k1 ))z j,1 (k + 1) + 2z j,2 (k)z j,1 (k + 1)

T ×S j,1 (Z j,1 (k1 ))z 2j,1 (k + 1) + 2τ j,1 σˆ j,1 (k1 ) j,1 T ×S j,1 (Z j,1 (k1 ))z j,1 (k + 1)+τ 2j,1σˆ j,1 (k1 ) j,1 σˆ j,1 (k1 ).

From the definition of  j,1 (k1 ) and Assumption 1, it has −2z 2j,1 (k + 1)/ j,1 (k1 ) ≤ −2z 2j,1 (k + 1)/g¯ j,1. Using Young’s inequality, we have S Tj,1 (Z j,1(k1 )) j,1 S j,1 (Z j,1(k1 ))z 2j,1 (k + 1) ≤ λ¯ j,1l j,1 z 2j,1 (k + 1) − 2ε∗j,1 z j,1 (k + 1) ≤ λ¯ j,1 z 2j,1 (k + 1)/g¯ j,1 + g¯ j,1ε¯ 2j,1 /λ¯ j,1 T 2τ j,1 σˆ j,1 (k1 ) j,1 S j,1 (Z j,1 (k1 ))z j,1 (k + 1) ≤ λ¯ j,1l j,1 z 2j,1 (k + 1)/g¯ j,1 + g¯ j,1 τ 2j,1 λ¯ j,1 σˆ j,1 (k1 )2 T 2σ˜ j,1 (k1 )σˆ j,1 (k1 ) ∗ 2  = σ˜ j,1 (k1 )2 + σˆ j,1 (k1 )2 − σ j,1 2z j,2 (k)z j,1 (k + 1) ≤ λ¯ j,1 z 2j,1 (k +1)/g¯ j,1 + g¯ j,1 z 2j,2 (k)/λ¯ j,1

where l j,1 is the NN node number, λ¯ j,1 is the maximum eigenvalue of the matrix j,1 . Then, we can obtain v j,1 ≤ −A j,1 z 2j,1 (k + 1)/g¯ j,1 − z 2j,1 (k)/g¯ j,1 + η j,1 +g¯ j,1 z 2j,2 (k)/λ¯ j,1 − τ j,1 B j,1σˆ j,1 (k1 )2 (18) where A j,1 = 1 − 2λ¯ j,1 − λ¯ j,1l j,1 − g¯ j,1λ¯ j,1l j,1 , η j,1 = 2 and B ¯ j,1 − g¯ j,1τ j,1 λ¯ j,1 . g¯ j,1ε¯ 2j,1 /λ¯ j,1 +τ j,1 σ¯ j,1 j,1 = 1 − τ j,1 λ Step i j (i j = 2, . . . , n j − 1): The (n j − i j + 1)th difference of z j,i j (k) = ξ j,i j (k) − β j,i j −1 (k − n j + i j − 1) is z j,i j (k + n j − i j + 1) = ξ j,i j (k + n j − i j + 1) − β j,i j −1 (k) =  j,i j (k) − β j,i j −1 (k) +  j,i j (k)ξ j,i j+1 (k +n j −i j ). (19) Likewise,  j,i j (k) and  j,i j (k) are unknown. Then, define the unknown function

q=0

× −1 ˜ j,1 (k j,1 σ

T (k )S (Z (k )) = It can be obtained from (15) that σ˜ j,1 1 j,1 j,1 1 ∗ z j,1 (k + 1)/ j,1 (k1 )+ε j,1 (Z j,1(k1 ))− z j,2 (k). Thus, (17) can be rewritten as

v j,1 = z 2j,1 (k + 1) − z 2j,1 (k) /g¯ j,1 − 2z 2j,1 (k + 1)/ j,1 (k1 ) T (k1 )σˆ j,1 (k1 ) + S Tj,1 (Z j,1(k1 )) j,1 −2τ j,1 σ˜ j,1

Using the RBFNNs, U j,1 (k) can be approximated as ∗T U j,1 (k) = σ j,1 S j,1 (Z j,1 (k)) + ε∗j,1(Z j,1 (k))

Using (14), the above equation can be rewritten as

T (k1 ) v j,1 = z 2j,1 (k + 1) − z 2j,1 (k) /g¯ j,1 − 2σ˜ j,1

× S j,1 (Z j,1 (k1 ))z j,1 (k + 1) + τ j,1 σˆ j,1 (k1 )

T + S j,1 (Z j,1 (k1 ))z j,1 (k + 1) + τ j,1 σˆ j,1 (k1 )

× j,1 S j,1 (Z j,1 (k1 ))z j,1 (k + 1) + τ j,1 σˆ j,1 (k1 ) . (17)

T − σ˜ j,1 (k1 ) −1 ˜ j,1 (k1 ). j,1 σ

U j,i j (k) = −[ j,i j (k) − β j,i j −1 (k)]/ j,i j (k).

(20)

Using the NNs, U j,i j (k) can be approximated as ∗T U j,i j (k) = σ j,i S (Z j,i j (k)) + ε∗j,i j (Z j,i j (k)) j j,i j

(21)

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∗ where σ j,i ∈ R j,i j , S j,i j (Z j,i j (k)), and ε∗j,i j (Z j,i j (k)) j are the ideal constant weight, the basis function vector, and the optimal approximation error, respectively, and ∗  ≤ σ they satisfy σ j,i ¯ j,i j , S j,i j (Z j,i j (k))2 ≤ l j,i j , j ∗ and |ε j,i j (Z j,i j (k))| ≤ ε¯ j,i j ; σˆ j,i j (k) is the estimation ∗ ∗ ; Z of σ j,i and let σ˜ j,i j (k) = σˆ j,i j (k) − σ j,i j,i j (k) = j j T T [ξ¯ j,n j (k), β j,i j −1 (k)] ∈  j,i j is the input vector of the NNs. The virtual controller β j,i j (k) is designed as l

T β j,i j (k) = σˆ j,i S (Z j,i j (k)) j j,i j

(22)

and the corresponding adaptation law is designed as σˆ j,i j (k + 1) = σˆ j,i j (ki j ) − j,i j

× S j,i j (Z j,i j (ki j ))z j,i j (k + 1)+τ j,i j σˆ j,i j (ki j ) (23) where ki j = k − n j + i j . Equation (23) can be rewritten as σ˜ j,i j (k + 1) = σ˜ j,i j (ki j ) − j,i j

× τ j,i j σˆ j,i j (ki j )+ S j,i j (Z j,i j (ki j ))z j,i j (k +1) .

(24)

By introducing z j,i j +1 (k + n j − i j ) = ξ j,i j+1 (k + n j − i j ) −β j,i j (k) and substituting (20)–(22) into (19), it has z j,i j (k + n j − i j + 1) T (k)S j,i j (Z j,i j (k)) − ε∗j,i j (Z j,i j (k)) =  j,i j (k) σ˜ j,i j

(25) +z j,i j +1 (k + n j − i j ) Equation (25) can be rewritten as z j,i j (k + 1)

T (ki j )S j,i j (Z j,i j (ki j )) − ε∗j,i j =  j,i j (ki j ) σ˜ j,i j

× (Z j,i j (ki j ))+z j,i j +1 (k) .

(26)

Define the Lyapunov function v j,i j (k) = z 2j,i j (k)/g¯ j,i j n j −i j

+



T σ˜ j,i (ki j + q) −1 ˜ j,i j (ki j + q) j,i j σ j

(27)

q=0

The first difference of (27) is obtained

T v j,i j = z 2j,i j (k + 1) − z 2j,i j (k) /g¯ j,i j + σ˜ j,i (k + 1) j T ˜ j,i j (k + 1)− σ˜ j,i (ki j ) −1 ˜ j,i j (ki j ). × −1 j,i j σ j,i j σ j

Substituting (24) into the above equation yields

T (ki j ) v j,i j = z 2j,i j (k +1) − z 2j,i j (k) /g¯ j,i j − 2σ˜ j,i j

× S j,i j (Z j,i j (ki j ))z j,i j (k +1) + τ j,i j σˆ j,i j (ki j )

T + S j,i j (Z j,i j (ki j ))z j,i j (k +1) + τ j,i j σˆ j,i j (ki j ) j,i j

× S j,i j (Z j,i j (ki j ))z j,i j (k +1)+τ j,i j σˆ j,i j (ki j ) . (28) T (k ) can be obtained from (26) that σ˜ j,i ij j ∗ S j,i j (Z j,i j (ki j )) = z j,i j (k + 1)/ j,i j (ki j ) + ε j,i j (Z j,i j (ki j )) − z j,i j +1 (k). Similar to Step 1 and based on Assumption 1, it has

 −2 z 2j,i j (k + 1) / j,i j ki j ≤ −2z 2j,i j (k + 1) /g¯ j,i j .

It

Using Young’s inequality, we have S Tj,i j (Z j,i j (ki j )) j,i j S j,i j (Z j,i j (ki j ))z 2j,i j (k + 1) ≤ λ¯ j,i j l j,i j z 2j,i j (k + 1) 2ε∗j,i j z j,i j (k + 1) ≤ λ¯ j,i j z 2j,i j (k + 1)/g¯ j,i j + g¯ j,i j ε¯ 2j,i j /λ¯ j,i j T 2τ j,i j σˆ j,i (ki j ) j,i j S j,i j (Z j,i j (ki j ))z j,i j (k + 1) j

≤ λ¯ j,i j l j,i j z 2j,i j (k + 1)/g¯ j,i j + g¯ j,i j τ 2j,i j λ¯ j,i j σˆ j,i j (ki j )2 T 2σ˜ j,i (ki j )σˆ j,i j (ki j ) j ∗ 2 = σ˜ j,i j (ki j )2 + σˆ j,i j (ki j )2 − σ j,i j

2z j,i j +1 (k)z j,i j (k + 1) ≤ λ¯ j,i j z 2j,i j (k + 1)/g¯ j,i j + g¯ j,i j z 2j,i j +1 (k)/λ¯ j,i j where l j,i j is the NN node number, and λ¯ j,i j is the maximum eigenvalue of the matrix j,i j . Based on the above facts, we obtain v j,i j ≤ −A j,i j z 2j,i j (k + 1)/g¯ j,i j −z 2j,i j (k)/g¯ j,i j + η j,i j +g¯ j,i j z 2j,i j +1 (k)/λ¯ j,i j −τ j,i j B j,i j σˆ j,i j (ki j )2 (29) where α j,i j = 1 − 2λ¯ j,i j − λ¯ j,i j l j,i j − g¯ j,i j λ¯ j,i j l j,i j , η j,i j = 2 τ j,i j σ¯ j,i + g¯ j,i j ε¯ 2j,i j /λ¯ j,i j and B j,i j = 1 − τ j,i j λ¯ j,i j − j g¯ j,i j τ j,i j λ¯ j,i j . Remark 3: In the following step, the unknown dead zone will appear in the input of the systems. To compensate for the optimal neural weight and the unknown parameter in the dead zone, two adaptation laws will be designed. Only one adaptation law for the optimal neural weight is designed in the previous results without considering the dead zone. In this paper, owing to the existence of the dead zone, it is inevitable to make a completed design procedure. Step n j : Define z j,n j (k) = ξ j,n j (k) − β j,n j −1 (k − 1) and the first difference of z j,n j (k) is z j,n j (k + 1) = ξ j,n j (k + 1) − β j,n j −1 (k) = f j,n j (k) + g j,n j (k)u j (k) + d j,n j (k) − β j,n j −1 (k). (30) Because u j (k) = D[v j (k)] is the output of the dead zone, which contains unavailable information, it is not directly designed. Then, substituting (3) into (30) yields z j,n j (k + 1) = f j,n j (k) + g j,n j (k)m j (k)v j (k) +g j,n j (k)b j (k)+d j,n j (k)−β j,n j−1 (k).(31) Define the following unknown function: U j,n j (k) = −[ f j,n j (k) − β j,n j −1 (k)]/g j,n j (k)m j (k). (32) Using the NNs, U j,n j (k) can be approximated as ∗T U j,n j (k) = σ j,n S (Z j,n j (k)) + ε∗j,n j (Z j,n j (k)) j j,n j

(33)

∗ ∈ R j,n j , S j,n j (Z j,n j (k)), and ε∗j,n j (Z j,n j (k)) where σ j,n j are the ideal constant weight, the basis function vector, and the optimal approximation error, respectively, and they l

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∗  ≤ σ satisfy σ j,n ¯ j,n j , S j,n j (Z j,n j (k))2 ≤ l j,n j and j ∗ |ε j,n j (Z j,n j (k))| ≤ ε¯ j,n j ; σˆ j,n j (k) is the estimation of ∗ ∗ ; Z σ j,n and let σ˜ j,n j (k) = σˆ j,n j (k) − σ j,n j,n j (k) = j j T [ξ(k), β j,n j −1 (k)] ∈  j,n j is the input vector of the NNs. Using (32) and (33), (31) becomes

z j,n j (k + 1) ∗T = g j,n j (k)m j (k)[v j (k) − σ j,n × S j,n j (Z j,n j (k)) j

−ε∗j,n j (Z j,n j (k))+ b j (k)/m j (k)] + d j,n j (k).

(34)

Using (39) and (40), the above equation becomes

T (k) v j,n j = z 2j,n j (k + 1) − z 2j,n j (k) /g¯ j,n j m¯ j − 2σ˜ j,n j

× S j,n j (Z j,n j (k))z j,n j (k + 1) + τ j,n j σˆ j,n j (k)

T + S j,n j (Z j,n j (k))z j,n j (k + 1) + τ j,n j σˆ j,n j (k)

× j,n j S j,n j (Z j,n j (k))z j,n j (k + 1) + τ j,n j σˆ j,n j (k) −2ρ˜ j (k) z j,n j (k + 1) + δ j ρˆ j (k)]

2 +γ j z j,n j (k + 1) + δ j ρˆ j (k) . (43) It follows from (41) that:

It follows that: b j (k)/m j (k) ≤ b¯ j /m j = ρ j .

(35)

Because ρ j is an unknown constant parameter, ρˆ j (k) is used to denote the estimation of ρ j and let ρ˜ j (k) = ρˆ j (k) − ρ j . The actual controller v j (k) is designed as T v j (k) = σˆ j,n S (Z j,n j (k)) + ρˆ j (k). j j,n j

(36)

The adaptation laws are designed as

T σ˜ j,n S (Z j,n j (k)) + ρ˜ j (k) j j,n j

= z j,n j (k + 1)/g j,n j (k)m j (k)

+ ε∗j,n j (Z j,n j (k)) − b j (k)/m j (k) −ρ j − d j,n j (k)/g j,n j (k)m j (k).

Combining (43) with the above equation, we have

v j,n j = z 2j,n j (k +1)−z 2j,n j (k) /g¯ j,n j m¯ j + 2z j,n j (k + 1)ρ j T −2z 2j,n j (k + 1)/g j,n j (k)m j (k) − 2τ j,n j σ˜ j,n (k) j

σˆ j,n j (k + 1) = σˆ j,n j (k) − j,n j

× τ j,n j σˆ j,n j (k)+S j,n j (Z j,n j (k))z j,n j (k + 1)

ρˆ j (k + 1) = ρˆ j (k) − γ j z j,n j (k + 1) + δ j ρˆ j (k) .

×σˆ j,n j (k) − 2z j,n j (k + 1)ε∗j,n j (Z j,n j (k)) + 2z j,n j (k + 1) (37)

×b j (k)/m j (k) + 2z j,n j (k + 1)d j,n j (k)/g j,n j (k)m j (k)

(38)

2 + j,n j τ 2j,n j σˆ j,n (k) + γ j z 2j,n j (k + 1) + S Tj,n j (Z j,n j (k)) j

Remark 4: Since the adaptation law for ρˆ j (k) is used to compensate for uncertain parameters of the dead-zone input, the effect of the dead zone is avoided. If the controller does not contain ρˆ j (k), it is not easy to ensure the stability of the closed-loop system. This fact will be verified later in the simulation section. Equations (37) and (38) can be rewritten as σ˜ j,n j (k + 1)

(39) (40)

+ d j,n j (k).

Based on Assumption 1, it has −z 2j,n j (k + 1)/g j,n j (k)m j (k) ≤ −z 2j,n j (k + 1)/g¯ j,n j m¯ j . S Tj,n j (Z j,n j (k)) j,n j S j,n j (Z j,n j (k))z 2j,n j (k + 1) ≤ λ¯ j,n j l j,n j z 2j,n j (k + 1) +g¯ j,n j m¯ j ε¯ 2j,n j /λ¯ j,n j

z j,n j (k + 1)

− ε∗j,n j (Z j,n j (k)) + ρ j + b j (k)/m j (k)

+2γ j δ j ρˆ j (k)z j,n j (k + 1) + γ j δ 2j ρˆ 2j (k).

−2ε∗j,n j z j,n j (k + 1) ≤ λ¯ j,n j z 2j,n j (k + 1)/g¯ j,n j m¯ j

Based on (36), (34) can be expressed as T = g j,n j (k)m j (k) σ˜ j,n S (Z j,n j (k)) + ρ˜ j (k) j j,n j

×S j,n j (Z j,n j (k))z j,n j (k + 1) − 2δ j ρ˜ j (k)ρˆ j (k)

Using Young’s inequality, we have

= σ˜ j,n j (k) − j,n j τ j,n j σˆ j,n j (k)

+S j,n j (Z j,n j (k))z j,n j (k + 1)

ρ˜ j (k + 1) = ρ˜ j (k) − γ j z j,n j (k + 1) + δ j ρˆ j (k) .

× j,n j S j,n j (Z j,n j (k))z 2j,n j (k + 1) + 2τ j,n j j,n j σˆ j,n j (k)

(41)

Define the following Lyapunov function:

T 2τ j,n j σˆ j,n (k) j,n j S j,n j (Z j,n j (k))z j,n j (k + 1) j

≤ λ¯ j,n j l j,n j z 2j,n j (k + 1)/g¯ j,n j m¯ j +g¯ j,n j m¯ j τ 2j,n j λ¯ j,n j σˆ j,n j (k)2 T 2σ˜ j,n (k)σˆ j,n j (k) j

v j,n j (k) T (k) −1 ˜ j,n j (k) + γ j−1 ρ˜ 2j (k). = z 2j,n j (k)/g¯ j,n j m¯ j + σ˜ j,n j,n j σ j

(42) The first difference of (42) is obtained

v j,n j = z 2j,n j (k + 1) − z 2j,n j (k) /g¯ j,n j m¯ j T T + σ˜ j,n (k + 1) −1 ˜ j,n j (k + 1) − σ˜ j,n (k) j,n j σ j j −1 −1 2 −1 2 × j,n j σ˜ j,n j (k) + γ j ρ˜ j (k + 1) − γ j ρ˜ j (k).

∗ 2 = σ˜ j,n j (k)2 + σˆ j,n j (k)2 − σ j,n j

2z j,n j (k + 1)b j (k)/m j (k) + 2z j,n j (k + 1)ρ j ≤ 4|z j,n j (k + 1)|ρ j ≤ 2λ¯ j,n j z 2j,n j (k + 1)/g¯ j,n j m¯ j + 2 g¯ j,n j m¯ j ρ 2j /λ¯ j,n j 2z j,n j (k + 1)d j,n j (k)/g j,n j (k)m j (k) ≤ 2|z j,n j (k + 1)|d¯ j,n j /g j,n j m j

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≤ λ¯ j,n j z 2j,n j (k + 1)/g¯ j,n j m¯ j

IV. S IMULATION E XAMPLES

+g¯ j,n j m¯ j d¯2j,n j /(g 2j,n j m 2j λ¯ j,n j )

Example 1: In order to verify the feasibility of the design results, the proposed approach is used to control the systems ⎧ ⎪ ξ1,1 (k + 1) = f 1,1 (ξ¯1,1 (k)) + g1,1(ξ¯1,1 (k))ξ1,2 (k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ξ1,2 (k + 1) = f 1,2 (ξ(k)) + g1,2 (ξ(k))u 1 (k)+d1 (k) (45) ξ2,1 (k + 1) = f 2,1 (ξ¯2,1 (k)) + g2,1(ξ¯2,1 (k))ξ2,2 (k) ⎪ ⎪ ⎪ (k + 1) = f (ξ(k)) + g (ξ(k))u (k)+d (k) ξ ⎪ 2,2 2,2 2,2 2 2 ⎪ ⎪ ⎩ y (k) = ξ (k), y (k) = ξ (k) 1 1,1 2 2,1

2γ j δ j ρˆ j (k)z j,n j (k + 1) ≤ λ¯ j,n j γ j z 2j,n j (k + 1)/g¯ j,n j m¯ j +g¯ j,n j γ j δ 2j m¯ j ρˆ 2j (k)/λ¯ j,n j 2ρ˜ j (k)ρˆ j (k) = ρ˜ j (k)2 + ρˆ j (k)2 − ρ j 2 where l j,n j is the NN node number and λ¯ j,n j is the maximum eigenvalue of the matrix j,n j . Using these facts, we can obtain v j,n j ≤ −A j,n j z 2j,n (k + 1)/g¯ j,n j m¯ j j

−τ j,n j B j,n j σˆ j,n j (k)2 + η j,n j − δ j C j ρˆ j (k)2 −z 2j,n (k)/g¯ j,n j m¯ j j

(44)

where A j,n j = 1 − 4λ¯ j,n j − λ¯ j,n j l j,n j − λ¯ j,n j γ j − m¯ j g¯ j,n j λ¯ j,n j l j,n j B j,n j = 1 − λ¯ j,n j τ j,n j − g¯ j,n j m¯ j τ j,n j λ¯ j,n j C j = 1 − γ j δ j − g¯ j,n j γ j δ j m¯ j /λ¯ j,n j 2 η j,n j = g¯ j,n j m¯ j ε¯ 2j,n j /λ¯ j,n j + τ j,n j σ¯ j,n + 2 g¯ j,n j m¯ j ρ 2j /λ¯ j,n j j

 +g¯ j,n j m¯ j d¯ 2j,n j / g 2j,n j m 2j λ¯ j,n j + δ j ρ j 2 .

Remark 5: From the above derivations, we can find that the design strategy and the technical way used in this paper have many differences from the previous works. The main differences contain: 1) due to the presence of the dead zone, the different Lyapunov function form (42) has been defined; 2) because ρˆ j (k) is needed to be tuned online, it will result in additional complexities of the first difference for v j,n j (k); and 3) Several different Young’s inequalities are skillfully employed to analyze the stability of the systems in the proof. B. Stability Analysis Theorem 1: Consider the nonlinear MIMO system (1) with the dead-zone inputs described in (2). On the compact sets  j,i j , if we initialize ξ(0) ∈  with  being an intersection of  j,i j , i j = 1, . . . , n j , j = 1, . . . , N and Assumptions 1 and 2 are satisfied, by constructing the actual controller (36), the virtual controllers (12), (22), and the adaptive laws (13), (23), (37), (38), if the design parameters are chosen appropriately, then, the proposed NN control algorithm can guarantee that the system state ξ(k), the variables z j,i j (k), the control inputs v j (k), the adaptation laws σˆ j,i j (k), ρˆ j (k), i j = 1, . . . , n j are SGUUB and the tracking errors z j,1 (k), j = 1, . . . , N converge to a small compact set denoted by z j,1 := {z j,1 | lim z j,1 (k) ≤ j .} where j is a positive k→∞ constant. Proof: See the Appendix.

where ⎧ 2 (k)/(1 + ξ 2 (k)), g (ξ¯ (k)) = 0.3 f (ξ¯ (k)) = ξ1,1 ⎪ 1,1 1,1 1,1 ⎪ ⎨ 1,1 1,1 2 (k) ξ1,1 f 1,2 (ξ(k)) = 1+ξ 2 (k)+ξ 2 (k)+ξ 2 (k) , ⎪ 1,2 2,1 2,2 ⎪ ⎩ g1,2(ξ(k)) = 1, d1 (k) = 0.1 cos(0.05k) cos(ξ1,1 (k)) ⎧ 2 (k)/(1 + ξ 2 (k)), g (ξ¯ (k)) = 0.2 f (ξ¯ (k)) = ξ2,1 ⎪ 2,1 2,1 ⎪ 2,1 ⎨ 2,1 2,1 2 (k) ξ1,1 2 f 2,2 (ξ(k)) = 2 (k)+ξ 2 (k)+ξ 2 (k) u 1 (k), 1+ξ1,2 ⎪ 2,1 2,2 ⎪ ⎩ g2,2 (ξ(k)) = 1, d2 (k) = 0.1 cos(0.05k) cos(ξ2,1 (k)) and u(k) = [u 1 , u 2 ]T ∈ R 2 is the input vector of the systems and the outputs of the dead zone to be described as ⎧ ⎪ ⎨0.3(v1 (k) − 0.5), if v1 (k) ≥ 0.5 u 1 (k) = D(v1 (k)) = 0, if − 0.6 < v1 (k) < 0.5 ⎪ ⎩ 0.2(v1 (k) + 0.6), if v1 (k) ≤ −0.6 ⎧ ⎪ ⎨0.3(v2 (k) − 0.3), if v2 (k) ≥ 0.3 u 2 (k) = D(v2 (k)) = 0, if − 0.4 < v2 (k) < 0.3 ⎪ ⎩ 0.1(v2 (k) + 0.4), if v2 (k) ≤ −0.4. The reference signals are chosen as yd,1(k) = 0.5 + 0.25 cos(kT π/4) + 0.25 cos(kT π/2) yd,2(k) = 0.5 + 0.25 sin(kT π/4) + 0.25 sin(kT π/2) T = 0.01. The control objective is to design the controllers for the system (45) such that: 1) the outputs y j (k) track yd, j (k) to a certain neighborhood of the origin and 2) the boundedness of all the signals in the closed-loop system is guaranteed. According to the design procedure in the above section, the controllers and the adaptation laws are designed as

 T S j,2 Z j,2 (k) + ρˆ j (k) v j (k) = σˆ j,2 σˆ j,1 (k + 1) = σˆ j,1 (k) − j,1 τ j,1 σˆ j,1 (k) 

+S j,1 Z j,1 (k) z j,1 (k + 1) σˆ j,2 (k + 1) = σˆ j,2 (k) − j,2 τ j,2 σˆ j,2 (k)



+S j,2 Z j,2 (k) z j,2 (k + 1)

ρˆ j (k + 1) = ρˆ j (k) − γ j z j,2 (k + 1) + δ j ρˆ j (k) . (46) The initial conditions for the states are set as ξ1,1 (0) = ξ1,2 (0) = ξ2,1 (0) = ξ2,2 (0) = 0. The initial values of the adaptation parameters are set as σˆ j,1 (0) = 0.01, σˆ j,2 (0) = 0.01 and ρˆ j (k) = 0.1. The parameters are selected as γ j = 0.2, δ j = 0.2, 1,1 = 1,2 = 2,1 = 0.025I, 2,2 = 0.2I , τ j,1 = τ j,2 = 0.01.

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Fig. 1. (a) y1 (k) = x1,1 (k) (solid line) and yd,1 (k) (dashed line). (b) y2 (k) = x2,1 (k) (solid line) and yd,2 (k) (dashed line) in this paper.

Fig. 3. (a) σˆ 1,1 (k) (solid line), σˆ 1,2 (k) (dashed line), σˆ 2,1 (k) (dot-anddash line), and σˆ 2,2 (k) (dot line). (b) ρˆ1 (k) (solid line) and ρˆ2 (k) (dashed line) in this paper.

Fig. 2. (a) z 1,2 (k) (solid line) and z 2,2 (k) (dashed line). (b) v1 (solid line) and v2 (dashed line) in this paper.

Fig. 4. (a) y1 (k) = x1,1 (k) (solid line) and yd,1 (k) (dashed line). (b) y2 (k) = x2,1 (k) (solid line) and yd,2 (k) (dashed line) in [28].

The node numbers of the NNs are chosen as l1,1 = 12, l1,2 = 20, l2,1 = 12, l2,2 = 30. In the simulation studies, the centers and widths are selected on a regular lattice in the T (k)S (Z (k)) contains respective compact sets. The NN σˆ 1,1 1,1 1,1 12 nodes with the centers μ1,1 evenly spaced in [−3, 3] × T (k) × [−2, 2] × [−1.5, 1.5] and widths υ1,1 = 5. The NN σˆ 1,2 S1,2 (Z 1,2 (k)) contains 12 nodes with the centers μ1,2 evenly spaced in [−2, 2] × [−1, 1] × [−2, 2] × [−3, 3] × [−1.5, 1.5], T (k)S (Z (k)) contains and the widths υ1,2 = 3. The NN σˆ 2,1 2,1 2,1 20 nodes with the centers μ2,1 evenly spaced in [−1, 1] × [−1.5, 1.5] × [−2.5, 2.5] and the widths υ2,1 = 2. The NN T (k)× S (Z (k)) contains 30 nodes with the centers μ σˆ 2,2 2,2 2,2 2,2 evenly spaced in [−1, 1] × [−3, 3] × [−2.5, 2.5] × [−2, 2] × [−1, 1] and the widths υ2,2 = 3. Figs. 1–3 show the simulation results which are obtained by applying the controller (46) to the system (45). Fig. 1 shows the tracking trajectories and it can be seen that the good tracking performances can be obtained. The trajectories of the variables z 1,2 (k), z 2,2 (k), v1 (k), and v2 (k) are shown in

Fig. 2. It can be seen that z 1,2 (k), z 2,2 (k), v1 (k), and v2 (k) are bounded. In Fig. 3, the boundedness of σˆ 1,1 (k), σˆ 1,2 (k), σˆ 2,1 (k), σˆ 2,2 (k), ρˆ1 (k), and ρˆ2 (k) is given. To compare with the control performances between our control approach and that in [26], we apply the control approach in [26] to control the system (45) with the deadzone input. In this simulation, the initial values for the states, the NNs, and the design parameters apart from the parameters γ j , δ j are chosen the same as our control method. The tracking performances for the approach in [26] are shown in Fig. 4. It can be seen that a good tracking performance is not achieved. This implies that the approach in [26] cannot achieve the same good tracking performance as this paper. Remark 6: From the above comparison, it is known that the approach in [26] is not effective to compensate for the dead zone when the initial values for the states and the NNs are chosen the same as the control method of this paper. The main reason is that the approach in [26] is developed for the systems without considering the dead zone. Because the right

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and left slopes of the dead zone are different and the controller v j is expressed as a piecewise function, it is necessary to avoid the effect of the dead zone. The approach in [26] ignored the effect of the dead zone. From this comparison, we can see that the approach in this paper can be reliable to compensate for the dead-zone inputs. Example 2: To further validate the effectiveness of the proposed method, we consider the cascade chemical reactor systems given in [50]. The process is described by the following differential equations:  q0 q+q R qR dC A1 −(E a /RT1 ) dt = V C A0 − V C A1 + V C A2 − αC A1 e (47) q+q R q+q R dC A2 −(E /RT a 2) dt = V C A1 − V C A2 − αC A2 e ⎧ dT q0 q+q R qR 1 ⎪ ⎪ dt = V T0 − V T1 + V T2 ⎪ ⎪ αλ ⎨ − ρc C A1 e−(Ea /RT1 ) − p

UA ρc p V (T1 − T j 1 ) αλ −(E a /RT2 ) ρc p C A2 e

q0 q+q R d T2 ⎪ ⎪ dt = V T1 − V T2 − ⎪ ⎪ ⎩ − ρcUpAV (T2 − T j 2) ⎧ ⎨ d T j 1 = q j 1 (T j 10 − T j 1 ) − dt vj ⎩ d T j 2 = q j 2 (T j 20 − T j 2 ) − dt

vj

UA ρ j c j v j (T1 UA ρ j c j v j (T2

− T j 1) − T j 2)

(48)

(49)

where

g3,2 = u1 = u3 = C A1 =

φ21 =

φ22 =

 UA ξ2,1 + T2d − ξ2,2 − T jd2 ρjcjvj  αλ d q0 d q +q R T0 − C A1 e−(Ea /R(ξ3,1 +T1 )) ξ3,1 +T1d − V V ρc p   UA qR d ξ2,1 + T2 − ξ3,1 + T1d − T jd1 + V ρc p V  q j1 d T j 10 − ξ3,2 − T jd1 vj  UA + ξ3,1 + T1d − ξ3,2 − T jd1 ρjcjvj q + qR qR C A1 + C A2 − αC Ai e−(Ea /RTi ) , i = 1, 2 V V q + qR qR αλ T1 − T2 − C A2 e−(Ea /RT2 ) V V ρc p UA (T2 − T j 2 ) − ρc p V   q + qR q + qR F1 − + αe−(Ea /RT2 ) F2 V V Ea α Ea − RT − C A2 e 2 F3 . RT22 +

where q j 1 and q j 2 are flow rates, T j 1 and T j 2 are temperatures, v j 1 = v j 2 = v j are the volume of the cooling jackets, v1 = v2 = V are the volume of the reactor, and the flow of reactants are q0 = q2 = q and q1 = q + q R . Let ξ1,1 = C A2 − C dA2 , ξ1,2 = q2 − q2d , ξ2,1 = T2 − T2d , ξ2,2 = T j 2 − T jd2, ξ3,1 = T1 − T1d , ξ3,2 = T j 1 − T jd1. Then, the systems (47)–(49) can be changed as ⎧ ξ˙1,1 = g1,1ξ1,2 , ξ˙1,2 = g1,2 u 1 , y1 = ξ1,1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξ˙ = g2,1 ξ2,2 +φ2,1 +ξ3,1 , ξ˙2,2 ⎪ ⎨ 2,1 (50) = g2,2 u 2 +φ2,2 , y2 = ξ2,1 ⎪ ⎪ ⎪ ˙ ˙ ⎪ ξ3,1 = g3,1ξ3,2 + φ3,1 + ω, ξ3,2 ⎪ ⎪ ⎪ ⎩ = g3,2 u 3 + φ3,2 , y3 = ξ3,1

q j2 UA UA , g2,2 = , g3,1 = ρc p V vj ρc p V q j1 q + qR q0 , ω = T0 − T0d , = , = vj V V (q + q R ) q0 C A0 − F4 , u 2 = T j 20 − T jd20 V2 T j 10 − T jd10 

d  V ξ1,2 + α ξ1,1 + C dA2 e−(Ea /R(x21+T2 )) q + qR 

+ ξ1,1 + C dA2   αλ q + qR d q + qR T1 − ξ2,1 + T2d − ξ1,1 + C dA2 V V ρc p

 d U A ×e−(Ea /R(x21+T2 )) − ξ2,1 + T2d − T jd2 ρc p V  q j2 d T j 20 − ξ2,2 − T jd2 vj

g1,1 = 1, g1,2 = 1; g2,1 =

Fig. 5. (a) y1 (k) (solid line),y2 (k) (dashed line), and y3 (k) (dash-dot line). (b) v1 (solid line),v2 (dashed line), and v3 (dash-dot line).

φ31 =

φ32 =

Fi = F3 =

F4 =

Similar to [22], using first-order Taylor expansion, (50) can be discretized by a discrete-time model as ⎧ ξ1,1 (k + 1) = ξ1,1 (k) + g1,1ξ1,2 (k) T, ⎪ ⎪ ⎪ ⎪ ⎪ ξ1,2 (k + 1) = ξ1,2 (k) + g1,2 u 1 T, y1 = ξ1,1 (k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ξ2,1 (k + 1) = ξ2,1 (k) + (g2,1 ξ2,2 (k) + φ2,1 +ξ3,1 (k)) T, ⎪ ⎪ ⎪ ξ2,2 (k + 1) = ξ2,2 (k) + (g2,2 u 2 + φ2,2 ) T, y2 = ξ2,1 (k) ⎪ ⎪ ⎪ ⎪ ⎪ ξ3,1 (k + 1) = ξ3,1 (k) + (g3,1ξ3,2 (k) + φ3,1 + ω) T ⎪ ⎪ ⎩ ξ3,2 (k + 1) = ξ3,2 (k) + (g3,2 u 3 + φ3,2 ) T, y3 = ξ3,1 (k) (51) where T is sampling period. The parameters of the dead-zone inputs for u 1 and u 3 are the same as u 1 in Example 1. The parameters of the deadzone input for u 2 are the same as u 2 in Example 1. The

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control objective is to design the controllers for the system (51) such that the outputs y j (k), j = 1, 2, 3 track the reference signals yd, j (k) = 0, j = 1, 2, 3 to a certain neighborhood of the origin. For the system (51) with the initial values ξ1, j (0) = −1, ξ2, j (0) = 1, and ξ3, j (0) = 0.1, based on the design method, the simulation results are obtained in Fig. 5. Fig. 5(a) shows the tracking trajectories and it can be seen that the good tracking performances can be obtained. The trajectories of the control signals v1 (k), v2 (k), and v3 (k) are shown in Fig. 5(b). It can be seen that they are also bounded. V. C ONCLUSION This paper has proposed an adaptive NN control algorithm for a class of MIMO nonlinear discrete-time systems in strict-feedback form. The unknown functions, the external disturbance, and the unknown dead-zone input are considered in the systems. To solve the noncausal problem, the coordinate transformations are used to transform the controlled systems into a new special form which is suitable for backstepping design technique. The adaptation laws and the controllers are designed based on the transformed systems. It has been proved that the tracking errors and the adaptation laws are SGUUB based on Lyapunov method. The simulation examples and the comparisons with the previous approaches have been provided to illustrate the effectiveness of the control approach. A PPENDIX Proof of Theorem 1: In order to guarantee that A j,n j , B j,n j , C j are positive constants, the design parameters are chosen as λ¯ j,n j < 1/(4 + l j,n j + γ j + m¯ j g¯ j,n j l j,n j ) τ j,n j < 1/(λ¯ j,n j + g¯ j,n j m¯ j λ¯ j,n j ) γ j δ j < 1/(1 + g¯ j,n j m¯ j /λ¯ j,n j ). Then, (44) becomes v j,n j ≤ −z 2j,n (k)/(g¯ j,n j m¯ j ) + η j,n j . j

(52)

(53)

Using (41), (53) can be rewritten as σ˜ j,n j (k + 1) = α j,n j (k)σ˜ j,n j (k) + ω j,n j (k)

(54)

where α j,n j (k) = (I − j,n j τ j,n j − j,n j g j,n j (k)m j (k)S j,n j (.)2 ) ω j,n j (k) = − j,n j S j,n j (.)g j,n j (k)m j (k)[ρ˜ j (k) + ε∗j,n j (.) − j,n j ρ j − b j (k)/m j (k)]

∗ (k) + j,n j S j,n j (.)d j,n j (k). − j,n j τ j,n j σ j,n j

v j,n j−1 ≤ −A j,n j−1 z 2j,n j−1 (k + 1)/g¯ j,n j −1 −z 2j,n j −1 (k)/g¯ j,n j −1 + g¯ j,n j−1 z 2j,n j (k)/λ¯ j,n j −1 + η j,n j −1 −τ j,n j−1 B j,n j −1 σˆ j,n j −1 (kn j −1 )2 . (55) If the design parameters are chosen as λ¯ j,n j −1 < 1/(2 + l j,n j −1 + g¯ j,n j −1l j,n j −1 ) τ j,n j −1 < 1/(λ¯ j,n j −1 + g¯ j,n j −1 λ¯ j,n j −1 ) it leads to A j,n j −1 > 0 and B j,n j −1 > 0. Subsequently, it has v j,n j −1 ≤ −z 2j,n j −1 (k)/g¯ j,n j −1 +η j,n j −1 + g¯ j,n j −1 z 2j,n j (k)/λ¯ j,n j −1 . (56) We have proved that z j,n j (k) is bounded and let g¯ j,n j −1 z 2j,n j (k)/λ¯ j,n j −1 ≤ g¯ j,n j −1 b j,n j /λ¯ j,n j −1 = μ j,n j −1 .  If |z j,n j −1 (k)| > g¯ j,n j −1 m¯ j η j,n j −1 + μ j,n j is satisfied, then, v j,n j −1 ≤ 0. This implies that v j,n j −1 (k) is bounded. From the definition of v j,n j −1 (k), it leads to the boundedness of z j,n j −1 (k). The boundedness of σˆ j,n j −1 (k), or equivalently σ˜ j,n j −1 (k) can be proved by using the same procedure and methods for σ˜ j,n j (k). Following the same procedure as in the above, we can prove in turn z j,i j (k) and σˆ j,i j (k), i j = n j − 2, . . . , 2 are bounded by choosing the design parameters as λ¯ j,i j < 1/(2 + l j,i j + g¯ j,i j l j,i j ), τ j,i j < 1/(λ¯ j,i j + g¯ j,i j λ¯ j,i j ).

 Similar to [22]–[26], if |z j,n j (k)| > g¯ j,n j m¯ j η j,n j is satisfied, then, v j,n j ≤ 0. This implies that v j,n j (k) is bounded. From the definition of v j,n j (k), it has that z j,n j (k), ρ˜ j (k) and ρˆ j (k) = ρ˜ j (k) + ρ j are SGUUB. The adaptation dynamic (39) can be expressed as σ˜ j,n j (k + 1) = (I − j,n j τ j,n j )σ˜ j,n j (k) − j,n j

∗ × τ j,n j × σ j,n (k)+ S j,n j (Z j,n j (k))z j,n j (k +1) . j

Because 0 < λ¯ j,n j , τ j,n j < 1, and g j,n j (k), m j (k) are bounded, we know from the proof in [22] that the transition matrix α j,n j (k) always satisfies (k, k0 ) < 1. Furthermore, owing to the boundedness of S j,n j (.), d j,n j (k), ε∗j,n j (.) and ρ˜ j (k), ω j,n j (k) must be bounded. Thus, according to [26, Lemma 2], σ˜ j,n j (k) is bounded, and hence, σˆ j,n j (k) = σ˜ j,n j (k) + σ j,n j (k) is bounded. When i j = n j − 1 in Step i j , (29) can be expressed as

In Step 1, in order to guarantee that A j,1 , B j,1 are positive constants, the design parameters are chosen as λ¯ j,1 < 1/(2 + l j,1 + g¯ j,1l j,1 ), τ j,1 < 1/(λ¯ j,1 + g¯ j,1λ¯ j,1 ). Then, (18) becomes v j,1 ≤ −z 2j,1 (k)/g¯ j,1 + η j,1 + g¯ j,1 z 2j,2 (k)/λ¯ j,1 .

(57)

Owing to the boundedness of z j,2 (k), we can define g¯ j,1 z 2j,2 (k)/λ¯ j,1 ≤ g¯ j,1b j,2 /λ¯ j,1 = μ j,1 . Then, v j,1 ≤ 0  once the error z j,1 (k) is larger than g¯ j,1 η j,1 + μ j,1 . This implies that v j,1 (k) is bounded. From the definition of v j,1 (k) for all k ≥ 0. In a similar way, the boundedness of σˆ j,n j −1 (k), or equivalently σ˜ j,n j −1 (k) can be proved. Furthermore, the tracking error z j,1 (k)  will converge to a compact set denoted by  j,1 := {χ|χ ≤ g¯ j,1η j,1 + μ j,1 .}. Based on the procedure above, we can conclude that v j (k) in (36) is bounded because S j,n j (Z j,n j (k)), σˆ j,n j (k) and ρˆ j (k) are bounded. Similar to the results in [26], if we initialize ξ¯ j,i j (0) ∈  and choose the appropriate design parameters, there exists a k ∗ , such that all the errors z j,i j (k) asymptotically converge to the compact sets  j,i j . This implies that the closed-loop system is SGUUB. The proof is completed.

LIU et al.: ADAPTIVE NN CONTROLLER DESIGN

R EFERENCES [1] L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Englewood Cliffs, NJ, USA: Prentice-Hall, 1994. [2] C. L. P. Chen and Y. H. Pao, “An integration of neural network and rule-based systems for design and planning of mechanical assemblies,” IEEE Trans. Syst., Man, Cybern., vol. 23, no. 5, pp. 1359–1371, Sep./Oct. 1993. [3] C. L. P. Chen, S. R. LeClair, and Y. H. Pao, “An incremental adaptive implementation of functional-link processing for function approximation, time-series prediction, and system identification,” Neurocomputing, vol. 18, nos. 1–3, pp. 11–31, 1998. [4] Q. Zhou, P. Shi, J. Lu, and S. Xu, “Adaptive output feedback fuzzy tracking control for a class of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 972–982, Oct. 2011. [5] W. S. Chen and L. C. Jiao, “Adaptive tracking for periodically time varying and nonlinearly parameterized systems using multilayer neural networks,” IEEE Trans. Neural Netw., vol. 21, no. 2, pp. 345–351, Feb. 2011. [6] S. C. Tong, Y. Li, Y. M. Li, and Y. J. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [7] Z. J. Li, X. Q. Cao, and N. Ding, “Adaptive fuzzy control for synchronization of nonlinear teleoperators with stochastic time-varying communication delays,” IEEE Trans. Fuzzy Syst., vol. 19, no. 4, pp. 745–757, Aug. 2011. [8] W. S. Chen, L. C. Jiao, J. Li, and R. Li, “Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 939–950, Jun. 2010. [9] Q. Zhou, P. Shi, S. Y. Xu, and H. Li, “Observer-based adaptive neural network control for nonlinear stochastic systems with time delay,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 1, pp. 71–80, Jan. 2013. [10] C. L. P. Chen, Y. J. Liu, and G. X. Wen, “Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems,” IEEE Trans. Cybern., vol. 44, no. 5, pp. 583–593, May 2014. [11] J. Na, X. M. Ren, and D. D. Zheng, “Adaptive control for nonlinear purefeedback systems with high-order sliding mode observer,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 3, pp. 370–382, Mar. 2013. [12] H. G. Han and J. F. Qiao, “An adaptive computation algorithm for RBF neural network,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 342–347, Feb. 2012. [13] W. S. Chen, S. Y. Hua, and H. G. Zhang, “Consensus-based distributed cooperative learning from closed-loop neural control systems,” IEEE Trans. Neural Netw. Learn. Syst., doi: 10.1109/TNNLS.2014.2315535, in press. [14] H. Y. Li, J. Y. Yu, H. H. Liu, and C. Hilton, “Adaptive sliding mode control for nonlinear active suspension vehicle systems using T–S fuzzy approach,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 3328–3338, Aug. 2013. [15] C. L. P. Chen, G. X. Wen, Y. J. Liu, and F. Y. Wang, “Adaptive consensus control for a class of nonlinear multiagent time-delay systems using neural networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 6, pp. 1217–1226, Jun. 2014. [16] F. L. Lewis, K. Liu, and A. Yesildirek, “Neural net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Netw., vol. 6, no. 3, pp. 703–715, May 1995. [17] J. P. Yu, P. Shi, W. J. Dong, B. Chen, and C. Lin, “Neural network-based adaptive dynamic surface control for permanent magnet synchronous motors,” IEEE Trans. Neural Netw. Learn. Syst., doi: 10.1109/TNNLS.2014.2316289, in press. [18] W. S. Chen, C. Y. Wen, S. Y. Hua, and C. Y. Sun, “Distributed cooperative adaptive identification and control for a group of continuoustime systems with a cooperative PE condition via consensus,” IEEE Trans. Autom. Control, vol. 59, no. 1, pp. 91–106, Jan. 2014. [19] Z. J. Li and C. Y. Su, “Neural-adaptive control of single-master– multiple-slaves teleoperation for coordinated multiple mobile manipulators with time-varying communication delays and input uncertainties,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 9, pp. 1400–1413, Sep. 2013. [20] Z. J. Li, L. Ding, H. Gao, G. R. Duan, and C. Y. Su, “Trilateral teleoperation of adaptive fuzzy force/motion control for nonlinear teleoperators with communication random delays,” IEEE Trans. Fuzzy Syst., vol. 21, no. 4, pp. 610–624, Aug. 2013.

1017

[21] C. G. Yang, S. S. Ge, and T. H. Lee, “Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions,” Automatica, vol. 45, no. 1, pp. 270–276, 2009. [22] S. S. Ge, C. G. Yang, and T. H. Lee, “Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time,” IEEE Trans. Neural Netw., vol. 19, no. 9, pp. 1599–1614, Sep. 2008. [23] C. G. Yang, S. S. Ge, C. Xiang, T. Chai, and T. H. Lee, “Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1873–1886, Nov. 2008. [24] S. Jagannathan and F. L. Lewis, “Discrete-time neural net controller for a class of nonlinear dynamical systems,” IEEE Trans. Autom. Control, vol. 41, no. 11, pp. 1693–1699, Nov. 1996. [25] S. Jagannathan, M. W. Vandegrift, and F. L. Lewis, “Adaptive fuzzy logic control of discrete-time dynamical systems,” Automatica, vol. 36, no. 2, pp. 229–241, 2000. [26] S. S. Ge, J. Zhang, and T. H. Lee, “Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discretetime,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 34, no. 4, pp. 1630–1645, Aug. 2004. [27] S. S. Ge, G. Y. Li, J. Zhang, and T. H. Lee, “Direct adaptive control for a class of MIMO nonlinear systems using neural networks,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 2001–2006, Nov. 2004. [28] A. Y. Alanis, E. N. Sanchez, and A. G. Loukianov, “Discrete-time adaptive backstepping nonlinear control via high-order neural networks,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1185–1195, Jul. 2007. [29] S. Jagannathan, “Control of a class of nonlinear discrete-time systems using multilayer neural networks,” IEEE Trans. Neural Netw., vol. 12, no. 5, pp. 1113–1120, Sep. 2001. [30] Y. N. Li, C. G. Yang, S. S. Ge, and T. H. Lee, “Adaptive output feedback NN control of a class of discrete-time MIMO nonlinear systems with unknown control directions,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 41, no. 2, pp. 507–517, Apr. 2011. [31] Y. J. Liu, C. L. P. Chen, G. X. Wen, and S. C. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discretetime nonlinear systems,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1162–1167, Jul. 2011. [32] A. Y. Alanis, E. N. Sanchez, A. G. Loukianov, and M. A. Perez-Cisneros, “Real-time discrete neural block control using sliding modes for electric induction motors,” IEEE Trans. Control Syst. Technol., vol. 18, no. 1, pp. 11–21, Jan. 2010. [33] A. Y. Alanis, E. N. Sanchez, and A. G. Loukianov, “Real-time discrete backstepping neural control for induction motors,” IEEE Trans. Control Syst. Technol., vol. 19, no. 2, pp. 359–366, Mar. 2011. [34] H. Zhang, Q. Wei, and Y. Luo, “A novel infinite-time optimal tracking control scheme for a class of discrete-time nonlinear systems via the greedy HDP iteration algorithm,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 38, no. 4, pp. 937–942, Aug. 2008. [35] D. R. Liu, H. Javaherian, O. Kovalenko, and T. Huang, “Adaptive critic learning techniques for engine torque and air–fuel ratio control,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 38, no. 4, pp. 988–993, Aug. 2008. [36] H. G. Zhang, Y. H. Luo, and D. R. Liu, “Neural network-based nearoptimal control for a class of discrete-time affine nonlinear systems with control constraint,” IEEE Trans. Neural Netw., vol. 20, no. 9, pp. 1490–1503, Sep. 2009. [37] H. Zhang, R. Song, Q. Wei, and T. Zhang, “Optimal tracking control for a class of nonlinear discrete-time systems with time delays based on heuristic dynamic programming,” IEEE Trans. Neural Netw., vol. 22, no. 12, pp. 1851–1862, Dec. 2011. [38] D. Liu, D. Wang, D. Zhao, Q. Wei, and N. Jin, “Neural-network-based optimal control for a class of unknown discrete-time nonlinear systems using globalized dual heuristic programming,” IEEE Trans. Autom. Sci. Eng., vol. 9, no. 3, pp. 628–634, Jul. 2012. [39] D. Liu and Q. Wei, “Finite-approximation-error-based optimal control approach for discrete-time nonlinear systems,” IEEE Trans. Cybern., vol. 43, no. 2, pp. 779–789, Apr. 2013. [40] D. Liu and Q. Wei, “Policy iterative adaptive dynamic programming algorithm for discrete-time nonlinear systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 3, pp. 621–634, Mar. 2014. [41] X. S. Wang, C. Y. Su, and H. Hong, “Robust adaptive control of a class of nonlinear systems with unknown dead-zone,” Automatica, vol. 40, no. 3, pp. 407–413, 2004. [42] S. Ibrir, W. F. Xie, and C. Y. Su, “Adaptive tracking of nonlinear systems with non-symmetric dead-zone input,” Automatica, vol. 43, no. 3, pp. 522–530, 2007.

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[43] D. J. Li, “Adaptive neural network control for a class of continuous stirred tank reactor systems,” Sci. China Inform. Sci., doi: 10.1007/s11432-013-4824-7, in press. [44] M. Chen, S. S. Ge, and B. B. Ren, “Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints,” Automatica, vol. 47, no. 3, pp. 452–465, 2011. [45] F. L. Lewis, W. K. Tim, L. Z. Wang, and Z. X. Li, “Deadzone compensation in motion control systems using adaptive fuzzy logic control,” IEEE Trans. Control Syst. Technol., vol. 7, no. 6, pp. 731–741, Nov. 1999. [46] R. R. Selmic and F. L. Lewis, “Dead-zone compensation in motion control systems using neural networks,” IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 602–613, Apr. 2000. [47] J. Na, X. M. Ren, G. Herrmann, and Z. Qiao, “Adaptive neural dynamic surface control for servo systems with unknown dead-zone,” Control Eng. Pract., vol. 19, no. 11, pp. 1328–1343, 2011. [48] B. B. Ren, S. S. Ge, T. H. Lee, and C. Y. Su, “Adaptive neural control for a class of nonlinear systems with uncertain hysteresis inputs and time-varying state delays,” IEEE Trans. Neural Netw., vol. 20, no. 7, pp. 1148–1164, Jul. 2009. [49] M. Chen, S. S. Ge, and B. B. Ren, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010. [50] D. J. Li, “Neural network control for a class of continuous stirred tank reactor process with dead-zone input,” Neurocomputing, vol. 131, pp. 453–459, May 2014. [51] S. C. Tong and Y. M. Li, “Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones,” IEEE Trans. Fuzzy Syst., vol. 20, no. 1, pp. 168–180, Feb. 2012. [52] G. Tao and P. V. Kokotovic, “Discrete-time adaptive control of plants with unknown output dead-zones,” Automatica, vol. 31, no. 2, pp. 287–291, 1995. [53] J. Campos and F. L. Lewis, “Deadzone compensation in discrete time using adaptive fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 7, no. 6, pp. 697–707, Dec. 1999. [54] X. Zhang, H. G. Zhang, D. R. Liu, and Y. S. Kim, “Neural-networkbased reinforcement learning controller for nonlinear systems with non-symmetric dead-zone inputs,” in Proc. IEEE Symp. Adapt. Dyn. Program. Reinforcement Learn., Nashville, TN, USA, Apr. 2009, pp. 124–129. [55] V. K. Deolia, S. Purwar, and T. N. Sharma, “Backstepping control of discrete-time nonlinear system under unknown dead-zone constraint,” in Proc. Int. Conf. Commun. Syst. Netw. Technol., Jun. 2011, pp. 344–349.

Yan-Jun Liu received the B.S. degree in applied mathematics and the M.S. degree in 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 an Associate 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, and reinforcement learning.

Li Tang received the B.S. degree in information and computing science and the M.S. degree in applied mathematics from the Liaoning University of Technology, Jinzhou, China, in 2010 and 2014, respectively. She is currently pursuing the Ph.D. degree with the College of Information Science and Engineering, Northeastern University, Shenyang, China. Her current research interests include nonlinear control, neural network control, reinforcement learning, and switching control.

Shaocheng Tong received the B.S. degree from the Department of Mathematics, Jinzhou Normal College, Jinzhou, China, in 1982, the M.S. degree from the Department of Mathematics, Dalian Marine University, Dalian, China, in 1988, 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. 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. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, and Purdue University, West Lafayette, IN, USA, in 1985 and 1988, respectively. He is currently a Chair Professor with the Department of Computer and Information Science and the Dean of the Faculty of Science and Technology at the University of Macau, Macau, China. Dr. Chen is a fellow of the American Association for the Advancement of Science. He is currently the Junior Past President of the IEEE Systems, Man, and Cybernetics Society, and has served as an Associate Editor of the IEEE T RANSACTIONS ON C YBER NETICS since 2013 and the Editor-in-Chief of the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS : S YSTEMS since 2014.