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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 6, JUNE 2015

A Direct Self-Constructing Neural Controller Design for a Class of Nonlinear Systems Honggui Han, Member, IEEE, Wendong Zhou, Junfei Qiao, Member, IEEE, and Gang Feng, Fellow, IEEE

Abstract— This paper is concerned with the problem of adaptive neural control for a class of uncertain or ill-defined nonaffine nonlinear systems. Using a self-organizing radial basis function neural network (RBFNN), a direct self-constructing neural controller (DSNC) is designed so that unknown nonlinearities can be approximated and the closed-loop system is stable. The key features of the proposed DSNC design scheme can be summarized as follows. First, different from the existing results in literature, a self-organizing RBFNN with adaptive threshold is constructed online for DSNC to improve the control performance. Second, the control law and adaptive law for the weights of RBFNN are established so that the closed-loop system is stable in the term of Lyapunov stability theory. Third, the tracking error is guaranteed to uniformly asymptotically converge to zero with the aid of an additional robustifying control term. An example is finally given to demonstrate the design procedure and the performance of the proposed method. Simulation results reveal the effectiveness of the proposed method. Index Terms— Adaptive control, asymptotically stability, neural networks (NNs), nonlinear systems, self-organizing.

I. I NTRODUCTION

O

VER the last decade, there has been tremendous progress in development of various controllers for uncertain dynamical systems [1]. Based on a biological prototype of the human brain, the neural network (NN) methods have attracted considerable attention for modeling and controlling uncertain, Manuscript received March 29, 2014; revised November 21, 2014; accepted February 3, 2015. Date of publication February 19, 2015; date of current version May 15, 2015. This work was supported in part by the National Science Foundation of China under Grant 61034008, Grant 61203099, and Grant 61225016, in part by the Beijing Municipal Natural Science Foundation under Grant 4122006, in part by the Beijing Science and Technology Project under Grant Z141100001414005 and Grant Z141101004414058, in part by the Hong Kong Scholar Program under Grant XJ2013018, in part by the Beijing Municipal Education Commission Foundation under Grant KZ201410005002 and Grant km201410005001, in part by the Beijing Nova Program under Grant Z131104000413007, in part by the China Post-Doctoral Science Foundation under Grant 2014M550017, and in part by the Ph.D. Program Foundation through the Ministry of Education, China, under Grant 20121103120020 and Grant 20131103110016. H. Han is with the Beijing Key Laboratory of Computational Intelligence and Intelligent System, the Beijing Laboratory for Urban Mass Transit, Engineering Research Center of Digital Community, College of Electronic and Control Engineering, Ministry of Education, Beijing University of Technology, Beijing 100081, China, and also with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). W. Zhou and J. Qiao are with the Beijing Key Laboratory of Computational Intelligence and Intelligent System, College of Electronic and Control Engineering, Beijing University of Technology, Beijing 100081, China (e-mail: [email protected]; [email protected]). G. Feng is with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong (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.2015.2401395

nonlinear, and complex systems, owing to their learning and adaptation capabilities [2]–[4]. The substantial advantage of designing a control system using NNs is that it does not require an exact mathematical model of the system to be controlled [5], [6]. In general, adaptive neural control methods can be divided into two categories: 1) direct adaptive neural control and 2) indirect adaptive neural control. In direct adaptive neural control, the output of an NN becomes the control input to the system to be controlled, and the neural controller is obtained by NN adaptive laws [7], [8]. In indirect adaptive neural control, an NN is normally used to estimate the unknown nonlinearities of the system to be controlled. Then, a control input is generated using the estimated information [9], [10]. In the earlier development stage, the so-called backpropagation method has been widely used to derive parameter adaptation laws with few analytical results for stability or robustness of the concerned closed-loop control system [11]. Current research works often use Lyapunov stability theory [5], [6] in development of adaptive neural control. A key advantage of adaptive neural control based on the Lyapunov stability theory is that the stability of the concerned closed-loop control system can be readily established and at the same time the adaptation laws can be easily obtained. Nevertheless, in neural adaptive control, many issues still remain to be further addressed. An approximation error will occur if an NN is used to approximate the nonlinear function of a given system, and it would inevitably degrade the performance of the closedloop control system or even worse destabilize it. Recently, various new neural adaptive control approaches have been developed [12]–[16]. Li et al. [12] used a nonminimal set of NN weights to relax the restriction on the control gain function. But, the determination of virtual control terms and their time derivatives requires tedious and complex analysis [13], [14]. On the other hand, the method of adding a sliding mode input term is widely used [15], [16]. In [16], neural adaptive control has been developed for nonlinear systems in block-triangular form. The suggested adaptive neural controller guarantees that the tracking error converges to a small neighborhood of the origin, and other signals in the closed-loop system are bounded. However, the sliding mode control is computed from the information of the bounding constant of the system uncertainty, which is difficult to obtain in real applications. In general, how to achieve perfect tracking of the closed-loop neural adaptive control system with the inevitable approximation error is still a challenge.

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HAN et al.: DSNC DESIGN FOR A CLASS OF NONLINEAR SYSTEMS

In most existing schemes of neural adaptive control, the number of adaptation parameters depends on the number of NN neurons [17], [18]. In most cases of adaptive neural control for affine nonlinear systems, the structure of NNs and thus the number of neurons are fixed after they are initially determined [19], [20]. Neural adaptive control schemes might run into trouble due to unnecessarily large scale or too small scale of structure [21]–[23]. If a system contains a large number of unknown nonlinear functions, or more NN neurons are used to improve the approximation precision, there will be a large number of adaptation parameters that need to be updated online simultaneously [24], [35]–[39]. In this case, the learning/convergence time is normally very large and thus unacceptable in applications [25]. On the other hand, if the number of hidden neurons is too small, the learning performance may not be good enough to achieve the desired control objective. Ideally, when an NN is used as an adaptive controller, the number of NN neurons should be self-organized online to get the sufficient approximation accuracy and at the same time to ensure the learning to converge fast enough. Recently, the self-organizing NNs or fuzzy NNs are widely used for controller design [27], [28], [35], [43], [44]. In these controllers, self-organizing NNs or fuzzy NNs with concurrent both structure and parameter learning are developed to achieve better performance. However, most of these NN or fuzzy NN controllers are supposed to have fixed structure. In other words, adaptive neural control with self-organizing structure is still to be investigated. Motivated by the above review and analysis, a direct self-constructing neural controller (DSNC) using a self-organizing radial basis function NN (RBFNN) is proposed for nonlinear systems in this paper. The advantages of the proposed control scheme can be summarized as follows. First, the approximation error can be perfectly compensated by utilizing the proposed DSNC. Second, the neurons in the hidden layer of the RBFNN are self-organized through adaptation. That is, the number of hidden neurons can be increased or pruned depending on the particular condition of the concerned system as time goes on. Third, the bounding constant of the lumped uncertainty due to approximation and external disturbances is estimated online, and the estimate is used to design the additional robustifying control term. The proposed control law and adaptation law for all the parameters are designed such that the closed-loop adaptive control system is asymptotically stable. The outline of this paper is as follows. Section II gives the problem formulation and preliminaries for control of uncertain nonlinear systems. In Section III, the basic idea of the control scheme, the self-organizing structure of RBFNN, and the stability analysis are presented. The parameter adaptive laws are also derived. The simulation results are presented to demonstrate the effectiveness of the proposed DSNC in Section IV. Finally, the conclusion is drawn in Section V. Notations: The following notations and definitions will be used throughout this paper. Let R be the real number, R n and R n×m represent the real vectors and the real n × m matrices, respectively. || · || denotes the usual Euclidean norm of a vector. In the case where y is a scalar, ||Y || means

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Fig. 1.

RBFNN structure.

Frobenious norm defined as Y  = trace {·} stands for the trace operator.



trace{Y T Y } where

II. P ROBLEM F ORMULATION AND P RELIMINARIES A. Problem Formulation The single-input single-output (SISO) nonlinear dynamical systems considered in this paper are described by the following differential equation: x˙1 = x 2 x˙ 2 = x 3 ··· x˙n = F(x, u) y = x1

(1)

where y ∈ R is the measured output, u ∈ R is the control input, F(·) is assumed to be the unknown nonlinear function of the system, and x = [x 1 , x 2 , . . . , x n ]T = [y, y (1), . . . , y (n−1) ]T ∈ R n is the state vector. It should be noted that the nonlinearity F(·) is an unknown smooth function and may not be linearly parameterized. It may also contain various uncertainties. Without loss of generality, we assume that F(0, 0) = 0. The aim of this paper is to design a DSNC for the nonlinear SISO systems (1) with uncertainties such that all the signals of the closed-loop system are bounded and the output tracking error with respect to a given reference output yd (t) converges to zero asymptotically. The reference signal yd (t) and its time derivatives yd(1)(t), . . . , yd(n) (t) are assumed to be bounded and let xd = [yd (t), yd(1) (t), . . . , yd(n−1) (t)]T ∈ R n . The tracking error is defined as e¯ = yd − y the corresponding error vector is e¯ e¯(n−1) ]T ∈ R n .

(2) =

[e, ¯ e¯(1), . . . ,

B. Radial Basis Function Neural Network (RBFNN) Fig. 1 shows the structure of a basic RBFNN that consists of one input layer, one output layer, and one hidden layer.

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Fig. 2.

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 6, JUNE 2015

Overall block diagram of DSNC scheme.

A single-output RBFNN with K hidden neurons can be described by f (X) =

K 

wk θk (X)

where X = [X 1 , X 2 , . . . , X M ]T denotes the input of the network, f () is the output of the single-output RBFNN, M is the number of input variables, W = [w1 , w2 , . . . , w K ] is the connecting weights between the hidden neurons and the output layer, and θk (X) is the output value of the kth hidden neuron   2 (4) θk (X) = e −X−µk /σk where µk denotes the center vector of the kth hidden neuron, ||X − µk || is the Euclidean distance between X and µk , and σk is the radius or width of the kth hidden neuron. In DSNC, an RBFNN is used as an approximation model for the unknown nonlinearity (5)

where θ(X) = [θ1 (X), θ2 (X), . . . , θ K (X)]T ∈ R K, ε(X) is the approximation error and g() is the unknown nonlinearity of a nonlinear system. As shown in [26], an RBFNN can be used to approximate any continuous smooth function g(X) over a compact set  X ∈ R M to arbitrary accuracy, that is, there exists an RBFNN such that g(X) = W∗ θ(X) + ε∗ (X) ∀X ∈ X ⊂ R M

(7)

where Wmax is a positive constant, then the optimal weight value of the RBFNN in this case can be described as W∗ = arg min [ sup |g(X) − f (X|W)|]. W∈g X∈ X

It is noted that Wmax is a parameter as suggested in [33].

y (n) = F(x, u) = υu + [F(x, u) − υu]

(9)

where υ is a design constant. Let g(x, u) ≡ F(x, u) − υu. Motivated from [27] and [28], the control input is defined as 1 (u dc − u ad + u sl ) (10) υ where u dc is a control input to stabilize linearized dynamics, u ad is an adaptive control input to deal with g(x, u) based on an RBFNN and u sl is an additional robustifying control input to be determined in sequel. In the DSNC scheme, as shown in Fig. 2, a self-organizing RBFNN controller is directly designed to imitate a predetermined model-based stabilizing control law. Substituting (10) into (9) yields u=

y (n) = u dc + [g(x, u) − u ad ] + u sl

(11)

where u dc is defined as (n)

u dc = yd + kT e¯

(12)

and k = [k1 , k2 , . . . , kn ]T is chosen so that n n−1 + · · · + kn ) is Hurwitz. Thus, if g(x, u) is (s + k1 s perfectly canceled by u ad , i.e., g(x, u) = u ad , and u sl = 0, the closed-loop dynamic system is asymptotically stable.

(6)

where W∗ is the ideal constant weight vector of the RBFNN and ε∗ is the approximation error for the case when W = W∗ . Defining the compact set g = {W : W ≤ Wmax } ∈ R M

A. Direct Self-Constructing Neural Controller (DSNC) It is noted that (1) can be rewritten as

(3)

k=1

g(X) = f (X) + ε(X) = Wθ(X) + ε(X)

III. N EURAL C ONTROLLER D ESIGN

(8)

B. RBFNN Approximation As previously mentioned an RBFNN is usually employed to approximate g(x, u). The inputs of the RBFNN are x and u. However, to avoid recurrent formulation of the RBFNN, we apply x and u ∗ (:= u dc + u sl ) to be the inputs of RBFNN instead. In order to ensure that the RBFNN with these inputs (x and u ∗ ) can approximate u ad properly, the new variable u ∗ad satisfying the following equation is taken into consideration:     u ∗ − u ∗ad g ∗ x, u ∗ , u ∗ad ≡ g x, (13) − u ∗ad = 0 υ which is a function of x and u ∗ .

HAN et al.: DSNC DESIGN FOR A CLASS OF NONLINEAR SYSTEMS

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Assumption 1: The following assumption is made: ∂ F(x, u) >0 (14) ∂u for all (x, u) ∈ x × R, where x is the state compact region of interest. Lemma 1: Let the constant υ satisfy the condition   1 ∂F . (15) υ> 2 ∂u Then, there exists a unique u ∗ad that is a function of x and u ∗ such that u ∗ad (x, u ∗ ) satisfies (13) for all (x, u ∗ ) ∈ x × R. Proof: The sufficient condition for the existence of u ∗ad is that the function g is a contraction over the entire input domain, that is, the following inequality must hold [27]:    ∂g    (16)  ∂u ∗  < 1. ad

In this paper, a self-organizing RBFNN is proposed for designing the controller. The self-organizing mechanism derives from the average firing rate of hidden neurons and the mutual information (MI) intensities between the hidden and the output layers. The hidden neurons with high firing rates will be divided into new neurons. In addition, the connections with a small MI value will be pruned to simplify the structure. Two aspects of the self-organizing method for designing the structure of RBFNN are discussed as follows. 1) Neuron Splitting Mechanism: Each neuron in the nerve system is assumed to be independently conscious with a component of the electrical activity [41]. To split the hidden neurons of the RBFNN, the firing rates are introduced. Consider the active firing (AF) of the hidden neurons by the following equation: θi (X) , (i = 1, 2, . . . , K ) A f i = 10e−X−μi   K i=1 θi (X)

(19)

It is easily observed that this inequality holds if the condition (15) is satisfied. So, the solution u ∗ad to (13) exists. Then, the function g ∗ is nonsingular around u ∗ad . Differentiating the left-hand side of (13) with respect to u ∗ad yields  ∂g ∗ (x, u ∗ , u ad )   ∂u ad u ad =u ∗ad  ∂{g(x, u) − u ad }  =  ∂u ad u ad =u ∗ad  ∂{F(x, u) − vu} ∂u  = −1 ∂u ∂u ad u ad =u ∗ ad 

  ∂ F(x, u)  1 = − v − −1 ∂u u ad =u ∗ v ad  1 ∂ F(x, u)  (17) =− v ∂u u ad =u ∗

where Afi is the AF value of the i th hidden neuron, K is the number of hidden neurons, and θi is the output value of the i th hidden neuron. When Afi is larger than the activity threshold Afo , the hidden neuron i is the active neuron. In addition, this hidden neuron will be divided into new neurons in the hidden layer. One key parameter is the activity threshold [41], [45]. In this paper, an adaptive activity threshold is chosen and defined as

which is nonzero by Assumption 1. Thus, according to the implicit function theorem [29], there is a unique u ∗ad (x, u ∗ ) satisfying (13) for all (x, u ∗ ) ∈ x × R. Lemma 1 makes it possible to employ the RBFNN with inputs (x and u ∗ ) to approximate u ad . In this paper, u ∗ad (x, u ∗ ) is approximated by the RBFNN shown in Fig. 1. The output of RBFNN is represented by

where µi and σi are the center and radius of the i th hidden neuron. The weights between the new neurons and the output layer are chosen as wi · θi (X) − ε(X) (22) wi, j = r j θi, j (X)

ad

∗

u ad (x, u ) =

K 

wk θk (X)

(18)

k=1

where K is the number of hidden neurons, X = [X 1 , X 2 , . . . , X M ]T is the input vector of the RBFNN denoted as X = [x, u ∗ ]T . C. Self-Organizing RBFNN It is noted that the number of adaptation parameters in neural adaptive control depends on the number of hidden neurons. In most existing neural adaptive control approaches, the number of hidden neurons in RBFNNs is often assumed to be fixed and their locations and weights are determined by the training data set, which usually result in a large network size.

A f0 = 1 − e

K − M

(20)

where M is the number of input variables and K is the number of hidden neurons. The centers and the radii of the newly divided hidden neurons will be designed as µi, j = (1 + 0.1 × A f i )µi + (0.1 × A f i )x σi, j = (1 + 0.1 × A f i )σi , j = 1, 2

(21)

where ri is the allocated parameters for the new neurons (r1 = 1/K , r2 = 1 − 1/K ), θi (x) is the output value of the i th hidden neuron, θi, j (x) is the output value of the newly inserted j th hidden neuron, wi is weight of the i th hidden neuron, and ε(X) is the current approximation error of RBFNN. 2) Neuron Adjusting Mechanism: An MI, free of assumptions about the nature of the underlying joint distribution, provides a natural quantitative measure of the degree of statistical dependence between the stochastic variables [30]. The definition of MI between two stochastic variables X i and Y is  p(X i , Y ) (23) p(X i , Y ) log2 M(X i ;Y ) = p(X i ) p(Y ) x,y where p(X i , Y ) is the joint distribution, p(X i ) and p(Y ) are the single-variable marginal, X i is the output value of the

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 6, JUNE 2015

unknown nonlinear function g can be described as g(x, u) = u ∗ad (x, u ∗ ) + ε∗ (X) = W∗ θ(X) + ε∗ (X). Fig. 3.

It follows from (18) and (28) that the approximation error is

Correlation between neurons X i and Y . (a) m > m 0 . (b) m ≤ m 0 .

i th hidden neuron, and Y is the network’s output. If and only if the two variables X i and Y are statistically independent then p(X i , Y ) = p(X i ) p(Y ). The magnitude of MI between two neurons is dependent on the information content of each neuron, in terms of their Shannon entropy H (X i ) and H (Y ). MI is expressed as M(X i ;Y ) = H (X i ) − H (X i |Y ) = H (Y ) − H (Y |X i )

ε(X) = g(x, u) − u ad (x, u ∗ ) = u ∗ad (x, u ∗ ) − u ad (x, u ∗ ) + ε∗ (X) = W∗ θ(X) − Wθ(X) + ε∗ (X) = θ(X) + ε∗ (X)

(n)

e¯(n) = yd − y (n)

= −kT e¯ − [g(x, u) − u ad (x, u ∗ )] − u sl

ln(2πe) K |COV(X

M(X i ;Y ) ≤ min(H (X i ), H (Y )).

(25)

This bound, together with M(X i ; Y ) ≥ 0, implies that the normalized MI defined as M(X i ;Y ) (26) M(X i ;Y ) = min(H (X i ), H (Y )) satisfies 0 ≤ M(X i ;Y ) ≤ 1. In the RBFNN structure, the neurons X i and Y in the hidden and output layers connect if and only if the normalized MI m(X i , Y ) is larger than the threshold m 0 [Fig. 3(a)]. If the normalized MI m(X i , Y ) equals to or is less than the threshold m 0 , it means that neurons X i and Y are independent [Fig. 3(b)]. The details can be observed in Fig. 3, and the adjusting threshold m 0 is defined as m 0 = 0.1 × e

K − M

.

(27)

It is now clear that the self-organizing RBFNN can insert or prune hidden neurons depending on the hidden neurons’ activities and the connecting MI. This indicates that the self-organizing RBFNN does not have a fixed structure, instead it has an adaptive structure. Remark 1: Essentially, the proposed self-organizing mechanism is similar to the adaptive search strategy in brains [31], which depends on hidden neurons’ activities and connecting MI. This feature is useful to nonlinear modeling and control. Remark 2: Although a neuron adjusting mechanism is proposed for self-organizing RBFNNs in [32]. However, the appropriate thresholds are difficult to be determined [46]. To overcome this problem, novel adaptive methods are proposed to tune the thresholds in this paper. D. Adaptive Laws and Stability Analysis In this section, we derive the robustifying control input u sl and adaptation laws for parameter W. It is noted that the

(29)

where = W∗ − W is the parameter error. From (1) and (11), the error dynamics can be defined as

(24)

where H (X i ) = i )|/2, COV(X i ) is the standard covariance of X i , e is a mathematical constant here (2.718), and H (Y ) = ln(2πe)|COV(Y )|/2. The MI is semipositive definite and equal to zero if and only if the two variables X i and Y are statistically independent, and moreover

(28)

= −kT e¯ − [ θ(X) + ε∗ (X)] − u sl

(30)

e˙¯ = A¯e + b{[ θ(X) + ε∗ (X)] − u sl }

(31)

or

where ⎡

0 ⎢ 0 ⎢ A=⎣ ··· −kn

1 0 ··· −kn−1

0 1 ··· ···

··· ··· ··· ···

0 0 ··· ···

⎤ ⎡ ⎤ 0 0 ⎢ ⎥ 0 ⎥ ⎥, b = ⎢ 0 ⎥. ⎣· · ·⎦ ··· ⎦ −k1 1 (32)

Because A is a stable matrix, there exists a unique positive definite symmetric matrix P satisfying the following Lyapunov equation: A T P + P A = −q I

(33)

where q is a positive constant indicating the degree of stability of the matrix A and I is the identity matrix. Now, the robustifying control is determined as u sl = sgn(¯eT Pb)ψ.

(34)

The adaptation laws for W and the estimation of the parameter ψ are defined as ˙ = γ1 e¯ T PbθT W ψ˙ = γ2 ¯eT Pb

(35) (36)

where γ1 and γ2 are positive learning rates that are used to tune the convergence rate of the adaptation process. It has been shown in [27] that the following inequality holds: |ε(t)| ≤ ε1

(37)

where ε1 is an unknown finite constant. Theorem 1: Consider the error dynamics defined by (30). Suppose the number of hidden neurons of the RBFNN is K , and the update laws for W and ψ are given as in (34) and (35). Under Assumption 1, the tracking error vector e¯ can uniformly asymptotically converge to 0. Proof: Consider the Lyapunov function candidate   1 1 1 T T T (38) e¯ P e¯ + trace( ) + trace(ϕϕ ) V = 2 γ1 γ2 where ϕ = ψ − ε1 .

HAN et al.: DSNC DESIGN FOR A CLASS OF NONLINEAR SYSTEMS

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The time derivative V along the solutions of (30) is 1 V˙ = − q ¯e2 + e¯ T Pb{[ θ(X) + ε∗ (X)] − u sl } 2 1 1 ˙ T ) + trace(ϕ ϕ˙ T ). (39) + trace( γ1 γ2 By substituting (35) and (36) into (39), one has 1 V˙ = − q¯e2 + e¯ T Pb θ(X) 2 + e¯ T Pbε∗ (X) − e¯ T Pbu sl − e¯ T Pb θ(X) + ¯eT Pbϕ 1 = − q¯e2 + e¯ T Pbε∗ (X) − e¯ T Pbu sl + ¯eT Pbϕ. (40) 2 By considering (34), one further has 1 V˙ ≤ − q¯e2 + ¯eT Pbε1 − e¯ T Pbsgn(¯eT Pb)ψ 2 1 + ¯eT Pb(ψ − ε1 ) = − q¯e2 + ¯eT Pbε1 2 1 T T −¯e Pbψ + ¯e Pb(ψ − ε1 ) = − q¯e2 . 2

Fig. 4. Simulation results of DSNC (x(0) = [−1, 0]T ). (a) Output y(t) and reference yd (t). (b) Error between y(t) and yd (t).

(41)

It follows from (38) and (41) that V is not increasing, W and ψ are bounded, and ||¯e|| is bounded. Integrating both sides of (41) yields  ∞ 2 (42) ¯e(t)2 dt ≤ (V (0) − V (∞)). q 0 Since the right side of (42) is bounded, one can further conclude that lim e¯ (t) = 0

t →∞

(43)

and the stability result of Theorem 1 is thus completed. Remark 3: In many existing works, the robustifying control u sl in (34) is dependent on the tracking error, Taylor series expansion, and input signals [27]. However, the robustifying control of DSNC is only dependent on the tracking error and the outputs of hidden neurons, and the controller is thus simplified. Theorem 1 is stated for the RBFNN with a fixed structure. The stability analysis of RBFNN with a self-organizing structure will be presented in sequel. Theorem 2: Consider the error dynamics defined by (30). Suppose the number of hidden neurons of the RBFNN is selforganized in the learning process, and the update laws for W and ψ are given as in (34) and (35). Under Assumption 1, the tracking error vector e¯ can also uniformly asymptotically converge to 0. Proof: When the activity neuron has been divided into new hidden neurons (there are K + 1 hidden neurons), the approximation error ε (t) after the neuron splitting is given by ε (t) =

K +1 

wk θk (X(t)) − g(X(t)).

(44)

k=1

Based on (21) and (22), one has ε (t) = 0.

(45)

Meanwhile, as shown in [32], the approximation error ε (t) in the neuron adjusting step is bounded, that is |ε (t)| ≤ ε2

(46)

where ε2 is an unknown finite constant (it may be different from ε1 defined for different number of neurons for the fixed structure as in Theorem 1). To prove Theorem 2, the following Lyapunov function candidate V¯ is chosen:   1 1 1 T T T ¯ (47) V = e¯ P e¯ + trace( ) + trace(ϕ¯ ϕ¯ ) 2 γ1 γ2 where ϕ¯ = ψ − ε2 . It is easily seen that the time derivative of V¯ is 1 V˙¯ ≤ − q¯e2 + ¯eT Pbε2 2 − e¯ T Pbsgn(¯eT Pb)ψ + ¯eT Pb(ψ − ε2 ).

(48)

Following the similar arguments as in the proof of Theorem 1, one concludes that V˙¯ is negative definite. The stability result of Theorem 2 can thus be established. IV. S IMULATION S TUDIES The performance and effectiveness of the DSNC was demonstrated by applying it to control a typical nonlinear dynamic system. In addition, the proposed DSNC algorithm was compared with two existing schemes. All the simulations are programmed with MATLAB version 7.01, and were run on a Pentium 4 with a clock speed of 2.6 GHz and 1 GB of RAM, under a Microsoft Windows XP environment. Consider a nonlinear plant [27], [40] x˙1 = x 2

  x˙2 = x 12 + 0.15u 3 + 0.1 1 + x 22 u + sin(0.1u) y = x1.

(49)

Since the nonlinearities in the plant is an implicit function with respect to u, it is impossible to get an explicit controller

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 6, JUNE 2015

Fig. 6.

Number of hidden neurons of DSNC (x(0) = [−1, 0]T ).

Fig. 5. Control inputs (x(0) = [−1, 0]T ). (a) Whole control input u(t). (b) Control input u sl (t). (c) Control input u dc (t). (d) Control input u ad (t).

Fig. 7.

Value of ||W|| (x(0) = [−1, 0]T ).

for system feedback linearization. In this example, the system nonlinearities are not assumed to be known a priori. However, it can be verified that ∂ F/∂u > 0 and Assumption 1 is satisfied. The design constant υ is chosen as υ = 2. The control objective is to make the output y(t) track the desired reference yd (t) = sin(t) + cos(0.5t). Two different initial conditions x(0) = [−1, 0]T and x(0) = [0.6, 0.5]T are considered in simulation. The design parameters are specified as follows: 1) γ1 = 0.5; 2) γ2 = 0.001; 3) k1 = 6; and 4) k2 = 9. The initial structure of the RBFNN consists of three input variables, one output variable, and a hidden layer with two initial neurons. The initial value σ = 2.5 is applied to every hidden neuron. The initial weights W are taken randomly in the interval [−0.5, 0.5]. With the proposed DSNC, first, the simulations were performed for the initial condition x(0) = [−1, 0]T . Figs. 4–7 show the results. Figs. 4 and 5 show the results of the output response and the control input, respectively. Figs. 6 and 7 show

the simulation results of the variation of the number of neurons and the values of ||W||, respectively. It can be observed from the results that the output of the closed-loop control system can track the reference output well by generating an appropriate number of neurons automatically, and the weights are bounded. Then, the simulation is carried out for the initial condition x(0) = [0.6, 0.5]T . Figs. 8 and 9 show the output and input of the controlled system, respectively. Fig. 10 shows the number of the hidden neurons. Fig. 11 shows the simulation results of ||W||. It can also be observed from the results that the track performance can be achieved by automatically generated number of neurons, and the weights are bounded. To further evaluate the performance of the DSNC method against external disturbances, it was compared with two other schemes: 1) the direct adaptive controller using self-structuring NNs [27] and 2) the direct adaptive NN controller [40].

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Fig. 8. Simulation results of DSNC (x(0) = [0.6, 0.5]T ). (a) Output y(t) and reference yd (t). (b) Error between y(t) and yd (t). TABLE I P ERFORMANCE C OMPARISON OF D IFFERENT A LGORITHMS

In this case, the nonlinear plant in (49) is supposed to be subject to an external disturbance. That is, the second equation in (49) is replaced by the following one:   x˙2 = x 12 +0.15u 3 +0.1 1 + x 22 u +sin(0.1u) + 0.5 sin(2πt).

Fig. 9. Control inputs (x(0) = [0.6, 0.5]T ). (a) Whole control input u(t). (b) Control input u sl (t). (c) Control input u dc (t). (d) Control input u ad (t).

(50) Moreover, to show the robustness of the proposed DSNC with respect to the measurement noise the presence of random measurement noise r (t) is considered in this experiment. It is r (t) = 0 for t ≤ 10, and for t > 10, |r (t)| ≤ 0.1. The output responses under three control methods are shown in Fig. 12. Fig. 13 shows the number of hidden neurons in the controlling process. The control input of the system is displayed in Fig. 14, and Fig. 15 shows the norms of the weights W. Figs. 12–15 testify the effectiveness of the developed controller. In addition, the tracking performances of the three different schemes are summarized in Table I, where three measures, the overshoot defined as max |y(t) − yd (t)|, the  number of hidden neurons, and the mean-error defined 30 as t =0 |y(t) − yd (t)|/30 are used. The results reveal that the performance of the DSNC is better than the other schemes. Through the simulation results, it can be observed that the proposed adaptive neural control algorithm—DSNC does have a self-organizing capability, that is, the number of hidden

Fig. 10.

Number of hidden neurons of DSNC (x(0) = [0.6, 0.5]T ).

neurons adapts based on the operating condition of the controlled system. Moreover, one can observe that the DSNC has the following advantages.

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Fig. 11.

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 6, JUNE 2015

Value of ||W||(x(0) = [0.6, 0.5]T ).

Fig. 13.

Number of hidden neurons of DSNC (x(0) = [1, 0]T ).

Fig. 12. Simulation results of three control algorithms (x(0) = [1, 0]T ). (a) Output response. (b) Errors.

Fig. 14. Control inputs with bounded external disturbance (x(0) = [1, 0]T ). (a) Whole control input u(t). (b) Control input u sl (t). (c) Control input u dc (t). (d) Control input u ad (t).

1) The proposed self-organizing RBFNN can enable the closed-loop control to track the reference output well by generating an appropriate number of hidden neurons automatically. Figs. 6, 10, and 13 show that the number

of hidden neurons can be adjusted online according to the different conditions of the nonlinear systems. 2) Although the dynamics and operation regions of the plant are quite different in the above experiment tests,

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ACKNOWLEDGMENT The authors would like to thank Dr. R. Dale-Jones for reading the manuscript and providing his valuable comments. They would like to thank the Editor-in-Chief and the reviewers for their valuable comments and suggestions, which helped to improve this paper greatly. R EFERENCES

Fig. 15.

Value of ||W||(x(0) = [1, 0]T ).

the simulation results indicate that the developed DSNC has strong adaptability and achieves good control performance for all of the simulations. 3) It is noticed that DSNC is highly structural. Such a structural property is particularly suitable for hardware implementation in practical applications. In addition, the controller, once configured, can be applied to other similar plants without repeating the complex controller design procedure for different system nonlinearities. 4) The proposed DSNC guarantees the stability of the closed-loop adaptive control system and also zero tracking error. V. C ONCLUSION In this paper, a new neural controller, named DSNC is proposed, which is capable of learning and controlling nonlinear systems. Numerical simulation results are provided to demonstrate its superior performance. The major contributions of this paper are summarized as follows. 1) DSNC is proposed to deal with uncertain nonaffine nonlinear systems based on a self-organizing RBFNN with online adaptation of the number of hidden neurons. In contrast to control schemes using NNs with fixed number of hidden neurons, the neurons in the hidden layer are generated or pruned online based on the underlying system conditions. 2) The adaptive laws and control input are designed based on Lyapunov stability theory. The tracking error is guaranteed to uniformly asymptotically converge to zero with the aid of an additional robustifying control term. 3) The proposed DSNC is particularly suitable for parallel processing and hardware implementation in practical applications. In addition, the controller, once configured, can be applied to other similar plants without repeating the complex controller design procedure for different system nonlinearities.

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[46] Q. Shen, P. Shi, T. Zhang, and C.-C. Lim, “Novel neural control for a class of uncertain pure-feedback systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 4, pp. 718–727, Apr. 2014. Honggui Han (M’11) received the B.S. degree in automatic from the Civil Aviation University of China, Tianjin, China, in 2005, and the M.E. and Ph.D. degrees from the Beijing University of Technology, Beijing, China, in 2007 and 2011, respectively. He has been with the Beijing University of Technology since 2011, where he is currently a Professor. His current research interests include neural networks, fuzzy systems, intelligent systems, modeling and control in process systems, and civil and environmental engineering. Prof. Han is a member of the IEEE Computational Intelligence Society. He is currently a reviewer of the IEEE T RANSACTIONS ON F UZZY S YSTEMS , the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS , the IEEE T RANSACTIONS ON C YBERNETICS , the IEEE T RANSACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY, C ONTROL E NGINEERING P RACTICE, and the Journal of Process Control. Wendong Zhou received the B.S. degree in control engineering from Shenyang Ligong University, Shenyang, China, in 2013. He is currently pursuing the M.S. degree with the College of Electronic and Control Engineering, Beijing University of Technology, Beijing, China. His current research interests include neural networks, intelligent systems, modeling and control in process systems.

Junfei Qiao (M’11) received the B.E. and M.E. degrees in control engineering from Liaoning Technical University, Huludao, China, in 1992 and 1995, respectively, and the Ph.D. degree from Northeast University, Shenyang, China, in 1998. He was a Post-Doctoral Fellow with the School of Automatics, Tianjin University, Tianjin, China, from 1998 to 2000. He joined the Beijing University of Technology, Beijing, China, where he is currently a Professor. He is also the Director of the Intelligence Systems Laboratory. His current research interests include neural networks, intelligent systems, self-adaptive/learning systems, and process control systems. Prof. Qiao is a member of the IEEE Computational Intelligence Society. He is currently a reviewer of over 20 international journals, such as the IEEE T RANSACTIONS ON F UZZY S YSTEMS and the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS . Gang Feng (F’09) received the B.E. and M.E. degrees in automatic control from Nanjing Aeronautical Institute, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Melbourne, Parkville, VIC, Australia, in 1992. He was a Lecturer/Senior Lecturer with the School of Electrical Engineering, University of New South Wales, Kensington, NSW, Australia, from 1992 to 1999. He has been with the City University of Hong Kong, Hong Kong, since 2000, where he is currently a Chair Professor and the Associate Provost. His current research interests include hybrid systems and control, modeling and control of energy systems, and intelligent systems and control. Prof. Feng received the Changjiang Chair Professorship of the Nanjing University of Science and Technology from the Ministry of Education in China. He was an Associate Editor of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART C: A PPLICATIONS AND R EVIEWS , and the Journal of Control Theory and Applications. He is also an Associate Editor of the IEEE T RANSACTIONS ON F UZZY S YSTEMS AND IEEE/ASME Transactions on Mechatronics.