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Adaptive Neural Control of MIMO Nonlinear Systems With a Block-Triangular Pure-Feedback Control Structure Zhenfeng Chen, Shuzhi Sam Ge, Fellow, IEEE, Yun Zhang, and Yanan Li, Member, IEEE

Abstract— This paper presents adaptive neural tracking control for a class of uncertain multiinput-multioutput (MIMO) nonlinear systems in block-triangular form. All subsystems within these MIMO nonlinear systems are of completely nonaffine pure-feedback form and allowed to have different orders. To deal with the nonaffine appearance of the control variables, the mean value theorem is employed to transform the systems into a block-triangular strict-feedback form with control coefficients being couplings among various inputs and outputs. A systematic procedure is proposed for the design of a new singularityfree adaptive neural tracking control strategy. Such a design procedure can remove the couplings among subsystems and hence avoids the possible circular control construction problem. As a consequence, all the signals in the closed-loop system are guaranteed to be semiglobally uniformly ultimately bounded. Moreover, the outputs of the systems are ensured to converge to a small neighborhood of the desired trajectories. Simulation studies verify the theoretical findings revealed in this paper. Index Terms— Adaptive neural control, backstepping, coupling, multiinput-multioutput (MIMO) nonlinear systems, neural networks (NNs).

I. I NTRODUCTION

N

EURAL NETWORKS (NNs) can approximate continuous functions to any desired accuracy by learning and parallel processing [1]. Owing to such a property, a lot of effort has been invested on adaptive NN control for single-input-single-output (SISO) nonlinear systems in recent

Manuscript received January 8, 2013; accepted January 12, 2014. The work was supported in part by the National Natural Science Foundation of China under Grant 60974047, in part by the Natural Science Foundation of Guangdong Province under Grant S2012010008967, in part by the Science Fund for Distinguished Young Scholars under Grant S20120011437, in part by the Ministry of Education of New Century Excellent Talent under Grant NCET-12-0637, in part by the Doctoral Fund of Ministry of Education of China under Grant 20124420130001, and in part by the Basic Research Program of China (973 program) under Grant 2011CB707005. Z. Chen is with the College of Automation, Guangdong Polytechnic Normal University, Guangzhou 510635, China (e-mail: [email protected]). S. S. Ge is with Department of Electrical and Computer Engineering, and the Social Robotics Laboratory, Interactive Digital Media Institute, National University of Singapore, Singapore 117576, and also with the Robotics Institute, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]). Y. Zhang is with the School of Automation, Guangdong University of Technology, Guangzhou 510006, China (e-mail: [email protected]). Y. Li is with the Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore 138632 (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.2014.2302856

years (see [2]–[4] and references therein). For multiinputmultioutput (MIMO) nonlinear systems, where couplings, usually with uncertainties, exist among various inputs and outputs, the control problem becomes much more complex and attracts a growing number of research interest [5]–[10]. For example, in [5]–[8], adaptive control was proposed for MIMO nonlinear systems with parametric uncertainties in the input coupling matrix. To decouple the couplings among system inputs, these methods require the estimate of the “decoupling matrix” to be invertible during parameter adaptation period. The possible singularity problem thus has to be handled when inverting the estimated decoupling matrix. To avoid the difficulty of dealing with low-rank decoupling matrices, some researchers adopted different methodologies. In [9], an integral Lyapunov-based adaptive NN controller was developed for MIMO nonlinear systems with nonparametric uncertainties in both the input coupling matrix and the last equation of each subsystem within system interconnections. Because this method does not try to cancel the decoupling matrix when linearizing the system, the necessity of matrix inversion vanishes and the singularity problem is thus removed. In the followup work [10], the authors further considered the control problem of MIMO block-triangular strictfeedback nonlinear systems. Within these systems, the plants to be controlled contain couplings with unknown nonlinearities and/or parametric uncertainties. Besides the coupling terms in the input matrices, system interconnections are allowed in every equation of each subsystem, rather than only in the last equation. By exploring the special structure of the MIMO nonlinear systems, the adaptive NN control developed in [10] avoids the singularity problem completely without using projection algorithms [5]. It is noteworthy that the aforementioned adaptive NN control is applicable only for MIMO affine nonlinear systems. To control MIMO nonaffine nonlinear systems containing nonaffine appearances, it is much more difficult to find the explicit virtual control and actual control to stabilize the systems under study. Moreover, when the desired virtual control and actual control are approximated using NN in the backstepping design, as carried out in [9] and [10], the actual control will be generally involved as the input of the NN approximation, whereas the NN approximation is a part of the actual control. As mentioned in [2], the extension of controls designed for affine systems [9], [10] to nonaffine systems will lead to a circular construction of the actual control.

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The problem of circular construction in controlling SISO nonaffine pure-feedback systems has been solved in [2], [11]–[14]. In [2] and [11], the main idea is to refrain from constructing an overall Lyapunov function for the closed-loop system, which can be realized by integrating the backstepping method, inputto-state stability analysis and the small-gain theorem in the control system design. In [12] and [13], a filtered signal was introduced to circumvent the potential circular control problem as most actuators have low-pass properties. In the followup work [14], by introducing a set of alternative state variables and the corresponding transformation, state-feedback control of the pure-feedback system can be viewed as output-feedback control of a canonical system. Consequently, the previously encountered circular control problem was also circumvented. Unfortunately, it is nontrivial to extend the design method of SISO to MIMO nonlinear systems owing to various couplings involved. Therefore, it remains an open problem to establish an effective design procedure that can simultaneously deal with couplings and the possible circular control construction problem in MIMO nonaffine nonlinear systems. Motivated by the aforementioned problems, in this paper we consider the design procedure for a class of MIMO nonlinear continuous-time systems which are more general than those studied in [10]. Specifically, these systems possess a blocktriangular control structure, with each subsystem being of the completely nonaffine pure-feedback form, and couplings in the forms of unknown nonlinearities in every equation of each subsystem. Using the mean value theorem (MVT), the MIMO block-triangular pure-feedback systems are firstly transformed into a MIMO block-triangular strict-feedback form similar to that considered in [10], whereas the control coefficients are allowed to be nonaffine rather than affine appearances required in [10]. With the transformed systems, a systematic design procedure is then developed for the design of a new singularity-free adaptive neural control. All the signals in the closed-loop system are guaranteed to be semiglobally uniformly ultimately bounded (SGUUB) and the outputs of the systems are proven to converge to a small neighborhood of the desired trajectories. The control performance of the closedloop system is guaranteed by suitably choosing design parameters. Simulation results are finally presented to demonstrate the effectiveness of the proposed control. The main contributions of this paper are as follows. 1) To the best of our knowledge, it is the first time, in the literature, that the tracking control problem of blocktriangular MIMO nonlinear continuous-time systems with each subsystem having the completely nonaffine purefeedback form is investigated. 2) A systematic procedure is developed for the design of an adaptive NN control such that, for the derivatives of Lyapunov function candidates with respect to the control variables, the affine parts can be guaranteed to be stabilized and the nonaffine parts, which are couplings of system inputs and outputs among subsystems, can be guaranteed to be nonpositive. Because of their negative semidefiniteness, the nonaffine parts can be removed in the derivatives of Lyapunov function candidates, which simplifies the control design process and provides the

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following advantages: the couplings among various inputs and outputs have been completely removed without estimating the “decoupling matrix” as carried out in [5]–[8], and subsequently, the aforementioned circular control construction problem has been avoided. 3) Despite the interconnections between the subsystems, the stability of the whole closed-loop system can be established by analyzing individual subsystems separately, much simpler than the analysis on the basis of a complex nested iterative manner in [10]. The rest of this paper is organized as follows. Section II presents the problem formulation and preliminaries. In Section III, we describe the proposed adaptive neural control along with the main theoretical results. Section IV provides a simulation example to illustrate the effectiveness of the proposed approach. Finally, in Section V we draw our conclusion. Throughout this paper, A := B denotes that B is defined as A,  ·  denotes the Euclidean norm of vectors and induced norm of matrices, λmax (M) denotes the largest eigenvalue of a square matrix M, i, j, and l denote integer indices, and i j denotes the subscription of the i j th component of the corresponding items in the j th subsystem. II. P ROBLEM S TATEMENT AND P RELIMINARIES A. Problem Statement Consider the following MIMO nonlinear systems with each subsystem having the completely nonaffine pure-feedback form: ⎧ x˙ j,i j = f j,i j (x¯1,(i j − j 1 ) , x¯2,(i j − j 2 ) , . . . , ⎪ ⎪ ⎨ x¯m,(i j − j m ) , x j,i j +1 ), i j = 1, 2, . . . , ρ j − 1 (1) x˙ j,ρ j = f j,ρ j (X, u¯ j , d j (t)) ⎪ ⎪ ⎩ y j = x j,1 , j = 1, 2, . . . , m where x j,i j x¯ j,i j uj u¯ j yj f j,i j ρj  jl d j (t) X

∈ R, the i j th state of the j th subsystem; [x j,1, x j,2 , . . . , x j,i j ]T ∈ R i j ; ∈ R, the input of the j th subsystem; [u 1 , u 2 , . . . , u j ]T ∈ R j ; ∈ R, the output of the j th subsystem; the unknown nonlinear functions; the order of the j th subsystem; ρ j − ρl , l = 1, 2, . . . , m; ∈ R, the external disturbance; m [x¯1,ρ1 , x¯2,ρ2 , . . . , x¯m,ρm ]T ∈ R k=1 ρk , the vector of all state variables in the complete system. Assume that f j,i j (·) and f j,ρ j (·, ·, 0), i j = 1, 2, . . . , ρ j − 1, j = 1, 2, . . . , m, are smooth functions of their arguments, with f j,ρ j (·, 0, ·) satisfying the Lipschitz condition, and d j (t), j = 1, 2, . . . , m, are bounded by unknown positive constants d ∗j , that is, |d j (t)| ≤ d ∗j . Remark 1: For the order differences  j l , as introduced in [10], there exist three cases to be considered: 1) when j = l, then  j l = 0, and accordingly state vector x¯ j,(i j − jl ) = x¯ j,i j exists in (1); 2) when j = l and i j −  j l ≤ 0, then the corresponding vector x¯l,(i j − jl ) does not exist, and does not appear in (1); and 3) when j = l and i j −  j l > 0, then state vector x¯l,(i j − jl ) exists in (1).

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Remark 2: Compared with MIMO nonlinear system studied in [10], where each subsystem is limited to the affine strictfeedback form, (1) is more general in the sense that it includes not only the system inputs, u¯ j , the control signals of the first to the j th subsystem, but also the completely nonaffine properties, which represent a large class of nonlinearities including the affine strict-feedback form. These properties imply that there exist strong couplings among the system states and inputs, and accordingly cause the difficulty in finding stable controllers for (1). In reality, many practical systems possess these features, such as biochemical processes [1], [15], flight control systems [16], mechanical systems [17], etc. Recent examples of practical systems falling into this category are dynamic models for a small-scale autonomous helicopter [18]. The control objective is to synthesize an adaptive neural tracking control for (1) such that all the signals in the closedloop system remain SGUUB, while the output y j tracks the reference signal yr j ∈ R, the output of the following reference model:

g j,i j (·), i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, are unknown and smooth as well. Moreover, since f j,ρ j (·, 0, ·) satisfies the Lipschitz condition, there exists a constant d¯ ∗j > 0 such that |δ j (t)| ≤ d¯∗j . From (5) and (6), (1) can be rewritten as ⎧ c ⎪ ⎪ ⎪x˙ j,i j = h j,i j ( j,i j ) + g j,i j ( j,i j , x j,i j +1 )x j,i j +1 ⎪ ⎨ i j = 1, 2, . . . , ρ j − 1 (7) ⎪x˙ j,ρ j = h j,ρ j (X, u¯ j −1 ) + g j,ρ j (X, u¯ j −1, u cj )u j + δ j (t) ⎪ ⎪ ⎪ ⎩y = x , j = 1, 2, . . . , m. j j,1

x˙ri = fri (xr ), 1 ≤ i ≤ n yr j = xr j , 1 ≤ j ≤ m ≤ n

(2)

where xr = [xr1 , xr2 , . . . , xrm ]T ∈ R m is the measured state and fri , i = 1, 2, . . . , n are known smooth nonlinear functions. Note that the states xr are assumed bounded in (2), that is, xr ∈ xr , ∀t ≥ 0, where xr ⊂ R m is a compact set. B. Transformation of the System Representation For the control of (1), define g j,i j (·) :=

∂ f j,i j (·)

, i j = 1, 2, . . . , ρ j − 1 ∂ x j,i j +1 ∂ f j,ρ j (·) g j,ρ j (·) := , j = 1, 2, . . . , m. ∂u j

(3) (4)

For convenience, denote x j,ρ j +1 = u j and

Note that since g j,i j (·), i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, are smooth functions, they are bounded within some compact set. As commonly done in the literature, the following assumptions are made for (7). Assumption 1: The signs of g j,i j (·) are known, and there exist positive constants g j,i and g j,i j such that for i j = j 1, 2, . . . , ρ j , j = 1, 2, . . . , m: 1) |g j,i j (·)| > g j,i , ∀( j,i j , x j,i j +1 ); j 2) |g j,i j (·)| ≤ g j,i j , ∀( j,i j , x j,i j +1 ) ∈ x j,i j +1 where x j,i j +1 is a compact subset of the appropriate dimension space. Assumption 1 means that g j,i j is strictly either positive or negative definite. Without loss of generality, assume that g j,i j > g j,i > 0. It should be emphasized that the j bounds g j,i and g j,i j are not necessarily known. j Remark 3: In (7), although functions g j,i j are similar to the affine terms in MIMO systems [10], major differences lie in that: g j,ρ j , j = 1, 2, . . . , m, are fully interconnection terms, that is, they include all the state variables X; and g j,i j , i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, are functions of x j,i j +1 (x j,ρ j +1 := u j ). Therefore, (7) is still a nonaffine nonlinear system, and is more general than the MIMO system considered in [10]. C. Gaussian Radial Basis Networks

T

 j,i j = [x¯1,(i j − j 1 ) , x¯2,(i j − j 2 ) , . . . , x¯m,(i j − j m ) ] .

With its great capabilities in function approximation, the following Gaussian radial basis function (RBF) NN [20] is used to approximate a smooth function h(Z ) : R q → R

Using the MVT [19], it follows that f j,i j ( j,i j , x j,i j +1 ) = h j,i j ( j,i j ) + g j,i j ( j,i j , x cj,i j +1 )x j,i j +1

(5)

f j,ρ j (X, u¯ j , d j (t)) = h j,ρ j (X, u¯ j −1 )+g j,ρ j (X, u¯ j −1 , u cj )u j + δ j (t)

(6)

gnn (Z ) = W T S(Z )

where h j,i j ( j,i j ) = f j,i j ( j,i j , 0) h j,ρ j (X, u¯ j −1 ) = f j,ρ j (X, u¯ j −1 , 0)   x cj,i j +1 ∈ min{x j,i j +1 , 0}, max{x j,i j +1 , 0}   u cj ∈ min{u j , 0}, max{u j , 0} δ j (t) = f j,ρ j (X, u¯ j , d j (t)) − f j,ρ j (X, u¯ j , 0). Consider f j,i j (·) and f j,ρ j (·, ·, 0), i j = 1, 2, . . . , ρ j − 1, j = 1, 2, . . . , m, which are unknown smooth nonlinear functions of their arguments. Accordingly, functions h j,i j (·) and

(8)

where Z ∈  Z ⊂ R q is the input vector, W ∈ Rl is the weight vector, l > 1 is the NN nodes number, and S(Z ) = [s1 (Z ), s2 (Z ), . . . , sl (Z )]T ∈ Rl is the basis function vector with si (Z ) being the commonly used Gaussian functions, that is  −(Z − μi )T (Z − μi ) si (Z ) = exp , i = 1, 2, . . . , l (9) υ2 where μi = [μi1 , μi2 , . . . , μiq ]T is the center of the receptive field, and υ is the width of the Gaussian functions. It has been shown that any continuous function over a compact set Z ∈  Z ⊂ R q can be approximated to any arbitrary accuracy by using (8), that is h(Z ) = W ∗T S(Z ) + ε(Z )

∀Z ∈  Z

(10)

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where W ∗ denotes ideal constant weights, and ε(Z ) is the approximation error. For clarity, write ε := ε(Z ). Assumption 2: For a given continuous function h(Z ) and NN approximator (8), there exist ideal constant weights W ∗ such that |ε| ≤ ε∗ , ∀Z ∈  Z with constant ε∗ > 0. For the Gaussian RBF NN approximator (8) and (9), the following lemma shows that there exists an upper bound on the two-norm of vector S(Z ), which is useful in stability analysis of the closed-loop system. Lemma 1: For the Gaussian RBF NN approximator (8) and (9), there exists a constant cnn > 0 such that S(Z ) ≤ cnn

(11)

where cnn is the limited value of the infinite series {3q(k + 2 2 2 2)q−1 e−2ρ k /υ }(k = 0, 1, . . . , +∞), with υ being the width of the Gaussian function, q the dimension of input Z , and ρ := 12 mini= j μi = μ j  [2], [21]. Remark 4: The positive constant cnn is independent of the NN input variables Z and the NN node number l. Note that in the following control system design, NN approximation is only guaranteed with some compact sets. Accordingly, the stability results on the considered systems are semiglobal since there exists controller(s) with a sufficiently large number of NN nodes such that all signals in the closedloop system remain bounded, as long as the input variables of the NN remain within some compact sets that can be made as large as desired.

In this section, adaptive neural control for MIMO system (7) is presented based on the backstepping approach and the following coordinate transformation: z j,1 = x j,1 − xr1

(12a)

 z j,2 = x j,2 − α j,1  j,1 , xr , Wˆ j,1

(12b)

.. .

z j,i j = x j,i j − α j,i j −1 ( j,i j −1 , xr , Wˆ j,i j −1 ) 2 ≤ i j ≤ ρj

(12c)

z j,ρ j = x j,ρ j − α j,ρ j −1 ( j,ρ j −1 , xr , Wˆ j,ρ j −1 )

(12d)

where α j,i j (·), i j = 1, 2, . . . , ρ j − 1, are virtual controls to be T  determined later, and Wˆ j,i j = Wˆ T , Wˆ T , . . . , Wˆ T with j,2

j,i j

Wˆ j,i j being the estimates of the ideal constant weights W ∗j,i j . The design of α j,i j and Wˆ j,i j is achieved by constructing appropriate Lyapunov functions at the recursive i j th step. The actual control u j appears at the ρ j th step and the design of u j and Wˆ j,ρ j is performed to stabilize (7). Step 1: Differentiating both sides of (12a) yields z˙ j,1 = h j,1 ( j,1 ) + g j,1 x j,2 − x˙r1

where ν j,1 = (1/g j,1 )[h j,1 ( j,1 ) − x˙r1 ] and g + j,1 = (g j,1/g j,1 ) − 1 > 0. Constructing a Lyapunov function candidate Vz j,1 = (1/2)z 2j,1 and differentiating it, we have

(15) V˙z j,1 = z j,1 z˙ j,1 = g j,1 z j,1 ν j,1 + x j,2 + g + j,1 x j,2 . The basic idea of the control design in this paper is to guarantee Vz j,1 to be a Lyapunov function by setting the terms involved in (15) suitably. This can be accomplished by choosing α ∗j,1 := x j,2 as a virtual control input such that: 1) α ∗j,1 = −c j,1 z j,1 − ν j,1 , where c j,1 > 0 is a design ∗ constant and 2) g + j,1 z j,1 α j,1 ≤ 0. After these manipulations, Vz j,1 becomes a Lyapunov function, and z j,1 = 0 is thus asymptotically stable. In (14), ν j,1 is an unknown smooth function of  j,1 and x˙r1 . Thus, ν j,1 can be approximated by employing a Gaussian T S (Z ), where Z T RBF NN W j,1 j,1 j,1 j,1 = [ j,1 , x˙r1 ] ∈  Z j,1 , that is ν j,1 = W ∗T j,1 S j,1 (Z j,1 ) + ε j,1

∀Z j,1 ∈  Z j,1

W ∗j,1

is the ideal constant weights and |ε j,1| ≤ where the approximation error with constant ε∗j,1 > 0. Choose the virtual control α j,1 as T α j,1 = −c j,1 z j,1 −  j,1 Wˆ j,1 S j,1 (Z j,1)

III. A DAPTIVE N EURAL C ONTROL

j,1

g j,1 is a function of x j,2 . From Assumption 1, we get that g j,1 > g j,1 > 0, and (13) can be rewritten as

z˙ j,1 = g j,1 ν j,1 + x j,2 + g + (14) j,1 x j,2

(13)

where  j,1 = [x¯1,(1− j 1) , . . . , x¯ j,1 , . . . , x¯m,(1− j m ) ]T with  j l =  j − l , l = 1, 2, . . . , m. As mentioned in Section II-A, for 1 −  j l ≤ 0, the state x¯l,(1− jl ) vanishes in (14). Note that

where Wˆ j,1 is the estimate of neural weights W ∗j,1 and   ω j,1 T , ω j,1 = Wˆ j,1 S j,1 (Z j,1)z j,1  j,1 = tanh  j,1

(16) ε∗j,1

is

(17)

(18)

where  j,1 > 0 is a small constant. According to (12b), (16), and (17), (14) becomes  T S j,1 (Z j,1 ) z˙ j,1 = −g j,1 c j,1 z j,1 +  j,1 Wˆ j,1  g j,1 + ∗T − W j,1 S j,1 (Z j,1 )−ε j,1 − z j,2 −g j,1α j,1 . g j,1 In light of Assumption 1, the following inequality holds:    ω j,1 + 2 g+ c ω z α = −g z +tanh j,1 ≤ 0. (19) j,1 j,1 j,1 j,1 j,1 j,1  j,1 Remark 5: Note that for the control of the dynamics in (13), if g j,1 is independent of the state x j,2 , then the most commonly used control structure is α ∗j,1 = (−h j,1 + x˙r1 + υ ∗ )/g j,1 with υ ∗ being a new control; and if g j,1 is a function of x j,2 , and α ∗j,1 is unknown and is approximated by NN, then the circular control construction problem will arise since x j,2 has to be chosen as an input of the NN approximation, which is one part of the virtual control x j,2 . On the other hand, by guar∗ anteeing the coupling term g + j,1 z j,1 α j,1 ≤ 0 in (15) instead of ∗ approximating α j,1 , such problem can be completely handled using (17), as the coupling term is removed. Moreover, to use T S (Z ) less neurons, x˙r1 ∈ R is chosen as an input to W j,1 j,1 j,1

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rather than xr ∈ R m since fr1 and xr are known, and then x˙r1 = fr1 (xr ) is available. Thus, the online computation load is lightened. The same ideas of constructing the adaptive NN control and choosing the inputs of NN are also used in the following design steps. Consider a Lyapunov function candidate V j,1 as g j,1 1 T ˜ V j,1 = z 2j,1 + W˜ j,1  −1 (20) j,1 W j,1 2 2 where W˜ j,1 = Wˆ j,1 − W ∗j,1 and  j,1 =  Tj,1 > 0 is an adaptation gain matrix. Using (19), the derivative of V j,1 is  T ˙ V j,1 = g j,1 z j,1 − c j,1 z j,1 −  j,1 Wˆ j,1 S j,1 (Z j,1 )  g j,1 + +W ∗T S (Z )+ε + z + g α j,1 j,1 j,1 j,2 j,1 j,1 j,1 g j,1 ˙ˆ T +g j,1 W˜ j,1  −1 j,1 W j,1   g j,1 2 ≤ g j,1 − c j,1 z j,1 + z j,1 z j,2 + z j,1 ε j,1 +  j,1 (21) g j,1

5

Substituting (25)–(28) into (21), we have V˙ j,1 ≤ − where g ∗j,1 = η j,1

˙ˆ T ˜T  j,1 = ω j,1 −  j,1 ω j,1 + W˜ j,1  −1 j,1 W j,1 − W j,1 S j,1 (Z j,1 )z j,1   ω j,1 ω j,1 = ω j,1 − tanh   j,1  T +W˜ j,1  −1 W˙ˆ j,1 −  j,1 S j,1 (Z j,1)z j,1 (22) j,1

and the following nice property of function tanh(·) [22]:

ω 0 ≤ |ω| − ω tanh ≤ 0.2785 ∀ > 0, ∀ω ∈ R. (23)  Design adaptation law for Wˆ j,1 as   S (Z )z − σ Wˆ (24) W˙ˆ =  j,1

j,1

j,1

j,1

j,1

j,1

j,1

where σ j,1 > 0 is a design parameter, and the σ -modification term σ j,1 Wˆ j,1 is designed to improve the controller robustness [23]. Without such a modification term, it will result in variation of a high-gain control since the NN weight estimates Wˆ j,1 might drift to very large values in the presence of the NN approximation errors [24]. Using (23) and (24), then (22) becomes T ˆ W j,1 .  j,1 ≤ 0.2785 j,1 − σ j,1 W˜ j,1

(25)

From Young’s inequality [25], we have T ˆ T ˜ T −σ j,1 W˜ j,1 W ∗j,1 W j,1 = −σ j,1 W˜ j,1 W j,1 − σ j,1 W˜ j,1  2    ≤ −σ j,1  W˜ j,1  + σ j,1 W˜ j,1 W ∗j,1   2  2 σ j,1 W ∗j,1  σ j,1  W˜ j,1  + (26) ≤− 2 2 g j,1 g j,1 z j,1 z j,2 ≤ |z j,1 ||z j,2 | g j,1 g j,1

≤ z j,1 ε j,1 ≤

c j,1 2 z + 4 j,1

g 2j,1 2 z c j,1 g 2j,1 j,2

c j,1 2 1 ∗2 z j,1 + ε . 4 c j,1 j,1

(27) (28)

g2j,1 c j,1 g j,1 ⎛

 ∗ 2 ⎞   σ j,1 W j,1 1 ∗2 ⎠. = g j,1 ⎝0.2785 j,1 + ε j,1 + c j,1 2

Let

 β j,1 := min g j,1 c j,1 ,

σ j,1



λmax ( −1 j,1 )

−1 with λmax ( −1 j,1 ) being the largest eigenvalue of matrix  j,1 , then

V˙ j,1 ≤ −β j,1 V j,1 + g ∗j,1 z 2j,2 + η j,1 .

˜ T −1 ˙ˆ W ∗T j,1 S j,1 (Z j,1 )z j,1 −  j,1 ω j,1 + W j,1  j,1 W j,1 .

where  j,1 = Consider the facts that

g j,1

 2  c j,1 z 2j,1 + σ j,1 W˜ j,1  +g ∗j,1 z 2j,2 + η j,1 (29) 2

(30)

Step i j (2 ≤ i j ≤ ρ j − 1): Considering (12c), its derivative is z˙ j,i j = [h j,i j ( j,i j ) + g j,i j x j,i j +1 ] − α˙ j,i j −1 = −g ∗j,i j −1 z j,i j

(31) +g j,i ν j,i j + x j,i j +1 + g + j,i j x j,i j +1 j

where g ∗j,i j −1 = ν j,i j

g 2j,i j −1

, g+ j,i j =

g j,i j

−1>0 c j,i j −1 g j,i −1 g j,i j j  1  h j,i j ( j,i j ) − α˙ j,i j −1 + g ∗j,i j −1 z j,i j = g j,i

(32)

j

T α j,i j −1 = −c j,i j −1 z j,i j −1 −  j,i j −1 Wˆ j,i S (Z j,i j −1 ) j −1 j,i j −1

  j,i j −1 = tanh

ω j,i j −1



(33)

 j,i j −1

T with ω j,i j −1 = Wˆ j,i S j,i j −1 (Z j,i j −1 )z j,i j −1 , Wˆ j,i j −1 being j −1 ∗ the estimate of W j,i j −1 ,  j,i j −1 being a small positive constant,  j,i j = [x¯1,(i j − j 1 ) , . . . , x¯ j,i j , . . . , x¯m,(i j − j m ) ]T , and  j l =  j − l , l = 1, 2, . . . , m. Again, as mentioned in Section II-A, if i j −  j l ≤ 0, then the state vector x¯l,(i j − jl ) does not appear in (31). Note that g j,i j is a function of x j,i j +1 . Constructing a Lyapunov function candidate Vz j,i j = (1/2)z 2j,i j , its derivative is

V˙z j,i j = −g ∗j,i j −1 z 2j,i j

 +g j,i z j,i j ν j,i j + x j,i j +1 + g + j,i j x j,i j +1 . j

(34)

If we choose α ∗j,i j := x j,i j +1 as a virtual control input such that: 1) α ∗j,i j = −c j,i j z j,i j − ν j,i j , where c j,i j > 0 is a design ∗ constant and meanwhile and 2) g + j,i j z j,i j α j,i j ≤ 0, then, Vz j,i j is a Lyapunov function, and z j,i j = 0 is asymptotically stable.

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From (33), α j,i j −1 is a function of  j,i j −1 , xr and Wˆ j,1 , . . . , Wˆ j,i j −1 . Thus, α˙ j,i j −1 can be written as α˙ j,i j −1 =

m i j −1−ρ   jl ∂α j,i j −1 l=1

k=1

∂ xl,k

(h l,k + gl,k xl,k+1 ) + ζ j,i j −1

Using (39) and (40), the derivative of V j,i j is V˙ j,i j ≤ V˙ j,i j −1 − g ∗j,i j −1 z 2j,i j + g j,i j   g j,i j × − c j,i j z 2j,i j + z j,i j z j,i j +1 +z j,i j ε j,i j + j,i j g j,i j

(42)

where ζ j,i j −1 =

∂α j,i j −1 ∂ xr i j −1

x˙r

 ∂α j,i j −1 

 +  j,k S j,k (Z j,k )z j,k − σ j,k Wˆ j,k ∂ Wˆ j,k k=1

(35) is computable. From (32), ν j,i j is an unknown smooth function of  j,i j , α˙ j,i j −1 and α j,i j −1 . Considering (35), ν j,i j can be approximated by employing a Gaussian RBF NN T S W j,i j,i j (Z j,i j ), that is, ν j,i j can be expressed as j ν j,i j = W ∗T j,i j S j,i j (Z j,i j ) + ε j,i j ∀Z j,i j ∈  Z j,i j

(36)

where W ∗j,i j denotes the ideal constant weights, |ε j,i j | ≤ ε∗j,i j is the approximation error with constant ε∗j,i j > 0, and  T  T ∂α j,i j −1 , ζ j,i j −1 , α j,i j −1 ∈  Z j,i j . Z j,i j =  j,i j , ∂ j,i j −1 Choose the virtual control α j,i j as T S (Z j,i j ) α j,i j = −c j,i j z j,i j −  j,i j Wˆ j,i j j,i j

(37)

where Wˆ j,i j is the estimate of neural weights and   ω j,i j T , ω j,i j = Wˆ j,i S (Z j,i j )z j,i j (38)  j,i j = tanh j j,i j  j,i j W ∗j,i j ,

with  j,i j > 0 being a small constant. From (12c), (36), and (37), (31) becomes z˙ j,i j = −g ∗j,i j −1 z j,i j + g j,i j  T × − c j,i j z j,i j −  j,i j Wˆ j,i S (Z j,i j ) j j,i j + W ∗T j,i j S j,i j (Z j,i j ) + ε j,i j + +

g+ j,i j α j,i j

 .

g j,i j g j,i

˜ T −1 where  j,i j = W ∗T j,i j S j,i j (Z j,i j )z j,i j −  j,i j ω j,i j + W j,i j  j,i j W˙ˆ . j,i j

Consider the fact that   j,i j = ω j,i j − tanh

ω j,i j  j,i j

 ω j,i j

˙  T ˆ +W˜ j,i  −1 j,i j W j,i j −  j,i j S j,i j (Z j,i j )z j,i j . j Design adaptation law for Wˆ j,i j as   ˙ Wˆ j,i j =  j,i j S j,i j (Z j,i j )z j,i j − σ j,i j Wˆ j,i j

(43)

(44)

where σ j,i j > 0 is a design parameter. Using the property of function tanh(·) in (23) and combining (44), then (43) becomes T ˆ  j,i j ≤ 0.2785 j,i j − σ j,i j W˜ j,i W j,i j . j

Using Young’s inequality [25], we have  T 2  ∗ 2  W  σ σ j,i j W˜ j,i j,i j j,i j j T ˆ + −σ j,i j W˜ j,i W ≤ − j,i j j 2 2 g2j,i j g j,i j c j,i j 2 z z j,i j z j,i j +1 ≤ + z2 g j,i 4 j,i j c j,i j g 2j,i j,i j +1 j

(45)

(46) (47)

j

z j,i j ε j,i j ≤

c j,i j 4

z 2j,i j +

1 ∗2 ε . c j,i j j,i j

(48)

Substituting (45)–(48) into (42), we have g j,i

 T 2  j  c j,i j z 2j,i j + σ j,i j W˜ j,i V˙ j,i j ≤ −β j,i j −1 V j,i j −1 − j 2 +g ∗j,i j z 2j,i j +1 + η j,i j (49) where

 2⎞ σ j,i j W ∗j,i j  ⎠+η j,i j −1 . η j,i j = g j,i ⎝0.2785 j,i j + ε∗2 + j c j,i j j,i j 2 ⎛

1

z j,i j +1

j

Let (39)

According to Assumption 1, the following inequality holds:    ω j,i j + + 2 ω j,i j ≤ 0. g j,i j z j,i j α j,i j = −g j,i j c j,i j z j,i j +tanh  j,i j (40) Consider a Lyapunov function candidate V j,i j as g j,i 1 j T ˜ W˜ j,i V j,i j = V j,i j −1 + z 2j,i j +  −1 (41) j,i j W j,i j j 2 2 where W˜ j,i j = Wˆ j,i j − W ∗j,i j and  j,i j =  Tj,i j > 0 is an adaptation gain matrix.

β j,i j

⎫ ⎬ σ j,i j

 := min β j,i j −1 , g j,i c j,i j , j ⎭ ⎩ λmax  −1 j,i j ⎧ ⎨

−1 with λmax ( −1 j,i j ) being the largest eigenvalue of matrix  j,i j , then

V˙ j,i j ≤ −β j,i j V j,i j + g ∗j,i j z 2j,i j +1 + η j,i j .

(50)

Step ρ j : Considering (12d), its derivative is   z˙ j,ρ j = h j,ρ j (X, u¯ j −1 ) + g j,ρ j u j + δ j (t) − α˙ j,ρ j −1   = −g ∗j,ρ j −1 z j,ρ j + g j,ρ ν j,ρ j + u j + g + j,ρ j u j + δ j (t) j

(51)

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where

7

Then, the dynamics of z j,ρ j are governed by

g ∗j,ρ j −1 = ν j,ρ j =

g2j,ρ j −1 c j,ρ j −1 g j,ρ 1  g j,ρ

j −1

, g+ j,ρ j =

g j,ρ j g j,ρ

−1>0

j

 h j,ρ j (X, u¯ j −1 ) − α˙ j,ρ j −1 + g ∗j,ρ j −1 z j,ρ j .

(57)

j

It should be noticed that g j,ρ j is a function of u j . Constructing a Lyapunov function candidate Vz j,ρ j (1/2)z 2j,ρ j , its derivative is V˙z j,ρ j = −g ∗j,ρ j −1 z 2j,ρ j

+g j,ρ z j,ρ j ν j,ρ j + u j + g + j,ρ j u j +z j,ρ j δ j (t). j

=

m ρ l −1  ∂α j,ρ j −1

∂ xl,k

l=1 k=1

According to Assumption 1, the following inequality holds:    ω j,ρ j + 2 g+ z u = −g z + tanh c ω j,ρ j j,ρ j,ρ j j j,ρ j j ≤ 0. j,ρ j j,ρ j  j,ρ j

(58) (52)

If we construct the actual control input u j such that: 1) u j = −c j,ρ j z j,ρ j − ν j,ρ j , where c j,ρ j > 0 is a design constant and meanwhile and 2) g + j,ρ j z j,ρ j u j ≤ 0, then, Vz j,ρ j converges to a small neighborhood of Vz j,ρ j = 0. From the design at the former step, it can be seen that α j,ρ j −1 is a function of  j,ρ j −1 , xr and Wˆ j,1 , . . . , Wˆ j,ρ j −1 . Thus, α˙ j,ρ j −1 can be written as α˙ j,ρ j −1 =

z˙ j,ρ j = −g ∗j,ρ j −1 z j,ρ j T +g j,ρ − c j,ρ j z j,ρ j −  j,ρ j Wˆ j,ρ S (Z j,ρ j ) j j,ρ j j

+ +W ∗T j,ρ j S j,ρ j (Z j,ρ j ) + ε j,ρ j + g j,ρ j u j + δ j (t).

(h l,k + gl,k xl,k+1 ) + ζ j,ρ j −1

Consider a Lyapunov function candidate V j,ρ j as g j,ρ 1 j T ˜ W˜ j,ρ  −1 V j,ρ j = V j,ρ j −1 + z 2j,ρ j + j,ρ j W j,ρ j j 2 2

(59)

where W˜ j,ρ j = Wˆ j,ρ j − W ∗j,ρ j and  j,ρ j =  Tj,ρ j > 0 is an adaptation gain matrix. Using (57) and (58), the derivative of V j,ρ j is V˙ j,ρ j ≤ V˙ j,ρ j −1 − g ∗j,ρ j −1 z 2j,ρ j

+g j,ρ − c j,ρ j z 2j,ρ j + z j,ρ j ε j,ρ j +  j,ρ j j

+z j,ρ j δ j (t)

(60)

where

where ζ j,ρ j −1 =

∂α j,ρ j −1 ∂ xr

−1 ˙ˆ ˜T  j,ρ j = W ∗T j,ρ j S j,ρ j (Z j,ρ j )z j,ρ j − j,ρ j ω j,ρ j +W j,ρ j  j,ρ j W j,ρ j .

x˙r

ρ j −1

 ∂α j,ρ j −1 

  j,k S j,k (Z j,k )z j,k −σ j,k Wˆ j,k + ∂ Wˆ j,k k=1

(53) is computable. In (51), ν j,ρ j is an unknown smooth function of X, u¯ j −1 , α˙ j,ρ j −1 and α j,ρ j −1 . Considering (53), ν j,ρ j can be approximated by employing a Gaussian RBF NN T S W j,ρ j,ρ j (Z j,ρ j ), that is, ν j,ρ j can be written as j ν j,ρ j =

W ∗T j,ρ j S j,ρ j (Z j,ρ j ) + ε j,ρ j

∀Z j,ρ j ∈  Z j,ρ j (54)

where W ∗j,ρ j denotes the ideal constant weights, |ε j,ρ j | ≤ ε∗j,ρ j is the approximation error with constant ε∗j,ρ j > 0, and  Z j,ρ j = X, u¯ j −1 ,



∂α j,ρ j −1 ∂ j,ρ j −1

T

T

, ζ j,ρ j −1 , α j,ρ j −1

∈  Z j,ρ j .

Design the actual control input u j as T S (Z j,ρ j ) u j = −c j,ρ j z j,ρ j −  j,ρ j Wˆ j,ρ j j,ρ j

(55)

where Wˆ j,ρ j is the estimate of neural weights W ∗j,ρ j , and   ω j,ρ j T  j,ρ j = tanh S (Z j,ρ j )z j,ρ j (56) , ω j,ρ j = Wˆ j,ρ j j,ρ j  j,ρ j with  j,ρ j > 0 being a small constant.

Consider that  j,ρ j = ω j,ρ j − tanh



ω j,ρ j  j,ρ j

 ω j,ρ j

˙  T ˆ +W˜ j,ρ  −1 j,ρ j W j,ρ j −  j,ρ j S j,ρ j (Z j,ρ j )z j,ρ j . (61) j Design adaptation law for Wˆ j,ρ j as   W˙ˆ j,ρ j =  j,ρ j S j,ρ j (Z j,ρ j )z j,ρ j − σ j,ρ j Wˆ j,ρ j

(62)

where σ j,ρ j > 0 is a design parameter. Using the property of function tanh(·) in (23) and combining (62), then (61) becomes T Wˆ j,ρ j .  j,ρ j ≤ 0.2785 j,ρ j − σ j,ρ j W˜ j,ρ j

(63)

Using Young’s inequality [25], we have T T T −σ j,ρ j W˜ j,ρ Wˆ j,ρ j = −σ j,ρ j W˜ j,ρ W˜ j,ρ j −σ j,ρ j W˜ j,ρ W ∗j,ρ j j j j  T 2  ∗ 2 σ j,ρ j W j,ρ j  σ j,ρ j  W˜ j,ρ j  + (64) ≤− 2 2 c j,ρ j 2 1 ∗2 z j,ρ j + z j,ρ j ε j,ρ j ≤ ε (65) 4 c j,ρ j j,ρ j c j,ρ j g j,ρ 1 j 2 z j,ρ j + d¯ ∗2 . (66) z j,ρ j δ j (t) ≤ 4 c j,ρ j g j,ρ j j

Recall that at Step ρ j − 1, V˙ j,ρ j −1 can be expressed as V˙ j,ρ j −1 ≤ −β j,ρ j −1 V j,ρ j −1 + g ∗j,ρ j −1 z 2j,ρ j + η j,ρ j −1 . (67)

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Substituting (63)–(67) into (60), we have V˙ j,ρ j ≤ −β j,ρ j −1 V j,ρ j −1    g j,ρ   T 2 j c j,ρ j z 2j,ρ j + σ j,ρ j W˜ j,ρ −  + η j,ρ j j 2 where

⎛

⎜ η j,ρ j = g j,ρ ⎝0.2785 j,ρ j + j +η j,ρ j −1 +

(68)

 2 ⎞   σ j,ρ j W ∗j,ρ j  ⎟ + ⎠ 2

1 ε∗2 2c j,ρ j j,ρ j

1 d¯ ∗2 . c j,ρ j g j,ρ j j

Let

⎧ ⎨

⎫ ⎬

σ j,ρ j

 −1 ⎭  max j,ρ j

β j,ρ j := min β j,ρ j −1 , g j,ρ c j,ρ j , j ⎩ λ

−1 with λmax ( −1 j,ρ j ) being the largest eigenvalue of matrix  j,ρ j , then

V˙ j,ρ j ≤ −β j,ρ j V j,ρ j + η j,ρ j .

(69)

According to the above analysis, the following theorem states the stability and control performance of the closed-loop system. Theorem 1: Consider the closed-loop system consisting of (1) satisfying Assumptions 1, the reference model (2), the controller (55) and the NN weight updating laws (24), (44), and (62). Assume that there exists a sufficiently large compact set  Z j,i j , i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, such that Z j,i j ∈  Z j,i j , ∀t ≥ 0. Then, for bounded initial conditions: 1) all signals in the closed-loop system are bounded, and for the state vector X and the neural weights Wˆ j =  T  T ,...,W ˆ T T , j = 1, 2, . . . , m, they eventuWˆ j,1 , Wˆ j,2 j,ρ j ally converge to the compact set !   ! η s := X, Wˆ 1 , Wˆ 2 , . . . , Wˆ m !!V ≤ , xr ∈ xr (70) β " # where β = min β1,ρ1 , β2,ρ2 , . . . , βm,ρm and η = $ m j =1 η j,ρ j are positive constants; 2) the tracking error E = [z 1,1 , z 2,1 , . . . , z m,1 ]T ∈ R m converges to a small neighborhood around zero by appropriately choosing design parameters. Proof: 1) Consider the Lyapunov function candidate V =

m 

V j,ρ j .

(71)

j =1

From (69), differentiating V yields V˙ ≤

m  (−β j,ρ j V j,ρ j + η j,ρ j ) ≤ −βV + η j =1

(72)

" # where $m β = min β1,ρ1 , β2,ρ2 , . . . , βm,ρm and η = j =1 η j,ρ j are positive constants.

Let χ(t) := X (t), Wˆ 1 (t), . . . , Wˆ m (t) . If the initial values χ(0) ∈ s , where s is defined in (70), from

[26, Th. 2.14], signals X and Wˆ j stay inside s , that is, χ(t) ∈ s , ∀t ≥ 0; if χ(0) ∈ cs , where cs denotes the complimentary set of s , then (72) drives X and Wˆ j to enter and remain inside s . In summary, all z j,i j and Wˆ j,i j , i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, are uniformly ultimately bounded for bounded initial conditions. From (12), (17), and (37), system state variables x j,i j , i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, can be expressed as ⎧ = z j,1 + xr1 x ⎪ ⎪ j,1 ⎨ x j,i j = z j,i j − c j,i j −1 z j,i j −1 (73) T −  j,i j −1 Wˆ j,i S j,i j −1 (Z j,i j −1 ) ⎪ ⎪ j −1 ⎩ i j = 2, 3, . . . , ρ j , j = 1, 2, . . . , m. From Lemma 1, we have S j,i j (Z j,i j ) ≤ cnn j,i j with finite constant cnn j,i j > 0. Moreover, from the facts that the reference signals xr are bounded and  j,i j ∈ (−1, 1), we obtain from (73) that x j,i j , i j = 1, 2, . . . , ρ j , j = 1, 2, . . . , m, remain bounded. Using (55), control u j , j = 1, 2, . . . , m, are bounded as well. Therefore, all signals in the closed-loop system remain bounded. 2) Denote ς := η/β > 0, then (72) satisfies 0 ≤ V (t) < ς + V (0) exp(−βt).

(74)

From (74), we can obtain ρj

1 2 z j,i j < ς + V (0) exp(−βt). 2 m

(75)

j =1 i j =1

Furthermore, we have m 

z 2j,1 < 2ς + 2V (0) exp(−βt)

(76)

j =1

√ which implies that, given γ > 2ς , there exists T > 0 such that ⎛ ⎞1 2 m  2 E = ⎝ z j,1 ⎠ < γ ∀t ≥ T (77) j =1

where γ is the size of a small residual set which depends on the NN approximation error ε j,i j and controller parameters c j,i j , σ j,i j ,  j,i j . Remark 6: With Lemma 1 and the coordinate transformation (12), although there exist interconnections between the subsystems, the stability of the whole closed-loop system can be concluded by stability analysis of individual subsystem separately without complex analysis in a nested iterative manner as in [10]. Remark 7: From (68)–(70), it can be seen that the size of s depends on W ∗j,i j , ε∗j,i j , d¯∗j and all design parameters. Because there is no analytical result in the NN literature to give an explicit expression of the NN node numbers l j , the ideal constant weights W ∗j,i j , and the approximation error ε∗j,i j , we here point out the following relationships: 1) increasing l j will help to decrease ε∗j,i j , and therefore decrease s , and 2) increasing c j,i j and decreasing σ j,i j and  j,i j might lead

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to smaller s . However, in practical applications, there is a certain tradeoff between the choice of the design parameters and the numerical precision of the tools involved in the MIMO control design [27], [28]. Remark 8: In the above systematic design procedure, by ∗ ≤ 0 in guaranteeing the coupling terms g + j,i j z j,i j α j,i j the derivatives of Lyapunov function candidates rather than approximating α ∗j,i j , the coupling terms have been removed, and consequently, both the circular control construction problem and the singularity problem have been handled by the developed adaptive NN control.

In this section, simulation examples are presented to illustrate the effectiveness of the proposed control approach. Example 1: Consider the following MIMO nonlinear system with each subsystem having the completely nonaffine purefeedback form: = x 1,1 + x 1,2 +

3 x 1,2 5

= x 1,1 x 1,2 + x 2,1 + u 1 +

u 31 7

+ d1 (t)

= x 1,1 x 1,2 + x 2,1 + u 1 + u 2 + = x j,1 , j = 1, 2

u 32 7

with T α1,1 = −c1,1 z 1,1 − 1,1 Wˆ 1,1 S1,1 (Z 1,1 ) ∂α1,1 ∂α1,1 ∂α1,1 ˙ˆ x˙r1 + x˙r2 + ζ1,1 = W1,1 ∂ xr1 ∂ xr2 ∂ Wˆ 1,1   T Wˆ 1,i1 S1,i1 (Z 1,i1 )z 1,i1 i 1 = 1, 2 1,i1 = tanh 1,i1   T S (Z )z Wˆ 2,1 2,1 2,1 2,1 2,1 = tanh 2,1

and NN weights Wˆ j,i j are updated by W˙ˆ j,i j =  j,i j [S j,i j (Z j,i j )z j,i j − σ j,i j Wˆ j,i j ].

IV. S IMULATION S TUDIES

⎧ ⎪ ⎪ x˙1,1 ⎪ ⎪ ⎨ x˙1,2 ⎪ ⎪ ⎪ x˙2,1 ⎪ ⎩ yj

9

(78)

+ d2 (t)

where d j (t) = 0.1 cos(0.01t) cos(x j,1 ), j = 1, 2. Clearly, system (78) consists of two subsystems (ρ1 = 2; ρ2 = 1), with each subsystem in the nonaffine pure-feedback form. Since 1 − ρ12 = 0, the state vector x¯2,(1−ρ12 ) dose not appear in (78). The control objective is to make the outputs y1 and y2 track the desired reference trajectories yr1 and yr2 , which are the outputs of the famous van der Pol oscillator [29] ⎧ ⎨ x˙r1 = xr2 2 )x x˙r2 = −xr1 + β(1 − xr1 r2 ⎩ yr j = xr j , j = 1, 2

(79)

where the output trajectories of the van der Pol oscillator approach a limit cycle when β > 0. The adaptive NN controllers and the design parameters for (78) are chosen as follows: T S j,2 (Z j,2 ), j = 1, 2 u j = −c j,2 z j,2 −  j,2 Wˆ j,2

where z 1,1 = x 1,1 − yr1 , z 1,2 = x 1,2 − α1,1 , z 2,1 = x 2,1 − yr2 , Z 1,1 = [x 1,1, x˙r1 ]T ∈ R 2 ∂α1,1 Z 1,2 = [x 1,1 , x 1,2 , x 2,1 , , ζ1,1 , α1,1 ]T ∈ R 6 ∂ x 1,1 Z 2,1 = [x 1,1 , x 1,2 , x 2,1 , u 1 , x˙r2 ]T ∈ R 5

(80)

(81)

In practice, the selection of the centers and widths of RBF NN has a great influence on the performance of the designed controller. According to [10], Gaussian RBF NN arranged on a regular lattice on R n can uniformly approximate sufficiently smooth functions on closed, bounded subsets. In T S (Z ) contains the following simulation studies, NN Wˆ 1,1 1,1 1,1 nine nodes (i.e., l1,1 = 9), with widths υ1,1,k = 2 (k = 1, 2, . . . , l1,1 ) and centers μ1,1,k (k = 1, 2, . . . , l1,1 ) evenly T S (Z ) contains 729 spaced in [−2.5, 2.5] × [−7, 7]; Wˆ 1,2 1,2 1,2 nodes (i.e., l1,2 = 729), with widths υ1,2,k = 4 (k = 1, 2, . . . , l1,2 ) and centers μ1,2,k (k = 1, 2, . . . , l1,2 ) evenly spaced in [−2.5, 2.5] × [−3, 2] × [−2, 2] × [−1.9, −1.6] × T S (Z ) contains 243 nodes (i.e., [−4, 4] × [−4, 4]; Wˆ 2,1 2,1 2,1 l2,1 = 243), with widths υ2,1,k = 2 (k = 1, 2, . . . , l2,1 ) and centers μ2,1,k (k = 1, 2, . . . , l2,1 ) evenly spaced in [−2.5, 2.5] × [−3, 2] × [−2, 2] × [−3.5, 3.5] × [−4, 4]. Figs. 1–7 show the simulation results of applying controller (80) and the NN weight updating laws (81) to (78) for tracking reference signals yr j , j = 1, 2 with β = 0.001 and the initial conditions x = [0.5; 2; 1.3], xr = [1.5; 0.8]. According to [27] and [28], there is a certain tradeoff between the choice of the design parameters and the control action. The design parameters of the above controller are chosen as c1,1 = 3, c1,2 = 6, c2,1 = 5, 1,1 = 1,2 = 2,1 = diag{2.0}, σ1,1 = σ1,2 = σ2,1 = 1, 1,1 = 1,2 = 2,1 = 0.1. Figs. 1 and 2 show the fairly good tracking performance. From Figs. 3 and 4, it follows that the control signals u 1 and u 2 are bounded and become periodic signals after 2s. Figs. 5–7 illustrate the learning ability of NNs by plotting the nonlinear function as well as its estimate. Note that the tracking performance improves with increase of matching between the nonlinear function and its estimate. Hence, the proposed adaptive controller possesses the abilities of learning and controlling the unknown MIMO nonlinear system. Example 2: Consider a SISO nonaffine pure-feedback system as in [2] ⎧ x3 ⎪ ⎨ x˙1 = x 1 + x 2 + 52 3 (82) x˙2 = x 1 x 2 + u + u7 + d(t) ⎪ ⎩ y = x1 where d(t) = 0.1 cos(0.01t) cos(y). The control objective is to design a controller for (82) such that the output y tracks a desired reference trajectories yr ,

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Fig. 1.

Output y1 (“—”) follows yr1 (“- -”).

Fig. 2.

Output y2 (“—”) follows yr2 (“- -”).

T S (Z ) Fig. 5. Unknown function ν1,1 (“- -”) and its estimate 1,1 W1,1 1,1 1,1 (“–”).

T S (Z ) Fig. 6. Unknown function ν1,2 (“- -”) and its estimate 1,2 W1,2 1,2 1,2 (“–”).

Fig. 3.

Control input u 1 .

T S (Z ) Fig. 7. Unknown function ν2,1 (“- -”) and its estimate 2,1 W2,1 2,1 2,1 (“–”).

with

Fig. 4.

α1 = −c1 z 1 − 1 Wˆ 1T S1 (Z 1 ) ∂α1 ζ1 = x˙r1 + ∂α1 /∂ xr2 x˙r2 + ∂α1 /∂ Wˆ 1 W˙ˆ 1 ∂ xr1 i = tanh(Wˆ iT Si (Z i )z i /i ), i = 1, 2

Control input u 2 .

which is the output yr1 of the famous van der Pol oscillator (79), where β = 0.2 in this simulation. According to (82), the adaptive NN controller is chosen according to (55) (i.e., j = 1, ρ1 = 2) as follows: u = −c2 z 2 − 2 Wˆ 2T S2 (Z 2 ) where z 1 = x 1 − xr1 , z 2 = x 2 − α1 Z 1 = [x 1 , xr2 ]T Z 2 = [x, ∂α1 /∂ x 1 , ζ1 , α1 ]T

(83)

and NN weights Wˆ i are updated by Wˆ˙ i = i [Si (Z i )z i −σi Wˆ i ]. NN Wˆ 1T S1 (Z 1 ) contains nine nodes, with widths υ1,k = 2 (k = 1, 2, . . . , l1 ) and centers μ1,k (k = 1, 2, . . . , l1 ) evenly spaced in [−2.5, 2.5] × [−7, 7]; Wˆ 2T S2 (Z 2 ) contains 243 nodes, with widths υ2,k = 4 (k = 1, 2, . . . , l2 ) and centers μ2,k (k = 1, 2, . . . , l2 ) evenly spaced in [−2.5, 2.5]×[−3, 2]× [−4, 0] × [−6, 6] × [−4, 5]. The design parameters of the above controller are chosen as c1 = 9, c2 = 3, 1 = 2 = diag{5.0}, σ1 = σ2 = 0.1, 1 = 2 = 0.1, and the initial conditions x = [0.5; 1.8]. From Fig. 8, we can see that fairly good tracking performance is obtained. The boundedness of control signal u and

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Fig. 8.

Output y (“—”) follows yr (“- -”).

Fig. 12.

Output y (“—”) follows yr (“- -”).

Fig. 9.

Control input u.

Fig. 13.

Control input u.

Fig. 14.

L 2 norm of the NN weights: 1 W1 (“—”) and 2 W2 (“- -”).

Fig. 10.

L 2 norm of the NN weights: 1 W1 (“—”) and 2 W2 (“- -”).

Fig. 11. Comparative tracking errors: the proposed approach (“—”) in this paper and the ISS-modular approach (“- -”) in [2].

Fig. 15. Comparative tracking errors: the proposed NN control (“—”) in this paper and the ISS-modular approach (“- -”) in [2].

NN weights 1 W1 and 2 W2 are shown in Figs. 9 and 10, respectively. Comparative tracking errors of the ISS-modular approach in [2] and the proposed approach in this paper are given in Fig. 11. Keeping all design parameters as before, Figs. 12–15 show the simulation results of applying (83) to (82) for tracking reference signal yr = 0.5[sin(t) + sin(0.5t)], and confirm the effectiveness of the developed approach. Fig. 12 shows fairly good tracking performance obtained by the same adaptive controller (83). Figs. 13 and 14 show that control signal u

and NN weights 1 W1 and 2 W2 are bounded, and Fig. 15 gives comparative tracking errors of the ISS-modular approach in [2] and the proposed approach in this paper. It is shown that the convergence of the ISS-modular approach in [2] is slower compared with the developed approach in this paper. V. C ONCLUSION In this paper, we have proposed adaptive neural tracking control for a class of uncertain MIMO block-triangular nonaffine pure-feedback systems in the continuous-time form.

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Theoretical analysis and simulation studies suggest that our approach can tackle the difficulties in controlling MIMO block-triangular nonaffine pure-feedback systems and simplify the control design process. All signals in the closed-loop system are guaranteed to be semiglobal uniform ultimate bounded, and the system outputs are proven to converge to a small neighborhood of the desired trajectory. The adaptive NN scheme can be applied to a large number of uncertain MIMO pure-feedback nonlinear systems without repeating the complex controller design procedure. ACKNOWLEDGMENT The authors would like to thank the constructive comments from the anonymous Associate Editor and reviewers, and Z. Chen would like to thank Prof. Weichao Xu of Guangdong University of Technology for his dedication in improving the quality and presentation of this paper. R EFERENCES [1] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, MA, USA: Kluwer, 2002. [2] C. Wang, D. J. Hill, S. S. Ge, and G. Chen, “An ISS-modular approach for adaptive neural control of pure-feedback systems,” Automatica, vol. 42, no. 5, pp. 723–731, 2006. [3] S. Jagannathan and P. He, “Neural-network-based state feedback control of a nonlinear discrete-time system in nonstrict feedback form,” IEEE Trans. Neural Netw., vol. 19, no. 12, pp. 2073–2087, Dec. 2008. [4] Z. Chen and S. Jagannathan, “Generalized Hamilton–Jacobi–Bellman formulation-based neural network control of affine nonlinear discretetime systems,” IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 90–106, Jan. 2008. [5] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Autom. Control, vol. 34, no. 11, pp. 1123–1131, Nov. 1989. [6] K. S. Narendra and S. Mukhopadhyay, “Adaptive control of nonlinear multivariable system using neural networks,” Neural Netw., vol. 7, no. 5, pp. 737–752, 1994. [7] F. C. Chen and H. K. Khalil, “Adaptive control of a class of nonlinear discrete-time systems using neural networks,” IEEE Trans. Autom. Control, vol. 40, no. 5, pp. 791–801, May 1995. [8] S. S. Ge, “Robust adaptive NN feedback linearization control of nonlinea systems,” Int. J. Syst. Sci., vol. 27, no. 12, pp. 1327–1338, 1996. [9] S. S. Ge, C. C. Hang, and T. Zhang, “Stable adaptive control for nonlinear multivariable systems with a triangular control structure,” IEEE Trans. Autom. Control, vol. 45, no. 6, pp. 1221–1225, Jun. 2000. [10] S. S. Ge and C. Wang, “Adaptive neural control of uncertain MIMO nonlinear systems,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 674–692, May 2004. [11] M. Wang, C. Wang, and S. S. Ge, “A novel ISS-modular adaptive neural control of pure-feedback nonlinear systems,” Int. J. Innov. Comput., Inf. Control, vol. 7, no. 11, pp. 6559–6570, 2011. [12] A. M. Zou, Z. G. Hou, and M. Tan, “Adaptive control of a class of nonlinear pure-feedback systems using fuzzy backstepping approach,” IEEE Trans. Fuzzy Syst., vol. 16, no. 4, pp. 886–897, Aug. 2008. [13] J. Na, X. Ren, C. Shang, and Y. Guo, “Adaptive neural network predictive control for nonlinear pure feedback systems with input delay,” J. Process Control, vol. 22, no. 1, pp. 194–206, 2012. [14] J. Na, X. Ren, and 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. [15] M. Krsti´c, I. Kanellakopoulos, and P. V. Kokotovi´c, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [16] L. R. Hunt and G. Meyer, “Stable inversion for nonlinear systems,” Automatica, vol. 33, no. 8, pp. 1549–1554, 1997. [17] A. Ferrara and L. Giacomini, “Control of a class of mechanical systems with uncertainties via a constructive adaptive/second order VSC approach,” J. Dyn. Syst., Meas. Control, vol. 122, no. 1, pp. 33–39, 2000. [18] B. Ren, S. S. Ge, C. Chen, C.-H. Fua, and T. H. Lee, Modeling, Control and Coordination of Helicopter Systems. New York, NY, USA: SpringerVerlag, 2011.

[19] S. C. Malik, Mathematical Analysis. New York, NY, USA: Wiley, 1984. [20] S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 1999. [21] A. J. Kurdila, F. J. Narcowich, and J. D. Ward, “Persistency of excitation in identification using neural networks,” SIAM J. Control Optim., vol. 33, no. 2, pp. 625–642, 1995. [22] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 32, no. 3, pp. 423–427, 1996. [23] P. A. Ioannou and J. Sun, Robust Adaptive Control. Englewood Cliffs, NJ, USA: Prentice-Hall, 1995. [24] T. Zhang, S. S. Ge, and C. C. Hang, “Design and performance analysis of a direct adaptive controller of nonlinear systems,” Automatica, vol. 35, no. 11, pp. 1809–1817, 1999. [25] W. H. Young, “On the multiplication of successions of Fourier constants,” Proc. R. Soc. London. Ser. A, vol. 87, no. 596, pp. 331–339, 1912. [26] Z. Qu, Robust Control of Nonlinear Uncertain Systems. New York, NY, USA: Wiley, 1998. [27] F. Pozo, F. Ikhouane, and J. Rodellar, “Numerical issues in backstepping control: Sensitivity and parameter tuning,” J. Franklin Inst., vol. 345, no. 8, pp. 891–905, 2008. [28] Y. Chang, “Block backstepping control of MIMO systems,” IEEE Trans. Autom. Control, vol. 56, no. 5, pp. 1191–1197, May 2011. [29] M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 1993.

Zhenfeng Chen received the B.Sc. degree in information and computing science from Zhaoqing University, Zhaoqing, China, and the M.Sc. degree in applied mathematics and the Ph.D. degree in control science and engineering from the Guangdong University of Technology, Guangzhou, China, in 2006, 2009, and 2013, respectively. He was with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, from 2011 to 2012. He is currently a Faculty Member with the College of Automation, Guangdong Polytechnic Normal University, Guangzhou. His current research interests include adaptive neural control, intelligent control, and robotics.

Shuzhi Sam Ge (S’90–M’92–SM’99–F’06) received the B.Sc. degree from the Beijing University of Aeronautics and Astronautics, Beijing, China, and the Ph.D. degree from the Imperial College of Science, Technology, and Medicine, University of London, London, U.K., in 1986 and 1993, respectively. He is the Director of the Institute of Intelligent Systems and Information Technology and the Robotics Institute, University of Electronic Science and Technology of China, Chengdu, China, and the Founding Director of the Social Robotics Laboratory, Interactive Digital Media Institute, National University of Singapore, Singapore. He is a Professor with the Department of Electrical and Computer Engineering, National University of Singapore. He has authored or co-authored three books, edited a book, and published more than 140 international journal papers. His current research interests include social robotics, multimedia fusion, adaptive control, and intelligent systems. Dr. Ge has been serving as the Vice President of Technical Activities, the IEEE Control Systems Society from 2009 to 2010. He serves and has served as an Associate Editor for a number of flagship journals. He serves as an Editor of the Taylor and Francis Automation and Control Engineering Series. He is an Editor-in-Chief of the International Journal of Social Robotics.

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

Yanan Li (S’10–M’14) received the B.E. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, the M.E. degree in control and mechatronics engineering from the Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, China, and the Ph.D. degree from the Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore, in 2006, 2009, and 2013, respectively. He is currently a Research Scientist with the Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore. His current research interests include physical human–robot interaction and robot control.