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Observer-Based Adaptive Neural Network Control for Nonlinear Systems in Nonstrict-Feedback Form Bing Chen, Huaguang Zhang, Senior Member, IEEE, and Chong Lin, Senior Member, IEEE Abstract— This paper focuses on the problem of adaptive neural network (NN) control for a class of nonlinear nonstrictfeedback systems via output feedback. A novel adaptive NN backstepping output-feedback control approach is first proposed for nonlinear nonstrict-feedback systems. The monotonicity of system bounding functions and the structure character of radial basis function (RBF) NNs are used to overcome the difficulties that arise from nonstrict-feedback structure. A state observer is constructed to estimate the immeasurable state variables. By combining adaptive backstepping technique with approximation capability of radial basis function NNs, an output-feedback adaptive NN controller is designed through backstepping approach. It is shown that the proposed controller guarantees semiglobal boundedness of all the signals in the closed-loop systems. Two examples are used to illustrate the effectiveness of the proposed approach. Index Terms— Adaptive neural control, backstepping, nonlinear systems, nonstrict-feedback structure.

I. I NTRODUCTION N THE past decade, there has been an increasing interest in approximation-based adaptive control for nonlinear systems. With the inherent approximation capability of neural networks (NNs) or fuzzy logic systems, some adaptive neural/fuzzy controllers were proposed for nonlinear systems [1]–[14]. In [1]–[11], the problem of stabilization or tracking control was addressed for single-input and single-output (SISO) nonlinear strictfeedback systems, and the corresponding adaptive neural/fuzzy controllers were developed via state feedback control strategy. Furthermore, adaptive neural/fuzzy control was discussed for multi-input and multioutput (MIMO) nonlinear strict-feedback systems in [12]–[14], respectively. Notice that state variables are usually unknown or partly known in practice, and thus the aforementioned control strategies via state feedback are difficult to be implemented. Therefore, some output-feedback control strategies were developed in recent years. Adaptive NN output-feedback

I

Manuscript received August 13, 2014; revised December 25, 2014 and March 1, 2015; accepted March 6, 2015. Date of publication March 25, 2015; date of current version December 17, 2015. This work was supported by the National Natural Science Foundation of China under Grant 61473160, Grant 61174033, and Grant 61034005. B. Chen and C. Lin are with the Institute of Complexity Science, Qingdao University, Qingdao 266071, China (e-mail: chenbing1958@ 126.com; [email protected]). H. Zhang is with the School of Information Science and Engineering, Northeastern University, Shenyang 110006, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2412121

control was addressed in [15]–[19]. Leu et al. [15] addressed observer-based adaptive fuzzy-neural control for a class of SISO nonlinear systems, and an indirect adaptive fuzzyneural controller was proposed. An observer-based direct adaptive fuzzy-neural controller was developed in [16]. In [17], the problem of adaptive NN output-feedback control was investigated for two classes of nonlinear discrete-time systems with unknown control directions, and an unified approach of control design was proposed. Tong et al. [18] addressed adaptive NN backstepping control for a class of delayed large-scale systems, and decentralized adaptive NN output-feedback controllers were developed. It was provided that under the action of suggested neural controllers, the adaptive closed-loop systems were semiglobally uniformly ultimately bounded (SGUUB). In [19], adaptive output-feedback neural control was first considered for stochastic nonlinear delayed systems with strict-feedback structure. The proposed control scheme employed only one NN to compensate for all unknown nonlinear terms. In those works, unknown nonlinear terms in systems were assumed to be the functions of system’s outputs only. In [20]–[22], observer-based output-feedback adaptive fuzzy backstepping control scheme was proposed for SISO nonlinear strict-feedback systems. The corresponding work was considered for discrete-time nonlinear systems in [23]. The constructed adaptive fuzzy controller guaranteed that the closed-loop systems were uniformly ultimately bounded. The work was further extended from SISO systems to MIMO systems in [24] and [25], where the effect from unknown dead zone on controlled systems was considered. In [26], an input-driven filter was first introduced to estimate the immeasurable state variables; furthermore, an adaptive output-feedback fuzzy tracking controller was proposed for nonlinear systems in lower triangle form. This work was further extended to nonlinear stochastic strict-feedback systems in [27]. In [28], using NNs to approximate unknown nonlinear functions, an output-feedback adaptive neural controller was presented for nonlinear strict-feedback systems with time delays. In [29], adaptive output-feedback fuzzy control was discussed. An output-feedback fuzzy controller was developed for nonlinear delayed strict-feedback systems with unknown control direction. Although adaptive NN/fuzzy backstepping control has been one of the most popular design approaches to a class of nonlinear systems, the existing adaptive backstepping approaches suffer from a major limitation of

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strict-feedback form. On the other hand, some real controlled systems appear in nonstrict-feedback form, such as the ball and beam system [30] and the helicopter model [31]. Theoretically, these control strategies may be infeasible for the systems without strict-feedback form. To relax such a restriction of system structure, in [32]–[34], the aforementioned adaptive neural/fuzzy schemes from nonlinear strict-feedback systems to nonlinear pure-feedback systems were extended, which included the strict-feedback form as a special case. The adaptive neural/fuzzy controllers proposed in the literature guaranteed that all the signals of the resulting adaptive closedloop systems were SGUUB. In [34], a backstepping-based procedure was developed to design an adaptive NN controller for affine-in control pure-feedback systems. Differently from the method in [34], a key lemma was established in [35] to guarantee the existence of solution to multivariate functions; furthermore, adaptive NN control scheme was proposed via backstepping for affine-in control pure-feedback systems. This result was further extended to nonaffine pure-feedback systems in [36]. More recently, adaptive neural/fuzzy control for nonlinear nonstrict-feedback systems has been considered. In [37], the problem of adaptive fuzzy control for nonlinear nonstrict-feedback systems was considered. Adaptive fuzzy backstepping approach was extended to nonstrict-feedback systems. In [38], a novel direct adaptive NN control strategy was proposed for nonlinear nonstrict-feedback systems. The proposed adaptive NN controller guarantees that all the closed-loop signals are SGUUB. Wang et al. [39] further considered adaptive neural tracking control for stochastic nonlinear nonstrict-feedback systems. A stochastic adaptive neural control strategy was proposed. However, these control schemes are all developed via state feedback. So far, no results on the adaptive neural output-feedback control strategy have been reported for the nonlinear nonstrict-feedback systems. Motivated by the aforementioned observation, we consider the more challenging problem in this paper, i.e., the output-feedback adaptive NN control for nonlinear systems in nonstrict-feedback form. An observer-based adaptive NN output-feedback control scheme is proposed for a class of nonlinear systems in nonstrict-feedback form. Science input saturation often occurs in practice. In addition, it maybe degenerate the performance of closed-loop systems. Therefore, actuator saturation nonlinearity is considered. It is shown that the proposed controller guarantees semiglobal boundedness of all the signals in the closed-loop systems. The main contribution of this paper lies in that it is first time to extend adaptive NNs output-feedback control strategy from strictfeedback systems to nonstrict-feedback systems in theory. So, the proposed adaptive NN backstepping-design approach can be used for control design of nonstrict feedback systems that include the strict-feedback systems as a special case. The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section II. A novel adaptive neural control scheme is presented in Section III. The simulation examples are given in Section IV. Finally, the conclusion is drawn in Section V.

II. P ROBLEM F ORMULATION AND P RELIMINARIES To express the control problem clearly, some preliminaries are first formulated as follows. A. Nonlinear Control Problem Consider the following nonlinear system in nonstrictfeedback form: x˙ i = x i+1 + f i (x), 1 ≤ i ≤ n − 1 x˙n = u(v) + f n (x) y = x1

(1)

where x = [x 1 , x 2 , . . . , x n ]T ∈ R n and y ∈ R denote state vector and output variable of the system, respectively, and the variable y = x 1 is measured directly only. f i (.)s, for (1 ≤ i ≤ n), are the unknown smooth nonlinear functions with f i (.) = 0. v ∈ R stands for the desired control input to be designed and u(v) denotes the plant input subject to saturation nonlinearity described by  |v| ≥ u M sign(v)u M , (2) u(v) = sat(v) = v, |v| < u M where u M is a parameter of input saturation. In this paper, we introduce input saturation in control deign to avoid high-gain controller. Therefore, the parameter u M is assumed to be known. Remark 1: System (1) is said to be in nonstrict-feedback form, because each subsystem function f i (.) is the function of whole state variable x. Thus, (1) is no longer in strict-feedback form or pure-feedback form. Since the existing adaptive neural/fuzzy backstepping methods depend on the assumption that the controlled system must be in strict-feedback form or pure-feedback form, they cannot be used to control (1) in theory. In addition, as shown later, the nonstrict-feedback form brings considerable difficulty for adaptive backsteppingcontrol design. From (2), it can be seen that there exists a sharp corner when |v| = u M . Therefore, backstepping technique cannot be directly applied to construct control input signal. To solve this problem, the method proposed in [40] will be used. As done in [40], a smooth function is defined as g(v) = u M ∗ tanh(v/u M ) ev/u M − e−v/u M = u M ∗ v/u . (3) e M + e−v/u M Then, sat(v) in (2) can be expressed in the following form: sat(v) = g(v) + d(v)

(4)

where d(v) = sat(v) − g(v) is a bounded function and its bound can be obtained as |d(v)| = |sat(v) − g(v)| ≤ u M (1 − tanh(1)) = D.

(5)

According to the mean-value theorem [41], there exists a constant μ with 0 < μ < 1 such that g(v) = g(v 0 ) + gv μ (v − v 0 )

(6)

CHEN et al.: OBSERVER-BASED ADAPTIVE NN CONTROL FOR NONLINEAR SYSTEMS IN NONSTRICT-FEEDBACK FORM

where gv μ = (∂g(v)/∂v)|v=v μ = (4/(ev/u M + e−v/u M )2 )|v=v μ , v μ = μv + (1 − μ)v 0 . Thus, by choosing v 0 = 0, (6) can be written as g(v) = gv μ v.

(7)

To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments. Assumption 1: For nonlinear function fi (.), there exist known constants h i j and h i j such that hi j ≤

∂ fi ≤ h i j , 1 ≤ i, j ≤ n. ∂x j

Remark 2: Note that f i (x) = [(∂ f i /∂ x 1 ), . . . , (∂ fi /∂ x n )]x. By Assumption 1, it is easy to prove that there exists constant h i such that | f i (x)| ≤ h i ||x||. Thus, smooth functions φi (s) = h i s with s ∈ R is a bounding function of f i (.). Assumption 2 [40]: The plant is input-to-state stable. Remark 3: As pointed out in [40], Assumption 2 is reasonable. The following example was given in [40] to show that an unstable pant cannot be globally stabilized in the presence of input saturation. Consider a simple system x˙ = ax + u(v) with x ∈ R is state variable and u(v) ∈ R denotes the plant input subject to saturation described by (2). When a > 0 and the initial value x(0) > u M /a, there does not exist any control that satisfies the saturation constraint to stabilize the system. According to Assumption 2, the state variables of (1) are bounded; furthermore, the desired control input signal v is also bounded. Therefore, there exists a positive constant g0 such that ∂g(v) 4 4 = v/u ≥ ≥ g0 . (8) −v/u 2 |v|/u M M M )2 ∂v (e +e ) (2e In this research, radial basis function (RBF) NNs will be used to approximate the unknown functions. As pointed out in [42], for a continuous nonlinear function f (Z ) over a compact set  Z ⊂ R q , there exists a RBF NN W ∗T S(Z ) such that for a desired level of accuracy ε > 0 f (Z ) = W ∗T S(Z ) + δ(Z ), |δ(Z )| ≤ ε

(9)

where W ∗ is the ideal constant weight vector and defined as W ∗ := arg min { sup | f (Z ) − W T S(Z )|} W ∈ R¯ l Z ∈ Z

and δ(Z ) denotes the approximation error, W = [w1 , w2 , . . . , wl ]T ∈ Rl is weight vector and S(Z ) = [s1 (Z ), s2 (Z ), . . . , sl (Z )]T means the basis function vector, si (Z ) is basis function vector with l being the number of RBF NN nodes, and l > 1. The RBF is chosen as   (Z − μi )T (Z − μi ) si (Z ) = exp − η2 where μi = [μi1 , μi2 , . . . , μiq ]T , i = 1, . . . , l, is the center of the receptive field and η is the width of the Gaussian function. Lemma 1 [43]: For any w ∈ R and ε > 0, the following holds:   w ≤ δε, δ = 0.2785. 0 ≤ |w| − w tanh ε

91

Similar to [37, Lemma 3] and [39, Lemma 3], for coordinate transformations z i = xˆ i − αi−1 , for i = 1, 2, . . . , n, the following lemma can be obtained easily. Lemma 2: For the coordinate transformations z i = xˆi − αi−1 , for i = 1, 2, . . . , n, the following result holds: n  |z i |ϕi (θi ) (10) ||x|| ˆ ≤ i=1

where ϕi = (ki + 0.5) + (1/2ai2 )θi s with s being a constant for 1 ≤ i ≤ n − 1, ϕn = 1, and the functions αi s are defined by (13). The proof of this lemma is very similar to that of [37, Lemma 3] or [39, Lemma 3]. So, it is omitted. Lemma 3: Let A, B, and D be symmetrical matrices, and μ(t) a real-valued function with 0 ≤ μ(t) ≤ 1. Then A + μB + (1 − μ)D < 0 holds if and only if A + B < 0 and A + D < 0. Lemma 3 is obvious. Therefore, its proof is omitted. III. M AIN R ESULT In this section, we will develop an output-feedback adaptive NN backstepping-design scheme for (1). An observer is first constructed to estimate the unknown state variables, and then an adaptive NN controller will be proposed. A. Observer Design To estimate the immeasurable state variables, as done in [26] and [27], the following observer is adopted: x˙ˆi = xˆi+1 − li (y − xˆ1 ), 1 ≤ i ≤ n − 1 (11) xˆ˙n = u(v) − ln (y − xˆ1 ) where xˆi is the estimation of x i , 1 ≤ i ≤ n. The design parameter li is chosen such that the polynomial p(s) = s n + l1 s n−1 + · · · + ln−1 s + ln is Hurwizts. In general, two cases maybe result in the control input saturation phenomenon in practical control issue. One is that the designer proposes the saturation control strategy for some special control objective. In this case, the saturation nonlinearity is known. Another is from the control equipment itself. Some physical characteristics of control equipment lead to that the plant input subjects to saturation nonlinearity when the desired control signal passes through the actuator. In this case, the saturation nonlinearity is usually unknown. In practical engineering, the observers have the same actuator and the same control signal as corresponding ones in systems. So, it is reasonable to consider the observer has the same saturation nonlinearity. Define the estimation error as ei = x i − xˆ i , 1 ≤ i ≤ n. From (1) and (11), the error dynamic can be expressed as follows: e˙ = Ao e + F(x)

(12)

where e = [e1 , . . . , en ]T , F(x) = [ f 1 (x), . . . , fn (x)]T and   L n−1 In−1 Ao = , L n−1 = [l1 , . . . , ln−1 ]T . ln 01×(n−1) A block diagram of control system is shown in Fig. 1.

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ˆ one has For the term 2e T P(F(x) − F(x)), ˆ = 2e T P J e 2e T P(F(x) − F(x)) where J is a Jacobin matrix with its element at the of i th row and the j th column being ∂ fi /∂ x j . Furthermore, by Assumption 1, ∂ f i /∂ x j can be expressed as a convex combination of h i j and h i j . Namely, there exists a function 0 ≤ μi j (t) ≤ 1 such that ∂ fi = μi j h i j + (1 − μi j )h i j . ∂x j Substituting this equality into the above equality gives Fig. 1.

Block diagram of controlled system.

ˆ = 2e T P(H + H )e 2e T P(F(x) − F(x))

B. Control Design and Stability Analysis Adaptive neural/fuzzy backstepping design is usually carried out based on a set of state transformation, z i = xˆi − αi−1 , for 1 ≤ i ≤ n, with α0 = 0. The function αi is usually reviewed as the virtual control signal for the i th subsystem, and it is required to be in the following form: αi = −(ki + 0.5)z i −

1 z i θi SiT (Z i )Si (Z i ) 2ai2

with ri and σi being the positive design parameters. Remark 4: As pointed out in [1], if the initial condition θi (0) is nonnegative, then θi (t) ≥ 0 holds for all t ≥ 0. Therefore, in this paper, it is always assumed that θi ≥ 0. Consider the Lyapunov function candidate as V = Ve + Vz + Vθ

n 2 where Ve = e T Pe, Vz = i=1 (1/2)z i , and n 2 ∗ ˜ ˜ Vθ = i=1 (1/2ri )θi with θi = θi − θi . According to (12), one has

(15) V˙e = e T P Ao + AoT P e + 2e T P F(x). Remark 5: It can be seen clearly from (15) that a main difficulty for output-feedback control design of nonlinear systems in nonstrict-feedback form comes from the term F(x). In some works on adaptive neural/fuzzy output-feedback control for strict-feedback nonlinear systems, the procedure tackling such a nonlinear function generates some terms in form ||P||2 e T e. As a result, these terms make it become very difficult to verify the stability conditions of error dynamics. In the following, we will develop a new approach to deal with this nonlinear function F(x). In addition, the proposed stability conditions for error dynamics are more easy to be tested. For the crossing term 2e T P F(x), it can be rewritten as ˆ + 2e T P F(xˆ ). 2e T P F(x) = 2e T P(F(x) − F(x))

where the matrices H and H are defined as H (i j ) = μi j h i j and H (i j ) = (1 − μi j )h i j , and the symbol A(i j ) stands for the element at the i th row and the j th column of matrix A. Using Remark 2 and (10), one has ˆ 2 F(x) ˆ 2e T P F(xˆ ) ≤ ε0 e T e + ε0−1 F T (x)P ≤ ε0 e T e + ε0−1 ||P 2 ||||F(x)|| ˆ 2

≤ ε0 e T e + c0 ||x|| ˆ 2  n 2  T ≤ ε 0 e e + c0 |z i |ϕi (θi )

(13)

where ki and ai are the positive design parameters, Z i = [xˆ1 , . . . , xˆi , θ1 , . . . , θi ]T , θi satisfies (14) and is an estimation of the unknown constant θi∗ = ||wi ||2 with ||wi || being the norm of the ideal weight vector of the i th NN ri (14) θ˙i = 2 z i2 SiT Si − σi θi 2ai

(16)

(17)

 i=1 n  T 2 2 ≤ ε0 e e + c z i ϕi (θi )

(18)

i=1

 with c0 = ε0−1 ||P 2 || ni=1 h 2i and c = nc0 . Consequently, substituting (16)–(18) into (15) gives

V˙e ≤ e T P Ao + AoT P + P(H + H ) + (H + H )T P + ε0 I e n  +c z i2 ϕi2 (θi ). (19) i=1

The last term at the right-hand side of the above equality is just the function of z and θ only. This means that it can be well handled in the control design procedure. For backstepping-based control design, at the i th step, for 1 ≤ i ≤ n−1, we chose a control Lyapunov function candidate as Vi = (1/2)z i2 . Notice that z i = xˆ i − αi−1 . A simple calculation gives V˙i = z i (x˙ˆi − α˙ i−1 ) = z i (xˆi+1 − li e1 − α˙ i−1 ) = z i (αi − li e1 − α˙ i−1 ) + z i z i+1

(20)

with z i+1 = xˆ i+1 − αi and α˙ i−1 =

i−1   ∂αi−1 j =1

=

∂ xˆ j

i−1   ∂αi−1 j =1

∂ xˆ j +

∂αi−1 x˙ˆ j + θ˙ j ∂θ j



(xˆ j +1 − l j e1 )

  ∂αi−1 r j 2 T . z S S − σ θ j j j ∂θ j 2a 2j j j

(21)

CHEN et al.: OBSERVER-BASED ADAPTIVE NN CONTROL FOR NONLINEAR SYSTEMS IN NONSTRICT-FEEDBACK FORM

Furthermore, V˙i can be expressed as

with αn = u(v), and

V˙i = z i (x˙ˆi − α˙ i−1 ) = z i (αi − li e1 ) + z i z i+1  i−1  ∂αi−1 − zi (xˆ j +1 − l j e1 ) ∂ xˆ j j =1  ∂αi−1 r j 2 T z j Sj Sj − σj θj + ∂θ j 2a 2j ⎛ i−1  ∂αi−1 = z i ⎝αi − xˆ j +1 ∂ xˆ j j =1  ⎞ i−1  ∂αi−1 r j 2 T − z S S − σjθj ⎠ 2 j j j ∂θ j 2a j j =1 ⎛ ⎞ i−1  ∂αi−1 + zi ⎝ l j e1 − li e1 ⎠ + z i z i+1 . (22) ∂ xˆ j

f¯i (Z i ) = −

The virtual control signal α i cannot directly compensate for the effect from the term z i ( i−1 j =1 (∂αi−1 /∂ xˆ j )l j e1 − l i e1 ). Otherwise, it will contain variable e1 , further, contain the variable x 1 . Thus, x˙1 , further f 1 (x), will appear in the expression of α˙ i . This will lead to much difficulty for the next step control design. To overcome this difficulty, it is handled to get the following inequality: ⎛ ⎞ ⎛ ⎞2 i−1 i−1  z i2  ∂αi−1 ∂α i−1 ⎝ zi ⎝ l j e1 − l i e1 ⎠ ≤ l j − li ⎠ ∂ xˆ j 2βi ∂ xˆ j j =1

1 + βi e12 . 2

(23)

Herein, αi can counteract the effect of the first term at the right side of (23). The performance of error dynamics can compensate for the effect of the second term. Now, taking (22) with (23) into account gives ⎛ i−1  ∂αi−1 xˆ j +1 V˙i ≤ z i ⎝αi − ∂ xˆ j j =1  ⎞ i−1  ∂αi−1 r j 2 T − z S S − σjθj ⎠ 2 j j j ∂θ j 2a j j =1 ⎛ ⎞2 i−1 1 2 ⎝ ∂αi−1 + z l j − li ⎠ 2βi i ∂ xˆ j j =1

1 + z i z i+1 + βi e12 . 2

(24)

In particular, in the last design step, for Vn = (1/2)z n2 , its time derivative is given by

V˙z ≤

i=1

z i (αi + f¯i ) +

i=1

2

βi e12 −

1 ≤ i ≤ n − 1, z 0 = 0 ¯ fn (Z n ) = z n−1 − α˙ n−1 − ln e1 + cz n ϕn2 (θn ). NNs WiT Si (Z i ) is now utilized to approximate the unknown function f¯i such that for given εi > 0 f¯i = WiT Si (Z i ) + δi (Z i ) with δi being an approximation error and satisfying |δi | ≤ εi . As a result, for 1 ≤ i ≤ n − 1, one has

z i f¯i = z i WiT Si (Z i ) + δi (Z i ) 1 2 ∗ T 1 1 1 ≤ z i θi Si Si + ai2 + z i2 + εi2 (27) 2 2 2 2ai2 where θi∗ = ||Wi ||2 . Notice that the ideal weighting vector Wi is unknown. As the square of its norm, θi∗ is thus an unknown constant, and cannot be used for control design. So, in the following, the adaptive technique will be utilized to estimate it. Particularly, we have g0 1 g0 1 2 ε z n f¯n ≤ 2 z n2 θn∗ SnT Sn + an2 + z n2 + 2an 2 2 2g0 n

(28)

with θn∗ = ||Wn ||2 /g0 . In addition, constructing the desired control input signal v as v = −(kn + 0.5 + )z n +

1 z n θn SnT Sn . 2an2

(29)

Then, from (8) and Remark 4, it follows that: z n u(v) = z n gv μ v + z n d(v)   1 T = −z n gv μ (kn + 0.5 + )z n + 2 z n θn Sn Sn 2an +z n d(v) ≤ −gv μ (kn + 0.5 + )z n2 gv μ 1 1 − 2 z n2 θn SnT Sn + g0 z n2 + D2 2an 2 2 g0 ≤ −g0 (kn + 0.5 + )z n2 g0 1 1 − 2 z n2 θn SnT Sn + g0 z n2 + D2 . (30) 2an 2 2 g0 n 

k¯i z i2 −

i=1

i=1



j =1 2 + cz i ϕi (θi ),

V˙z ≤ − n 

xˆ j +1

rj 2 T − z S Sj − σj θj ∂θ j 2a 2j j j j =1 ⎛ ⎞2 i−1  1 ∂αi−1 zi ⎝ l j − li ⎠ + z i−1 + 2βi ∂ xˆ j i−1  ∂αi−1

(25)

Then, a straightforward computing gives n  1

∂ xˆ j

Substituting (13), (27)–(30) into (26) yields

V˙n = z n (u(v) − α˙ n−1 − ln e1 ).

n 

i−1  ∂αi−1 j =1

j =1

j =1

93

cz i2 ϕi2 (θi )

(26)

+

n n  

bi 2 T 1 2 ai + εi2 z i θi Si Si + 2 2 2ai i=1 i=1

1 2 1 2  2 2 D + βi e − cz i ϕi (θi ) 2 2 1 n

n

i=1

i=1

(31)

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where k¯i = ki , bi = 1 for i ≤ n − 1 and k¯n = g0 kn , bn = g0 . Furthermore, one has V˙ = V˙e + V˙z + V˙θ ≤ e T (P Ao + AoT P + P(H + H ) +( H + H )T P + ε0 I )e + c

n 

z i2 ϕi2 (θi )

i=1

−

n 

k¯i z i2 −

i=1

+

1 2 D + 2

= e T (P Ao +

n 

i=1 n 

 1

1 2 T ai2 + εi2 z i θi Si Si + 2 2 2ai i=1 n

 bi 1 2  2 2 βi e1 − cz i ϕi (θi ) − θ˜i θ˙i 2 2ri

i=1 AoT P

n

n

i=1

i=1 T

+ P(H + H ) + (H + H ) P)e

+ e (ε0 I + β) e n n  

1 2 1 2 − ai + εi2 + D ki z i2 + 2 2 i=1 i=1  n  ri 2 T bi + z θi Si Si − θ˙i (32) θ˜ 2ri 2ai2 i i=1  with β = diag[ ni=1 (1/2)βi , 0, . . . , 0]. At the present stage, we can summarize out main result in the following theorem. Theorem: Consider (1) satisfying Assumption 1, and the observer (11). Suppose that the packaged nonlinear function f¯i can be approximated by NNs in the sense that the approximate errors are bounded. If for given constant ε0 , the matrices β, A0 , and all matrices Mα ∈ , there exist definitive positive matrices P and L such that T

P A0 + A0T P + P Mα + MαT P + ε0 I + β < 0

(33)

where A0 is defined by (12) and  is a matrix set and defined as   (34)  = M|M(i j ) = h i j or h i j , 1 ≤ i, j ≤ n 2

and α = 1, 2, . . . , 2n , the control law v = −(kn + 0.5 + )z n +

1 z n θn SnT Sn 2an2

associated with the virtual control signals αi is defined by (13) and the adaptive law ri θ˙i = 2 z i2 SiT Si − σi θi 2ai with ai , ri , ki , , and σi being positive constants, guarantees that the closed-loop signals are SGUUB, and moreover, all the variables of the closed-loop system can converge to a small enough neighborhood around the origin by choosing appropriate design parameters. Proof: By the definition of θ˙i , the following inequality holds:  n n   ri 2 T bi bi σi ˙ ˜θ z i θi Si Si − θi = θ˜i θi . 2ri 2ri 2ai2 i=1

i=1

Notice that

1 1 θ˜i θi = θ˜i θi∗ − θ˜i ≤ − θ˜i2 + θi∗2 . 2 2

(35)

Thus, taking (32) into account with (35), we have V˙ ≤ e T P Ao + AoT P + P(H + H ) + (H + H )T P n n  

bi σi 2 k¯i z i2 − θ˜ + ε0 I + β e − 2ri i i=1 i=1   n 1 2 1 2 bi σi ∗2 2 + D + θ (36) a + εi + 2 2 i 2ri i i=1

where H and H are the time-varying matrices, and they lead to the stability analysis difficult. To overcome this difficulty, we will develop the time-independent linear matrix inequality (LMI) stability conditions for the error dynamic system in the following. For notation simplicity, let Ii j denote the n-order square matrix, its element at the i th row and the j th column is 1 and the others are 0. Thus, according to the definitions of H and H , one has the following formulation: H+H=

n 

(μi j h i j + (1 − μi j )h i j )Ii j

i, j =1

where μi j is a function and satisfies 0 ≤ μi j ≤ 1. As a result P A0 + A0T P + P(H + H ) + (H + H )T P + ε0 I + β < 0 (37) is equivalent to P A0 + A0T P + ε0 I + β +

n 

(μi j h i j + (1 − μi j )h i j )P Ii j

i, j =1

+

n 

(μi j h i j + (1 − μi j )h i j )IiTj P < 0.

(38)

i, j =1

It follows immediately from applying Lemma 3 to (38) repeatedly that (38) holds only if the following is true: P A0 + A0T P + P Mα + MαT P + ε0 I + β < 0

(39)

where Mα ∈ , which is defined by (34). Since each element of M ∈  takes its value as h i j or h i j for 1 ≤ i , 2 j ≤ n. The set  thus includes 2n different matrices. 2 So, one has α = 2n . In addition, when (39) holds, there exists a constant α0 > 0 such that for all α P Ao + AoT P + P Mα + MαT P + ε0 I + β < −α0 I holds, which implies that

e T P Ao + AoT P + P(H + H ) + (H + H )T P + ε0 I + β e α0 λ M (P)e T e ≤ −α0 e T e = − λ M (P) α0 e T Pe ≤− λ M (P)

CHEN et al.: OBSERVER-BASED ADAPTIVE NN CONTROL FOR NONLINEAR SYSTEMS IN NONSTRICT-FEEDBACK FORM

95

where λ M (P) denotes the maximal eigenvalue of matrix P. Now, taking     α0  ¯ a0 = min , 2ki , bi σi ,  i = 1, 2, . . . , n λ M (P)   n  1 bi σi ∗2 1 2 b0 = θi + ai2 + εi2 + D 2 2ri 2 i=1

then (36) can be rewritten as V˙ ≤ −a0 (Ve + Vz + Vθ ) + b0 = −a0 V + b0 . Therefore, the following inequality holds:   b0 −a0 t b0 V ≤ V (0) − + e a0 a0

Fig. 2.

x1 (solid line) and xˆ1 (dashed-dotted line) for example 1.

Fig. 3.

x2 (solid line) and xˆ2 (dashed-dotted line) for example 1.

Fig. 4.

u (solid line) and v (dashed-dotted line) for example 1.

which implies that all the closed-loop signals are bounded. In particular, for given η > 0, by appropriately choosing ai , εi , and σi to be sufficiently small, as well as ri and to be sufficiently large, it is possible to let b0 /a0 ≤ η. The proof is thus completed. IV. S IMULATION E XAMPLE In this section, two examples are used to test the feasibility of the proposed method. Example 1: Consider a simple nonlinear second-order system x˙1 = x 2 + 0.25 sin2 (x 1 ) cos2 (x 2 ) x˙ 2 = u(v) y = x1 . This system is in nonstrict-feedback form because the first subsystem contains the term sin2 (x 1 ) cos2 (x 2 ). As a second-order system, it is also a pure-feedback system. Since (∂ f 1 /∂ x 2 ) = −0.5 sin2 (x 1 ) cos(x 2 ) sin(x 2 ), the inequality, (∂ f 1 /∂ x 2 ) > 0 cannot be assured, the existing control methods for pure-feedback systems cannot be used to control this system. A simple calculation shows that −0.5 ≤ (∂ f 1 /∂ x i ) ≤ 0.5 and (∂ f 2 /∂ x i ) = 0. In this case, α = 4. For given ε0 = 0.01, β = 0.01I , all Mα , solve LMIs (33) and obtain the feasible solutions L = [−18.6508, −6.4185]T , and   0.1514 −0.3565 P= . −0.3565 1.0596 The control parameters are chosen as a1 = a2 = 1, r1 = r2 = 7.5, k1 = k2 = 1, σ1 = σ2 = 0.05, and = 1. The simulation is run at the initial conditions x 1 (0) = 1, x 2 (0) = 1.5, xˆ1 (0) = xˆ2 (0) = 0, and θ1 (0) = θ2 (0) = 0. The bounding of input saturation is Umax = 0.5. According to [42], Gaussian RBF NNs arranged on a regular lattice on R n can uniformly approximate sufficiently smooth functions on closed bounded subsets. Therefore, in the following, the centers and widths are chosen on a regular lattice in the respective compact sets. Concretely, we apply seven nodes for each input dimension of W1T S1 (.) and W2T S2 (.). Therefore, W1T S1 (.) contains 49 nodes with centers spaced evenly in

the interval [−1.5, 1.5] × [−1.5, 1.5] and widths being equal to 2; NN W2T S2 (.) contains 2401 nodes with centers spaced evenly in the interval [−1.5, 1.5] × [−1.5, 1.5] × [−1.5, 1.5] × [−1.5, 1.5] and widths being still equal to 2. The simulation results are displayed by Figs. 2–5. Example 2: Consider the electromechanical system that is from [20] and described by the following differential equations: x˙1 = x 2 N x3 B B − sin x 1 − x 2 + cos x 2 sin x 3 x˙2 = M M M M K u R x˙3 = − x 2 − x 3 L L L y = x1

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 1, JANUARY 2016

Fig. 5.

θˆ1 (solid line) and θˆ2 (dashed-dotted line) for example 1.

Fig. 6.

x1 (solid line) and xˆ1 (dashed-dotted line) for example 2.

with the parameters M = (J/K t ) + (m L 20 /3K t ) + (M0 L 20 /K t ) + (2M0 R02 /5K t ), N = (m L 0 G/2K t ) + (M0 L 0 G/K t ), B = (B0 /K t ) and J = 0.001625, m = 0.506, R0 = 0.023, M0 = 0.434, L 0 = 0.305, B0 = 0.01625, L = 0.025, K t = 0.9, K = 0.9, R = 0.5, and G = 9.8. Since the error between the math model and the real physical system is inevitable, and thus the term (B/M) cos x 2 sin x 3 has been added to the second equations to denote the model errors. For this case, the method proposed in [20] cannot be used to control this system theoretically, since the systems are no longer in strict-feedback form. In this example, nonlinear functions f 1 = 0 and f 3 = 0. And (−N /M) ≤ (∂ f 2 /∂ x 1) ≤ (N/M) and (−R/M) ≤ (∂ f 2 /∂ x i ) ≤ (R/M) for i = 1, 2. In this case, we have α = 8. For given ε0 = 0.01, β = 0.01I , and Mα , we solve LMIs (33) and obtain the feasible solutions L = [−30.8862, −274.3514, 793.1504]T, and ⎡ ⎤ 393.7789 −0.4270 15.1147 P = ⎣ −0.4270 1.0473 0.5371 ⎦. 15.1147 0.5371 1.1654 Chose the control parameters as k1 = k2 = k3 = 2, a1 = a2 = a3 = 0.5, σ1 = σ1 = σ1 = 0.5, r1 = r2 = r3 = 7.5, and = 1, the simulation is carried out with the same initial conditions in [20], i.e., x 1 (0) = −0.2, x 2 (0) = −0.1, x 3 = 0.1, xˆ1 (0) = 0.1, xˆ2 (0) = xˆ 3 (0) = 0, and θi (0) = 0, i = 1, 2, 3. The bounding of input saturation is u M = 0.35. In the simulation, we still apply seven nodes for each input dimension of W1T S1 (.), W2T S2 (.), and W3T S3 (.). The simulation results are shown in Figs. 6–10.

Fig. 7.

x2 (solid line) and xˆ2 (dashed-dotted line) for example 2.

Fig. 8.

x3 (solid line) and xˆ3 (dashed-dotted line) for example 2.

Fig. 9.

u (solid line) and v (dashed-dotted line) for example 2.

Fig. 10.

Solid line: θˆ1 . Dashed-dotted line: θˆ2 . Dotted line: θˆ3 .

Figs. 2, 3, and 6–8 show the state and observed state responses for examples 1 and 2, respectively. Figs. 4 and 9 show the desired control input curves and real control input curves for examples 1 and 2, respectively. Figs. 5 and 6 show

CHEN et al.: OBSERVER-BASED ADAPTIVE NN CONTROL FOR NONLINEAR SYSTEMS IN NONSTRICT-FEEDBACK FORM

the adaptive parameter curves. The simulation results illustrate the effectiveness of the proposed adaptive NN output-feedback control scheme. Under the action of the proposed controllers, all the signals in the closed-loop systems converge to a small neighborhood around the origin though the plant input subjects to an unknown saturation nonlinearity. V. C ONCLUSION In this paper, adaptive NN output-feedback control for nonlinear nonstrict-feedback systems was investigated. Adaptive NN output-feedback control scheme was first proposed for nonlinear nonstrict-feedback systems, which include the strict-feedback systems and some pure-feedback systems as the special cases. Therefore, the proposed adaptive NN control strategy in this paper is more universal than the existing results. Finally, the simulations illustrate the effectiveness of the proposed method. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the reviewers for their valuable comments and helpful suggestions. R EFERENCES [1] B. Chen, X. Liu, K. Liu, and C. Lin, “Direct adaptive fuzzy control of nonlinear strict-feedback systems,” Automatica, vol. 45, no. 6, pp. 1530–1535, 2009. [2] S. Zhou, G. Feng, and C.-B. Feng, “Robust control for a class of uncertain nonlinear systems: Adaptive fuzzy approach based on backstepping,” Fuzzy Sets Syst., vol. 151, no. 1, pp. 1–20, 2005. [3] J. Zhang, P. Shi, and Y. Xia, “Robust adaptive sliding-mode control for fuzzy systems with mismatched uncertainties,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 700–711, Aug. 2010. [4] Y. Xia, H. Yang, P. Shi, and M. Fu, “Constrained infinite-horizon model predictive control for fuzzy-discrete-time systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 2, pp. 429–436, Apr. 2010. [5] W. Chen, L. Jiao, R. Li, and J. Li, “Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 674–685, Aug. 2010. [6] Y.-J. Liu, S.-C. Tong, and W. Wang, “Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems,” Fuzzy Sets Syst., vol. 160, no. 19, pp. 2727–2754, 2009. [7] T.-S. Li, D. Wang, G. Feng, and S.-C. Tong, “A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 4, pp. 915–927, Jun. 2010. [8] Y. G. Niu, J. Lam, D. W. C. Ho, and X. Wang, “Adaptive H∞ control using backstepping and neural networks,” J. Dyn. Syst., Meas., Control, vol. 127, no. 3, pp. 478–485, 2005. [9] Z. Li, X. Cao, and N. Ding, “Adaptive fuzzy control for synchronization of nonlinear teleoperators with stochastic time-varying communication delays,” IEEE Trans. Fuzzy Syst., vol. 19, no. 4, pp. 745–757, Aug. 2011. [10] Z. Liu, G. Lai, Y. Zhang, X. Chen, and C. L. P. Chen, “Adaptive neural control for a class of nonlinear time-varying delay systems with unknown hysteresis,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 12, pp. 2129–2140, Dec. 2014. [11] G. Chowdhary, H. A. Kingravi, J. P. How, and P. A. Vela, “Bayesian nonparametric adaptive control using Gaussian processes,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 3, pp. 537–550, Mar. 2015. [12] T.-S. Li, S.-C. Tong, and G. Feng, “A novel robust adaptive-fuzzytracking control for a class of nonlinear multi-input/multi-output systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 1, pp. 150–160, Feb. 2010. [13] 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. [14] A. Theodorakopoulos and G. A. Rovithakis, “A simplified adaptive neural network prescribed performance controller for uncertain MIMO feedback linearizable systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 3, pp. 589–600, Mar. 2015.

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Bing Chen received the B.A. degree in mathematics from Liaoning University, Jinzhou, China, the M.A. degree in mathematics from the Harbin Institute of Technology, Harbin, China, and the Ph.D. degree in electrical engineering from Northeastern University, Shenyang, China, in 1982, 1991, and 1998, respectively. He is currently a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. His current research interests include nonlinear control systems, robust control, and adaptive fuzzy control.

Huaguang Zhang (SM’04) received the B.S. and M.S. degrees in control engineering from the Northeast Dianli University of China, Jilin, China, in 1982 and 1985, respectively, and the Ph.D. degree in thermal power engineering and automation from Southeast University, Nanjing, China, in 1991. He joined the Department of Automatic Control, Northeastern University, Shenyang, China, in 1992, as a Post-Doctoral Fellow, for two years, where he has been a Professor and the Head of the School of Information Science and Engineering, Institute of Electric Automation, since 1994. He has authored or co-authored over 200 journal and conference papers, four monographs, and co-invented 20 patents. His current research interests include fuzzy control, stochastic system control, neural networks based control, nonlinear control, and their applications. Prof. Zhang is the Chair of the Adaptive Dynamic Programming and Reinforcement Learning Technical Committee on the IEEE Computational Intelligence Society. He is an Associate Editor of the Automatica, the IEEE T RANSACTIONS ON C YBERNETICS , and the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS , respectively. He was a recipient of the Outstanding Youth Science Foundation Award from the National Natural Science Foundation Committee of China in 2003. He was named the Cheung Kong Scholar from the Ministry of Education, China, in 2005. Chong Lin (SM’06) received the B.Sc. and M.Sc. degrees in applied mathematics from Northeastern University, Shenyang, China, in 1989 and 1992, respectively, and the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 1999. He was a Research Associate with the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, in 1999. He was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, from 2000 to 2006. He has been a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, China, since 2006. He has authored over 60 research papers and co-authored two monographs. His current research interests include systems analysis and control, robust control, and fuzzy control.