Robust Adaptive Neural Backstepping Control for a Class of Nonlinear Systems with Dynamic Uncertainties

and Applied Analysis 3 For all t ≥ 0, the solutions are defined. Without loss of generality, this paper takes μ(⋅) as μ(s) = sμ 0 (s), where μ(⋅) is a nonnegative smooth function. Therefore, the dynamical r defined by (5) becomes ̇ r = −cr + x 2 1 μ 0 ( 󵄨󵄨󵄨󵄨 x 2 1 󵄨󵄨󵄨󵄨 ) + d 0 , r (0) = r0, (7) where μ 0 is a nonnegative smooth function. Throughout this paper, RBF neural networks are applied to model the unknown continuous nonlinear functions. In [60], it has been indicated that, with enough node number l, the RBF neural networks φξ(X) can model the continuous function f(X) within a compact set Ω X ⊂ R to arbitrary accuracy ε > 0 as f (X) = φ ∗T ξ (X) + δ (X) , ∀X ∈ ΩX ∈ R q , (8) in which φ denotes the ideal weight vector and is specified as


Introduction
In the past decades, much attention has been paid on the control design of complex nonlinear systems [1][2][3][4][5][6][7][8][9][10][11].Many remarkable control approaches in this area have been developed, including adaptive backstepping technique [1][2][3], fault tolerant control [12][13][14][15][16][17], and and fuzzy control [18][19][20][21][22][23][24][25][26][27][28][29].In particular, adaptive backstepping approach has played an important role in the control of strict-feedback nonlinear systems.Generally, adaptive backstepping provides a systematic control approach to solve the tracking or regulation control problems of uncertain nonlinear systems, in which the classic adaptive control is applied to deal with the unknown parameter and backstepping technique is used to construct controller.The main feature of adaptive backstepping control is that it can handle the control problems of nonlinear systems without the requirement of matching condition.Adaptive backstepping technique was provided in [1] to obtain global stability and asymptotic tracking performance for parametric strict-feedback systems with overparameterization, and the overparameterization was overcome by applying the tuning functions in [2].Then, a backstepping-based design was extensively utilized to control different types of nonlinear systems [30][31][32][33][34][35].All the above control methods, however, assume that the nonlinear functions of the control systems are either known or bounded by known functions multiplying uncertain parameters.This restriction makes the aforementioned methods inapplicable to the control of the systems with unknown continuous nonlinear functions.
On the other hand, approximation-based adaptive neural (or fuzzy) backstepping control has received increasing attention in recent years.In general, approximation-based adaptive backstepping technique is an effective control approach for handling the control problem of highly uncertain complex nonlinear strict-feedback systems, in which neural networks or fuzzy systems are utilized to model the unknown nonlinear functions.So far, there exist some elegant results; see, for example, [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] and the references therein.By applying adaptive neural control together with backstepping, in [36][37][38][39][40][41][42][43], many control approaches are developed for single-input and single-output (SISO) nonlinear systems or multi-input and multioutput (MIMO) nonlinear systems.Alternatively, several fuzzy adaptive control strategies [19,[44][45][46][47][48][49][50][51][52][53][54][55] were developed to deal with the control problem of uncertain nonlinear systems with strict-feedback form.However, the above adaptive neural or fuzzy backstepping control approaches required the controlled strict-feedback nonlinear systems to be free of the unmodeled dynamics and dynamic disturbances.As stated in [56,57], the unmodeled dynamics and dynamic disturbances often appear in practical systems [58,59] due to the measurement noise, modeling errors, external disturbances, modeling simplifications, or changes with time variations, and they are also the resources of the instability of the considered systems.Therefore, some researchers have concentrated on the problem of control design for nonlinear systems with unmodeled dynamics and dynamic disturbances.In [56,57], the problem of adaptive backstepping control was investigated for a class of nonlinear systems with dynamics uncertainties, in which the nonlinear functions were assumed to be linear combinations of the known functions with unknown parameters.Furthermore, by using the approximation properties of fuzzy logic systems, Tong et al. [58,59] developed several fuzzy adaptive control approaches for nonlinear systems in strict-feedback form, where the number of adaptation laws depends on the number of fuzzy base functions.The more fuzzy rules are applied to improve approximation accuracy, the more adaptive parameters will be needed, and, in this way, the online learning time may be very large.
Inspired by previous works, this paper focuses on the problem of adaptive neural control for nonlinear strictfeedback systems with unmodeled dynamics and dynamic disturbances.During the controller design, a dynamic signal is introduced to handle the unmodeled dynamics and RBF neural networks are used to approximate the unknown nonlinearities, and then an adaptive neural control scheme is systematically derived via backstepping.The proposed controller guarantees that all the signals in the closedloop systems are semiglobally uniformly ultimately bounded (SGUUB) in the sense of mean square.Compared with the control approaches [58,59], the main contributions of this paper are summarized as follows: (1) the strict limitation to the dynamic disturbances is relaxed, which can refer to Remark 3; (2) by estimating the norm of the weight vector of neural networks basis functions, the number of adaptive parameters is not more than the order of the considered nonlinear system.As a result, the burdensome computation is significantly alleviated, which makes our control design more suitable for the practical applications.
The remainder of the paper is organized as follows.Section 2 begins with the problem formulation and some preliminaries.A backstepping-based adaptive control scheme is design in Section 3. In Section 4, a numerical example is given.Finally, the conclusion of this paper is shown in Section 5.

Problem Formulation and Preliminaries
In this paper, we consider a class of nonlinear strict-feedback systems described by ż =  (, ) , where  = [ Remark 1.It is worth noting that many practical systems such as the electromechanical system [59] transformable into (1) have been investigated extensively during the last decades from both theoretical and practical viewpoints; see, for example, [56][57][58][59].
In order to facilitate the control design later, the below assumptions are imposed on the system (1).( Remark 3.This assumption is similar to the one in [58,59] in which  1 (⋅) and  2 (⋅) are known.Assumption 2, however, does not require them to be known.Therefore, Assumption 2 relaxes the restriction in the existing results.
Assumption 5 (see [50]).For 1 ≤  ≤ , the signs of   (  ) are known, and there exists unknown positive constant  such that Remark 6. Equation (4) implies that   (  ) are either strictly positive or negative.Without loss of generality, it is supposed that 0 <  ≤   (  ).In addition, since  is not required in the designed controller, its true value is not required to be known.
Throughout this paper, RBF neural networks are applied to model the unknown continuous nonlinear functions.In [60], it has been indicated that, with enough node number , the RBF neural networks  *  () can model the continuous function () within a compact set Ω  ⊂   to arbitrary accuracy  > 0 as in which  * denotes the ideal weight vector and is specified as () depicts the approximation error satisfying |()| ≤ ,  * = [ 1 ,  2 , . . .,   ]  ∈   is the weight vector, and () = [ 1 (),  2 (), . . .,   ()]  is the basis function vector with  being the number of the neural networks nodes and  > 1.
The basis function   () is taken as the Gaussian function in the below form: where   = [ 1 ,  2 , . . .,   ]  and   are the center of the receptive field and the width of the Gaussian function, respectively.

Adaptive Neural Control Design
In this section, the adaptive backstepping control design for system (1) is proposed.As usual, in the backstepping approach, the following coordinate transformation is made: where  0 = 0,   is the virtual control signal and will be constructed at Step , and the actual controller  will be designed at Step .Now, we begin the controller design procedure.
Step 1.Consider the following subsystem: Based on  1 =  1 , then choose Lyapunov functions as where ( 1 ) =  2 1  0 ( 2 1 ),  0 and  1 are positive design parameters, and θ1 =  1 − θ1 is the parameter error with θ1 being the estimation of  1 which is defined later.
Next, the following result can be obtained by substituting (20) into (18): By using one has with  1 =  −1 ‖ 1 ‖ 2 being an unknown parameter.Construct the virtual control signal  1 as where  1 and  1 are positive design constants.By taking Assumption 5 into account, one has Further, by substituting ( 25) into (23), we obtain Next, we choose the adaptive law in the following form: where  1 and  1 are design parameters.By using (27), we can rewrite (26) as Noting then the following inequality holds: where  1 =  Step 2. Based on  2 =  2 −  1 , then the time derivative of  2 is given by where Construct the Lyapunov function The derivative of  2 is By Assumption 2, we have Substituting ( 35) into (33) gives where  3 =  3 −  2 and the function f2 ( 2 ) is specified as with where  23 denotes approximation error.Then, substituting (38) into (36), one has By using it can be easily verified that with  2 =  −1 ‖ 2 ‖ 2 being an unknown constant.Furthermore, the virtual control  2 is constructed as where  2 > 0 and  2 are the design constants.
Then, the following result can be easily obtained: By applying ( 43), ( 41) can be rewritten as Define the adaptive law as where  2 ,  2 , and  2 are design parameters.
Combining (44) with ( 45) produces where   =  Step i (3 ≤  ≤ −1).According to   =   − −1 , the dynamics of   is where Consider the Lyapunov function   as By using the derivations similar to those used in the former steps, we can obtain Similar to (34) Furthermore, the following inequalities can be easily verified by repeating the same arguments as (35): where 2 , noting that   () ≥ 0 for all  ≥ 0. Substituting ( 51) and ( 52) into (49) results in where the function f (  ) is defined by where and Currently, a neural network      (  ) is utilized to model f (  ) such that, for a given  3 > 0, f (  ) can be expressed as Further, similar to (40), we can obtain where   =  −1 ‖  ‖ 2 is an unknown constant.Now, construct the virtual control signal   as with   > 0 and   being design constants.Then, by substituting ( 55)-( 57) into (53), choosing the adaptive law with   ,   , and   being the design parameters, and then following the same line as the procedures from ( 43) to (46), we have where   =  Step n.In this step, the actual controller  is designed.According to   =   −  −1 , then we have where Similarly, choose the following Lyapunov function as From ( 53) and ( 54), we have Using the same estimation methods as ( 42)-( 44), we have where φ1 (  , θ−1 , ),   1 , φ2 (  , θ−1 , ),   2 , and   () are defined in (51) or (52) with  = .By substituting (63) into (62), one has where   =  −1 ‖  ‖ 2 denotes an unknown constant and   is a design constant.Subsequently, by combining (64) together with (66), the inequality below holds: At the present stage, construct the real controller  and adaptive law θ  in the following forms: where   ,   ,   , and   are design constants.Then, repeating the similar procedures as ( 43)-( 46), we can obtain where   =  Proof.To give the stability analysis for the closed-loop system, consider the Lyapunov function in the form  =   , and define Furthermore, we can rewrite (70) as Next, from (72), the following inequality can be easily verified:

Conclusion
In this research, a backstepping-based adaptive neural control scheme has been developed for strict-feedback nonlinear systems with unmodeled dynamics and dynamic disturbances.The proposed adaptive neural controller guarantees that all the signals of the resulting closed-loop system remain semiglobally uniformly ultimately bounded in the sense of mean square.Simulation results have been provided to illustrate the effectiveness of the proposed control scheme.It should be pointed out that the work in this paper does not consider the problem of input nonlinearity and time-delay.Then, they may occur in practical engineering.So, how to control a nonlinear system with input nonlinearity and time-delay is our future research direction.

Theorem 8 .
(58) the main result of this research is summarized as follows.Consider the system (1) consisting of Assumptions 2-5, the control input (68), and the adaptive laws(58)and (69).Assume that the packaged unknown functions f (  ) ( = 1, 2, . . ., ) could be modeled by neural networks      (  ) with the bounded approximation errors.Then, for bounded initial values with θ (0) ≥ 0, all the signals in the closed-loop system are semiglobally boundedness in mean square.