Direct Neural Network Adaptive Tracking Control for Uncertain Non-Strict Feedback Systems With Nonsymmetric Dead-Zone

In this paper, combined with the approximation of neural network, a novel direct adaptive alleviating tracking control algorithm is presented for a class of non-strict feedback uncertain nonlinear systems. Here, both nonlinear uncertainties and nonsymmetric dead-zone inputs are considered. First, according to some coordinate transforms and variable separation methods, the non-strict feedback form is converted into the normal form. Second, the relationship of state vector and error functions are established, and the inputs of dead-zone are compensated with adaptive approaches. This novel direct scheme assumes that the approximation error and optimal approximation norms of NN are to be bounded by unknown constants and can alleviate the number of online adjusted parameters so as to improve the robust control performance of the systems. At last, under Lyapunov theorem analysis, the uniformly ultimately boundness of all the signals in the closed-loop systems can be guaranteed and the dead-zone inputs can be compensated, the effectiveness of this algorithm is well demonstrated by simulation results.


I. INTRODUCTION
During the last decades, stability theory for uncertain systems with nonlinearities were discussed constantly [1]- [11], diverse adaptive approximation-based fuzzy or NN control schemes have been designed for uncertain systems with nonlinearities [5], [9]- [19]. Note that many of the these mentioned approximation-based fuzzy [7]- [10], [14], [17] or NN [3], [5], [6], [11]- [13] approaches were based on strict-feedback uncertain nonlinear systems [5], [12], [13] or pure-feedback nonlinear systems [8], [14]- [16], rather than uncertain non-strict feedback systems. In fact, the functions of non-strict feedback uncertain systems contain all the state variables of the system, that is to say, the above two structures strict feedback and pure-feedback forms are included in the The associate editor coordinating the review of this manuscript and approving it for publication was Choon Ki Ahn . non-strict feedback ones. So, the non-strict ones are more challenge and general for practical control systems.
Recently, many adaptive researches and control strategies based on backstepping techniques and approximation of fuzzy or NN have been proposed for non-strict feedback uncertain nonlinear systems [17]- [23]. Combing with input saturation and output constraint, [17] discussed fuzzy control for non-strict feedback systems. [18] extended NN scheme to the non-strict feedback with backlashlike hysteresis uncertain systems. Considering a class of discrete-time systems, [19]established states NN reinforcement learning adaptive control approach. Based on finite-time adaptive control approaches, [20], [21] analyzed fuzzy states feedback control and output feedback dynamic surface control for non-strict feedback respectively. [22] extended NN adaptive command filter control to stochastic time-delayed systems with unknown input saturation. Neural control methods for VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ full-state constraints and unmodeled dynamics in non-strict feedback uncertain systems are designed [23]. But, many of these papers did not consider the unknown dead-zone inputs, especially for the more complex uncertain nonlinear nonsymmetric dead-zones. Unknown dead-zone input as one of the nonlinearities often occurs in the process of the practical engineering, which is a source of instability and limitation of performance of systems. Recently, the investigations of input dead-zone has attracted a great deal of attention [24]- [30]. Decentralized control for large-scale systems with actuator faults and tracking control for switched stochastic actuator deadzone systems were discussed in [24], [25]. [26] studied the non-backstepping VUFC algorithm for pure-feedback form. Based on switched nonlinear systems, [27] [28] extended time-varying tan-type barrier Lyapunov function adaptive fuzzy control and adaptive neural quantized control for states constrained systems and MIMO asymmetric actuator systems. Adaptive neural control [29] and fuzzy decentralized control [30] were proposed for unknown control directions systems and strong interconnected nonlinear systems in unmodeled dynamics. Based on robust optimal control method, [36] discussed the event-triggered physically interconnected mobile Euler-Lagrange systems.
Although many researchers have extensively studied for non-strict feedback for nonlinear systems or for systems with unknown dead-zones, to the authors' best knowledge, very few investigators concentrated on non-strict feedback systems with uncertain nonlinearities and non-symmetric unknown nonlinear dead-zone inputs, and many adaptive parameters need to be adjusted in the recursive process of these backstepping or approximation-based approaches, due to updating parameters of NN optimal weight vector or the optimal approximation vector of fuzzy logic systems, which would affect the systems control performance and the online computation burden. As far as we know, for non-strict feedback nonlinear systems, no reports on novel alleviating computation NN control approach in the literature can be found. All of these motivate this paper.
Motivated by the above considerations, aiming at alleviating the computation, this paper consider a novel adaptive NN tracking control for a class of non-strict feedback systems with nonsymmetric dead-zone inputs. Neural networks (NN) are utilized to approximate the unknown nonlinearities and nonlinear functions, and a robust NN state-feedback tracking control method is developed in the framework of backstepping design technique. This approach can not only compensate the effect of the non-symmetric dead-zone inputs but also improve the robust performance of the system by updating estimations of unknown bounds. Compared with the related existing literature, the main advantages and contributions of this paper proposed are listed below.
1) This established control scheme can compensate nonsymmetric dead-zone inputs, uncertainties and solve the problems of included non-affine structure states non-strict feedback simultaneously. Although the previous results in [17]- [23] also studied the same control design problem for non-strict feedback nonlinear systems, they do not consider uncertain non-symmetric dead-zones and have computing burden problem.
2) Based on NN novel alleviating computation control approach, at each design step, F-norm parameters and unknown constants are used to approximate the bound of optimal weight vector of NN and the approximation error. Thus, this approach needs to adjust only one parameter rather than the elements of the optimal approximation vectors of NN. As a result, compared with the traditional back-steppingbased and approximation-based scheme for nonlinear systems [4]- [14], [17], [23], [27], [29], [37], [38], the approach needs to adjust fewer parameters and the computational burden is significantly alleviated.
The rest of this paper is organized as follows. Preliminaries and problem formulation and are explained in Sect. 2. A novel adaptive NN tracking control design procedure is presented in Sect. 3. Simulation is demonstrated in Sect. 4 to illustrate the availability of the approach, Sect. 5 gives the conclusion.

II. PROBLEM STATEMENTS AND PRELIMINARIES A. PRELIMINARIES FORMULATION AND SYSTEM DESCRIPTIONS
In this paper, we focus on a class of uncertain nonlinear timevarying non-strict feedback systems with unknown nonlinearities and non-symmetrical dead-zone inputs as follows: where the non-symmetrical dead-zone with input v(t) and output u(t) as shown in Fig. 1, and the dynamic model of unknown non-strict feedback dead-zone nonlinear systems [26] can be described as: (2) 220796 VOLUME 8, 2020 ∈ R n , and y ∈ R are the state vector and output of the systems respectively. u(t) ∈ R is the input of the system (output of the dead-zones ); v(t) ∈ R is the input to dead-zone. In this paper, f i (·), i = 1, 2, . . . , n and g i (·), i = 1, 2, . . . , n are unknown smooth nonlinear functions with f i (0) = 0, g i (0) = 0; i (·), i = 1, 2, . . . , n are smooth uncertain disturbance. m r (·), m l (·) for dead-zone are unknown nonlinear smooth functions; b r , b l represent unknown right and left slopes of dead-zone and dead-zone breakpoint parameters respectively.
The control objective is to design robust adaptive NN controllers v(t) for the non-strict feedback systems (1), such that the following can be observed: 1) The system output y(t) = x 1 can track the desired trajectory reference signal y d (t) very small; 2) All the signals in the closed-loop systems are uniformly ultimately bounded. Where y d (t) and its kth order derivative y (k) d (t) (k = 1, 2, . . . , n) are assumed to be bounded and continuous.
Similar to [32], [32], to facilitate control system design, we need the following Assumptions for the dead-zone of the control problem investigated in this paper.
Asusmption 1 [26], [28]: The dead-zone outputs u(t) is assumed to be not available and the parameters b r and b l are assumed to be unknown constants, but their signs are known, i.e., b r > 0 and b l < 0.
Remark 1: As stated in [25], [27], [32], [33], this non-strict feedback nonlinear model with unknown dead-zone input is a typical model for a hydraulic servo valve or a servo motor in many practical industrial mechanical processes. However, many results in these papers were based on traditional backstepping technique as well as the approximation features of FlSs or NN [17], [23], as we known that in the recursive process of these approximation and backstepping-based approaches, as the order increased, the design procedure can cause 'explosion of complexity' [25], [27], [33], many adaptive parameters were needed to be adjusted [29]- [33] even together with dynamic surface control (DSC) method [12], therefore, the online computation burden is rather heavy, especially in dealing with MIMO or non-strict feedback nonlinear systems [17], [23]. Different from these results [29]- [33], or the optimal control method to compensate the dead-zone [36], in this paper, we will explore a direct novel alleviating computation NN control method for nonlinear non-strict feedback systems.
Asusmption 2 [26]: Assume that the dead-zones' left and right growth functions m r (·), m l (·) are smooth, and there exist unknown positive constants k l0 , k l1 , k r0 , k r1 , such that In general, for convenience, m r (v(t)) and m l (v(t)) in above Eqs. are assumed to be true for v(t) ∈ (−∞, m l ] and for v(t) ∈ [m r , +∞) respectively. According to the differential mean value theorem, there exist ξ l ∈ (−∞, b l ] and ξ r ∈ [b r , +∞) such that Now define vectors (t) and (t) as follows: and where Based on Assumption 2, the dead-zone model (2) can be redefined as follows: d(v) can be calculated from Assumption 2 and above equations: where Asusmption 3: Assume that the signs of g i (x i ), (i = 1, 2, . . . , n) are known, and there exist positive parameters g i0 and g i1 , satisfying Remark 2: In this paper, dead-zone output u(t) is assumed to be not available, parameters b l and b r are assumed to be unknown but with b r > 0 and b l < 0 [33]- [35]. In addition, according to Assumption 2, we conclude that |d(v)| p * , and p * is an unknown positive constant and can be chosen as p [35].

B. RADIAL BASIS FUNCTION NEURAL NETWORK(RBF NN)
In this paper, we will exploit RBF neural networks to approximate the unknown nonlinearities for system (1).
Such as, an unknown smooth nonlinear function ψ(Z ) : R → R will be approximated on a compact set by the following RBF neural network . . , m are constant vectors called the center of the basis function, and η > 0 is a real number called the width of basis function.
As pointed out in [5] and [6], according to the approximation property of the RBF network, for a continuous realvalued function ψ(Z ) : → R, is a compact, and any ς H > 0, by appropriately choosing i ∈ and η, i = 1, . . . , m, for some sufficiently large integer m, there exists an ideal weight vector W * ∈ R m such that the RBF network W T ξ (Z ) can approximate the given function ψ(Z )with the approximation error bounded by ς H .
Remark 4: In this paper, based on F-norm approximation of NN, the proposed direct novel alleviating NN tracking control could algorithm guarantee that the adaptive adjusted parameters here are only one no matter how many states in the design procedure. Thus, this new approach can alleviate the online computation burden and improve the robust control performance.

III. ADAPTIVE ROBUST RBF NN CONTROL DESIGN AND PERFORMANCE ANALYSIS
Different from the similar backstepping-based results in feedback form with unknown dead-zone inputs in [1]- [6], [12]- [16], [29]- [34], [38], in this section, we will discuss a novel alleviating computation adaptive NN approximationbased tracking control approach in details for the nonlinear non-strict feedback plant in (1). The concrete design procedure contains n steps. First, from step 1 to step n − 1, virtual controllers α i and adaptive lawsθ i ,δ i , (i = 1, 2, . . . , n − 1) will be constructed, in step n, actual controller v(t) will be designed to ensure that the whole system is stable and the adaptive lawsθ i ,δ i will be given in the following design procedure.

A. ADAPTIVE NN DESIGNING PERFORMANCE
The coordinate transformation is given as follows: where α i−1 (i = 1, 2, . . . , n) are virtual controllers, which will be determined in i − 1th steps. To make the system achieve the desired performance, the system (1) is considered to be a series of subsystems. Different from designing a fractional order controller, here, based on backstepping design technique, NN approximation and the alleviating algorithm, we will give the detailed feasible virtual control signals controller, NN adaptive laws and actual controller design procedure in the following steps. The first feasible virtual control signal α 1 and adaptation lawsθ 1 ,δ 1 are considered as follows: where parameters c 1 > 0, τ 1 > 0 and γ 1 > 0 are positive design constants to be designed.θ 1 ,δ 1 are adaptive adjusted parameters to be designed later.
The ith feasible virtual control signal α i and adaptation lawsθ i ,δ i are considered as follows: where parameters c i > 0, τ > 0 are positive design constants to be designed later.θ i ,δ i are adaptive adjusted parameters to be designed later.
Finally, the independent actual controller v(t) and the nth adaptive lawsθ n ,δ n are designed as follows: where parameters c n > 0, τ > 0 are positive constants to be designed later.θ n ,δ n are adaptive adjusted parameters to be designed later.
The following four lemmas will be used for control design in this Section.
Remark 4: Lemma 4 gives the relationship between x and error signals z i , (i = 1, 2, . . . , n), together with (5), plays an important role in this paper, due to the nonlinear function f i (x) contains the whole state variables in the ith differentiate equation, which cannot be estimated by RBF NN directly. Then, it provide a variable separation approach to decompose the function f i (x) into a sum bounded functions with respect to z i , (i = 1, 2, , . . . , n).
The main results are presented by the following theorem. Theorem 1: Consider the closed-loop system with unknown dead-zone input of the plant (1) and (2), the virtual controllers α 1 in (8),α i in (11) and adaptive lawsθ 1 in (9), δ 1 in (10),θ i in (12),δ i in (13),θ n in (15),δ n in (16), and the actual controller v(t) in (14), under Assumptions 1-4. Suppose that for i = 1, 2, . . . , n, the unknown functions H i (Z i ) can be approximated by RBF NN system W T i ξ (Z i ) in the sense that the approximation error ς i is bounded, then based on the bounded initial conditions, according to the Lyapunov stable analysis methods.
1) It can guarantee that all the signals in the closed-loop system are ultimately uniformly bounded(UUB).
2) The output y = x 1 can track the reference signals y d and make sure that the tracking error convergence to a small neighborhood of zero.
Proof: There will contain n steps. Step1: Consider the first part in plant (1)ẋ 1 = g 1 (x 1 )x 2 + f 1 (x) + 1 (t). Define the first tracking error variable z 1 = x 1 − y d , and along its trajectory, we haveż 1 Define the first smooth Lyapunov function as follows: will be designed in the following analysis. The time derivative of V 1 iṡ According to Assumption 4 and Lemma 1-4, we conclude that, whereφ 1 (|z j ψ j |) = (n + 1)|ψ j |h 1 ((n + 1)z j ψ j ) VOLUME 8, 2020 And together with Lemma 1, we conclude another inequality.
Define the ith smooth Lyapunov function as follows: i will be designed in the following analysis. where The time derivative of V i at t is.

Based on Lemma 1, we have
, and ε i , σ are positive constants to be designed.
Substitute these inequalities (33), (34) intoV n (32), we have: Now, we choose the whole Lyapunov function for the plant (1), V = n i=1 V i , the derivative of V is concluded based on the above analysis.
where c(n, k) = (n − k + 1)/2 Substituting this equality in to aboveV , we get, To facilitate the adaptive controller design, we will use RBF neural networks to approximate the nonlinearities, now define H 1 (·), H i (·), H n (·) as follows: For H i , (i = 2, 3, . . . , n), we define, Then, substituting H i (Z i ), (i = 1, 2, . . . , n) intoV in (37), we obtain: According to the definition of H i (Z i ) and the Lemma1-4 and Assumption 1-4, it can conclude H i (Z i ) are also smooth functions, then, based on the universe approximation lemma, we can use RBF NN to approximate the unknown smooth function H i (Z i ) on the compact space 1 , and H i (Z i ) can be rewritten as where Z i is the input of the NN system, W * T i and ς i denote the ideal optimal approximation parameter vector and the NN approximator error, respectively. For simplification, we define Throughout this paper, in order to alleviate online approximation parameters, we assume the following Assumption: Assumption 5: Based on the definition of θ i , on the compact i , we assume that the optimal approximation parameter vector W * T i and the NN approximator errors ς i , satisfy: where i = 1, 2, . . . , n, parameters θ i 0 and δ i 0 are unknown constants. Z i , W * i and ς i will be defined later. θ i 0,δ i 0 will be used to denote estimations of the θ i and δ i respectively. Throughout this paper,( ·) =( ·) − (·).
Remark 5: There are a lot of significant results regarding adaptive fuzzy or NN control or FNN control algorithms for nonlinear systems with unknown dead-zones. However, many of these approximation control methods go through updating the estimations of each optimal parameter of FLSs [7]- [11], [20]- [23] NN, FNN directly, resulting the heavy online computation burden due to the rules of fuzzy, the hidden nodes of NN, or FNN are rather large generally. In this paper, Assumption 5 relaxes the conditions that the approximation errors or external disturbance are bounded with only unknown constants rather than known constants or satisfying square integrable condition. Only estimationsθ i ,δ i , (i = 1, 2, . . . , n, of parameters θ i , δ i need to adaptively adjusted. Thus, this novel proposed approach reduces the adjusted parameters and alleviate the on-line computation burden. According to approximation functions, H i , (i = 1, 2, . . . , n), virtute controllers α 1 , . . . , α n−1 , and actual controller v(t), and the adaptive lawsθ i ,δ i , (i = 1, 2, . . . , n) back intoV . Based on Young's inequalities, we obtain the following inequalities: For i = 1, 2, . . . , n, based on the Lemma 4, we could conclude the following inequalities hold: Similarly, based on the adaptation laws (24), (25) and Young's inequality, we have By substituting inequality (43), (44) back into (40), we acquire, If we choose the appropriate adjusted parameters and constants τ (1) i , τ (2) i , σ i , ε i , ρ (1) i , ρ (2) i , γ (1) i , γ (2) i , p * , d * , θ i , δ i and based on the Assumptions 1-4, Lemmas 1-4 and the RBF NN approximations, together with the virtual and actual controllers, we will have the following inequalities.
where µ = min i , 2ρ (2) i } and Then, we obtainV Multiplying both sides of the above Eq. by e µt and it can be rewritten as Then, integrating the above equation over [0, t], we can obtain If we note that 0 < e −µt < 1 and ( /µ ) e −µt > 0, then, we can know the above Eq.holds as and we can conclude that Therefore, it can be shown that all the signals z i ,θ i ,δ i (i = 1, 2, . . . , n) in the closed-loop systems (1) are bounded. There exists T > 0, for T > √ 2µ/ , satisfying |z 1 | T for all t T , the tracking error z 1 = x 1 − y d converges to a neighborhood of zero. The proof is completed.
Remark 6: Compared with many approximation control approaches, which involve updating the estimations of each optimal parameter of FLSs NN, and FNN directly [2]- [4], [6]- [11], [19], [22], [29]- [31], due to the hidden nodes of NN, or FNN and the rules of fuzzy are rather large generally, which result in the heavy online computation burden. Based on Assumption 5, at each design procedure for each system in this paper, fewer parameters need to be adjusted, we only need to approximate the unknown constant for the norm of the optimal parameter. So, this new approach can improve the robust control performance and alleviate the online computation burden.

IV. SIMULATION EXAMPLE
In this section, based on a practical one-link robot simulation system and its figure model can be seen in [38], the effectiveness of the presented control technique will be illustrated. The dynamics of one-link manipulator with the inclusion of motor [20], [38] can be described by the following equations: where τ d is the torque disturbance, τ represents the torque produced by the electrical system [20], [38], q is the link position,q is velocity, andq is acceleration. D = 100kg/m 2 is the mechanical inertia. u is the control input used to represent the electromechanical torque. B = 1Nms/ras is the coefficient of viscous friction at the joint,K m = 2NM /A is the back-emf coefficient, H = 0.1F is the armature resistance, N = 10 is a positive constant related to the mass of the load and the coefficient of gravity [38], and M = 20 H is the armature inductance [38].
When we introduce the variable change x 1 = q, x 2 =q, and x 3 = τ , and assume that the system exist unknown disturbance and unknown functions, x = [x 1 , x 2 , x 3 ] T is the state of the system, and y = x 1 is system output, u is the input of the system and the output of the dead-zone. Then, above one-link system can be re-expressed as       ẋ where g 1 (x 1 ) = 1+0.6x 2 1 , f 1 (x) = ((B/D)x 1 +x 2 )x 3 , 1 (t) = exp (−(B/D)(x 1 + x 2 )), g 2 (x 2 ) = (N /D + cos(x 1 x 2 ))x 2 , 3 , and 3 (t) = 1 K m sin(x 3 ). Then, we obtain the following third-order uncertain non-strict feedback nonlinear system with unknown dead-zone input: Choose the initial values x 1 (0) = y(0) = x 2 (0) = 0.5, x 3 (0) = 0.7. The unsymmetrical dead-zone inputs satisfies the dead-zone break points are chosen as: The objective of simulation is to apply the proposed novel adaptive NN tracking control approach for this three-order system, satisfy 1) the whole signals in this closed-loop system are bounded, 2) the output y = x 1 can track the reference signal y d = 0.25 sin(t) very well.
Based on the novel adaptive robust NN tracking control approach in Sec3, the designed adaptive NN virtual controller α i adaptive laws θ i , δ i and actual controller v(t) are chosen as follows: where ], (j = 1, 2, . . . , 9).
From the above simulation results, it can be clearly shown that the proposed control method can guarantee that all the signals in the closed systems are UUB, the proposed controller design method is effective. Compared with related results [20], [22], [23], [38], we need only one adaptive law to compensate the unknown dead-zones in non-feedback form. This method can reduce the online computing burden and simplify the design procedure considerably.

V. CONCLUSION
In this paper, a novel NN alleviating tracking control approach has been proposed for a class of uncertain non-strict feedback systems with both asymmetrical deadzones inputs and unknown nonlinear functions. Compared with the existing results, we consider not only asymmetrical dead-zones, but also non-strict feedback structure. This presented scheme adopts variable separation technique and adaptive method to cope with the non-strict feedback structure and the unknown dead-zones, the unknown functions have been approximated by NN. By using two unknown parameters as the approximation error and the bound of the norm of the optimal approximation vectors of the NN, the number of adjusted parameters is alleviated. Furthermore, based on Lyapunov theorem analysis, it has been shown that all the signals in the closed-loop systems are UUB and the tracking error is controlled into a small compact set. Finally, simulation results illustrate the feasibility and effectiveness of this approach. In the future, we could explore these method to more complex systems, such as switched or stochastic nonlinear non-strict feedback system.