2. School of Electrical Engineering, Liaoning University of Technology, Jinzhou 121001, China
In recent decades, the control problems of various nonlinear systems have received wide attention. In the initial stage, scholars often used matching conditions to derive the asymptotic stability of systems. With the continuous progress of nonlinear systems control technology, the limitations of matching conditions have restricted further research of systems control. To overcome this problem, the backstepping technique has been introduced. This approach reduces the difficulty of the problem with decomposition of the systems to achieve control purposes. Over the past two decades, there has been an increased interest in adaptive control technology because it is a very effective tool for handling systems with uncertainties. In [1], the adaptive dynamic programming algorithm was successfully applied to a discretetime optimal control scheme. The backstepping technique was used in [2] and [3] to achieve a significant breakthrough in the adaptive control of a nonlinear timedelay system and nonlinear stochastic jump system. On the other hand, the backstepping method was also applied to nonlinear strict feedback systems [4][8] with unknown items. After obtaining a series of research results in strict feedback systems, researchers began to study corresponding nonlinear purefeedback systems. In [9], an uncertain purefeedback nonlinear system is transformed into a nonaffine form, and an adaptive control method was designed to solve the global asymptotic state stabilization problem. In modern control theory, neural networks (NNs) have been developed as approximators of unknown nonlinear functions. The work in [10] and [11] developed an adaptive dynamic programming algorithms with a value iteration and local value iteration for optimal control of discretetime nonlinear systems, while NNs were proposed to solve the problems faced by unknown items. For a nonlinear purefeedback system [12], based on the idea of backstepping, the author used the implicit function theorem to assert the existence of continuous desired virtual controls, and NNs were adopted to approximate the controllers. Adaptive NN control schemes were used on a class of output feedback nonlinear systems [13][16] with unknown functions. On account of backstepping algorithm and the Lyapunov method, adaptive NN approximation techniques were proposed in [17][19] for the ocean vessels, novel coaxial eightrotor unmanned aerial vehicles and networked multiagent systems, while their effectiveness have been proven. In addition, [20][33] also proposed an adaptive NN control design method for nonlinear systems. In fact, due to the physical limitations of the controller as well as the mechanical design and manufacture in the actual systems, various types of constraint characteristics exist in many typical industrial control systems. However, the system's constraints are ignored in these papers.
Under the influence of the actual background, the theoretical study of constraints becomes more and more important. In nonlinear control system [34] based on predictive control, the problem of satisfying pointwisein time input and/or state hard constraints was studied, and predictive control has also been applied in [35]. The control strategies of nonholonomic mechanical systems under various constraints were studied in [36]. Constraints can be roughly divided into the following types: output constraints, partial state constraints, full state constraints, etc. In many control applications, there are strong constraints on the control and state variables. For constrained systems, designing controllers based on unconstrained conditions will deteriorate the stability of the systems, produce some undesired responses such as oscillations, and even destroy the hardware equipment of systems. BLF is now a widely used approach to control constraints in design. In [37][46], the adaptive control method for nonlinear systems with output constraints and state constraints were investigated based on BLF. The adaptive control design based on a BLF framework not only can deal with the uncertainty of parameters, but also can deal with the uncertainty of functions by using NNs. The adaptive NN control design in [47][53] were applied to the BLFbased nonlinear systems with output or state constraints. The actual systems, state parameters and control parameters are timevarying and determined by the physical characteristics of the restraint systems, so the model cannot be described with the usual parameters. To deal with the aforementioned control problems of the systems, both the systems timevarying model and systems constraints must be taken into account. Motivated by these advancements in the constraints, many timevarying output constraints have been studied for various nonlinear systems [54][56]. In view of adaptive NN tracking control technology, state constraints are considered in nonlinear timevarying systems [57][59]. Based on the above theories, the authors applied timevarying state constraints to a robot system [60] and DC motor system [61], respectively. However, there is no work to deal with the adaptive NN control of purefeedback systems with timevarying constraints, which motivates our study.
Motivated by the aforementioned observations, we pay close attention to the problem of adaptive NN tracking control scheme for uncertain nonlinear purefeedback systems with timevarying full state constraints in this paper. The main contributions can be described as follows:
1) Many actual systems can be described as purefeedback systems. Meanwhile, control parameters are timevarying and determined by the physical characteristics in the actual systems. However, as far as we know, the control of purefeedback systems with timevarying state constraints are rarely mentioned in the literature.
2) For the sake of control of purefeedback systems, the implicit function theorem is employed to assert the existence of the smooth ideal control input. At the same time, NNs are applied to estimate both the virtual and actual controllers. Then, the mean value theorem is used to convert purefeedback systems to strict feedback systems.
3) Aimed at transformed systems, the BLFs are employed to ensure that the full state constraints were not violated in any step of backstepping design.
Furthermore, to obtain the stability of the purefeedback systems, a significant theorem is given and proved. The feasibility of the designed controller can be proved though two simulation examples.
Ⅱ. SYSTEM DESCRIPTIONSConsider nonlinear purefeedback systems with the following form:
$ \begin{align} \begin{cases} \dot {x}_i =f_i \left( {\bar {x}_i, x_{i+1} } \right), &i=1, \ldots, n1 \\ \dot {x}_n =f_n \left( {\bar {x}_n, u} \right) \\ y=x_1 \end{cases} \end{align} $  (1) 
where
The control objective of this paper is to construct an adaptive feedback controller
Remark 1: In this paper, nonlinear purefeedback systems with state constraints are researched. Compared with the nonpurefeedback systems [40][46], purefeedback systems have a wider range of applications as a class of more general lower triangular nonlinear systems. Moreover, in contrast to the control of nonlinear systems [40][46], [48][50] with constant state constraints, timevarying state constraints can relax the conditions of constant ones. At the same time, although [62] also addressed the purefeedback systems with timevarying state constraints, linear parameterization is necessary. This paper is more general in the case that the function is completely unknown.
Assumption 1 [60]: There exist the constants
Assumption 2 [55]: It is assumed that
Lemma 1 [12]: For
In this paper, for the systems (1), define unknown nonlinear functions
Assumption 3 [12]: The signs of functions
In [12], the radial basis function (RBF) NNs are employed to approximate the unknown nonlinear function
$ \begin{align} \Psi _{nn}( Y )=W^TS\left( Y \right) \end{align} $  (2) 
where
$ \begin{align} s_i ( Y )=\exp \left( {\frac{\left\ {Y\tau _i } \right\^2}{\varepsilon _i^2 }} \right), ~~~i=1, 2, \ldots, m \end{align} $  (3) 
where
It has been shown that any smooth function in a compact set
$ \begin{align} \Psi ( Y )=W^{\ast T}S( Y )+\eta ~~~ \forall Y\in \Omega _Y \subset \mathbb{R}^l \end{align} $  (4) 
where
$ \begin{align} W^\ast =\arg \mathop {\min }\limits_{W\in \mathbb{R}^m} \left\{ {\mathop {\sup }\limits_{Y\in \Omega _Y } \left {\Psi ( Y )W^TS( Y )} \right} \right\}. \end{align} $  (5) 
According to Assumption 3, we have
$ \begin{align} \begin{cases} \phi _1 =\dot {y}_d +k_1 z_1 \\ \phi _2 =\dot {\alpha }_1 +k_2 z_2 \\ ~~~~~~~~~\vdots\\ \phi _n =\dot {\alpha }_{n1} +k_n z_n \\ \end{cases} \end{align} $  (6) 
where
$ \begin{align} \frac{\partial \left( {f_i \left( {\bar {x}_i, x_{i+1} } \right)+\phi _i } \right)} {\partial x_{i+1} }>{g}_i >0. \end{align} $  (7) 
Using Lemma 1 which is an implicit function theorem obtained from [12], with
$ \begin{align} {\begin{cases} f_i \left( {\bar {x}_i, \alpha _i^\ast } \right)+\phi _i =0, &i=1, \ldots, n1 \\[2mm] f_n \left( {\bar {x}_n, \alpha _n^\ast } \right)+\phi _n =0. \\ \end{cases}} \end{align} $  (8) 
Combining the mean value theorem, there exist
$ \begin{align} \begin{cases} f_i \left( {\bar {x}_i, x_{i+1} } \right)=f_i \left( {\bar {x}_i, \alpha _i^\ast } \right)+g_{i\lambda _i } \left( {x_{i+1} \alpha _i^\ast } \right), \\ \qquad \qquad\qquad\qquad\qquad\qquad\qquad i=1, \ldots, n1 \\ f_n ( {\bar {x}_n, u} )=f_n \left( {\bar {x}_n, \alpha _n^\ast } \right)+g_{n\lambda _n } \left( {u\alpha _n^\ast } \right) \\ \end{cases} \end{align} $  (9) 
where
Lemma 2 [61]: For any
$ \begin{align} \log \frac{k_{b_i }^2 }{k_{b_i }^2 z_i^2 }\le \frac{z_i^2 }{k_{b_i }^2 z_i^2 } \end{align} $  (10) 
where
Proof: For any
$ \begin{align} \log &\frac{k_{b_i }^2 }{k_{b_i }^2 z_i^2 }=\log \left( {1+\frac{z_i^2 }{k_{b_i }^2 z_i^2 }} \right) \\[2mm] & \le \log \left( {1+\frac{z_i^2 }{k_{b_i }^2 z_i^2 }+\sum\limits_{n=2}^\infty {\frac{\left( {\frac{z_i^2 } {\left( {k_{b_i }^2 z_i^2 } \right)}} \right)^n} {n!}} } \right) \\[2mm] & =\log \left( {e^{\frac{z_i^2 } {k_{b_i }^2 z_i^2} }} \right) \\[2mm] & =\frac{z_i^2 }{k_{b_i }^2 z_i^2 }. \end{align} $  (11) 
Young's Inequality: It is assumed that if
$ \begin{align} x^Ty\le \frac{1}{pa^p}\left\ x \right\^p+\frac{a^q}{q}\left\ y \right\^q. \end{align} $  (12) 
Especially, if
$ \begin{align} x^Ty\le \frac{1}{2a^2}\left\ x \right\^2+\frac{a^2}{2}\left\ y \right\^2. \end{align} $  (13) 
In this paper, based on the backstepping method, an adaptive controller is designed for the nonlinear purefeedback systems with timevarying full state constraints. The specific process is as follows.
Step 1: Based on the systems (1), we know that differentiating
$ \begin{align} \dot {z}_1 =\dot {x}_1 \dot {y}_d =f_1 \left( {x_1 , x_2 } \right)\dot {y}_d . \end{align} $  (14) 
Substituting (6), (8) and (9) into (14) yields
$ \begin{align} \dot {z}_1 =k_1 z_1 +g_{1\lambda _1 } \left( {x_2 \alpha _1^\ast } \right). \end{align} $  (15) 
By defining the error variable
$ \begin{align} \dot {z}_1 =k_1 z_1 +g_{1\lambda _1 } \left( {z_2 +\alpha _1 \alpha _1^\ast } \right). \end{align} $  (16) 
Consider the following positive definite timevarying Lyapunov function
$ \begin{align} V_1 =\frac{1}{2g_{1\lambda _1 } }\log \left( {\frac{k_{b_1 }^2 (t)}{k_{b_1 }^2 (t)z_1^2 }} \right)+\frac{1}{2}\tilde {W}_1^T \Gamma _1^{\mbox{}1} \tilde {W}_1 \end{align} $  (17) 
where
Based on the definition of
Substituting (16) into the time derivative of
$ \begin{align} \dot {V}_1 =&\ \frac{z_1 }{g_{1\lambda _1 } (k_{b_1 }^2 (t)z_1^2 )}\left( {\dot {z}_1 z_1 \frac{\dot {k}_{b_1 } (t)}{k_{b_1 } (t)}} \right) \\[1mm] & \frac{\dot {g}_{1\lambda _1 } }{2g_{1\lambda _1 }^2 }\log \left( {\frac{k_{b_1 }^2 (t)}{k_{b_1 }^2 (t)z_1^2 }} \right)+\tilde {W}_1^T \Gamma _1^{\mbox{}1} \dot {\hat {W}}_1 \\[2mm] =&\ \frac{z_1 }{g_{1\lambda _1 } (k_{b_1 }^2 (t)z_1^2 )}\Bigg( {k_1 z_1 +g_{1\lambda _1 } \left( {z_2 +\alpha _1 \alpha _1^\ast } \right)} \\[1mm] & z_1 \frac{\dot {k}_{b_1 } (t)}{k_{b_1 } (t)} \Bigg)\frac{\dot {g}_{1\lambda _1 } }{2g_{1\lambda _1 }^2 }\log \left( {\frac{k_{b_1 }^2 (t)}{k_{b_1 }^2 (t)z_1^2 }} \right)\notag\\[1mm] &\ +\tilde {W}_1^T \Gamma _1^{\mbox{}1} \dot {\hat {W}}_1 . \end{align} $  (18) 
By using an NN
$ \begin{align} \alpha _1^\ast =W_1^{\ast T} S_1 \left( {Y_1 } \right)+\eta _1 \end{align} $  (19) 
where
Design the virtual control
$ \begin{align} \alpha _1 =& \left( {\rho _1 +\left( {\frac{1} {\bar {g}_{1\lambda _1 } }} \right)\bar {\rho }_1 ( t )} \right)z_1 +\hat {W}_1^T S_1 \left( {Y_1 } \right) \\[2mm] & \frac{1} {2\left( {\frac{z_1 } {\left( {k_{b_1 }^2 ( t )z_1^2 } \right)}} \right)} \end{align} $  (20) 
where
Substituting (19) and (20) into (18), on account of the fact that
$ \begin{align} \dot {V}_1 \le& \frac{k_1 z_1^2 }{g_{1\lambda _1 } \left( {k_{b_1 }^2 ( t )z_1^2 } \right)}+\frac{z_1 z_2 }{k_{b_1 }^2 ( t )z_1^2 }\frac{\rho _1 z_1^2 }{k_{b_1 }^2 \left( t \right)z_1^2 } \\[2mm] &\ +\frac{z_1 \tilde {W}_1^T S_1 \left( {Y_1 } \right)}{k_{b_1 }^2 ( t )z_1^2 }\frac{z_1^2 }{2\left( {k_{b_1 }^2 \left( t \right)z_1^2 } \right)^2}\frac{z_1 \eta _1 }{k_{b_1 }^2 \left( t \right)z_1^2 } \\[2mm] & \frac{\dot {g}_{1\lambda _1 } }{2g_{1\lambda _1 }^2 }\log \left( {\frac{k_{b_1 }^2 ( t )}{k_{b_1 }^2 \left( t \right)z_1^2 }} \right)+\tilde {W}_1^T \Gamma _1^{\mbox{}1} \dot {\hat {W}}_1 . \end{align} $  (21) 
From Young's inequality, it follows that
$ \begin{align} \frac{z_1 \eta _1 }{k_{b_1 }^2 \left( t \right)z_1^2 }\le \frac{z_1^2 }{2\left( {k_{b_1 }^2 \left( t \right)z_1^2 } \right)^2}+\frac{\eta _1^{\ast 2} }{2}. \end{align} $  (22) 
Further, (21) can be rewritten as
$ \begin{align} \dot {V}_1 \le& \frac{k_1 z_1^2 }{g_{1\lambda _1 } \left( {k_{b_1 }^2 ( t )z_1^2 } \right)}+\frac{z_1 z_2 }{k_{b_1 }^2 ( t )z_1^2 }\\[2mm] &\frac{\rho _1 z_1^2 }{k_{b_1 }^2 ( t )z_1^2 }+\frac{\eta _1^{\ast 2} }{2} +\frac{z_1 \tilde {W}_1^T S_1 \left( {Y_1 } \right)}{k_{b_1 }^2 ( t )z_1^2 }\\ &\frac{\dot {g}_{1\lambda _1 } }{2g_{1\lambda _1 }^2 }\log \left( {\frac{k_{b_1 }^2 ( t )}{k_{b_1 }^2 \left( t \right)z_1^2 }} \right)+\tilde {W}_1^T \Gamma _1^{\mbox{}1} \dot {\hat {W}}_1 . \end{align} $  (23) 
The adaptive law is designed as follows:%
$ \begin{align} \dot {\hat {W}}_1 \mbox{=}\Gamma _1 \left( {\frac{z_1 }{k_{b_1 }^2 ( t )z_1^2 }S_1 \left( {Y_1 } \right)\sigma _1 \hat {W}_1 } \right) \end{align} $  (24) 
where
Using Lemma 2, the following inequality is obtained
$ \begin{align} \dot {V}_1 \le& \frac{k_1 z_1^2 }{g_{1\lambda _1 } \left( {k_{b_1 }^2 ( t )z_1^2 } \right)}+\frac{z_1 z_2 }{k_{b_1 }^2 ( t )z_1^2 }\frac{\rho _1 z_1^2 }{k_{b_1 }^2 \left( t \right)z_1^2 } \\[2mm] &\ +\frac{\eta _1^{\ast 2} }{2}+\frac{g_{1d_1 } }{2\bar {g}_{1\lambda _1 }^2 }\frac{z_1^2 }{k_b^2 ( t )z_1^2 }\sigma _1 \tilde {W}_1^T \hat {W}_1 . \end{align} $  (25) 
Step
$ \begin{align} \dot {z}_i =\dot {x}_i \dot {\alpha }_{i1} =f_i \left( {\bar {x}_i, x_{i+1} } \right)\dot {\alpha }_{i1} . \end{align} $  (26) 
Substituting (6), (8) and (9) into (26) yields
$ \begin{align} \dot {z}_i =k_i z_i +g_{i\lambda _i } \left( {x_{i+1} \alpha _i^\ast } \right). \end{align} $  (27) 
By defining the error variable
$ \begin{align} \dot {z}_i =k_i z_i +g_{i\lambda _i } \left( {z_{i+1} +\alpha _i \alpha _i^\ast } \right). \end{align} $  (28) 
Because
$ \begin{align} \dot {\alpha }_{i1} =&\ \sum\limits_{l=1}^{i1} {\frac{\partial \alpha _{i1} }{\partial x_l }\dot {x}_l } +\frac{\partial \alpha _{i1} }{\partial y_d }\dot {y}_d +\sum\limits_{l=0}^1 {\frac{\partial \alpha _{i1} }{\partial k_{b_{i1} }^{\left( l \right)} }k_{b_{i1} }^{\left( {l+1} \right)} }\notag\\[2mm] &\ +\sum\limits_{l=1}^{i1} {\frac{\partial \alpha _{i1} }{\partial \hat {W}_l }\Gamma _l \left( {\frac{z_l }{k_{b_l }^2 ( t )z_l^2 }S_l \left( {Y_l } \right)\sigma _l \hat {W}_l } \right)} . \end{align} $  (29) 
For simplicity, the following definition is given
$ \begin{align} \Theta _{i1} =&\ \sum\limits_{l=0}^1 {\frac{\partial \alpha _{i1} }{\partial k_{b_{i1} }^{\left( l \right)} }k_{b_{i1} }^{\left( {l+1} \right)} } +\frac{\partial \alpha _{i1} }{\partial y_d }\dot {y}_d \notag\\[2mm] &\ +\sum\limits_{l=1}^{i1} {\frac{\partial \alpha _{i1} }{\partial \hat {W}_l }\Gamma _l \left( {\frac{z_l }{k_{b_l }^2 ( t )z_l^2 }S_l \left( {Y_l } \right)\sigma _l \hat {W}_l } \right)} . \end{align} $  (30) 
Consider the timevarying Lyapunov function candidate
$ \begin{align} V_i =V_{i1} +\frac{1}{2g_{i\lambda _i } }\log \left( {\frac{k_{b_i }^2 ( t )}{k_{b_i }^2 \left( t \right)z_i^2 }} \right)+\frac{1}{2}\tilde {W}_i^T \Gamma _i^{1} \tilde {W}_i \end{align} $  (31) 
where
Substituting (28) into the time derivative of
$ \begin{align} \dot {V}_i =&\ \dot {V}_{i1} +\frac{z_i }{g_{i\lambda _i } \left( {k_{b_i }^2 ( t )z_i^2 } \right)}\left( {\dot {z}_i z_i \frac{\dot {k}_{b_i } ( t )}{k_{b_i } \left( t \right)}} \right) \\[1mm] & \frac{\dot {g}_{i\lambda _i } }{2g_{i\lambda _i }^2 }\log \left( {\frac{k_{b_i }^2 ( t )}{k_{b_i }^2 \left( t \right)z_i^2 }} \right)+\tilde {W}_i^T \Gamma _i^{1} \dot {\hat {W}}_i \\[2mm] =&\ \dot {V}_{i1} +\frac{z_i }{g_{i\lambda _i } \left( {k_{b_i }^2 ( t )z_i^2 } \right)}\\[1mm] & \times\left( {k_i z_i +g_{i\lambda _i } \left( {z_{i+1} +\alpha _i \alpha _i^\ast } \right)} {z_i \frac{\dot {k}_{b_i } ( t )}{k_{b_i } \left( t \right)}} \right)\\[1mm] & \frac{\dot {g}_{i\lambda _i } }{2g_{i\lambda _i }^2 }\log \left( {\frac{k_{b_i }^2 ( t )}{k_{b_i }^2 \left( t \right)z_i^2 }} \right)+\tilde {W}_i^T \Gamma _i^{1} \dot {\hat {W}}_i . \end{align} $  (32) 
By using an NN
$ \begin{align} \alpha _i^\ast =W_i^{\ast T} S_i \left( {Y_i } \right)+\eta _i \end{align} $  (33) 
where approximation error
The virtual control
$ \begin{align} \alpha _i =& \left( {\rho _i +\left( {\frac{1} {\bar {g}_{i\lambda _i } }} \right)\bar {\rho }_i \left( t \right)} \right)z_i +\hat {W}_i^T S_i \left( {Y_i } \right) \\[2mm] & z_{i1} \frac{k_{b_i }^2 ( t )z_i^2 }{k_{b_{i1} }^2 ( t )z_{i1}^2 }\frac{z_i }{2\left( {k_{b_i }^2 ( t )z_i^2 } \right)} \end{align} $  (34) 
where design constant
Substituting (33) and (34) into (32), and due to the fact that
$ \begin{align} \dot {V}_i \le&\ \dot {V}_{i1} \frac{k_i z_i^2 }{g_{i\lambda _i } \left( {k_{b_i }^2 ( t )z_i^2 } \right)}+\frac{z_i z_{i+1} }{k_{b_i }^2 ( t )z_i^2 } \\[1mm] & \frac{\rho _i z_i^2 }{k_{b_i }^2 ( t )z_i^2 } \frac{z_{i1} z_i }{k_{b_{i1} }^2 ( t )z_{i1}^2 }+\frac{z_i \tilde {W}_i^T S_i \left( {Y_i } \right)}{k_{b_i }^2 ( t )z_i^2 } \\[1mm] & \frac{z_i^2 }{2\left( {k_{b_i }^2 \left( t \right)z_i^2 } \right)^2} \frac{z_i \eta _i }{k_{b_i }^2 \left( t \right)z_i^2 }\notag\\[1mm] & \frac{\dot {g}_{i\lambda _i } }{2g_{i\lambda _i }^2 }\log \left( {\frac{k_{b_i }^2 ( t )}{k_{b_i }^2 ( t )z_i^2 }} \right)+\tilde {W}_i^T \Gamma _i^{1} \dot {\hat {W}}_i . \end{align} $  (35) 
Using Young's inequality yields
$ \begin{align} \frac{z_i \eta _i }{k_{b_i }^2 \left( t \right)z_i^2 }\le \frac{z_i^2 }{2\left( {k_{b_i }^2 \left( t \right)z_i^2 } \right)^2}+\frac{\eta _i^{\ast 2} }{2}. \end{align} $  (36) 
Further, (35) becomes
$ \begin{align} \dot {V}_i \le&\ \dot {V}_{i1} \frac{k_i z_i^2 }{g_{i\lambda _i } \left( {k_{b_i }^2 ( t )z_i^2 } \right)}+\frac{z_i z_{i+1} }{k_{b_i }^2 ( t )z_i^2 }+\frac{\eta _i^{\ast 2} }{2} \\[1mm] & \frac{\rho _i z_i^2 }{k_{b_i }^2 ( t )z_i^2 }\frac{z_{i1} z_i }{k_{b_{i1} }^2 ( t )z_{i1}^2 }+\frac{z_i \tilde {W}_i^T S_i \left( {Y_i } \right)}{k_{b_i }^2 ( t )z_i^2 } \\[1mm] & \frac{\dot {g}_{i\lambda _i } }{2g_{i\lambda _i }^2 }\log \left( {\frac{k_{b_i }^2 ( t )}{k_{b_i }^2 \left( t \right)z_i^2 }} \right)+\tilde {W}_i^T \Gamma _i^{1} \dot {\hat {W}}_i . \end{align} $  (37) 
Design the adaptation law
$ \begin{align} \dot {\hat {W}}_i =\Gamma _i \left( {\frac{z_i }{k_{b_i }^2 ( t )z_i^2 }S_i \left( {Y_i } \right)\sigma _i \hat {W}_i } \right) \end{align} $  (38) 
where
Using Lemma 2, the following inequality is obtained:
$ \begin{align} \dot {V}_i \le &\ \dot {V}_{i1} \frac{k_i z_i^2 }{g_{i\lambda _i } \left( {k_{b_i }^2 ( t )z_i^2 } \right)}+\frac{z_i z_{i+1} }{k_{b_i }^2 ( t )z_i^2 }\\[1mm] & \frac{\rho _i z_i^2 }{k_{b_i }^2 ( t )z_i^2 } +\frac{\eta _i^{\ast 2} }{2}\frac{z_{i1} z_i }{k_{b_{i1} }^2 ( t )z_{i1}^2 }\\[1mm] &\ +\frac{g_{id_i } }{2\bar {g}_{i\lambda _i }^2 }\frac{z_i^2 }{k_{b_i }^2 \left( t \right)z_i^2 }\sigma _i \tilde {W}_i^T \hat {W}_i . \end{align} $  (39) 
Step
$ \begin{align} \dot {z}_n =\dot {x}_n \dot {\alpha }_{n1} =f_n \left( {\bar {x}_n, u} \right)\dot {\alpha }_{n1} . \end{align} $  (40) 
Substituting (6), (8) and (9) into (40) yields
$ \begin{align} \dot {z}_n =k_n z_n +g_{n\lambda _n } \left( {u\alpha _n^\ast } \right). \end{align} $  (41) 
Because
$ \begin{align} \dot {\alpha }_{n1} =&\ \sum\limits_{l=1}^{n1} {\frac{\partial \alpha _{n1} }{\partial x_l }\dot {x}_l } +\frac{\partial \alpha _{i1} }{\partial y_d }\dot {y}_d +\sum\limits_{l=0}^1 {\frac{\partial \alpha _{n1} }{\partial k_{b_{n1} }^{\left( l \right)} }k_{b_{n1} }^{\left( {l+1} \right)} } \\[2mm] &\ +\sum\limits_{l=1}^{n1} {\frac{\partial \alpha _{n1} }{\partial \hat {W}_l }\Gamma _l \left( {\frac{z_l }{k_{b_l }^2 ( t )z_l^2 }S_l \left( {Y_l } \right)\sigma _l \hat {W}_l } \right)} . \end{align} $  (42) 
For simplicity, the following definition is given
$ \begin{align} \Theta _{n1} =&\ \sum\limits_{l=0}^1 {\frac{\partial \alpha _{n1} }{\partial k_{b_{n1} }^{\left( l \right)} }k_{b_{n1} }^{\left( {l+1} \right)} } +\frac{\partial \alpha _{i1} }{\partial y_d }\dot {y}_d \\[2mm] &\ +\sum\limits_{l=1}^{n1} {\frac{\partial \alpha _{n1} }{\partial \hat {W}_l }\Gamma _l \left( {\frac{z_l }{k_{b_l }^2 ( t )z_l^2 }S_l \left( {Y_l } \right)\sigma _l \hat {W}_l } \right)} . \end{align} $  (43) 
Consider the following positive definite timevarying Lyapunov function
$ \begin{align} V_n =V_{n1} +\frac{1}{2g_{n\lambda _n } }\log \left( {\frac{k_{b_n }^2 ( t )}{k_{b_n }^2 \left( t \right)z_n^2 }} \right)+\frac{1}{2}\tilde {W}_n^T \Gamma _n^{1} \tilde {W}_n \end{align} $  (44) 
where
Substituting (41) into the time derivative of
$ \begin{align} \dot {V}_n =&\ \dot {V}_{n1} +\frac{z_n }{g_{n\lambda _n } \left( {k_{b_n }^2 ( t )z_n^2 } \right)}\left( {\dot {z}_n z_n \frac{\dot {k}_{b_n } ( t )}{k_{b_n } \left( t \right)}} \right) \\[1mm]& \frac{\dot {g}_{n\lambda _n } }{2g_{n\lambda _n }^2 }\log \left( {\frac{k_{b_n }^2 ( t )}{k_{b_n }^2 \left( t \right)z_n^2 }} \right)+\tilde {W}_n^T \Gamma _n^{1} \dot {\hat {W}}_n \\ =&\ \dot{V}_{n1}+\frac{z_n }{g_{n\lambda _n } \left( {k_{b_n }^2 ( t )z_n^2 } \right)}\Bigg(k_n z_n +g_{n\lambda _n } \left( {u\alpha _n^\ast} \right) \\[1mm] & z_n \frac{\dot{k}_{b_n } \left( t \right)}{k_{b_n } ( t )} \Bigg)\frac{\dot {g}_{n\lambda _n } }{2g_{n\lambda _n }^2 }\log \left( {\frac{k_{b_n }^2 ( t )}{k_{b_n }^2 ( t )z_n^2 }} \right)\\[1mm] &\ +\tilde {W}_n^T \Gamma _n^{1} \dot {\hat {W}}_n . \end{align} $  (45) 
By using an NN
$ \begin{align} \alpha _n^\ast =W_n^{\ast T} S_n \left( {Y_n } \right)+\eta _n \end{align} $  (46) 
where
The following actual controller
$ \begin{align} u=& \left( {\rho _n +\left( {\frac{1} {\bar {g}_{n\lambda _n } }} \right)\bar {\rho }_n ( t )} \right)z_n +\hat {W}_n^T S_n \left( {Y_n } \right) \\[2mm] & z_{n1} \frac{k_{b_n }^2 ( t )z_n^2 }{k_{b_{n1} }^2 ( t )z_{n1}^2 }\frac{z_n }{2\left( {k_{b_n }^2 ( t )z_n^2 } \right)} \end{align} $  (47) 
where design constant
Substituting (46) and (47) into (45), and based on the situation that
$ \begin{align} \dot {V}_n \le &\ \dot{V}_{n1} \frac{k_n z_n^2 }{g_{n\lambda _n } \left( {k_{b_n }^2 ( t )z_n^2 } \right)}\frac{\rho _n z_n^2 }{k_{b_n }^2 ( t )z_n^2 } \\[1mm] & \frac{z_n^2 }{2\left( {k_{b_n }^2 ( t )z_n^2 } \right)^2}\frac{z_{n1} z_n }{k_{b_{n1} }^2 \left( t \right)z_{n1}^2 }\\[1mm] &\ +\frac{z_n \tilde {W}_n^T S_n \left( {Y_n } \right)}{k_{b_n }^2 ( t )z_n^2 } \frac{z_n \eta _n }{k_{b_n }^2 \left( t \right)z_n^2 }\\[1mm] & \frac{\dot {g}_{n\lambda _n } }{2g_{n\lambda _n }^2 }\log \left( {\frac{k_{b_n }^2 ( t )}{k_{b_n }^2 \left( t \right)z_n^2 }} \right)+\tilde {W}_n^T \Gamma _n^{1} \dot {\hat {W}}_n . \end{align} $  (48) 
Based on the following inequality:
$ \begin{align} \frac{z_n \eta _n }{k_{b_n }^2 \left( t \right)z_n^2 }\le \frac{z_n^2 }{2\left( {k_{b_n }^2 \left( t \right)z_n^2 } \right)^2}+\frac{\eta _n^{\ast 2} }{2} \end{align} $  (49) 
(48) can be computed as follows:
$ \begin{align} \dot {V}_n \le &\ \dot {V}_{n1} \frac{k_n z_n^2 }{g_{n\lambda _n } \left( {k_{b_n }^2 ( t )z_n^2 } \right)}\frac{\rho _n z_n^2 }{k_{b_n }^2 ( t )z_n^2 } \\[1mm] & \frac{z_{n1} z_n }{k_{b_{n1} }^2 ( t )z_{n1}^2 }+\frac{\eta _n^{\ast 2} }{2}+\frac{z_n \tilde {W}_n^T S_n \left( {Y_n } \right)}{k_{b_n }^2 ( t )z_n^2 } \\[1mm] & \frac{\dot {g}_{n\lambda _n } }{2g_{n\lambda _n }^2 }\log \left( {\frac{k_{b_n }^2 ( t )}{k_{b_n }^2 \left( t \right)z_n^2 }} \right)+\tilde {W}_n^T \Gamma _n^{1} \dot {\hat {W}}_n . \end{align} $  (50) 
Design the adaptation law
$ \begin{align} \dot {\hat {W}}_n =\Gamma _n \left( {\frac{z_n }{k_{b_n }^2 ( t )z_n^2 }S_n \left( {Y_n } \right)\sigma _n \hat {W}_n } \right) \end{align} $  (51) 
where
Using Lemma 2, the following inequality is obtained:
$ \begin{align} \dot {V}_n \le&\ \dot {V}_{n1} \frac{k_n z_n^2 }{g_{n\lambda _n } \left( {k_{b_n }^2 ( t )z_n^2 } \right)}\frac{\rho _n z_n^2 }{k_{b_n }^2 ( t )z_n^2 }\\[1mm] & \frac{z_{n1} z_n }{k_{b_{n1} }^2 ( t )z_{n1}^2 } +\frac{\eta _n^{\ast 2} }{2}\frac{z_n \eta _n }{k_{b_n }^2 \left( t \right)z_n^2 }\\[1mm] &\ +\frac{g_{nd_n } }{2\bar {g}_{n\lambda _n }^2 }\frac{z_n^2 }{k_{b_n }^2 ( t )z_n^2 }\sigma _n \tilde {W}_n^T \hat {W}_n . \end{align} $  (52) 
With the first
$ \begin{align} \dot {V}_n \le& \sum\limits_{j=1}^n \left( \frac{k_j }{\bar {g}_{j\lambda _j } }+\rho _j \frac{g_{jd_j } }{2\bar {g}_{j\lambda _j }^2 } \right)\frac{z_j^2 }{k_{b_j }^2 ( t )z_j^2 } \\[1mm] & \sum\limits_{j=1}^n {\sigma _j \tilde {W}_j^T \hat {W}_j } +\sum\limits_{j=1}^n \frac{\eta _j^{\ast 2} }{2}. \end{align} $  (53) 
The following inequalities hold:
$ \begin{align} \sigma _j \tilde {W}_j^T \hat {W}_j &=\sigma _j \tilde {W}_j^T \left( {\tilde {W}_j +W_j^{\ast } } \right) \\[1mm] & \le \sigma _j \left\ {\tilde {W}_j } \right\^2+\sigma _j \left( {\frac{\left\ {\tilde {W}_j } \right\^2}{2}+\frac{\left\ {W_j^{\ast } } \right\^2}{2}} \right)\\[1mm] & \le \frac{\sigma _j \left\ {\tilde {W}_j } \right\^2}{2}+\frac{\sigma _j \left\ {W_j^{\ast } } \right\^2}{2}. \end{align} $  (54) 
By selecting the suitable
$ \begin{align} \left( {\frac{k_j }{\bar {g}_{j\lambda _j } }+\rho _j \frac{g_{jd_j } }{2\bar{g}_{j\lambda _j }^2 }} \right)\frac{z_j^2 }{k_{b_j }^2 ( t )z_j^2 }\le \frac{z_j^2 \rho _j^\ast }{k_{b_j }^2 ( t )z_j^2 }. \end{align} $  (55) 
Combining Assumption 3, Lemma 2 with the above inequality, we can get the following inequality
$ \begin{align} \dot {V}_n \le&\sum\limits_{j=1}^n {\rho _j^\ast \log \left( {\frac{k_{b_j }^2 ( t )}{k_{b_j }^2 \left( t \right)z_j^2 }} \right)} +\sum\limits_{j=1}^n {\frac{\eta _j^{\ast 2} }{2}}\notag \\[2mm] & \sum\limits_{j=1}^n {\frac{\sigma _j \left\ {\tilde {W}_j } \right\^2}{2}} +\sum\limits_{j=1}^n {\frac{\sigma _j \left\ {W_j^{\ast } } \right\^2}{2}} . \end{align} $  (56) 
Theorem 1: Under Assumptions 13, consider the systems (1) with timevarying full state constraints. If the initial states
Proof: According to (17), (31) and (44), we have
$ \begin{align} V_n =\sum\limits_{j=1}^n {\frac{1}{2g_{j\lambda _j } }\log \left( {\frac{k_{b_j }^2 ( t )}{k_{b_j }^2 \left( t \right)z_j^2 }} \right)} +\sum\limits_{j=1}^n {\frac{1}{2}\tilde {W}_j^T \Gamma _j^{1} \tilde {W}_j } . \end{align} $  (57) 
By choosing the
$ \begin{align} \dot {V}_n \le& \sum\limits_{j=1}^n {\rho _j^\ast \log \left( {\frac{k_{b_j }^2 ( t )}{k_{b_j }^2 \left( t \right)z_j^2 }} \right)} \sum\limits_{j=1}^n {\frac{\sigma _j \left\ {\tilde {W}_j } \right\^2}{2}} +B \\[2mm] \le& \sum\limits_{j=1}^n {\frac{\tau }{2\underline{g}_{j\lambda _j} }\log \left( {\frac{k_{b_j }^2 ( t )}{k_{b_j }^2 ( t )z_j^2 }} \right)} \sum\limits_{j=1}^n {\frac{\tau \tilde {W}_j^T \Gamma _j^{1} \tilde {W}_j }{2}} +B \\[2mm] \le& \tau \Bigg[\sum\limits_{j=1}^n {\frac{1}{2\underline{g}_{j\lambda _j} }\log \left( {\frac{k_{b_j }^2 ( t )}{k_{b_j }^2 \left( t \right)z_j^2 }} \right)} \\[2mm] & +\sum\limits_{j=1}^n {\frac{\tilde {W}_j^T \Gamma _j^{1} \tilde {W}_j }{2}} \Bigg]+B \le \tau V_n +B \end{align} $  (58) 
where
$ \begin{align*} B=\sum\limits_{j=1}^n \sigma _j \frac{\left\ {W_j^{\ast } } \right\^2} {2} +\sum\limits_{j=1}^n \frac{\eta _j^{\ast 2} } {2} . \end{align*} $ 
Multiplying
$ \begin{align} \frac{d( {V_n ( t )e^{\tau t}} )} {dt}\le Be^{\tau t}. \end{align} $  (59) 
Integrating (59) over
$ \begin{align} 0\le V_n ( t )\le \left( {V_n ( 0 )\frac{B} {\tau} } \right)e^{\tau t}+\frac{B} {\tau}\le V_n ( 0 )e^{\tau t}+\frac{B} { \tau }. \end{align} $  (60) 
Combining (57) with (60), the following inequality can be obtained
$ \begin{align} \frac{1}{2g^\ast }\sum\limits_{j=1}^n {\log \left( {\frac{k_{b_j }^2 ( t)}{k_{b_j }^2 ( t )z_j^2 }} \right)} \le V_n ( t ){\kern 1pt}\le V_n ( 0 )e^{\tau t}+\frac{B} {\tau} \end{align} $  (61) 
where
$ \begin{align} \frac{k_{b_j }^2 ( t )}{k_{b_j }^2 ( t )z_j^2 }\le e^{2g^\ast V_n ( 0 )e^{\tau t}+\frac{2g^\ast B} {\tau} }. \end{align} $  (62) 
Then we can get the tracking error which satisfies
$ \begin{align} \left {z_j } \right\le k_{b_j } ( t )\sqrt {1e^{\left[{2g^\ast V_n ( 0 )e^{\tau t}+\frac{2g^\ast B} {\tau} } \right]}} . \end{align} $  (63) 
According to Lemma 2,
In this section, two simulation studies are raised to demonstrate the effectiveness of the proposed adaptive NN control scheme in this paper.
Example 1: Consider the third order purefeedback system
$ \begin{align} \begin{cases} \dot {x}_1 =x_1^2 +x_2 \\ \dot {x}_2 =x_1^2 x_1 x_2 +\left( {0.1\sin x_2 +1.2} \right)x_3 \\ \dot {x}_3 =0.1x_1 x_2 0.5x_2^2 +\sin x_3 +\left( {e^{x_1^2 }+1} \right)u \\ y=x_1 \end{cases} \end{align} $  (64) 
where the states of system are constrained in
$ \begin{align} \begin{cases} \alpha _1 =\left( {\rho _1 +\left( {\frac{1} {\bar{g}_{1\lambda _1 } }} \right)\bar {\rho }_1 } \right)z_1 +\hat {W}_1^T S_1 ( {Y_1 } )\\[2mm] \qquad \ \frac{z_1 }{2\left( {k_{b_1 }^2 ( t )z_1^2 } \right)} \\[5mm] \alpha _2 =\left( {\rho _2 +\left( {\frac{1} {\bar{g}_{2\lambda _2 } }} \right)\bar {\rho }_2 } \right)z_2 +\hat {W}_2^T S_2 ( {Y_2 } )\\[2mm] \qquad\ z_1 \frac{k_{b_2 }^2 ( t )z_2^2 }{k_{b_1 }^2 ( t )z_1^2 } \frac{z_2 }{2\left( {k_{b_2 }^2 ( t )z_2^2 } \right)} \\[5mm] u=\left( {\rho _3 +\left( {\frac{1} {\bar{g}_{3\lambda _3 } }} \right)\bar {\rho }_3 } \right)z_3 +\hat {W}_3^T S_3 ( {Y_3 } )\\[2mm] \qquad z_2 \frac{k_{b_3 }^2 ( t )z_3^2 }{k_{b_2 }^2 ( t )z_2^2 } \frac{z_3 }{2\left( {k_{b_3 }^2 ( t )z_3^2 } \right)} \\ \end{cases} \end{align} $  (65) 
while the adaptation laws are designed as
$ \begin{align} \dot {\hat {W}}_i =\Gamma _i \left[{\frac{z_i }{k_{b_i }^2 ( t )z_i^2 }S_i \left( {Y_i } \right)\sigma _i \hat {W}_i } \right], ~~~i=1, 2, 3 \end{align} $  (66) 
where
$ \begin{align*} &z_1 =x_1 y_d, ~~z_2 =x_2 \alpha _1, ~~z_3 =x_3 \alpha _2\\ &Y_1 =\left[{x_1, \dot {y}_d, } {y_d } \right]^T\\ &Y_2 =\left[{\bar {x}_2^T, \frac{\partial \alpha _1 } {\partial x_1 }, \Theta _1 , \alpha _1 } \right]^T\\ &Y_3 =\left[{\bar {x}_3^T, \frac{\partial \alpha _2 } {\partial x_1 }, } {\frac{\partial \alpha _2 } {\partial x_2 }, \Theta _2, \alpha _2 } \right]^T. \end{align*} $ 
The initial conditions of the system are defined as
Figs. 14 are the simulation results of system (64). Fig. 1 clearly illustrates the excellent tracking performance of the system. At the same time, Figs. 1 and 2 are given to descript the trajectory of state and error variables, respectively, and it is easy to see that the timevarying constraints are not violated. The trajectories of the adaptive laws and actual controller are displayed in Figs. 3 and 4. From the above Figures, it can be concluded that all closedloop signals remain bounded and the timevarying constraints are never violated in the pure feedback system.
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Fig. 1 The trajectories of 
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Fig. 2 The trajectories of 
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Fig. 3 The trajectories of 
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Fig. 4 The trajectory of 
Example 2: Refer to [48], the control scheme proposed in this paper is applied to deal with the stability of the wing rock model with ailerons modelled by firstorder actuator dynamics under the condition of timevarying full state constraints. Meanwhile, the dynamics equation of the wing rock model is constructed as follows:
$ \begin{align} \begin{cases} \dot {x}_1 =x_2 \\[1mm] \dot {x}_2 =bx_3 +\theta _0 +\theta _1 x_1 +\theta _2 x_2 +\theta _3 \left {x_1 } \rightx_2\\\qquad\ + \theta _4 \left {x_2 } \rightx_2 +\theta _5 x_1^3 \\[1mm] \dot {x}_3 =\frac{x_3 } {\omega} +\frac{u} {\omega } \end{cases} \end{align} $  (67) 
where
The aerodynamic parameters of delta wing for
Figs. 58 are the simulation results of the practical system (67). Fig. 5 demonstrates the excellent tracking performance of the system, and shows that timevarying state constraints are never violated. The trajectories of the errors, the adaptive laws and the actual controller are plotted in Figs. 68. From the above Figures, it is easy to verify the effectiveness of the control scheme proposed in the practical experimental research.
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Fig. 5 The trajectories of 
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Fig. 6 The phase portrait of 
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Fig. 7 The trajectories of 
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Fig. 8 The trajectory of 
In this paper, we have proposed adaptive NN control scheme for the nonlinear purefeedback systems with timevarying full state constraints. The use of BLF prevents timevarying state constraints from being exceeded. The approximation property of NNs is applied to approximate the unknown nonlinear function generated by the controller design process. Though Lyapunov stability analysis and two simulation examples, the proposed control scheme ensures that all signals of the closedloop systems are uniformly ultimately bounded and asymptotic tracking is implemented without violation of constraints.
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