IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(5): 923-933 PDF
Adaptive Neural Network-Based Control for a Class of Nonlinear Pure-Feedback Systems With Time-Varying Full State Constraints
Tingting Gao1, Yan-Jun Liu1, Lei Liu1, Dapeng Li2
1. College of Science, Liaoning University of Technology, Jinzhou 121000, China;
2. School of Electrical Engineering, Liaoning University of Technology, Jinzhou 121001, China
Abstract: In this paper, an adaptive neural network (NN) control approach is proposed for nonlinear pure-feedback systems with time-varying full state constraints. The pure-feedback systems of this paper are assumed to possess nonlinear function uncertainties. By using the mean value theorem, pure-feedback systems can be transformed into strict feedback forms. For the newly generated systems, NNs are employed to approximate unknown items. Based on the adaptive control scheme and backstepping algorithm, an intelligent controller is designed. At the same time, time-varying Barrier Lyapunov functions (BLFs) with error variables are adopted to avoid violating full state constraints in every step of the backstepping design. All closedloop signals are uniformly ultimately bounded and the output tracking error converges to the neighborhood of zero, which can be verified by using the Lyapunov stability theorem. Two simulation examples reveal the performance of the adaptive NN control approach.
Key words: Adaptive control     neural networks (NNs)     nonlinear pure-feedback systems     time-varying constraints
Ⅰ. INTRODUCTION

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 pointwise-in time input and/or state hard constraints was studied, and predictive control has also been applied in [35]. The control strategies of non-holonomic 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 BLF-based nonlinear systems with output or state constraints. The actual systems, state parameters and control parameters are time-varying 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 time-varying model and systems constraints must be taken into account. Motivated by these advancements in the constraints, many time-varying 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 time-varying systems [57]-[59]. Based on the above theories, the authors applied time-varying 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 pure-feedback systems with time-varying 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 pure-feedback systems with time-varying full state constraints in this paper. The main contributions can be described as follows:

1) Many actual systems can be described as pure-feedback systems. Meanwhile, control parameters are time-varying and determined by the physical characteristics in the actual systems. However, as far as we know, the control of pure-feedback systems with time-varying state constraints are rarely mentioned in the literature.

2) For the sake of control of pure-feedback 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 pure-feedback 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 pure-feedback systems, a significant theorem is given and proved. The feasibility of the designed controller can be proved though two simulation examples.

Ⅱ. SYSTEM DESCRIPTIONS

Consider nonlinear pure-feedback 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, n-1 \\ \dot {x}_n =f_n \left( {\bar {x}_n, u} \right) \\ y=x_1 \end{cases} \end{align} (1)

where $\bar {x}_i =[{x_1, x_2, \ldots, x_i }]^T\in \mathbb{R}^i$ , $i=1, \ldots, n$ are the state vectors, $u\in \mathbb{R}$ and $y\in \mathbb{R}$ are the input and the output of the systems, respectively. $f_i ( \cdot ): \mathbb{R}^{i+1}\to \mathbb{R}$ are uncertain smooth functions. For the systems (1), the state variables $x_i$ , $i=1$ , $\ldots$ , $n$ are required to remain in the set $| x_i | < k_{c_i } ( t )$ , $\forall t\ge0$ , where $k_{c_i } ( t ):\mathbb{R}_+ \to \mathbb{R}$ .

The control objective of this paper is to construct an adaptive feedback controller $u$ which ensures that the system output $y$ is tracks the reference signal $y_d$ , while all the closed-loop signals are uniformly ultimately bounded and the time-varying full state constraints are not violated.

Remark 1: In this paper, nonlinear pure-feedback systems with state constraints are researched. Compared with the nonpure-feedback systems [40]-[46], pure-feedback 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, time-varying state constraints can relax the conditions of constant ones. At the same time, although [62] also addressed the pure-feedback systems with time-varying 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 $K_{c_i }^j$ , $i= 1,$ $\ldots,$ $n$ , $j=0, 1, \ldots, n$ such that $k_{c_i } ( t )\le K_{c_i }^0$ and $k_{c_i }^{ ( j )} ( t )$ $\le$ $K_{c_i }^j$ , $\forall t\ge 0$ , where $k_{c_i }^{ ( j )} ( t )$ denotes $j$ th time derivative of $k_{c_i } ( t )$ .

Assumption 2 [55]: It is assumed that $y_d ( t )$ satisfies $| {y_d ( t )} |$ $\le$ $Y_0 ( t ) < k_{c_i } ( t )$ and its $i$ th time derivative $y_d^{(i)} ( t )$ , satisfy $| {y_d^{(i)} ( t )}|$ $\le Y_i$ , $i=1, \ldots, n$ , $\forall t\ge 0$ , where $Y_0 ( t ):\mathbb{R}_+\to \mathbb{R}_+$ and $Y_1, \ldots, Y_n$ are positive constants.

Lemma 1 [12]: For $\forall( {x, u} )\in \mathbb{R}^n\times \mathbb{R}$ , assume that $f ( {x, u} ):$ $\mathbb{R}^n \times \mathbb{R}\to \mathbb{R}$ is continuous differentiable, and there is a constant $d$ such that $\frac{\partial f( {x, u} )} {\partial u}>d>0$ . Then there exists a smooth function $u^\ast =u( x )$ such that $f ({x, u^\ast } )=0$ .

In this paper, for the systems (1), define unknown nonlinear functions $g_i ( {\bar {x}_i, x_{i+1} } )=\frac{\partial f_i ( {\bar {x}_i, x_{i+1} } )} {\partial x_{i+1} }$ , $i=1, \ldots, n-1$ and $g_n ( {\bar {x}_n, u} )$ $= \frac{\partial f_n ( {\bar {x}_n, u} )} {\partial u}$ .

Assumption 3 [12]: The signs of functions $g_i ( \cdot )$ , $i=1, \ldots,$ $n$ are bounded, i.e., there exist the constants $\bar {g}_i \ge {g}_i >0$ such that ${g}_i \le | {g_i ( \cdot )} |\le \bar {g}_i$ . Without loss of generality, this paper assumes that ${g}_i \le g_i ( \cdot )\le \bar {g}_i$ .

In [12], the radial basis function (RBF) NNs are employed to approximate the unknown nonlinear function $\Psi ( Y ):\mathbb{R}^l\to \mathbb{R}$ as follows:

 \begin{align} \Psi _{nn}( Y )=W^TS\left( Y \right) \end{align} (2)

where $Y\in \Omega _Y \subset \mathbb{R}^l$ , $W= [{w_1, w_2, \ldots, w_m }]^T\in \mathbb{R}^m$ , and $m$ are the input vector, the weight vector and the node number $m$ $>$ $1$ of the NNs, respectively; while $S( Y )=[s_1 (Y), s_2 (Y),$ $\ldots,$ $s_m ( Y )]^T$ , where $s_i ( Y )$ being defined as the commonly used Gaussian functions, the form defined as

 \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 $\varepsilon _i$ is the width of the Gaussian function and $\tau _i =[\tau _{i1},$ $\tau _{i2},$ $\ldots, \tau _{il}]^T$ is the center of the receptive field.

It has been shown that any smooth function in a compact set $\Omega _Y \subset \mathbb{R}^l$ can be approximated by (2) to arbitrary accuracy as

 \begin{align} \Psi ( Y )=W^{\ast T}S( Y )+\eta ~~~ \forall Y\in \Omega _Y \subset \mathbb{R}^l \end{align} (4)

where $\eta$ is the approximation error and $W^\ast$ is the optimal weight vector which is constructed as the value of $W$ that minimizes $| \eta |$ for all $Y\in \Omega _Y \subset \mathbb{R}^l$ , i.e.,

 \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 $\frac{\partial f_i ( {\bar {x}_i, x_{i+1} } )} {\partial x_{i+1} }>{g}_i >0$ for all $\bar {x}_{i+1} \in \mathbb{R}^{i+1}$ . Introduce the function $\phi _i$ as

 \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 }_{n-1} +k_n z_n \\ \end{cases} \end{align} (6)

where $k_i$ , $i=1, 2, \ldots, n$ are positive constants, and define the tracking error $z_1 =x_1 -y_d$ and the variables $z_i =x_i -\alpha _{i-1}$ , $i$ $=2, \ldots, n$ , where $\alpha _{i-1}$ is a virtual controller to be designed by {Step} $i-1$ of the backstepping method. Because $\frac{\partial \phi _i } {\partial x_{i+1} }=0$ , the following inequality holds:

 \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 $x_{i+1}$ , $i=1, \ldots, n-1$ and $u$ as virtual control input, for every value of $\bar {x}_i$ , $i=1, \ldots, n$ and $\phi _i$ , $i$ $=1, \ldots, n$ , there exists the smooth ideal control inputs $\alpha _i^\ast ( \bar {x}_i, \phi _i )$ , $i=1, \ldots, n$ , such that

 \begin{align} {\begin{cases} f_i \left( {\bar {x}_i, \alpha _i^\ast } \right)+\phi _i =0, &i=1, \ldots, n-1 \\[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 $\lambda _i \in \left( {0, 1} \right)$ that makes

 \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, n-1 \\ 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 $g_{i\lambda _i } =g_i ( {\bar {x}_i, x_{( {i+1} )\lambda _i } } )=\frac{\partial f_i ( {\bar {x}_i, x_{( {i+1} )\lambda _i } } )} {\partial x_{ ( {i+1} )\lambda _i } }$ , $i=1, \ldots,$ $n$ are the defined unknown nonlinear functions with $x_{( {i+1})\lambda _i }$ $=$ $\lambda _i x_{i+1} + ( {1-\lambda _i } )\alpha _i^\ast$ . At the same time, we suppose that there exist a constant $g_{id_i } >0$ such that $| {\dot {g}_{i\lambda _i } ( \cdot )} |\le g_{id_i }$ , and similar to Assumption 3, we have ${\underline{g}}_{i\lambda _i } \le g_{i\lambda _i } ( \cdot )\le \bar {g}_{i\lambda _i }$ , where constants ${\underline{g}}_{i\lambda _i }$ , $\bar {g}_{i\lambda _i } >0$ .

Lemma 2 [61]: For any $k_{b_i }$ and $z_i$ , $i=1, 2, \ldots, n$ , satisfying $| {z_i } | < k_{b_i }$ , then there is

 \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 $k_{b_i }$ will be given later.

Proof: For any $z_i$ that satisfies $| {z_i } | < k_{b_i }$ , the term $\log ( {k_{b_i }^2 }/$ $( k_{b_i }^2-z_i^2 ) )$ in (10) can be rewritten as

 \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 $x\in \mathbb{R}^n$ , $y\in \mathbb{R}^n$ , $p,$ $q>1$ and $a$ is an arbitrary positive constant, and $1 / {p}$ $+$ $1 / q$ $=1$ , then

 \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 $p=q=2$ , $a$ is an arbitrary positive constant, then

 \begin{align} x^Ty\le \frac{1}{2a^2}\left\| x \right\|^2+\frac{a^2}{2}\left\| y \right\|^2. \end{align} (13)
Ⅲ. THE CONTROLLER DESIGN AND STABILITY ANALYSIS

In this paper, based on the backstepping method, an adaptive controller is designed for the nonlinear pure-feedback systems with time-varying full state constraints. The specific process is as follows.

Step 1: Based on the systems (1), we know that differentiating $z_1$ with respect to time yields

 \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 $z_2 =x_2 -\alpha _1$ , we express

 \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 time-varying 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 $\log ( \Delta )$ represents the natural logarithm of $\Delta$ , $\tilde {W}_1 =$ $\hat {W}_1-W_1^\ast$ , $\Gamma _1 =\Gamma _1^T >0$ is a constant gain matrix, and $k_{b_1 } ( t )$ $=$ $k_{c_1 } ( t )-y_d ( t )$ is a barrier function.

Based on the definition of $k_{b_1 } ( t )$ , and using Assumptions 1 and 2, we can obtain $k_{b_1 } ( t )\le K_{b_1 }$ with $K_{b_1 }$ being a constant.

Substituting (16) into the time derivative of $V_1$ leads to

 \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 $W_1^T S_1 ( {Y_1 } )$ to approximate $\alpha _1^\ast ( {x_1, \phi _1 } )$ , where $Y_1 = [{x_1, \dot {y}_d, y_d }]^T$ , $\alpha _1^\ast$ can be defined as

 \begin{align} \alpha _1^\ast =W_1^{\ast T} S_1 \left( {Y_1 } \right)+\eta _1 \end{align} (19)

where $W_1^\ast$ is an optimal weight vector of the NN, and approximation error $\eta _1$ is bounded over the compact set, i.e., $| {\eta _1 } |$ $\le$ $\eta _1^\ast$ with constant $\eta _1^\ast >0$ .

Design the virtual control $\alpha _1$ as

 \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 $\bar {\rho }_1 ( t )=\sqrt { ( {{\dot {k}_{b_1 } ( t )} /{k_{b_1 } ( t )}} )^2+\beta _1 }$ , $\beta _1 >0$ is a constant. Note that even if $\dot {k}_{b_1 } ( t )$ is zero, $\beta _1$ can ensure that the time derivative of $\alpha _1$ is bounded. $\hat {W}_1$ is the estimation of $W_1^\ast$ and $\rho _1$ is a design constant, $\rho _1 >0$ .

Substituting (19) and (20) into (18), on account of the fact that $\bar {\rho }_1 ( t )+{\dot {k}_{b_1 } ( t )} /{k_{b_1 } ( t )}\ge 0$ , (18) can be further 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 }-\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 $\sigma _1 >0$ is a design constant, and $\sigma _1$ -modification term $\sigma _1 \hat {W}_1$ is used to improve the robustness when there is an NN approximation error $\eta _1$ .

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 $i,$ ${\mathit{2}\le i\le n-\mathit{1}}$ : Based on the systems (1), the time derivative of variable $z_i =x_i -\alpha _{i-1}$ is

 \begin{align} \dot {z}_i =\dot {x}_i -\dot {\alpha }_{i-1} =f_i \left( {\bar {x}_i, x_{i+1} } \right)-\dot {\alpha }_{i-1} . \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 $z_{i+1} =x_{i+1} -\alpha _i$ , we express

 \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 $\alpha _{i-1}$ is the function of $\bar {x}_{i-1}$ , $y_d$ , $k_{b_{i-1} }$ , $\dot {k}_{b_{i-1} }$ and $\hat {W}_1,$ $\ldots, \hat {W}_{i-1}$ , $\dot {\alpha }_{i-1}$ can be described as

 \begin{align} \dot {\alpha }_{i-1} =&\ \sum\limits_{l=1}^{i-1} {\frac{\partial \alpha _{i-1} }{\partial x_l }\dot {x}_l } +\frac{\partial \alpha _{i-1} }{\partial y_d }\dot {y}_d +\sum\limits_{l=0}^1 {\frac{\partial \alpha _{i-1} }{\partial k_{b_{i-1} }^{\left( l \right)} }k_{b_{i-1} }^{\left( {l+1} \right)} }\notag\\[2mm] &\ +\sum\limits_{l=1}^{i-1} {\frac{\partial \alpha _{i-1} }{\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 _{i-1} =&\ \sum\limits_{l=0}^1 {\frac{\partial \alpha _{i-1} }{\partial k_{b_{i-1} }^{\left( l \right)} }k_{b_{i-1} }^{\left( {l+1} \right)} } +\frac{\partial \alpha _{i-1} }{\partial y_d }\dot {y}_d \notag\\[2mm] &\ +\sum\limits_{l=1}^{i-1} {\frac{\partial \alpha _{i-1} }{\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 time-varying Lyapunov function candidate

 \begin{align} V_i =V_{i-1} +\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 $\tilde {W}_i =\hat {W}_i -W_i^\ast$ , and $\Gamma _i =\Gamma _i^T$ is a positive constant gain matrix.

Substituting (28) into the time derivative of $V_i$ leads to

 \begin{align} \dot {V}_i =&\ \dot {V}_{i-1} +\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}_{i-1} +\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 $W_i^T S_i ( {Y_i } )$ to approximate $\alpha _i^\ast ( {\bar {x}_i, \phi _i } )$ , where $Y_i = [{\bar {x}_i^T, \frac{\partial \alpha _{i-1} } {\partial x_1 }, \ldots }, {\frac{\partial \alpha _{i-1} } {\partial x_{i-1} }, \Theta _{i-1}, \alpha _{i-1} }]^T$ , $\alpha _i^\ast$ can be expressed as

 \begin{align} \alpha _i^\ast =W_i^{\ast T} S_i \left( {Y_i } \right)+\eta _i \end{align} (33)

where approximation error $| {\eta _i } |\le \eta _i^\ast$ with constant $\eta _i^\ast >0$ .

The virtual control $\alpha _i$ is designed as

 \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_{i-1} \frac{k_{b_i }^2 ( t )-z_i^2 }{k_{b_{i-1} }^2 ( t )-z_{i-1}^2 }-\frac{z_i }{2\left( {k_{b_i }^2 ( t )-z_i^2 } \right)} \end{align} (34)

where design constant $\rho _i >0$ and the definition of $\bar {\rho }_i ( t )$ is similar to $\bar {\rho }_1 ( t )$ in Step 1.

Substituting (33) and (34) into (32), and due to the fact that $\bar {\rho }_i ( t )$ $+$ ${\dot {k}_{b_i } ( t )} / {k_{b_i } ( t )}\ge 0$ , then (32) can be described as

 \begin{align} \dot {V}_i \le&\ \dot {V}_{i-1} -\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_{i-1} z_i }{k_{b_{i-1} }^2 ( t )-z_{i-1}^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}_{i-1} -\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_{i-1} z_i }{k_{b_{i-1} }^2 ( t )-z_{i-1}^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 $\hat {W}_i$ 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) \end{align} (38)

where $\sigma _i$ is a positive constant, and $\sigma _i$ -modification term $\sigma _i \hat {W}_i$ is used to improve the robustness when there is a NN approximation error $\eta _i$ .

Using Lemma 2, the following inequality is obtained:

 \begin{align} \dot {V}_i \le &\ \dot {V}_{i-1} -\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_{i-1} z_i }{k_{b_{i-1} }^2 ( t )-z_{i-1}^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 $n$ : In the last step of backstepping, the actual controller $u$ will be obtained. The time derivative of $z_n =x_n -\alpha _{n-1}$ can be written as

 \begin{align} \dot {z}_n =\dot {x}_n -\dot {\alpha }_{n-1} =f_n \left( {\bar {x}_n, u} \right)-\dot {\alpha }_{n-1} . \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 $\alpha _{n-1}$ is the function of $\bar {x}_{n-1}$ , $y_d$ , $k_{b_{n-1} }$ , $\dot {k}_{b_{n-1} }$ and $\hat {W}_1,$ $\ldots,$ $\hat {W}_{n-1}$ , the derivative of $\alpha _{n-1}$ obtain

 \begin{align} \dot {\alpha }_{n-1} =&\ \sum\limits_{l=1}^{n-1} {\frac{\partial \alpha _{n-1} }{\partial x_l }\dot {x}_l } +\frac{\partial \alpha _{i-1} }{\partial y_d }\dot {y}_d +\sum\limits_{l=0}^1 {\frac{\partial \alpha _{n-1} }{\partial k_{b_{n-1} }^{\left( l \right)} }k_{b_{n-1} }^{\left( {l+1} \right)} } \\[2mm] &\ +\sum\limits_{l=1}^{n-1} {\frac{\partial \alpha _{n-1} }{\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 _{n-1} =&\ \sum\limits_{l=0}^1 {\frac{\partial \alpha _{n-1} }{\partial k_{b_{n-1} }^{\left( l \right)} }k_{b_{n-1} }^{\left( {l+1} \right)} } +\frac{\partial \alpha _{i-1} }{\partial y_d }\dot {y}_d \\[2mm] &\ +\sum\limits_{l=1}^{n-1} {\frac{\partial \alpha _{n-1} }{\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 time-varying Lyapunov function

 \begin{align} V_n =V_{n-1} +\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 $\tilde {W}_n =\hat {W}_n -W_n^\ast$ and $\Gamma _n =\Gamma_n^T >0$ is the constant gain matrix.

Substituting (41) into the time derivative of $V_n$ leads to

 \begin{align} \dot {V}_n =&\ \dot {V}_{n-1} +\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}_{n-1}+\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 $W_n^T S_n ( {Y_n } )$ to approximate $\alpha _n^\ast ( {\bar {x}_n, \phi _n } )$ , where $Y_n = [{\bar {x}_n^T , \frac{\partial \alpha _{n-1} } {\partial x_1 }, \ldots}, \frac{\partial \alpha _{n-1} } {\partial x_{n-1} }, \Theta _{n-1}, \alpha _{n-1}]^T$ , leading to the following equation:

 \begin{align} \alpha _n^\ast =W_n^{\ast T} S_n \left( {Y_n } \right)+\eta _n \end{align} (46)

where $\left| {\eta _n } \right|\le \eta _n^\ast$ with constant $\eta _n^\ast >0$ .

The following actual controller $u$ is constructed:

 \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_{n-1} \frac{k_{b_n }^2 ( t )-z_n^2 }{k_{b_{n-1} }^2 ( t )-z_{n-1}^2 }-\frac{z_n }{2\left( {k_{b_n }^2 ( t )-z_n^2 } \right)} \end{align} (47)

where design constant $\rho _n>0$ the definition of $\bar {\rho }_n ( t )$ is similar to $\bar {\rho }_1 ( t )$ in the Step 1.

Substituting (46) and (47) into (45), and based on the situation that $\bar {\rho }_n ( t )+{\dot {k}_{b_n } ( t )} / {k_{b_n } ( t )}\ge 0$ , while (45) can be further expressed as

 \begin{align} \dot {V}_n \le &\ \dot{V}_{n-1} -\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_{n-1} z_n }{k_{b_{n-1} }^2 \left( t \right)-z_{n-1}^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}_{n-1} -\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-1} z_n }{k_{b_{n-1} }^2 ( t )-z_{n-1}^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 $\hat {W}_n$ as

 \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 $\sigma _n >0$ is a design constant, and $\sigma _n$ -modification term $\sigma _n \hat {W}_n$ is used to improve the robustness when there is an NN approximation error $\eta _n$ .

Using Lemma 2, the following inequality is obtained:

 \begin{align} \dot {V}_n \le&\ \dot {V}_{n-1} -\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-1} z_n }{k_{b_{n-1} }^2 ( t )-z_{n-1}^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 $n-1$ steps, we have

 \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 $\rho _j$ , the inequality $( \rho _j^\ast=\rho _j+{k_j } /$ ${\bar {g}_{j\lambda _j } }-( {{g_{jd_j } } / {2\bar{g}_{j\lambda _j }^2 }} ) )>0$ is established. Then, we obtain

 \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 1-3, consider the systems (1) with time-varying full state constraints. If the initial states $| {x_j ( 0 )} |$ $<$ $K_{c_j }^0$ , $j=1, 2, \ldots, n$ are satisfied, then the following properties can be guaranteed: the error variables $z_j$ , $j=1, 2,$ $\ldots,$ $n$ are bounded; the time-varying full state constraints are never violated; and all closed-loop signals are bounded.

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 $\rho _j^\ast$ where $\rho _j^\ast \ge \tau / {2\underline{g}_{j0} }$ , and $\rho _j \ge \tau/ {2\underline{g}_{j0} }-{k_j } /{\bar {g}_{j\lambda _j } }+ ( {{g_{jd_j } } /{2\bar{g}_{j\lambda _j }^2 }} )$ where $\tau >0$ is a constant. Then, there is the inequality $\sigma _j \ge \tau \lambda _{\max } \{ {\Gamma _j^{-1} } \}$ by selecting $\tau$ and $\sigma _j$ . The derivative of $V_n$ along (56) and (57) satisfies

 \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 $e^{\tau t}$ by both sides of (58) yields

 \begin{align} \frac{d( {V_n ( t )e^{\tau t}} )} {dt}\le Be^{\tau t}. \end{align} (59)

Integrating (59) over $\left[{0, t} \right]$ , which leads to

 \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 $g^\ast ={\max }_{1\le j\le n} \{ {\bar{g}_{j\lambda _j } } \}$ . And taking the $e$ -exponential of both sides of (61), we have

 \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 {1-e^{-\left[{2g^\ast V_n ( 0 )e^{-\tau t}+\frac{2g^\ast B} {\tau} } \right]}} . \end{align} (63)

According to Lemma 2, $| {y_d ( t )} |\le Y_0$ and the fact $x_1 =z_1 +$ $y_d ( t )$ , we have an inequality $| {x_1 } |\le | {z_1 } |+ | {y_d ( t )} | < k_{b_1 } ( t )+Y_0$ . Then we have $| {x_1 } |\le k_{c_1 }$ , where $k_{c_1 } ( t )=k_{b_1 } ( t )+Y_0$ . Further, there is $| y |= | {x_1 } |\le k_{c_1 }$ . Thus, the output signal is bounded. It is obvious that the virtual controller which is defined in (20) is bounded $| {\alpha _1 } |\le \bar {\alpha }_1$ . From the fact that $x_2 =z_2 +\alpha _1$ and Lemma 2, it follows that $| {x_2 } |\le | {z_2 } |+ | {\alpha _1 } |\le k_{b_2 } +\bar {\alpha }_1$ . So, $| {x_2 } |$ $\le$ $k_{c_2 }$ , where $k_{c_2 } ( t )=k_{b_2 } ( t )+\bar {\alpha }_1$ . Similarly, we can prove that $| {x_{i+1} } |\le k_{c_{i+1} }$ , $i=2, \ldots, n-1$ , where $k_{c_{i+1} } ( t )=$ $k_{b_{i+1} } ( t )$ $+$ $\bar {\alpha }_i$ . Based on (60), we can get $( {\frac{1} {2}} )\tilde {W}_j^T \Gamma _j^{{-}1} \tilde {W}_j$ $\le$ $V_n ( 0 )e^{-\tau t}+{B}/ {\tau}$ and $\| {\tilde {W}_j } \|\le \sqrt {2\lambda _{\max } ( \Gamma ) ( {V_n ( 0 )e^{-\tau t}+{B}/ {\tau }} )}$ . Hence, the states $x_j$ , $j=1, 2, \ldots, n$ , the adaptive laws $\hat {W}_j$ , $j=1, 2,$ $\ldots,$ $n$ and the controller $u$ are bounded. In view of the above analysis, we can determine that all signals in the closed-loop systems are bounded and the time-varying full state constraints are never violated.

Ⅳ. SIMULATION EXAMPLE

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 pure-feedback 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 $| {x_1 } | < k_{c_1 } =1.4$ $+$ $0.2\sin ( {0.5t} )$ , $| {x_2 } | < k_{c_2 } =2.6+0.6\sin(t)$ and $| {x_3 } | < k_{c_3 }$ $=$ $5+0.8\sin(t)$ , the desired trajectory $y_d =0.8\sin( {{2t}} )$ , and the virtual controllers and the actual controller are designed as

 \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 $x_1( 0 )=0$ , $x_2 ( 0 )=1.4$ , and $x_3 ( 0 )=0$ , while the design parameters are given as follows $\rho _1 =40$ , $\rho _2 =15$ , $\rho _3 =18$ , ${\bar{g}}_{1\lambda _1 } =0.4$ , ${\bar{g}}_{2\lambda _2 }$ $=0.5$ , ${\bar{g}}_{3\lambda _3 } =0.6$ , $\Gamma _1 =7$ , $\Gamma _2 =8$ , $\Gamma _3 =4$ , $\sigma _1 =0.6$ , $\sigma _2$ $=0.5$ , $\sigma _3 =0.4$ , $\beta _1 =6$ , $\beta _2 =7$ and $\beta _3 =9$ .

Figs. 1-4 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 time-varying 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 closed-loop signals remain bounded and the time-varying constraints are never violated in the pure- feedback system.

 Download: larger image Fig. 1 The trajectories of $y_d$ , $y=x_1$ , $x_2$ and $x_3$
 Download: larger image Fig. 2 The trajectories of $z_1$ , $z_2$ and $z_3$
 Download: larger image Fig. 3 The trajectories of $\left\| {\hat {W}_1 } \right\|$ , $\left\| {\hat {W}_2 } \right\|$ and $\left\| {\hat {W}_3 } \right\|$
 Download: larger image Fig. 4 The trajectory of $u$

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 first-order actuator dynamics under the condition of time-varying 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 } \right|x_2\\\qquad\ + \theta _4 \left| {x_2 } \right|x_2 +\theta _5 x_1^3 \\[1mm] \dot {x}_3 =-\frac{x_3 } {\omega} +\frac{u} {\omega } \end{cases} \end{align} (67)

where $x_1$ , $x_2$ , $x_3$ and $u$ are the roll angle, the roll rate, the aileron deflection angle and the control input, respectively.

The aerodynamic parameters of delta wing for $25^{\circ}$ angle of attack are designed as $b=1.5$ , $\theta _0 =0$ , $\theta _1 =-0.01859521$ , $\theta _2$ $=0.015162375$ , $\theta _3 =-0.062445153$ , $\theta _4 =0.00954708$ , $\theta _5$ $= 0.02145291$ and $\omega ={1}/ {15}$ .

Figs. 5-8 are the simulation results of the practical system (67). Fig. 5 demonstrates the excellent tracking performance of the system, and shows that time-varying state constraints are never violated. The trajectories of the errors, the adaptive laws and the actual controller are plotted in Figs. 6-8. From the above Figures, it is easy to verify the effectiveness of the control scheme proposed in the practical experimental research.

 Download: larger image Fig. 5 The trajectories of $y_d$ , $y=x_1$ , $x_2$ and $x_3$
 Download: larger image Fig. 6 The phase portrait of $z_1$ , $z_2$ and $z_3$
 Download: larger image Fig. 7 The trajectories of $\left\| {\hat {W}_1 } \right\|$ , $\left\| {\hat {W}_2 } \right\|$ and $\left\| {\hat {W}_3 } \right\|$
 Download: larger image Fig. 8 The trajectory of $u$
Ⅴ. CONCLUSION

In this paper, we have proposed adaptive NN control scheme for the nonlinear pure-feedback systems with time-varying full state constraints. The use of BLF prevents time-varying 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 closed-loop systems are uniformly ultimately bounded and asymptotic tracking is implemented without violation of constraints.

REFERENCES